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0704.1219

Y-formalism and $b$ ghost in the Non-minimal Pure Spinor Formalism of Superstrings

arXiv:0704.1219v1 [hep-th] 10 Apr 2007 DPUR/TH/1 DFPD07/TH/06 April, 2007 Y-formalism and b ghost in the Non-minimal Pure Spinor Formalism of Superstrings Ichiro Oda 1 Department of Physics, Faculty of Science, University of the Ryukyus, Nishihara, Okinawa 903-0213, Japan. Mario Tonin 2 Dipartimento di Fisica, Universita degli Studi di Padova, Instituto Nazionale di Fisica Nucleare, Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy Abstract We present the Y-formalism for the non-minimal pure spinor quantization of super- strings. In the framework of this formalism we compute, at the quantum level, the explicit form of the compound operators involved in the construction of the b ghost, their normal- ordering contributions and the relevant relations among them. We use these results to construct the quantum-mechanical b ghost in the non-minimal pure spinor formalism. Moreover we show that this non-minimal b ghost is cohomologically equivalent to the non-covariant b ghost. 1E-mail address: [email protected] 2E-mail address: [email protected] http://arxiv.org/abs/0704.1219v1 1 Introduction Several years ago, a new formalism for the covariant quantization of superstrings was proposed by Berkovits [1]. Afterward, it has been recognized that this new formalism not only solves the longstanding problem of covariant quantization of the Green-Schwarz (GS) superstring, but also it is suitable to deal with problems that appear almost intractable in the Neveu-Schwarz- Ramond (NSR) approach, such as those involving space-time fermions and/or backgrounds with R-R fields. In this approach, the GS superstring action (let us say in the left-moving sector) is replaced with a free action for the bosonic coordinates Xa and their fermionic partners θα with their conjugate momenta pα, plus an action for the bosonic ghosts λ α and their conjugate momenta ωα, where λ α satisfy the ”pure spinor constraint” λΓaλ = 0. The ω − λ action looks like a free action but is not really free owing to the pure spinor constraint, which is necessary to have vanishing central charge and correct level of the Lorentz algebra. This formulation is nowadays called ”pure spinor formulation of superstrings” and many studies [2]-[24] were devoted to it in the recent years. 3 Another key ingredient in the pure spinor formulation is provided by the BRST charge λαdα where dα ≈ 0 contains the constraints generating a fermionic κ symmetry in the GS superstring and has the role of a spinorial derivative in superspace. The peculiar feature associated with this BRST charge is that Q is nilpotent only when the bosonic spinor λα satisfies the pure spinor condition. This peculiar feature is in fact expected since the constraint dα ≈ 0 in the GS approach involves both the first-class and the second-class constraints. Roughly speaking, the pure spinor condition is needed to handle the second-class constraint of the GS superstring, keeping the Lorentz covariance manifest. Since the BRST charge Q is nilpotent, one can define the cohomology and examine its physical content. Indeed, it has been shown that the BRST cohomology determines the physical spectrum which is equivalent to that of the RNS formalism and that of the GS formalism in the light-cone gauge [3]. Moreover, the BRST charge Q of the pure spinor formalism was found to be transformed to that of the NSR superstring [4] as well as that of the GS superstring in the light-cone gauge [18, 19]. Even if the pure spinor formalism provides a Lorentz-covariant superstring theory with manifest space-time supersymmetry even at the quantum level, there are some hidden sources of possible violation of Lorentz covariance. One of such sources is related to the b field defined by T = {Q, b} with T being the stress- energy tensor, which is necessary to compute higher loop amplitudes. Since the pure spinor formulation is not derived from a diffeomorphism-invariant action and does not contain the b− c ghosts of diffeomorphisms, the usual antighost b is not present in this approach. In [3] a compound b field whose BRST variation gives the stress energy tensor, was obtained. However this b field is not Lorentz-covariant. The same b field follows from an attempt [10] to derive, at the classical level, the pure spinor 3Alternative formalisms to remove the constraint were proposed in [25, 26]. formulation from a (suitably gauge-fixed and twisted) N = 2 superembedding approach. In this approach the b field is the twisted current of one of the two world-sheet (w.s.) supersymmetries whereas the integrand of the BRST charge Q is the twisted current of the other supersymmetry, suggesting an N = 2 topological origin of the pure spinor approach. This b field turns out to be proportional to the quantity Yα = where vα is a constant pure spinor, such that bY = YαG α where Gα is a covariant, spinor-like compound field, so that bY is not only Lorentz non-covariant but also singular at vλ = 0. A way to overcome the problem of the non-covariance and singular nature of bY was given in [14] where a recipe to compute higher loop amplitudes was proposed, in terms of a picture- raised b field constructed with the help of suitable covariant fields Gα, Hαβ, Kαβγ and Lαβγδ and some picture-changing operators Z ′s and Y ′s. 4 Recently, a very interesting formalism called ”non-minimal pure spinor formalism” has been put forward [27]. In this formalism, a non-minimal set of variables are added to that of the (minimal) pure spinor formulation. These non-minimal variables form a BRST quartet and have the role of changing the ghost-number anomaly from −8 to +3 without changing the central charge and the physical mass spectrum. A remarkable thing is that, in this formalism, one can define a Lorentz-covariant b ghost without the need of picture-changing operators. With the help of a suitable regulator, a recipe has been given to compute scattering amplitudes up to two-loop amplitudes. The OPE’s between the relevant operators that result in this approach show that the (non-minimal) pure spinor formulation is indeed a hidden, critical, N = 2 topological string theory. A significant improvement was obtained in [28]. Here a gauge invariant, BRST trivial regularization of the b field is proposed, that allows for a consistent prescription to compute amplitudes at any loop. A further source of possible non-covariance arises at intermediate steps of calculations, since the solution of the pure spinor constraint in terms of independent fields implies the breaking of SO(10) to U(5). 5 To be more precise, the space of (Euclidean) pure spinors in ten dimensions has the geometrical structure of a complex cone Q = SO(10) [21]. This space has been studied by Nekrasov [29] and the obstructions to its global definition are analyzed. It was shown that the obstructions are absent if the tip of the cone is removed. Then this complex cone is covered by 16 charts, U (α), (α) = 1, · · · , 16 and in each chart the local parametrization of the pure spinor, which breaks SO(10) to U(5), is taken such that the parameter that describes the generatrix of the cone is non-vanishing. This parametrization can be used to compute the relevant OPE’s [1, 3] (U(5)-formalism). In a previous work [30], we have proposed a new formalism named ”Y-formalism” for pur- poses of handling this unavoidable non-covariance stemming from the pure spinor condition. This Y-formalism is closely related to the U(5)-formalism, but has an advantage of treating all operators in a unified way without going back to the U(5)-decomposition. It is based on writing 4The picture-lowering operators YC , which are needed to absorb the zero modes of the ghost λ α, break the Lorentz-covariance but this breaking is BRST trivial and then harmless. 5In the extended pure spinor formalism [26], the same non-covariance can be found in the ghost sector where the ghosts are invariant under only U(5) group, but not S(10) group. the fundamental OPE between ω and λ in a form that involves Yα = . Strictly speaking, one needs 16, orthogonal, constant pure spinors v(α) (and 16 Y (α)) for each chart, such that U (α) (v(α)λ) 6= 0 in each chart. However, for our puposes it is sufficient to work in a given chart. Actually, it turned out that the Y-formalism is quite useful to find the full expression of b ghost [30]. More recently, the Y-formalism was also utilized to construct a four-dimensional pure spinor superstring [31]. The Y -field also arises in the regularization prescription proposed in [28]. The aim of the present paper is to extend the Y-formalism to the non-minimal case and to discuss in the framework of this formalism the non-minimal, covariant b field in addition to the fields Gα, Hαβ, Kαβγ and Lαβγδ, which are the building blocks of the b field. This will be done not only at the classical but also at the quantum level, by taking into account the subtleties of normal ordering. The consistent results which we will get in this article, could be regarded as a good check of the consistency of the Y-formalism. Moreover we shall show that the non-minimal, covariant b field is cohomologically equivalent to the non-covariant b field bY , improved by the term coming from the non-minimal sector. In section 2, we will review the Y-formalism for the minimal pure spinor formalism. In section 3, the operators Gα, Hαβ, Kαβγ and Lαβγδ, and their (anti-)commutation relations with the BRST charge, will be examined from the quantum-mechanical viewpoint. In section 4, we will construct the Y-formalism for the non-minimal pure spinor formalism. In section 5, based on the Y-formalism at hand, we will construct the Lorentz-covariant quantum b ghost, which satisfies the defining equation {Q, b} = T . We shall also show that it is cohomologically equivalent to the non-covariant b ghost bY (improved by the term coming from the non-minimal sector). Section 6 is devoted to conclusion and discussions. Some appendices are added. Appendix A contains our notation, conventions and useful identities. In Appendix B, we will review the normal-ordering prescriptions, the generalized Wick theorem and the rearrangement theorem which we will use many times in this article. Finally in Appendix C we give some details of the main calculations. 2 Review of the Y-formalism In this section, we start with a brief review of the (minimal) pure spinor formalism of super- strings [1], and then explain the Y-formalism [30]. For simplicity, we shall confine ourselves to only the left-moving (holomorphic) sector of a closed superstring theory. The generalization to the right-moving (anti-holomorphic) sector is straightforward. The pure spinor approach is based on the BRST charge dzλαdα, (2.1) and the action ∂Xa∂̄Xa + pα∂̄θ α − ωα∂̄λ α), (2.2) where λ is a pure spinor λΓaλ = 0. (2.3) This action is manifestly invariant under (global) super-Poincaré transformations. It is easily shown that the action I is also invariant under the BRST transformation generated by the BRST charge Q which is nilpotent owing to the pure spinor condition (2.3). Notice that in order to use Q as BRST charge it is implicit that the pure spinor condition is required to vanish in a strong sense. Moreover, the action I is invariant under the ω-symmetry δωα = Λa(Γ aλ)α, (2.4) where Λa are local gauge parameters. At the classical level the ghost current is J0 = ωλ, (2.5) and the Lorentz current for the ghost sector is given by Nab0 = ωΓabλ, (2.6) which together with T0λ = ω∂λ are the only super-Poincaré covariant bilinear fields involving ω and gauge invariant under the ω-symmetry. From the field equations it follows that p, θ, ω and λ are holomorphic fields. At the quantum level, one obtains the following OPE’s 6 involving the superspace coordinates ZM = (Xa, θα) and their super-Poincaré covariant momenta PM = (Πa, pα): < Xa(y)Xb(z) > = −ηab log(y − z), < pα(y)θ β(z) > = y − z δβα, (2.7) so that < dα(y)dβ(z) > = − y − z ΓaαβΠa(z), < dα(y)Π a(z) > = y − z (Γa∂θ)α(z), (2.8) where dα = pα − (∂Xa + θΓa∂θ)(Γaθ)α, Πa = ∂Xa + θΓa∂θ, Π̄a = ∂̄Xa + θΓa∂̄θ. (2.9) 6According to Appendix B, we should call them not the OPE’s but the contractions, but we have called ”OPE’s” since the terminology is usually used in the references of the pure spinor formulation. As for the ghost sector, the situation is a bit more complicated owing to the pure spinor condition (2.3). Namely, it would be inconsistent to assume a free field OPE between ω and λ. The reason is that since the pure spinor condition must vanish identically, not all the components of λ are independent: solving the condition, five of them are expressed nonlinearly in terms of the others. Accordingly, five components of ω are pure gauge. This problem is nicely resolved by introducing the Y-formalism. Let us first define the non-covariant object , (2.10) such that α = 1, (2.11) where vα is a constant pure spinor Y Γ aY = 0. Then it is useful to define the projector (Γaλ)α(Y Γa) β, (2.12) which, since TrK = 5, projects on a 5 dimensional subspace of the 16 dimensional spinor space in ten dimensions. The orthogonal projector is (1 − K)α β. Now the pure spinor condition implies β = 0. (2.13) Since K projects on a 5 dimensional subspace, Eq. (2.13) is a simple way to understand why a pure spinor has eleven independent components. Then we postulate the following OPE between ω and λ: < ωα(y)λ β(z) >= y − z (δβα −Kα β(z)). (2.14) It follows from Eq. (2.14) that the OPE between ω and the pure spinor condition vanishes identically. Moreover, the BRST charge Q is then strictly nilpotent even acting on ω. It is useful to notice that, with the help of the projector K, one can obtain a non-covariant but gauge-invariant antighost ω̃ defined as ω̃α = (1−K)α ωβ. (2.15) In the framework of this formalism one can compute [30] the OPE’s among the ghost current, Lorentz current and stress energy tensor and one can obtain the quantum version of these operators. Indeed, it has been shown in [30] that all the non-covariant, Y-dependent contributions in the r.h.s. of the OPE’s among these operators disappear if the stress energy tensor, the Lorentz current for the ghost sector, and the ghost current at the quantum level, are improved by Y -dependent correction terms, those are T = − ∂Xa∂Xa − pα∂θ α + Tλ ΠaΠa − dα∂θ α + ωα∂λ ∂(Y ∂λ), (2.16) Nab = ωΓabλ− ∂Y Γabλ− 2Y Γab∂λ , (2.17) J = ωλ+ Y ∂λ. (2.18) Then the OPE’s among T , Nab and J read < T (y)T (z) >= (y − z)2 T (z) + y − z ∂T (z), (2.19) < T (y)J(z) >= (y − z)3 (y − z)2 J(z) + y − z ∂J(z), (2.20) < T (y)Nab(z) >= (y − z)2 Nab(z) + y − z ∂Nab(z), (2.21) < J(y)J(z) >= − (y − z)2 , (2.22) < J(y)Nab(z) >= 0, (2.23) < Nab(y)N cd(z) >= − (y − z)2 ηd[aηb]c − y − z (ηa[cNd]b − ηb[cNd]a), (2.24) which are in full agreement with [1, 3]. Note that although the correction terms in the currents depend on the non-covariant Y-field explicitly, these can be rewritten as BRST-exact terms. Now a remark is in order. It appears at first sight that, due to the correction terms, the operators J , Nab and T are singular at vλ = 0 but the opposite is in fact true: it is clear from Eqs. (2.19)-(2.24) that the Y -dependent correction terms have just the rôle of cancelling the singularites which are present in the operators T0, N 0 and J0, owing to the singular nature of the OPE (2.14) between ω and λ. 7 It will be convenient to rewrite (2.17), (2.18) and Tλ as Nab = [ΩΓabλ− 2Y Γab∂λ], (2.25) J = Ωλ + 2Y ∂λ, (2.26) 7As anticipated in the notation , we will append a suffix ”0” when we refer to compound fields at the classical level, that is, given in terms of T0, N 0 and J0, and we will reserve the notation without suffix ”0” in denoting the corresponding quantities at the quantum level, given in terms of T , Nab and J . Tλ = Ω∂λ + 3∂Y ∂λ + Y ∂2λ, (2.27) where we have introduced the quantity Ωα = ωα − ∂Yα. (2.28) The Y-formalism explained thus far is also useful to deal with the b field which plays an important role in computing higher loop amplitudes. Its main property is {Q, b(z)} = T (z), (2.29) where T is the stress energy tensor. Since in the pure spinor formulation the reparametrization ghosts do not exist, b must be a composite field. Moreover, since the b ghost has ghost number −1 and the covariant fields, which include ωα and are gauge invariant under the ω-symmetry, always have ghost number zero or positive, one must use Yα (which also has ghost number −1) to construct the b ghost. Therefore b is not super-Poincaré invariant. The b ghost has been constructed for the first time in [3] in the U(5)-formalism in such a way that it satisfies Eq. (2.29). In the Y-formalism at hand, at the classical level it takes the form b0Y = ΠaY Γad+ ω(1−K)∂θ = YαG 0 , (2.30) where d)α − 0 (Γab∂θ) . (2.31) The last equality in (2.30) follows from the identity (A.3). The expression of bY at the quantum level will be derived in section 5. The non-covariance of bY is not dangerous since, as we shall show in section 5, the Lorentz variation of bY (or of its improvement at the non-minimal level) is BRST-exact. However, this operator cannot be accepted as insertion to compute higher loop amplitudes. Indeed, contrary to the operators T , Nab and J , it has a true singularity at vλ = 0 of the form (vλ)−1. The point is that there exists an operator ξ = Y θ, singular with a pole at vλ → 0, such that {Q, ξ} = 1 and the cohomology would become trivial if this operator is allowed in the Hilbert space, since for any closed operator V , V = {Q, ξV }. Then, for consistency, operators singular at vλ → 0 must be excluded from the Hilbert space. 3 Fundamental operators and normal-ordering effects When we attempt to construct a b ghost covariantly, either a picture-raised b ghost [14, 30] or a covariant b ghost in the framework of the non-minimal approach [27], we encounter several fundamental operators, Gα, Hαβ, Kαβγ and Lαβγδ [14, 30], which are a generalization of the constraints introduced by Siegel some time ago in [32]. Thus, in this section, we will consider those operators in order. We will pay a special attention to a consistent treatment of the normal-ordering effects. Let us notice that in addition to Gα, the totally antisymmetrized operatorsH [αβ], K [αβγ] and L[αβγδ] are the more fundamental objects and are of particular interest since they are involved in the construction of the b field in the non-minimal formulation. At the classical level, Gα is defined in (2.31) and H [αβ], K [αβγ] and L[αβγδ] are given by abc(dΓ abcd+ 24Nab0 Π [αβγ] 0 = − abc (Γ ad)γ]N bc0 , [αβγδ] 0 = − (Γabc) [αβ(Γade)γδ]N bc0 N0de. (3.1) They satisfy the following recursive relations: {Q,Gα0} = λ 0 ] = λ [αβγ] 0 } = λ [αβγδ] 0 ] = λ βγδρ] 0 = 0, (3.2) which one can verify easily. The full fields H 0 , K 0 and L 0 , which are involved in the construction of the picture-raised b ghost, can be obtained by adding new terms symmetric with respect to at least a couple of adjacent indices, and they satisfy the recursive relations 0 ] = λ 0 + · · · , 0 } = λ 0 + · · · , 0 ] = λ 0 + · · · , 0 = 0 + · · · , (3.3) where the dots denote ”Γ1-traceless terms”, i.e. terms that vanish if saturated with a Γ αiαi+1 between two adjacent indices. The fields H 0 , K 0 and L 0 are defined modulo Γ1-traceless terms. In this section we wish to discuss these operators and their recursive relations at the quantum level. A remark is in order. At the quantum level, in dealing with holomorphic operators composed of fields with singular OPE’s, a normal-ordering prescription is needed for their definition. As a rule, for the normal ordering of two operators A and B we shall adopt in this paper the generalized normal-ordering prescription, denoted by (AB) in [33] since it is convenient in carrying out explicit calculations. As explained in Appendix B, this prescription consists in subtracting the singular poles, evaluated at the point of the second entry and it is given by the contour integration (AB)(z) = w − z A(w)B(z). (3.4) Often, for simplicity, in dealing with this prescription the outermost parenthesis is suppressed and the normal ordering is taken from the right so that, in general, A1A2A3...An means (A1(A2(A3(· · ·An) · · ·))). A different prescription denoted as : AB :, that we shall call ”improved”, consists in sub- tracting the full contraction < A(y)B(z) >, included a possible finite term, as computed from the canonical OPE’s (2.7) and (2.14). In many cases the two prescriptions coincide but when they are different, it happens, as we shall see, that the final results look more natural if expressed in the improved prescription. 3.1 Gα Gα is obtained from (2.31) by replacing Nab0 and J0 with N ab and J as defined in Eqs. (2.17) and (2.18) and adding a normal-ordering term parametrized by a constant c1 Πa(Γad) Nab(Γ ab∂θ)α − J∂θα + c1∂ ≡ Gα1 +G 4 . (3.5) The constant c1 will be determined from the requirement that G α should be a primary field of conformal weight 2. Then we have to compute the OPE < T (y)Gα(z) >. The three terms Gα1 ≡ Πa(Γad) α, Gα2 ≡ − Nab(Γ ab∂θ)α and Gα3 ≡ − J∂θα are all products of two operators of conformal weight 1 so that their OPE’s with the stress energy tensor can be easily calculated. One finds that only Gα2 is a primary field. G 1 has a triple pole with residuum −5∂θ α and Gα3 has a triple pole with residuum −2∂θα. Moreover, the normal-ordering term Gα4 ≡ c1∂ 2θα also has a triple pole with residuum 2c1∂θ α. Therefore, putting them together, one has < T (y)Gα(z) >= −5− 2 + 2c1 (y − z)3 ∂θα(z) + (y − z)2 Gα(z) + y − z ∂Gα(z). (3.6) Hence, the requirement that Gα must be a primary field of conformal weight 2 is satisfied when we select the constant c1 to be In spite of the appearance, it turns out that this figure is in agreement with the result of [14] where the value −1 is indicated as the coefficient in front of the normal-ordering term ∂2θα in Gα. The difference is an artifact of the different normal-ordering prescriptions, the generalized normal-ordering prescription in (3.5) and the improved one. Whereas the two prescriptions coincide for Gα2 and G 3 , there appears a difference in G 1 . Indeed, since Πa(x)dα(z) = (x− z)2 [(Γaθ)α(z)− (Γ aθ)α(x)] (Γa∂θ)α(x)+ : Π a(z)dα(z) : + · · · , (3.7) we obtain (Πa(Γad) α) = − : Πadα : . (3.8) Substituting this result into Eq. (3.5), setting c1 = , we have Gα =: Πa(Γad) α : − Nab(Γ ab∂θ)α − J∂θα − ∂2θα, (3.9) which precisely coincides with the expression given in [14]. Next, we want to derive the quantum counterpart of the first (classical) recursive relations in (3.2) and, for that, we need to compute {Q,Gα}. In doing this calculation, one must be careful to deal with the order of the factors in the terms coming from the (anti)commutator among Q and Gα and use repeatedly the rearrangement theorem, reviewed in Appendix B, in order to recover the operator λαT . The details of this calculation are presented in Appendix C. As expected from the covariance of {Q,Gα}, the terms involving Y , coming from the rearrangement procedure, cancel exactly those coming from the Y -dependent correction terms of the operators Nab and J (see (2.17) and (2.18)) present in the definition of Gα. The final result is {Q,Gα} = λαT − ∂2λα. (3.10) The normal-ordering term −1 ∂2λα in (3.10) might appear to be strange at first sight, but it is indeed quite reasonable. The point is that it is not λαT but λαT − 1 ∂2λα that is a primary field of conformal weight 2 when we take account of the normal-ordering effects. In fact, since α(y)T (z) >≡ Rα1 (z) y − z ∂λα(z) y − z , (3.11) < T (y)(λαT )(z) > has a triple pole with residuum +∂2λ, and 1 ∂2λ has the same triple pole, it follows that Bα1 = λ ∂2λα, (3.12) is a BRST-closed primary operator of conformal weight 2. From now on, it is convenient to define α = Gα + , (3.13) so that (3.10) becomes {Q, Ĝα} = λαT. (3.14) Now we would like to study the operator λαGβ, that is expected to arise in the quantum counterpart of the second recursive relations in (3.2). As before, λαGβ is not primary since < λα(y)Gβ(z) > is different from zero. Indeed, < λα(y)Gβ(z) >≡ 2 (z) y − z , (3.15) where 2 = −∂θ αλβ + Γαβa (∂θΓ aλ). (3.16) Note that since ∂λα∂θβ is also primary, there is an ambiguity in defining a primary operator, say B 2 , associated to λ αGβ. Given (3.16), for the symmetric one, one has Γαβa [(λΓ aG) + ∂(λΓa∂θ) + c+(∂λΓ a∂θ)], (3.17) while, for the antisymmetric one, one has 2 = λ [αGβ] + ∂(λ[α∂θβ]) + c−∂λ [α∂θβ]. (3.18) Let us remark that λαGβ is not BRST-closed. Indeed {Q, λαGβ} = λαλβT− 1 λα∂2λβ. Whereas (λ[α(λβ]T )) = (T (λ[αλβ])) = 0, one has ((λαΓaαβ(λ βT )) = (T (λΓaλ)) − 2(λΓa∂2λ). Therefore the requirement that B 2 and B 2 are BRST-closed implies c+ = − and c− = − so that Γαβa [λΓ λΓa∂2θ + ∂(λΓa∂θ)], (3.19) 2 = λ [αGβ] + λ[α∂2θβ] = λ[αĜβ]. (3.20) 3.2 Hαβ A minimal choice for Hαβ is Hαβ = H(αβ) +H [αβ], (3.21) where (αβ) = Γαβa (N abΠb − JΠa + c2∂Π a), (3.22) H [αβ] = dΓabcd+ 6NabΠc). (3.23) First, we shall evaluate < T (y)Hαβ(z) > in order to fix the normal-ordering term. We can easily show that H [αβ] and the first term in H(αβ) are primary fields whereas − 1 Γαβa JΠ a and Γαβa ∂Π a have a triple pole with residua −4 1 Γαβa Π a and 2c2 Γαβa Π a, respectively. Thus, we obtain < T (y)Hαβ(z) >= −4 + 2c2 (y − z)3 Γαβa Π a(z) + (y − z)2 Hαβ(z) + y − z ∂Hαβ(z), (3.24) thereby taking c2 = 2 makes H αβ a primary field of conformal weight 2. This value agrees with the value in the Berkovits’ paper [14]. Next, we wish to evaluate [Q,Hαβ]: [Q,H(αβ)] = (λΓabd)Πb +N ab(λΓb∂θ) + (λd)Πa J(λΓa∂θ) + c2∂(λΓ , (3.25) [Q,H [αβ]] = ((Γdλ)ρΠd)(Γ abcd)ρ − (Γabcd)ρ(Γdλ)ρΠd + 3(λΓabd)Πc + 6Nab(λΓc∂θ) + 2c3(∂λΓ abc∂θ) . (3.26) Then, after some algebra and taking into account the normal-ordering terms by the rearrange- ment formula, we get for the symmetric part of Hαβ [Q,H(αβ)] = Γαβa [λΓ λΓa∂2θ + ∂(λΓa∂θ)], (3.27) and for the more interesting antisymmetric part H [αβ] [Q,H [αβ]] = λ[αGβ] + λ[α∂2θβ] = λ[αĜβ], (3.28) in agreement with (3.19) and (3.20). Notice that the Y -dependent contributions coming from rearrangement theorem cancel exactly those coming from the definitions (2.17) and (2.18) of Nab and J (For details see Appendix C). Since the term +∂(λΓa∂θ) in (3.27) is the BRST variation of ∂Πa, (3.27) can be rewritten [Q, Ĥ(αβ)] = Γαβa [λΓ λΓa∂2θ], (3.29) where we have defined as Ĥ(αβ) = H(αβ) − 1 Γαβa ∂Π Now let us consider the composite operator λαHβγ. Since Hαβ has conformal weight 2 but its contraction with λα does not vanish, one can expect that λαHβγ is not primary. Actually, using the fact < λα(y)Hβγ(z) >≡ 3 (z) y − z , (3.30) with R 3 being given by 3 = − Γβγa [(Γ abλ)αΠb − λ αΠa]− abc(Γ abλ)αΠc, (3.31) it turns out that a primary field of conformal weight 2 related to λαHβγ is 3 ≡ λ 3 . (3.32) Again there is an arbitrariness in choosing the primary field related to λαHβγ since ∂λαΠa is primary. As in previous cases we are especially interested in the antisymmetric part B [αβγ] 3 of B Since, in D = 10, a field which is totally antisymmetric in its three, spinor-like indices contains only the SO(10) irreducible representation (irrep.) 560 and R 3 in Eq. (3.31) does not contain such an irrep., it follows that [αβγ] 3 = 0, (3.33) so that B [αβγ] 3 simply becomes [αβγ] 3 = λ [αHβγ]. (3.34) From Eqs. (3.28), (3.15) and (3.16), it is then easy to show that λ[αHβγ] is BRST-closed. Indeed, one finds [Q, λ[αHβγ]] = λ[α(λβĜγ]) = Ĝ[γλαλβ] + λ[α∂(λβ∂θγ]) + ∂(λ[α∂θγ)λβ] = 0. (3.35) 3.3 Kαβγ A covariant expression of Kαβγ is Kαβγ = − Γαβa (Γbd) γNab − abc(Γ ad)γN bc (Γbd) αNab + (Γad)αJ + c3(Γ a∂d)α abc(Γ ad)αN bc 6 , (3.36) whereas the totally antisymmetric part is given by [αβγ] = − abc (Γ d)γ]N bc. (3.37) The term including a constant c3 describes the normal-ordering contribution. As before, we will calculate < T (y)Kαβγ(z) > in order to fix the normal-ordering term. One finds that all the terms K i are primary with conformal weight 2, except K (Γad)αJ and 5 ≡ c3 Γβγa (Γ a∂d)α which have triple poles in their OPE’s with T . In fact, < T (y)K 4 (z) > = (y − z)3 (Γad)α(z) + (y − z)2 4 (z) + y − z 4 (z), < T (y)K 5 (z) > = (y − z)3 (Γad)α(z) + (y − z)2 5 (z) + y − z 5 (z). (3.38) Therefore, one obtains < T (y)Kαβγ(z) >= 12 + 2c3 (y − z)3 (Γad)α(z) + (y − z)2 Kαβγ(z) + y − z ∂Kαβγ(z), (3.39) so that the condition of a primary operator of conformal weight 2 requires us to take c3 = −6, which is a new result. As for {Q,Kαβγ}, we will limit ourselves to considering only the antisymmetric part K [αβγ] of Kαβγ {Q,K [αβγ]} = ((ΓaΓdλ)γ]Πd)N (Γad)γ](λΓbcd) . (3.40) As before, the Y -dependent contributions coming from rearrangement theorem are exactly cancelled by those coming from the definition (2.17) of Nab, as expected from the covariance of the l.h.s of (3.40). Then from the rearrangement theorem and with a few algebra one gets {Q,K [αβγ]} = λ[αHβγ]. (3.41) Given that the Y -dependent terms are absent, (3.41) can also been argued as follows: coho- mology arguments based on Eq. (3.35) and the classical equivalence between {Q,K [αβγ]} and λ[αHβγ] imply {Q,K [αβγ]} = λ[αHβγ]+Λ [αβγ] 3 , where Λ [αβγ] 3 is a primary field of conformal weight 2 satisfying [Q,Λ [αβγ] 3 ] = 0. Then, notice that Λ [αβγ] 3 has ghost number +1 and involves ∂λ and Πa or ∂Πa and λα. However, using these fields, it is impossible to construct a 560 irrep. of SO(10), so Λ [αβγ] 3 vanishes identically. As before, let us construct a primary field of conformal weight 2 from λαKβγδ. We define < λα(y)Kβγδ(z) >≡ 4 (z) y − z , (3.42) where R 4 takes the form (Γabλ)αΓβγa (Γbd) (Γabλ)αΓ abc(Γ [(Γabλ)α(Γbd) β + 3λα(Γad)β]Γγδa + (Γabλ)α(Γcd)βΓ abc. (3.43) Provided that we define B 4 ≡ λ βγδ + 4 , (3.44) it is easy to get < T (y)B 4 (z) >= (y − z)2 4 (z) + y − z 4 (z), (3.45) which means that B 4 is a primary field of conformal weight 2 as expected. As before, there is an arbitrariness in the choice of the primary field related to λαKβγδ since the field ∂λαdβ is primary. If one considers the completely antisymmetric component B [αβγδ] 4 , one can notice that, in D = 10, a field antisymmetric in its four, spinor-like indices contains only the irreps. 770 and 1050 which are absent in the expression (3.43) of R [αβγδ] 4 so that one obtains [αβγδ] 4 = 0. (3.46) Consequently, we have [αβγδ] 4 = λ [αKβγδ]. (3.47) Furthermore, Eq. (3.41) together with (3.33) gives us the equation {Q, λ[αKβγδ]} = 0. (3.48) 3.4 Lαβγδ In this final subsection, we wish to consider Lαβγδ. In our previous paper [30], at the classical level, the form of Lαβγδ was fixed to be 0 = − λα(ω̃Γa)β[λγ(ω̃Γa) (ΓbΓaλ) γ(ω̃Γb)δ], (3.49) where ω̃ is defined in (2.15). One subtle point associated with this expression is that L cannot be entirely expressed in terms ofNab0 and J0. However, we have found that the dangerous terms involving ω̃Γa1a2a3a4λ cancel exactly in constructing the picture raised b ghost. On the other hand, when we consider the totally antisymmetrized part of L 0 , these dangerous terms never appear. In order to show that, let us notice that, given Eq. (3.49), one can write: 0 + L 0 = − (ω̃Λαβc λ)(ω̃Λ γδcλ)− λα(ω̃Γa)βλγ(ω̃Γa) δ, (3.50) where we have defined ω̃Λαβc λ = (ω̃Γc) αλβ − (ω̃Γb) α(ΓbΓcλ) β. (3.51) Then, taking the totally antisymmetrized part of Eq. (3.50) one gets [αβγδ] 0 = − (ω̃Λ[αβc λ)(ω̃Λ γδ]cλ). (3.52) Using (3.51) and (A.3), we can rewrite ω̃Λ[αβ]c λ as ω̃Λ[αβ]c λ = abc(ωΓ 0 . (3.53) Hence, we have shown that L [αβγδ] 0 is in fact expressed by N In order to have a covariant expression for L[αβγδ], at the quantum level, the classical Lorentz generator N bc0 must be replaced with N bc as given in (2.17) so that L[αβγδ] is [αβγδ] = − (Γabc) [αβ(Γade)γδ]N bcNde. (3.54) From the OPE’s < T (y)Nab(z) > and < Nab(y)N cd(z) >, one can easily verify that L[αβγδ] is a covariant, primary field of conformal weight 2. At the classical level one has the identities [αβγδ] 0 ] = λ βγδρ] 0 = 0, (3.55) where the last identity follows by noting that L [αβγδ] 0 is proportional to λ [α(ωΓa) β(ωΓb) γ(Γabλ)δ]. Since L[αβγδ] and λ[ǫLαβγδ] are covariant tensors and a possible quantum failure of these identities would involve Yα, thereby inducing violation of Lorentz covariance, one should expect that these identities hold at the quantum level as well. It is worthwhile to verify this result directly as a nice check of the consistency of the Y-formalism. The quantum counterpart of the former equation in Eq. (3.55) reads [Q,L[αβγδ]] = λ[αKβγδ]. (3.56) In this case there are no contributions from the rearrangement theorem and using (3.37) and (3.54) one finds that both sides of Eq. (3.56) are equal to 1 (Γabc) [αβ(Γade)γδ](dΓbcλ)Nde, thus showing that (3.56) is true. It is a little more cumbersome to verify the quantum analog of the latter equation in Eq. (3.55), which is given by λ[ǫLαβγδ] = 0. (3.57) To do that it is convenient to introduce the following notation that extends that in Eq. (3.51): if Ψα and Φ β are two spinors that (by the conventions which we adopt) belong to the 1̄6 and the 16 of SO(10), respectively, we define [αβ]Φ = (ΨΓc) [αΦβ] − (ΨΓb) [α(ΓbΓcΦ) β]. (3.58) Then, from Eqs. (2.17) and (3.54), L[αβγδ] can be rewritten as [αβγδ] = − , (3.59) where ab = ΩΛ[αβ]c λ− 2Y Λ c ∂λ, (3.60) and Ω is defined in (2.28). Using Eqs. (3.59) and (3.60), the l.h.s. of Eq. (3.57) splits in three parts: αβγδ] = − [λ[ǫL αβγδ] 1 + λ αβγδ] 2 + λ αβγδ] 3 ], (3.61) where we have defined αβγδ] 1 = λ [ǫ(ΩΛαβc λ)(ΩΛ λ), (3.62) αβγδ] 2 = −2 λ[ǫ(ΩΛαβc λ)(Y Λ γδ]c∂λ) + λ[ǫ(Y Λαβc ∂λ)(ΩΛ γδ]cλ) , (3.63) αβγδ] 3 = 4λ [ǫ(Y Λαβc ∂λ)(Y Λ γδ]c∂λ). (3.64) To compute the l.h.s. of Eq. (3.57), one must shift the fields Ω to the left using the rearrange- ment formula. Then αβγδ] 1 = Ωσ(Ωτ (λ [ǫ(Λαβc λ) σ(Λγδ]cλ)τ )) + ΩσA [ǫαβγδ]σ 1 + A [ǫαβγδ] 0 , (3.65) αβγδ] 2 = ΩσB [ǫαβγδ]σ [ǫαβγδ] 0 , (3.66) where A1, A0, B1 and B0 are Ω-independent, Y -dependent fields. The term quadratic in Ω in the r.h.s. of Eq. (3.65) vanishes since it contains the factor λ[ǫλβ(Γbcλ) δ]. An explicit calculation shows that the terms linear in Ω in (3.65) and (3.66) cancel each other: [ǫαβγδ]σ 1 + ΩσB [ǫαβγδ]σ 1 = 0, (3.67) and that the sum of the terms of zero-order in Ω in (3.65), (3.66) and (3.64) vanishes [ǫαβγδ] [ǫαβγδ] 0 + λ αβγδ] 3 = 0, (3.68) so that (3.57) is proved. The details of this calculation are given in Appendix C. 4 Y-formalism for the non-minimal pure spinor formal- In this section, we would like to construct the Y-formalism for the non −minimal pure spinor formalism which has been recently proposed by Berkovits [27]. Before doing that, we will first review the non-minimal pure spinor formalism briefly. The main idea is to add to the fields involved in the minimal formalism a BRST quartet of fields λ̄α, ω̄ α, rα and s α in such a way that their BRST variations are δλ̄α = rα, δrα = 0, δs α = ω̄α and δω̄α = 0. Here, λ̄α is a bosonic field, rα is a fermionic field, and ω̄ α and sα are the conjugate momenta of λ̄α and rα, respectively. These fields are required to satisfy the pure spinor conditions λ̄Γaλ̄ = 0, λ̄Γar = 0. (4.1) The action is then obtained by adding to the conventional pure spinor action I in Eq. (2.2), Ī given by the BRST variation of the ”gauge fermion” F = − (s∂̄λ̄) so that Inm ≡ I + Ī = ∂Xa∂̄Xa + pα∂̄θ α − ωα∂̄λ α + sα∂̄rα − ω̄ α∂̄λ̄α). (4.2) In addition to the ω-symmetry Eq. (2.4), due to the conditions Eq. (4.1), this action is invariant under new gauge symmetries involving ω̄ and s, δω̄α = Λ(1)a (Γ aλ̄)α − Λ(2)a (Γ ar)α, δsα = Λ(2)a (Γ aλ̄)α, (4.3) where Λ(1)a and Λ a are local gauge parameters. Let us note that the conditions Eq. (4.1) and these symmetries reduce the independent components of each field in the quartet to eleven com- ponents. It is easy to show that the action Inm is invariant under the new BRST transformation with BRST charge Qnm = dz(λαdα + ω̄ rα). (4.4) Of course the quartet does not contribute to the central charge and has trivial cohomology with respect to the (new) BRST charge. As a final remark, it is worthwhile to recall that this new formalism can be interpreted [27] as a critical topological string with ĉ = 3 and (twisted) N = 2 supersymmetry. Then it is possible to apply topological methods to the computation of multiloop amplitudes where a suitable regularization factor replaces picture-changing operators to soak up zero modes. The covariant b field and the regulator proposed in [27] allow to compute loop amplitudes up to g = 2. A more powerful regularization of b that allows to compute loop amplitudes at any g loop has been presented in [28]. This regularization is gauge invariant but Lorentz non-covariant since it involves the Y -field. However, this non-covariance is harmless since the regularized b field differs from the covariant one by BRST-exact terms. Now we are ready to present the Y-formalism for the non-minimal pure spinor quantization. As in Eqs. (2.10) and (2.12), we first introduce the non-covariant object Ȳ α = , (4.5) and the projector (Γaλ̄)α(Ȳ Γa)β, (4.6) where v̄α is a constant pure spinor so that we have Ȳ ΓaȲ = 0. (4.7) Let us note that the conditions (4.1) lead to relations λ̄αK̄ β = rαK̄ β = 0, which imply that λ̄α and rα have respectively eleven independent components. Next we postulate the following OPE’s among ω̄α, λ̄α, s α and rα: < ω̄α(y)λ̄β(z) >= y − z (δαβ − K̄ β(z)), (4.8) < sα(y)rβ(z) >= y − z (δαβ − K̄ β(z)), (4.9) < ω̄α(y)rβ(z) >= y − z [K̄α β(z)(Ȳ r)(z)− (Γar)α(z)(Ȳ Γa)β(z)], (4.10) < sα(y)λ̄β(z) >= 0. (4.11) Then, with these OPE’s it is easy to check that the OPE’s between the conjugate momenta ω̄α and sα, and the conditions (4.1) vanish identically: < ω̄α(y)(λ̄Γaλ̄)(z) > = 0, < ω̄α(y)(λ̄Γar)(z) > = 0, α(y)(λ̄Γaλ̄)(z) > = 0, < sα(y)(λ̄Γar)(z) > = 0. (4.12) Notice that (4.10) follows for consistency by acting with the BRST charge Qnm on (4.8) (or (4.9)). Following [27], the only holomorphic currents involving ω̄ and s and gauge invariant under (4.3) are: • i) the bosonic currents N̄ab = (ω̄Γabλ̄− sΓabr), J̄λ̄ = ω̄λ̄− sr, Tλ̄ = ω̄∂λ̄− s∂r, (4.13) those are, the Lorentz current, the ghost current and the stress energy tensor of the non-minimal fields, respectively. • ii) the fermionic currents Sab = sΓabλ̄, S = sλ̄, S(b) = s∂λ̄. (4.14) • iii) the doublet J0 = rs, Φ0 = ω̄r. (4.15) Using the fundamental OPE’s (4.8)-(4.11), one can compute the OPE’s among these operators. The OPE’s of N̄ab, Tλ̄ and J̄λ̄ with λ̄ and r and the ones among themselves are canonical, namely < N̄ab(y)λ̄α(z) > = y − z (Γabλ̄)α(z), < N̄ab(y)rα(z) >= y − z (Γabr)α(z), < J̄λ̄(y)λ̄α(z) > = y − z λ̄α(z), < J̄λ̄(y)rα(z) >= y − z rα(z), < Tλ̄(y)λ̄α(z) > = y − z ∂λ̄α(z), < Tλ̄(y)rα(z) >= y − z ∂rα(z), (4.16) < N̄ab(y)N̄cd(z) > = − y − z (ηc[bN̄a]d − ηd[bN̄a]c)(z), < N̄ab(y)J̄λ̄(z) > = 0, < N̄ab(y)Tλ̄(z) >= (y − z)2 N̄ab(z), < J̄λ̄(y)J̄λ̄(z) > = 0, < J̄λ̄(y)Tλ̄(z) >= (y − z)2 J̄λ̄(z), < Tλ̄(y)Tλ̄(z) > = (y − z)2 Tλ̄(z) + y − z ∂Tλ̄(z). (4.17) Notice that in contrast with the operators T , Nab and J in (2.16)-(2.18), the operators N̄ab, Tλ̄ and J̄λ̄ do not involve Ȳ -correction terms since the Ȳ -dependent terms which arise in their OPE’s are absent or cancel in the combinations (4.13). It is instructive to see explicitly how this cancellation arises. Let us write N̄ (ω̄λ̄) ω̄Γabλ̄ and N̄ sΓabr and consider for instance the OPE between N̄ab = N̄ (ω̄λ̄) ab − N̄ ab and rα. From Eq. (4.9), one obtains ab (y)rα(z) >= y − z (Γabr)α + y − z (Γf Ȳ )α(λ̄Γ fΓabr). (4.18) Then, the second term in the r.h.s. of (4.18) is exactly cancelled by the contribution of the OPE < N̄ (ω̄λ̄) ab (y)rα(z) > in terms of Eq. (4.10). As a second example, consider the OPE < N̄ab(y)N̄cd(z) >. The double poles coming from < N̄ (ω̄λ̄) ab (y)N̄ (ω̄λ̄) ab (z) > are cancelled by those coming from < N̄ ab (y)N̄ ab (z) >. As for the simple poles, one has (ω̄λ̄) ab (y)N̄ (ω̄λ̄) cd (z) > + < N̄ ab (y)N̄ cd (z) > y − z (ηc[bN̄a]d − ηd[bN̄a]c) + [(sΓabΓf Ȳ )(rΓ fΓcdλ̄) + (sΓcdΓf Ȳ )(rΓ fΓabλ̄)], (4.19) but the terms, which are independent of N̄ab in the r.h.s. of (4.19), are just cancelled by the contributions stemming from −(< N̄ (ω̄λ̄) ab (y)N̄ cd (z) > + < N̄ ab (y)N̄ (ω̄λ̄) cd (z) >). For all the remaining OPE’s in both (4.16) and (4.17), the spurious, Ȳ -dependent terms are absent or cancelled in a similar way. Moreover, the OPE’s among Sab, S and S(b) are regular and those of N̄ab, J̄λ̄ and Tλ̄ with S ab, S and S(b) are canonical so that S ab, S and S(b) are covariant primary fields with weight 1 and ghost number 2 with respect to the ghost current J̄λ̄. Thus, as for N̄ab, J̄λ̄ and Tλ̄, they do not have to include Ȳ -dependent corrections. The story is completely different for the currents Jr and Φ. Indeed, using the OPE’s (4.8)- (4.11), one finds < (rs)(y)N̄ab(z) >= (y − z)2 Ȳ Γabλ̄. (4.20) And since one has < (Ȳ ∂λ̄)(y)N̄ab(z) >= (y − z)2 Ȳ Γabλ̄, (4.21) the Ȳ -dependent term in < Jr(y)N̄ ab(z) > disappears if one assumes, as definition of Jr at quantum level, Jr = rs− 3Ȳ ∂λ̄. (4.22) With this definition, the OPE’s of Jr with N̄ab, J̄λ̄, Tλ̄, S ab, S and S(b) read < Jr(y)Jr(z) > = (y − z)2 < Jr(y)N̄ ab(z) > = 0, < J̄λ̄(y)Jr(z) > = (y − z)2 < Jr(y)Tλ̄(z) > = (y − z)3 (y − z)2 < Jr(y)S ab(z) > = y − z < Jr(y)S(z) > = y − z < Jr(y)S(b)(z) > = y − z S(b). (4.23) In particular, note that the coefficient 8 of the double pole in the contraction < J̄λ̄(y)Jr(z) > emerges from the arithmetic 8 = 11 − 3 where 11 comes from the first term and −3 from the second term in (4.22). In a similar manner, for Φ one has < (ω̄r)(y)N̄ab(z) > = − (y − z)2 [Ȳ Γabr − (Ȳ r)(Ȳ Γabλ̄)], < (ω̄r)(y)Sab(z) > = − (y − z)2 Ȳ Γabλ̄. (4.24) Therefore, at quantum level Φ must be defined as Φ = ω̄r + 3[Ȳ ∂r − (Ȳ r)(Ȳ ∂λ̄)] = ω̄r + 3∂(Ȳ r), (4.25) in order to avoid spurious Ȳ -dependent terms. With this new definition, one can also derive < Φ(y)N̄ab(z) > = 0, < Φ(y)J̄λ̄(z) > = 0, < Φ(y)Tλ̄(z) > = (y − z)2 < Φ(y)Sab(z) > = y − z < Φ(y)S(z) > = (y − z)2 y − z J̄λ̄, < Φ(y)S(b)(z) > = (y − z)3 (y − z)2 y − z < Φ(y)Jr(z) > = y − z Φ. (4.26) The operator Φ is part of the BRST current and S(b) is a contribution of the b ghost as will be seen in the next section. From the definitions (4.13) and (4.14), one finds that the operators N̄ab, J̄λ̄ and Tλ̄ are the BRST variations of the operators Sab, S, and S(b), respectively. Moreover, contrary to what happens for the operators in (2.16)-(2.18), the correction term of Jr in (4.22) is not BRST-exact but its BRST variation is just the correction term for −Φ in (4.25), so that Φ is just the BRST variation of −Jr. These properties are fully consistent with the OPE’s we have computed thus far. 8 As a final remark, let us note that in all the derivations of this section (but the second equality of (4.25)) we have never used the fact that v̄ in (4.5) is constant and therefore all the equations in this section remain true even if one replaces Ȳ α with Ỹ α ≡ λ 8Apart from a difference in the OPE < ΦS > where we find a double pole with residuum 8, not present in [27] (perhaps a misprint in [27]), our results agree with those computed in [27] by using the U(5)-formalism. 5 A quantum b ghost in the non-minimal pure spinor formalism In Ref. [27], Berkovits has obtained an expression for a covariant b ghost in the framework of non −minimal formalism. His idea was triggered by the observation that in this formalism the non-covariant Yα field can be replaced by a covariant field λ̃α (which will be defined soon) and then one can look for a new, covariant b ghost satisfying the defining equation {Qnm, bnm(z)} = T (z) + Tλ̄(z) ≡ Tnm(z), (5.1) by starting with bnm = λ̃αG α + sα∂λ̄α + · · ·. The result, given in [27], is bnm = s α∂λ̄α + λ̃αG α − 2λ̃β r̃αH + 6λ̃γ r̃β r̃αK [αβγ] − 24λ̃δ r̃γ r̃β r̃αL [αβγδ], (5.2) where we have defined λ̃α = (λ̄λ) r̃α = (λ̄λ) . (5.3) Note that λ̃α and r̃α are primary fields of conformal weight 0 with respect to Tnm. In this section, we will construct a covariant, quantum-mechanical b ghost in the non −minimal pure spinor formalism on the basis of our Y-formalism, taking care of normal-ordering effects. Furthermore, we shall show that this covariant b ghost is cohomologically equivalent to the non-covariant b̃Y ghost, improved by the non-minimal term s α∂λ̄α which takes the form at the classical level b̃0Y = YαG 0 + s α∂λ̄α. (5.4) It is now convenient to consider the following operators: ρ[αβ] ≡ (r̃αλ̃β − r̃βλ̃α) ≡ r̃[αλ̃β], ρ[αβγ] ≡ −r̃[αr̃βλ̃γ], ρ[αβγδ] ≡ −r̃[αr̃β r̃γ λ̃δ], ρ[αβγδǫ] ≡ r̃[αr̃β r̃γ r̃δλ̃ǫ], (5.5) that satisfy the recursive relations Qnm, λ̃α = λβρ[αβ], {Qnm, ρ[αβ]} = λ γρ[αβγ], Qnm, ρ[αβγ] = λδρ[αβγδ], {Qnm, ρ[αβγδ]} = λ ǫρ[αβγδǫ]. (5.6) Next, let us also recall the results which were obtained in section 3 and hold at the quantum level: {Q, Ĝα} = λαT, Q,H [αβ] = λ[αĜβ], {Q,K [αβγ]} = λ[αHβγ], Q,L[αβγδ] = λ[αKβγδ], λ[αLβγδρ] = 0, (5.7) where Ĝα is defined in (3.13). It is also useful to compute the contractions: β(y)ρ[βα](z) > = y − z < H [βγ](y)ρ[γβα](z) > = y − z < K [βγδ](y)ρ[δγβα](z) > = y − z < L[βγδǫ](y)ρ[ǫδγβα](z) > = (y − z)2 y − z . (5.8) After a simple calculation, it turns out that R1α is given by R1α = −2ρ[αβ][λ β(λ̃∂θ)− (Γaλ̃)β(λΓa∂θ)], (5.9) but the second term in the square bracket vanishes when contracted with ρ[αβ] due to the con- ditions (4.1). As for R2α, R3α, R4α and R̃4α, they all contain (at least) a factor abc(λ̃Γ abλ) ≡ λ̃Λ[αβ]c λ and therefore vanish when contracted with ρ[βγ···] by taking into account (5.5), (3.58) and (4.1). 9 To summarize, we have the following results: R1α = −2ρ[αβ]λ β(λ̃∂θ), R2α = R3α = R4α = R̃4α = 0. (5.10) As already noted, the non-minimal b field is expected to be of the form: bnm = S(b) + λ̃G+ · · · . (5.11) The anticommutator of Qnm with S(b) = s∂λ̄ is {Qnm, S(b)} = Tλ̄. (5.12) 9In R̃4α, there is also a term proportional to ρ[αβγδ](Γabc) αβ(Γdec)γδ(λ̃ΓabΓdeλ) that vanishes for the same reason. Now let us compute the anticommutator {Qnm, (λ̃αĜ {Qnm, (λ̃αĜ α)} = λ̃α(λ αT ) + (λβρ[αβ])Ĝ α. (5.13) Using the rearrangement theorem and some algebra, (5.13) can be rewritten as {Qnm, (λ̃αĜ α)} = T + ρ[αβ](λ βĜα) + {Qnm, ∂λ̃∂θ − (λ̃∂λ)(λ̃∂θ)}. (5.14) Here it is of interest to remark that the term ∂λ̃∂θ − (λ̃∂λ)(λ̃∂θ) that arises in the r.h.s. of (5.14) is just the difference between the generalized normal ordering (· · ·) in [33] and the improved one : · · · : of λ̃αĜ α, that is (λ̃αĜ α) =: λ̃αĜ α : +∂λ̃∂θ − (λ̃∂λ)(λ̃∂θ), (5.15) so that (5.14) becomes {Qnm, : λ̃αĜ α :} = T + ρ[αβ](λ βĜα). (5.16) With the help of the second recursive equations in (5.6) and (5.7) the last term in the r.h.s. of Eq. (5.16) reads ρ[αβ](λ [βĜα]) = ρ[αβ]([Qnm, H [βα]]) = {Qnm, ρ[αβ]H [αβ]} − (ρ[αβγ]λ [α)Hβγ]. (5.17) In this case, the rearrangement theorem does not give extra contributions since (ρ[αβγ]λ [α)Hβγ] − ρ[αβγ](λ βγ]) = R2γ∂λ γ + ∂ρ[αβγ]R [αβγ] 3 , (5.18) and the r.h.s. vanishes from (5.10) and (3.33) . Therefore, Eq. (5.17) can be rewritten as ρ[αβ](λ α]) = {Qnm, ρ[αβ]H [αβ]} − ρ[αβγ](λ βγ]). (5.19) For the last term in the r.h.s. of this equation, one can repeat the same procedure using the third recursive equations in (5.6) and (5.7). Again the contributions from the rearrangement theorem are absent since they involve the operators R [αβγδ] 4 and R3α that vanish according to (3.46) and (5.10). As a result, one obtains ρ[αβγ](λ [αHβγ]) = {Qnm, ρ[αβγ]K [αβγ]}+ (ρ[αβγδ]λ [α)Kβγδ]. (5.20) As a last step, one can express the last term in (5.20) in terms of {Qnm, ρ[αβγδ]L [αβγδ]} by using the fourth recursive equations in (5.6) and (5.7). Again the contributions from the rearrangement theorem are absent as before, so we have (ρ[αβγδ]λ [α)Kβγδ] = −{Qnm, (ρ[αβγδ]L [αβγδ])}, (5.21) where we have disregarded the term ρ[αβγδǫ]λ [ǫLαβγδ] that vanishes according to (3.57). Finally, using (5.12) and (5.16)-(5.21) we arrive at the result {Qnm, bnm} = Tnm, (5.22) where bnm = s α∂λ̄α+ : λ̃αĜ α : −2(λ̃β r̃α)H + 6(λ̃γ r̃β r̃α)K [αβγ] − 24(λ̃δ r̃γ r̃β r̃α)L [αβγδ]. (5.23) In conclusion, we have confirmed Eq. (5.2) provided that one interprets the compound field α as the operator : λ̃αĜ α : which is normal-ordered according to the improved prescription (For the other terms in (5.23) the generalized and the improved normal-ordering prescriptions coincide). Incidentally, we have also checked that this bnm possesses conformal weight 2 It might appear from (5.23) and the definition of λ̃ and r̃ that bnm is singular at λ̄λ → 0 with poles up to fourth order. However, as explained in [28], this singularity is not dangerous. Indeed in this case, the analogous of the operator ξ = Y θ that would trivialize the cohomology, ξnm = λ̃λ+ r̃θ = λ̄θ (−rθ)n−1 (λλ̄)n , (5.24) since {Qnm, ξnm} = 1. However, ξnm diverges like (λλ̄) −11 and to have a nontrivial cohomology it is sufficient to exclude from the Hilbert space operators that diverge like ξnm or stronger. Therefore bnm is allowed as insertion to compute higher loop amplitudes. To do actual calcula- tions at more than two loops [28], bnm must be regularized properly. In fact, in [28] a consistent regularization has been proposed. Now let us come back to the non-covariant b ghost b̃0Y in (5.4). As a first step, let us derive a quantum counterpart of b̃0Y , which is denoted as b̃Y . From the first equation in (5.7), one has {Qnm, YαĜ α} = Yα(λ αT ). (5.25) Moreover, since Yα(λ αT )− (Yαλ α)T = 2∂Y ∂λ from the rearrangement theorem, one obtains {Qnm, YαĜ α − 2∂Y ∂θ} = T. (5.26) As before, the term 2∂Y ∂θ is just the difference between (YαĜ α) and : YαĜ α : and therefore the quantum non-covariant b ghost takes the form b̃Y =: YαĜ α : +(s∂λ̄), (5.27) and it satisfies {Qnm, b̃Y } = Tnm. (5.28) Even if b̃Y is non-covariant, its Lorentz variation is BRST-exact. Actually, one has δLb̃Y = Qnm, 2(L αYβYγ)H , (5.29) where Lβα are (global) Lorentz parameters. From (5.22) and (5.28), it follows that b̃Y − bnm is BRST-closed and then it is plausible that it is also exact. Indeed in [34], we have shown that, at the classical level, the covariant non-minimal b ghost (5.2) and the non-covariant one (5.4) are cohomologically equivalent. In this respect, we wish to verify the cohomological equivalence between bnm and b̃Y even at the quantum level bnm = b̃Y + [Qnm,W ], (5.30) where W = 2(λ̃βYα)H [αβ] + 3!(λ̃γ r̃βYα)K [αβγ] + 4!(λ̃δr̃γ r̃βYα)L [αβγδ] +WR, (5.31) with WR being a quantum contribution coming from the rearrangement theorem, which will be determined later. In order to verify (5.30), let us compute the (anti)-commutators of Qnm with the first three terms in the r.h.s. of (5.31). We have 2[Qnm, (λ̃βYα)H [αβ]] = −ραβH [αβ] + (Y[γραβ]λ γ)H [αβ] + (λ̃βYα)(λ [αĜβ]) = −2(λ̃β r̃α)H [αβ] − 3!(Y[γ r̃αλ̃β])(λ γHαβ) + (λ̃αĜ α)− (YαĜ + RH +RG, (5.32) where RH and RG are the contributions coming from the rearrangement theorem of the last two terms in the first row of this equation. Then 3!{Qnm, (Yαr̃βλ̃γ)K [αβγ]} = 3!(r̃αr̃βλ̃γ)K [αβγ] − 4!(Yαr̃β r̃γ λ̃δ)(λ [δKαβγ]) + 3!(Yαr̃βλ̃γ)(λ βγ]) +RK , (5.33) where RK arises from rearrangement theorem. Finally, we have Qnm, (Yαr̃β r̃γ λ̃δL [αβγδ]) = 4!r̃αr̃β r̃γ λ̃δL [αβγδ] + 4!(Yαr̃β r̃γλ̃δ)(λ [αKβγδ]) +RL, (5.34) where RL comes from rearrangement formula. The quantum contributions RG, RH , RK and RL are explicitly given by RG = −[∂λ̃∂θ − (λ̃∂λ)(λ̃∂θ)− 2∂Y ∂θ] + 2[Qnm, (Yαλ̃β)W R1 ]− GΠa, (5.35) RH = 3![Qnm, (Yαr̃βλ̃γ)W [αβγ] R2 ]− AaHα(dΓa) AaGΠa, (5.36) RK = 4![Qnm, (Yαr̃β r̃γλ̃δ)W [αβγδ] R3 ] + AaHα(dΓa) AcKαβN c , (5.37) RL = − AcKαβN c +BL, (5.38) where ((Y + λ̃)Γa) [α∂λβ]Πa, [αβγ] ((Y + 2λ̃)Γa) β(Γad)γ], [αβγδ] ((Y + 3λ̃)Γa) [α∂λβNγδ]a, (5.39) AaG = 3!Y[αr̃βλ̃γ]λ γ((Y + 2λ̃)Γa)α∂λβ , AaHα = 4!Y[αr̃β r̃γλ̃δ]λ δ((Y + 3λ̃)Γa)β∂λγ , AaKαβ = 5!Y[αr̃β r̃γ r̃δλ̃ǫ]λ ǫ((Y + 4λ̃)Γa)δ∂λγ . (5.40) The Y -dependent operators AaG, A Hα and A Kαβ cancel when (5.35)-(5.38) are summed up. As for BL, it turns out that it is BRST-exact: BL = 4! Qnm, (Yαr̃β r̃γλ̃δ)W [αβγδ] Qnm, ∂((Yαr̃β r̃γ λ̃δ)W [αβγδ] , (5.41) where [αβγδ] [(ΓcY )[α(Γb(Y + 3λ̃))β(Γbc∂λ) γ∂λδ] − (ΓcY )[α(Γb(Y + 3λ̃))β(Γbcλ) γ∂2λδ] + 3(λ̃ΓcY )(Γb(Y + 2λ̃))[α∂λβ(ΓcΓ aλ)γ(ΓbΓa∂λ) δ]], (5.42) [αβγδ] (ΓcY )[α(Γb(Y + 3λ̃))β[(Γbcλ) γ∂λδ] − aλ)γ(ΓbΓa∂λ) δ]]. (5.43) Some details on the derivations of these results will be given in Appendix C. From (5.15), one finds that the term −∂λ̃∂θ+ (λ̃∂λ)(λ̃∂θ)− 2∂Y ∂θ transforms (λ̃αĜ α)− (YαĜ α) to : λ̃αĜ α : − : Collecting Eqs. (5.32)-(5.43), one recovers (5.30) where bnm and b̃Y are given in (5.23) and (5.27), respectively and W = 2(λ̃βYα)(H [αβ] +W R1 ) + 3!(λ̃γ r̃βYα)(K [αβγ] +W [αβγ] + 4!(λ̃δ r̃γ r̃βYα)(L [αβγδ] +W [αβγδ] R3 +W [αβγδ] R4 ) + 4!∂[(λ̃δ r̃γ r̃βYα)W [αβγδ] R5 ]. (5.44) 6 Conclusion In this article, using the Y-formalism [30], we have calculated the normal-ordering contributions existing in various composite operators in the pure spinor formalism of superstrings. These operators naturally appear when we try to construct a b ghost. Moreover, we have constructed the Y-formalism for the non-minimal sector. Using these information, we have presented a quantum-mechanical expression of the b ghost, bnm, in the non-minimal formulation and we have shown, in this case, that the non-covariant b field bY and bnm, are equivalent in cohomology. The consistent results we have obtained in this article could be regarded as a consistency check of the Y-formalism in the both minimal and non-minimal pure spinor formulation of superstrings. In the case of the non-minimal formulation, due to its field content and structure, it is natural to ask if it is possible to reach a fully covariant system of rules for the OPE’s in the minimal and non-minimal ghost sectors, by replacing the non-covariant fields Yα and Ȳ α with the covariant ones λ̃α = and Ỹ α = λ , respectively. As for the replacement of Ȳ α with Ỹ α, that is of v̄α with λα for the non-minimal sector, we do not see any problem, as noted at the end of section 4 because v̄α and λα are both BRST invariant and all the OPE’s among the currents of the non-minimal sector remain unchanged. On the contrary, a naive, straightforward replacement of Yα with λ̃α looks problematic. Indeed, even if the OPE’s among the Lorentz current Nab, the ghost current J , and the stress energy tensor Tλ of the minimal ghost sector are unchanged, those among these operators and that of the non-minimal sector become different from zero, since the correction terms in (2.16)- (2.18) now acquire a dependence from λ̄. Therefore the OPE’s among the total Lorentz current, ghost current and stress energy tensor of the (minimal and non-minimal) ghost sector do not close correctly. Moreover, the BRST variation of (2.14) appears to be inconsistent. We cannot exclude a possibility that these problems could be overcome by a smart modification of the basic OPE’s, but it is far from obvious that a consistent modification could be found. Thus, in this paper, we have refrained from exploring this possibility further and we hope to come back to this question in future. Acknowledgements The work of the first author (I.O.) was partially supported by the Grant-in-Aid for Scientific Research (C) No.14540277 from the Japan Ministry of Education, Science and Culture. The work of the second author (M.T.) was supported by the European Community’s Human Po- tential Programme under contract MRTN-CT-2004-005104 ”Constituents, Fundamental Forces and Symmetries of the Universe”. A Notation, Conventions and Useful identities In this appendix, we collect our notation, conventions and some useful formulae employed in this paper. As usual, in ten space-time dimensions, Γa are the Dirac matrices γa times the charge conjugation matrix C, that is, (Γa)βα = (γaC)αβ and (Γa)βα = (C −1γa)αβ; they are 16 × 16 symmetric matrices with respect to the spinor indices, and satisfiy the Clifford algebra {Γa,Γb} = 2ηab. Our metric convention is ηab = (−,+, · · · ,+). The square bracket and the brace respectively denote the antisymmetrization and the sym- metrization of p indices, normalized with a numerical factor 1 so that, for instance A[µBν] = (AµBν − AνBµ). As for the products of Γ a, Γa1···ap = Γ[a1···ap]. These antisymmetrized products of Γ have definite symmetry properties, which are given by (Γab)α β = −(Γ (Γabc)αβ = −(Γ abc)βα, (Γ abcd)α β = (Γ abcd)β α, (Γabcde)αβ = (Γ abcde)βα, etc. The product of generic spinors fα and gβ can be expanded in terms of the complete set of gamma matrices as fαgβ = Γaαβ(fΓag) + 16× 3! Γabcαβ (fΓabcg) + 16× 5! Γabcdeαβ (fΓabcdeg). (A.1) Similarly, for spinors fα and g β we have δβα(fg) + 16× 2! (Γab)α β(fΓabg) + 16× 4! (Γabcd)α β(fΓabcdg). (A.2) A useful identity, involving three spinor-like operators Aα, B β and Cγ is (BΓabA)(ΓabC) (BA)Cα = (BβA α)Cβ − ((ΓaB)αAβ)(ΓaC)β. (A.3) B Normal ordering, the generalized Wick theorem and rearrangement theorem In this appendix, we explain the prescription of normal ordering, the generalized Wick theorem and rearrangement theorem, which are used in this paper. The more detail of them can be seen in the texbook of conformal field theory [33]. B.1 Normal ordering In conformal field theory, we usually consider normal ordering for free fields where the OPE contains only one singular term with a constant coefficient. Then, normal ordering is defined as the subtraction of this singular term. This definition of normal ordering is found to be equivalent to the conventional normal ordering in the mode expansion where the annihilation operators are placed at the rightmost position. However, we sometimes meet the case for which the fields are not free in this sense. One of the well-known examples happens when we try to regularize the OPE between two stress enery tensors T (y)T (z). In this case, we have two singular terms where one singular term contains the quartic pole whose coefficient is proportional to the central charge while the other singular term contains the quadratic pole whose coefficient is not a constant but (2×) stress energy tensor itself. The ususal normal ordering prescription amounts to subtraction of the former, most singular term, but the latter singular term is still remained. Let us note that in the present context, the OPE between ω and λ is not free owing to the existence of the projection K reflecting the pure spinor constraint. From the physical point of view, we want to subtract all the singular terms in the OPE’s, so we have to generalize the definition of normal ordering. To this end, we introduce the generalized normal ordering which is usually denoted by parentheses, that is, explicitly, the generalized normal ordering of operators A and B is written as (AB)(z). A definition of the generalized normal ordering is given by the contour integration (AB)(z) = w − z A(w)B(z). (B.1) Then the OPE of A(z) and B(w) is described by A(z)B(w) =< A(z)B(w) > +(A(z)B(w)), (B.2) where < A(z)B(w) > denotes the contraction containing all the singular terms of the OPE and (A(z)B(w)) stands for the complete sequence of regular terms whose explicit forms can be extracted from the Taylor expansion of A(z) around w: (A(z)B(w)) = (z − w)k (∂kA ·B)(w). (B.3) Another definition of the generalized normal ordering is provided by the mode expansion. If the OPE of A and B is written as A(z)B(w) = {AB}k(w) (z − w)k , (B.4) where N is some positive integer, the definition of the generalized normal ordering reads (AB)(z) = {AB}0(z). (B.5) Incidentally, in this context, the contraction is expressed by < A(z)B(w) >= {AB}k(w) (z − w)k , (B.6) In this paper, we adopt the definition of the contour integration (B.1). Moreover, for simplicity, we do not write explicitly the outermost parenthesis representing the generalized normal ordering whenever we can easily judge from the context whether some operators are normal-odered or not. B.2 The generalized Wick theorem Relating to the generalization of the normal-ordering prescription, we also have to reformulate the Wick theorem for interacting fields. In general, the Wick theorem relates the time-ordered product to the normal-ordered product of free fields. However, such a relation cannot be generalized to interacting fields in a straightforward manner. Hence, the generalized Wick theorem is defined by generalizing a special form of the Wick theorem for the contraction of free fields. More explicitly, the generalized Wick theorem is simply defined as < A(z)(BC)(w) >= [< A(z)B(x) > C(w) +B(x) < A(z)C(w) >]. (B.7) From this definition, it is important to notice that the first regular term of the various OPE’s always contributes. If we would like to calculate < (BC)(z)A(w) >, we first calculate < A(z)(BC)(w) >, then interchange w and z, and finally expand the fields evaluated at z in the Taylor series around w. B.3 Rearrangement theorem We often encounter the situation where many of operators are normal-ordered, e.g., (A(BC))(z). With the generalized normal ordering, some complication occurs since there is no associativity in such normal-ordered operators (A(BC))(z) 6= ((AB)C)(z). (B.8) To deal with normal ordering of such composite operators, we make use of the rearrangement theorem. The useful formulae are given by (AB) = (BA) + ([A,B]), (B.9) (A(BC)) = (B(AC)) + (([A,B])C), (B.10) ((AB)C) = (A(BC)) + (A([C,B])) + (([C,A])B) + ([(AB), C]), (B.11) where A, B, and C are all the Grassmann-even quantities. Note that if the Grassmann-odd quantities are involved, we must change the sign and the commutator in a suitable manner. For instance, for the Grassmann-even A and the Grassmann-odd B and C, the last rearrangement theorem is modified as ((AB)C) = (A(BC))− (A({B,C}))− (([C,A])B) + ({(AB), C}). (B.12) In making use of these rearrangement theorems, we are forced to evaluate the generalized normal ordering of the (anti-)commutator ([A,B]). Then, we rely on the useful formula ([A,B])(z) = (−1)k+1 ∂k{AB}k(z). (B.13) Note that field-dependent singular terms contribute to the normal-ordering (anti-)commutator while the non-singular term {AB}0 does not. In this paper, we make heavy use of these formulae in evaluating various normal-ordered products of operators. C Some details about the calculations C.1 BRST variation of Gα To compute the BRST variation of Gα it is convenient to use the following notation gα(B,A,C) = − (BΓabA)(ΓabC) (BA)Cα = (BβA α)Cβ − ((ΓaB)αAβ)(ΓaC)β, (C.1) where Aα, Bβ , and C γ are generic spinors and the last step is the identity (A.3). Then, given (3.5), one has {Q,Gα1} = − λα(ΠaΠa) + (λΓa∂θ)(Γ ad)α. (C.2) Moreover, {Q,Gα2 +G 3} = −g α(d, λ, ∂θ) + gα(Ω, λ, ∂λ)− 2gα(Y, ∂λ, ∂λ)− (Y ∂λ)∂λα. (C.3) The last three terms come from the definitions (2.25) and (2.26) of Nab and J . Using the rearrangement formula (cf. (B.12)), one has gα(d, λ, ∂θ) = λα(d∂θ) + 8∂2λα + (λΓa∂θ)(Γ ad)α, (C.4) gα(Ω, λ, ∂λ) = (Ωβλ α)∂λβ − ((ΓaΩ)αλβ)(Γa∂λ)β . (C.5) Using the rearrangement theorem, the first term in the r.h.s. of Eq. (C.5) becomes α)∂λβ = λα(Ω∂λ)− (Y Γa)α(∂λΓa∂λ) + , (C.6) whereas the second term can be rewritten as ((ΓaΩ)αλβ)(Γa∂λ)β = − ∂2λα + (Y ∂2λ)λα + (Y ∂λ)∂λα + 3(∂Y ∂λ)λα + 2gα(Y, ∂λ, ∂λ) + (Y Γa)α(∂λΓa∂λ), (C.7) so that from (C.5)-(C.7), one obtains gα(Ω, λ, ∂λ) = λα(Ω∂λ) + 4∂2λα + (Y ∂2λ)λα + (Y ∂λ)∂λα + 3(∂Y ∂λ)λα + 2gα(Y, ∂λ, ∂λ). (C.8) Adding Eqs. (C.2), (C.3) and {Q,Gα4} = c1∂ 2λα with c1 = ,taking into account (C.4), (C.8) and using the definition (2.16) of the stress energy tensor T , we finally obtain {Q,Gα} = λαT − ∂2λα. (C.9) C.2 BRST variation of Hαβ Now let us consider the BRST variation of Hαβ. Eq. (3.25) can be rewritten as [Q,H(αβ)] = Γαβa h , (C.10) where (λΓaΓbd)Π b +Nab(λΓb∂θ)− J(λΓa∂θ) + 2∂(λΓa∂θ). (C.11) The first term in the r.h.s. of this equation can be rewritten as (λΓaΓbd)Π (λΓaΓbΠ bd) + 5∂(λΓa∂θ). (C.12) With the notation Λαβ ≡ ∂λ[αλβ], Λ̃[αβ] ≡ − (ΓcΛΓc)[αβ], (C.13) the vector V a = Nab(λΓb∂θ)− J(λΓa∂θ), (C.14) becomes V a = (ΩΓaΓbλ)(λΓb∂θ)− J(λΓ a∂θ) + 4(Y ΛΓa∂θ) + 4(Y ΓaΛ̃∂θ) + 2(∂λΓa∂θ). (C.15) The first term in the r.h.s. of (C.15) vanishes modulo a rearrangement contribution: (ΩΓaΓbλ)(λΓb∂θ) = −4(Y Γ aΛ̃∂θ)− 4(Y ΛΓa∂θ) + 4(∂λΓa∂θ), (C.16) so that ha becomes (λΓaΓbΠbd) + 5∂(λΓ a∂θ)− J(λΓa∂θ) − 2(∂λΓa∂θ) + 2∂(λΓa∂θ). (C.17) On the other hand, λΓaG = (λΓaΓbΠ bd) + (λΓa∂2θ) + Ṽ a, (C.18) where Ṽ a = −1 (λ̃ΓaN bcΓbc∂θ)− (λΓaJ∂θ). Then, using (2.25) and (2.26) Ṽ a = − (λΓc(ΩΓ aΓcλ)∂θ) + (λΓcΓb(ΩΓ bΓcλ)Γa∂θ)− (λΓa(Ωλ)∂θ) − 4(Y ΓaΛ̃∂θ) − 4(Y ΛΓ a∂θ) + 2(Y ΛΓa∂θ) + (∂λΓa∂θ). (C.19) But the first two terms in the r.h.s. of (C.19) vanish modulo the Y-dependent term 4[(Y ΓaΛ̃∂θ)+ (Y ΛΓa∂θ)] − 6(Y ΛΓa∂θ) coming from rearrangement theorem, so that we have Ṽ a = −(λΓa(Ωλ)∂θ) − 4(Y ΛΓa∂θ) + (∂λΓa∂θ) = −J(λΓa∂θ) + 4∂(λΓa∂θ)− 4(λΓa∂2θ), (C.20) and therefore λΓaG = (λΓaΓbΠ bd) + (λΓa∂2θ)− J(λΓa∂θ)− 4(λΓa∂2θ) + 4∂(λΓa∂θ). (C.21) Then comparing (C.17) with (C.21), one gets Eq. (3.27). Next let us consider the BRST variation of H [αβ]. Eq. (3.26) can be rewritten as [Q,H [αβ]] = (λΓabcΓdΠdd) + 6(λN abΓc∂θ) + 4(λΓabc∂2θ) + (∂λΓabc∂θ) , (C.22) where the last two terms in the r.h.s. of this equation come from normal ordering. On the other hand, λΓabcĜ = (λΓabcΓdΠdd) + 4(λΓ θ) + 6(λN [abΓc]∂θ) + 3(λΓfN f [aΓbc]∂θ) + (λΓfΓgΓ abcNfg∂θ)− (λΓabcJ∂θ). (C.23) Using (2.30), (2.31) and the notation introduced in (C.13) the quantity in the second row of (C.23), that is, [abc] = +3(λΓfN f [aΓbc]∂θ) + (λΓfΓgΓ (λΓabcJ∂θ), (C.24) can be rewritten as [abc] = − (λΓfΓ [ab(ΩΓc]Γfλ)∂θ) + (λΓfΓgΓ abc(ΩΓfΓgλ)∂θ)− 12(Y ΓaΛ̃Γbc∂θ) − 6(Y ΛΓabc∂θ) + (∂λΓabc∂θ). (C.25) On the other hand, by reordering, the sum of the first two, Ω-dependent terms in (C.25) yields 12(Y ΓaΛ̃Γbc∂θ) + 6(Y ΛΓabc∂θ) so that h[abc] = ∂λΓabc∂θ and (C.23) becomes λΓabcĜ = (λΓabcΓdΠdd) + 4(λΓ θ) + 6(λN [abΓc]∂θ) + (∂λΓabc∂θ). (C.26) By comparing (C.22) with (C.26) one gets Eq. (3.28). C.3 BRST variation of K [αβγ] Now let us check (3.41). Let us rewrite (3.40) as {Q,K [αβγ]} = k [αβγ] 1 + k [αβγ] 2 , (C.27) where [αβγ] 2 = − (Γad) (Γad)βλγ] − (Γbd) βΓbaλγ] abc(dΓ abcd), (C.28) [αβγ] aΓdλ)[αNβγ]a aΓdλ)[α[ (ΩΛβγ]a λ)− (Y Λ a ∂λ)]. (C.29) The first term in the r.h.s. of (C.29) can be elaborated as follows: aΓdλ)[α(ΩΛβγ]a λ) = Πd(ΩΓb)[αλβ(ΓbΓdλ) γ] +∆[αβγ] Πdλ[α(ΩΛ d λ) + ∆ [αβγ] + ∆̂[αβγ], (C.30) where ∆[αβγ] and ∆̂[αβγ] are the contributions of rearrangement theorem and are given by ∆[αβγ] = abc (Γ aΓd∂KΓ bcλ)γ] Πf (ΓaY )[α[∂λβ(ΓaΓfλ) γ] + (ΓaΓf∂λ) βλγ] − (ΓfΓb∂λ) β(ΓaΓ bλ)γ]] Πd(ΓdY ) [α∂λβλγ], (C.31) ∆̂[αβγ] = Πd(ΓfY )[α(∂λΓfΛ d λ). (C.32) Therefore, k [αβγ] 1 becomes [αβγ] Πdλ[αN d + {Π d(λ[αY Λ d ∂λ)− aΓdλ)[α(Y Λβγ]a ∂λ) + ∆̂[αβγ] +∆[αβγ]}. (C.33) With a little algebra, it is easy to show that the terms in the curly bracket in the r.h.s. of (C.33) vanish so that (C.33) becomes [αβγ] λ[αΠdN d . (C.34) Then, (C.27), together with (C.28), (C.34) and (3.23), reproduces Eq. (3.41). C.4 The vanishing of λ[ǫLαβγδ] Now let us consider λ[ǫLαβγδ] in order to verify that it vanishes. As discussed at the end of section 3, it consists of three terms: αβγδ] 1 = λ [ǫ(ΩΛαβc λ)(ΩΛ γδ]cλ) = ΩσA [ǫαβγδ]σ 1 + A [ǫαβγδ] 0 , (C.35) αβγδ] 2 = ΩσB [ǫαβγδ]σ [ǫαβγδ] 0 , (C.36) αβγδ] 3 = 4λ [ǫ(Y Λαβc ∂λ)(Y Λ γδ]c∂λ) (C.37) where ΩσB [ǫαβγδ]σ [ǫαβγδ]σ 1 = −4Ωσλ [ǫ(Y Λ[αβc ∂λ)(Λ γδ]cλ)σ, (C.38) and ΩσA 1 , A0 and B0 are rearrangement contributions coming when Ωσ is shifted to the left. The explicit calculation of ΩσA 1 gives [ǫαβγδ]σ 1 = Ωσ(Λ c[αβλ)σ(∂λΓfΛ c λ)(Γ fY )ǫ] + Ωσ(Λ c[αβΓfY )σ(∂λΓfΛ c λ)λ ǫ]. (C.39) The first term in the r.h.s. of this equation can be rewritten as 4Ωσλ [ǫ(Y Λ[αβc ∂λ)(Λ γδ]cλ)σ − 2Ωσ(Λ c[αβλ)σ(ΓcY ) γ∂λδλǫ] and the second one as 2Ωσ(Λ c[αβλ)σ(ΓcY ) γ∂λδλǫ] so that we have [ǫαβγδ]σ 1 = 4Ωσλ [ǫ(Y Λαβc ∂λ)(Λ γδ]cλ)σ. (C.40) Then, using (C.38) and (C.40), Eq. (3.67) can be derived. As for A [ǫαβγδ] 0 and B [ǫαβγδ] 0 , the explicit calculation gives [ǫαβγδ] (ΓfY )[ǫ(Y Λαβc λ)(∂λΓfΛ (ΓfY )[ǫ(Y Λαβc ∂λ)(∂λΓfΛ ∂((ΓfY )[ǫ(Y Λαβc λ)(∂λΓfΛ γδ]cλ)), (C.41) [ǫαβγδ] 0 = −2(Γ fY )[ǫ(Y Λαβc ∂λ)(∂λΓfΛ γδ]cλ) + 4λ[ǫ(Y Λαβc ∂λ)(Y Λ γδ]c∂λ) + 2∂λ[ǫ(Y Λαβc λ)(Y Λ γδ]c∂λ)− 2∂(λ[ǫ(Y Λαβc ∂λ)(Y Λ γδ]cλ)), (C.42) so that we obtain [ǫαβγδ] [ǫαβγδ] (ΓfY )[ǫ(Y Λαβc λ)(∂λΓfΛ γδ]c∂λ)− (ΓfY )[ǫ(Y Λαβc ∂λ)(∂λΓfΛ γδ]cλ) + 4λ[ǫ(Y Λαβc ∂λ)(Y Λ γδ]c∂λ) + 2∂λ[ǫ(Y Λαβc λ)(Y Λ γδ]c∂λ) ∂{(ΓfY )[ǫ(Y Λαβc λ)(∂λΓfΛ λ))− 4λ[ǫ(Y Λαβc ∂λ)(Y Λ λ)}. (C.43) In order to verify (3.68), one needs three useful identites: (ΓfY )[ǫ(Y Λαβc λ)(∂λΓfΛ λ) = 4λ[ǫ(Y Λαβc ∂λ)(Y Λ λ), (C.44) (ΓfY )[ǫ(Y Λαβc λ)(∂λΓfΛ γδ]c∂λ) = 5λ[ǫ(Y Λαβc ∂λ)(Y Λ γδ]c∂λ) + 10∂λ[ǫ(Y Λαβc λ)(Y Λ γδ]c∂λ), (C.45) (ΓfY )[ǫ(Y Λαβc ∂λ)(∂λΓfΛ γδ]cλ) = 3λ[ǫ(Y Λαβc ∂λ)(Y Λ γδ]c∂λ) + 2∂λ[ǫ(Y Λαβc λ)(Y Λ γδ]c∂λ).(C.46) From the identity (C.44), the derivative term in the last row of the r.h.s. of (C.43) vanishes. Then, removing the first two terms in the r.h.s. of (C.43) by means of the two identites (C.45) and (C.46), one gets [ǫαβγδ] [ǫαβγδ] 0 = −4λ [ǫ(Y Λαβc ∂λ)(Y Λ γδ]c∂λ), (C.47) which is nothing but Eq. (3.68). In this way, we have succeeded in proving Eq. (3.57). C.5 Equivalence in cohomology of bY and bnm As a last remark, let us report briefly about the derivation of the rearrangement terms RG, RH , RK , RL and BL, which appear at the end of section 5. In particular we shall show that BL is BRST-exact. From the recipe given in Appendix B.3 and using the OPE (3.15), one can compute RG = (λ̃βYα)(λ [αĜβ])− (λ̃βYαλ [α)Ĝβ] with the result RG = −[∂λ̃∂θ − (λ̃∂λ)(λ̃∂θ)− 2∂Y ∂θ] + R̂G, (C.48) where R̂G = Y[αλ̃β]((Y + λ̃)Γa) α∂λβ(λΓa∂θ). (C.49) With the replacement λΓa∂θ = [Qnm,Π a], (C.50) and some simple algebra, R̂G can be rewritten as R̂G = 2[Qnm, (Yαλ̃β)W R1 ]− AaGΠa, (C.51) where W R1 is defined in (5.39) and A G, defined in (5.40), comes from GΠa = Qnm, Y[αλ̃β]((Y + λ̃)Γa) Πa, (C.52) by using a simple algebra. In a similar way, RH is given by RH = − AaGΠa + R̂H , (C.53) where R̂H contains the factor (Γ aλ)γ which can be replaced by −{Qnm, (Γ cd)γ} and then, working as before, one arrives at R̂H = 3! Qnm, Y[αr̃βλ̃γ]W AaHα(dΓa) α, (C.54) where W [αβγ] R2 and A Hα are defined in (5.39) and (5.40), respectively. Moreover, AaHα(dΓa) α + R̂K , (C.55) where R̂K = (Y[αr̃β r̃γλ̃δ]((Y + 3λ̃)Γ c)α∂λβ [Qnm, N Qnm, (Y[αr̃β r̃γλ̃δ])W [αβγδ] + AcK[αβN c , (C.56) where again W [αβγδ] R3 and A Kαβ are defined in (5.39) and (5.40), respectively. Now let us move on to RL which, according to (5.34), is RL = 5! (Y[αr̃β r̃γ r̃δλ̃ǫ]λ ǫ)(N cαβNγδc ). (C.57) With rearrangement formula and using (3.58), RL becomes RL = 5! K[αβγδǫ]λ ǫ, N cαβNγδc K[αβγδǫ], N (−2)(K[αβγδǫ]((Y + 4λ̃)Λ c λ)∂λ ǫ)N cαβ + R̂L, (C.58) where we have defined K[αβγδǫ] = Y[αr̃β r̃γ r̃δλ̃ǫ]. (The expression of R̂L will be given below.) To the first term in the last row of Eq. (C.58), adding and subtracting the term defined by ((Y[αr̃β r̃γ r̃δλ̃ǫ])((Y + 4λ̃)Σ c λ)∂λ ǫ)N cαβ, (C.59) where we have also defined Ỹ Σ[αβ]c λ = (Ỹ Γc) [αλβ] + (Ỹ Γ) bΓcλ) β], (C.60) Ỹα = Yα + 4λ̃α, (C.61) RL is then reduced to RL = − AcKαβN c +R0 + R̂L. (C.62) Here we have introduced the quantity R̂L = 5! (∂R1 +R2 +R3), (C.63) where R1, R2 and R3 are defined by Y[αr̃β r̃γ r̃δλ̃ǫ]λ ǫ(ΓaY )α(ΓbỸ )β(ΓbΓ cλ)γ [(ΓaΓc∂λ) δ + 2δac∂λ R2 = − Y[αr̃β r̃γ r̃δλ̃ǫ]λ ǫ(ΓcY )α(ΓbỸ )β∂[(Γbcλ) γ∂λδ], R3 = Y[αr̃β r̃γ r̃δλ̃ǫ]λ ǫ(λ̃ΓfY )(Γc(2Y + 5λ̃))α(ΓcΓ bλ)β(ΓfΓb∂λ) γ∂λδ. (C.64) It is of importance that R1, R2 and R3 are all BRST-exact: Qnm, Y[αr̃β r̃γ λ̃δ](Γ aY )α(Γb(Y + 3λ̃))β(ΓbΓ cλ)γ[(ΓaΓc∂λ) δ + 2δac∂λ R2 = − Qnm, Y[αr̃β r̃γλ̃δ](Γ cY )α(Γb(Y + 3λ̃))β∂[(Γbcλ) γ∂λδ] Qnm, Y[αr̃β r̃γ λ̃δ](λ̃Γ fY )(Γc(Y + 2λ̃))α(ΓcΓ bλ)β(ΓfΓb∂λ) . (C.65) On the other hand, by rearrangement theorem, R0 can be rewritten as Ωσ(Y[αr̃β r̃γ r̃δλ̃ǫ](Ỹ Σ c λ)∂λ ǫ(Λαβcλ)σ) Y[αr̃β r̃γ r̃δλ̃ǫ]λ ǫ(ΓcỸ )α∂λβ(ΓbY )γ(ΓbΓc∂λ) δ. (C.66) The first term in the r.h.s. of (C.66) vanishes and the second one is BRST-exact. Indeed, one [Qnm, Y[αr̃β r̃γ λ̃δ](Γ c(Y + 3λ̃))α∂λβ(ΓbY )γ(ΓbΓc∂λ) δ]. (C.67) Eq. (C.62) is just Eq. (5.38) with BL = R0 + R̂L. Then, from Eqs. (C.63), (C.65) and (C.67), one can reproduce Eqs. (5.41)-(5.43). References [1] N. Berkovits, JHEP 0004 (2000) 018, hep-th/0001035. [2] N. Berkovits and B.C. Vallilo, JHEP 0007 (2000) 015, hep-th/0004171. [3] N. Berkovits, JHEP 0009 (2000) 046, hep-th/0006003. [4] N. Berkovits, JHEP 0108 (2001) 026, hep-th/0104247. [5] N. 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0704.1220

A multi-transition molecular line study of candidate massive young stellar objects associated with methanol masers

Astronomy & Astrophysics manuscript no. aa7289 c© ESO 2018 October 27, 2018 A multi-transition molecular line study of candidate massive young stellar objects associated with methanol masers ⋆ M. Szymczak1, A. Bartkiewicz1, and A.M.S. Richards2 1 Toruń Centre for Astronomy, Nicolaus Copernicus University, Gagarina 11, 87-100 Toruń, Poland 2 Jodrell Bank Observatory, University of Manchester, Macclesfield, Cheshire SK11 9DL, UK Received 13 February 2007 / Accepted 19 March 2007 ABSTRACT Aims. We characterize the molecular environment of candidate massive young stellar objects (MYSOs) signposted by methanol masers. Methods. Single pixel observations of 10 transitions of HCO+, CO and CS isotopomers were carried out, using the IRAM 30 m telescope. We studied a sample of 28 targets for which the 6.7 GHz maser emission positions are known with a sub-arcsecond accuracy. Results. The systemic velocity inferred from the optically thin lines agrees within ±3 km s−1 with the central velocity of the maser emission for most of the sources. About 64% of the sources show line wings in one or more transitions of CO, HCO+ and CS species, indicating the presence of molecular outflows. Comparison of the widths of line wings and methanol maser emission suggests that the 6.7 GHz maser line traces the environment of MYSO of various kinematic regimes. Therefore conditions conducive for the methanol maser can exist in the inner parts of molecular clouds or circumstellar discs as well as in the outer parts associated with molecular outflows. Calculations of the physical conditions based on the CO and HCO+ lines and the CS line intensity ratios refine the input parameters for maser models. Specifically, a gas number density of < 107 cm−3 is sufficient for strong maser emission and a high methanol fractional abundance (> 5× 10−7) is required. Key words. ISM: molecules − radio lines: ISM − stars: formation − masers 1. Introduction There is compelling evidence that methanol masers are a signature of recent or ongoing high-mass star formation (Menten 1991). However, it is not yet fully understood when they appear in an evolutionary sequence and what they actu- ally trace. The evaporation of grain mantles is postulated as the main process enhancing the fractional abundance of methanol molecules in the gas phase up to 10−6 (Dartois et al. 1999). This implies that methanol masers can emerge after the formation of an embedded heating source. Methanol maser sources rarely show strong (>100 mJy) free-free emission at centimetre wave- lengths implying that they precede the development of detectable ultra-compact HII (UCHII) region (Walsh et al. 1998; Codella & Moscadelli 2000). The estimated lifetime of methanol masers is a few ×104 yr (van der Walt 2005) which is similar to the typical dynamical timescales of molecular outflows. High spatial reso- lution observations revealed a variety of maser site sizes from 40−1200 AU (e.g. Norris et al. 1998; Walsh et al. 1998; Minier et al. 2000). The maser emission arises either from circumstel- lar discs or behind shocks tracing outflows from massive young stellar objects (MYSOs). No object was found that unequivo- cally confirms one of these scenarios. The non-linear nature of maser amplification means that it is difficult to relate the maser line intensity directly to the physical parameters of the active region. Theoretical models predict the formation of methanol maser lines under a rather wide range of gas and dust temperatures (30−200 K and 100−300 K, respec- tively) and hydrogen number densities (105− 108 cm−3) (Cragg ⋆ Figure A.1 and Table A.1 are only available in electronic form via http://www.edpsciences.org et al. 2002). Thus, it appears that a better understanding of the environments in which the masers arise is required in order to realise their full potential as probes of the formation of high- mass stars. In this paper we report our attempts to constrain the range of environments probed by methanol masers using observations of thermal emission from other molecular species and lines. Specifically, the ratios of the intensities of different transitions of CS and C34S molecules are used to obtain the temperature and density of the gas. The optically thin and thick lines of CO and HCO+ are used to constrain the column density. These tech- niques were successfully used to characterize other samples of MYSOs (e.g. Plume et al. 1997; Beuther et al. 2002a; Purcell et al. 2006). Additionally, the molecular line profiles yield informa- tion on the kinematics of various parts of the molecular clouds surrounding the high-mass protostars (e.g. Fuller et al. 2005; Purcell et al. 2006). A homogeneous and unbiased sample of MYSOs is neces- sary in order to address these issues properly. Our recent 6.7- GHz unbiased survey for methanol masers in selected regions of the Galactic plane (Szymczak et al. 2002) provides such a com- plete, sensitivity limited sample of candidate MYSOs. Objects identified in the survey probably represent a class of MYSOs in an early evolutionary phase. Some groups and individual sources in this class, selected using various diagnostics of high-mass star formation, have been studied in thermal molecular lines (Brand et al. 2001; Beuther et al. 2002a; Fuller et al. 2005), but this is the first published study of a homogeneous sample based solely on the presence of detectable methanol masers. http://arxiv.org/abs/0704.1220v1 http://www.edpsciences.org 2 M. Szymczak et al.: Molecular line study of high-mass protostars 2. The sample The 28 sources observed in this study (Table 1) were chosen from a sample of 100 methanol maser sources found in the Torun 32 m telescope blind survey for the 6.7 GHz methanol line in the Galactic plane area 20◦ ≤ l ≤ 40◦ and |b| ≤ 0.◦52 (Szymczak et al. 2002). This flux-limited (3σ ≃1.6 Jy) subsam- ple includes 25 out of 26 sources which were undetected prior to the Torun survey. Therefore, our subsample specifically excludes previously known sources associated with OH maser emission (Caswell et al. 1995) or with IRAS-selected bright UCHII can- didates (Schutte et al. 1993; van der Walt et al. 1995; Walsh et al. 1997). Assuming that CH3OH masing precedes the appear- ance of OH masers and detectable UCHII regions, the objects studied here represent sites of high-mass star formation at a very early stage. The average peak maser flux of the 28 targets is 17.3 Jy, a factor of 2.6 lower than that of the other 72 objects in the original sample, suggesting that distant or intrinsically faint objects may be over-represented in our subsample. The subsam- ple studied here is most certainly not complete. 2.1. Astrometric positions The coordinates and position uncertainties of the brightest 6.7- GHz maser component in each source are presented in Table 1. The LSR velocity of this component (Vp) and its peak flux den- sity (Sp) are given for each target. The positions and flux den- sities of all but three objects were measured with the Mark II − Cambridge baseline of MERLIN in two sessions between 2002 May and 2003 May. For the three objects not measured the peak velocities were taken from Szymczak et al. (2002). The observational setup and data reduction were de- scribed in Niezurawska et al. (2005). A primarily goal of those astrometric measurements was to determine the positions with sub-arcsecond accuracy for follow-up VLBI observations. Measurement errors mainly depended upon the ratio of the beam size to the signal to the noise ratio (Thompson et al. 1991). If the emission was complex we took the dispersion of neighbouring maxima as the position uncertainty. The N−S elongation of the synthesized beam close to declination 0◦ produces a split peak, in which case the position uncertainty in that direction was taken as half the separation of the maxima. Consequently, for sources with a single clear peak, the position errors in right ascension were as small as 0.′′02 but increased up to 0.′′90 for sources with complex emission. The respective errors in the declination were 0.′′14 and 1.′′6. Comparison with our unpublished VLBI observa- tions reveals position differences between MERLIN and VLBI measurements no worse than a few tens of milli-arcseconds. This implies that the values listed in Table 1 are maximal position er- rors for most of the targets. The flux densities listed in Table 1 are a factor of 2−3 lower than those measured with the single dish (Szymczak et al. 2002) and should be considered as lower limits. The exact flux scale and gain-elevation effects for low- declination sources are not yet fully investigated at 6 GHz but comparison of calibration sources in common with other exper- iments shows that the uncertainties are 10 − ≤50%. This sug- gests that about half the methanol flux arises on scales larger than the beam size of 50−100 mas. 2.2. Distances The distances were determined using the Galactic rotation curve of Brand & Blitz (1993) and the central velocity of each 6.7 GHz methanol maser profile as measured by Szymczak et al. (2002). Selection of this velocity as a reliable estimator of the systemic velocity is proven in Sect. 6.1. The sources are all in the first quadrant so that there is an ambiguity between the near and far kinematic distances. In most cases we are unable to resolve this ambiguity because there are no independent distance measure- ments in the literature for our sample. Based on the arguments discussed in Walsh et al. (1997), we adopted the near kinematic distances (Table 4). 3. Observations and data reduction Observations were carried out between 2004 September 28 and October 2 with the IRAM 30 m telescope. Ten transitions of HCO+, CO and CS isotopomers were observed. Two or three SIS receivers tuned to single sideband mode were used simulta- neously, in combination with the VESPA autocorrelator as well as with 100 kHz and 1 MHz filter banks. Table 2 lists the rest line frequencies, half power beam widths (HPBWs), velocity resolu- tions and typical system temperatures for each transition. The data were taken using the position switching mode. The off positions were typically 30′ away from the targets. In the few cases, especially for the C13O J=2−1 line, where emission was seen at the reference position, the offsets were increased up to 45′ in the direction away from the Galactic plane. The ob- servations were centered on the target positions listed in Table 1. Integration times on-source in total power mode were 12−18 min per frequency setting, resulting in rms noise levels ranging from ≈0.05 K at 87 GHz to ≈0.90 K at 245 GHz for a spectral resolution of 0.10−0.16 km s−1. Pointing was checked regularly on nearby continuum sources and was usually found to be within 2′′ and always within 3′′. The spectra were scaled to the main beam brightness temperature (Tmb) using the efficiencies sup- plied by the observatory1. Comparison of our data with those taken by Brand et al. (2001) with the same telescope and spec- tral resolutions for a source in common, 36.115+0.552, implies consistent intensity scales within less than 30%. The data reduction were performed using the CLASS soft- ware package. Low order polynomials were applied to remove baselines from the calibrated spectra. The line parameters were determined from Gaussian fits and are listed in Table A.1 where the following information is given: the rms (1σ) noise level, the extreme velocities Vs, Ve where the intensity drops below the 2σ level, the peak temperature Tmb, the velocity of the peak Vp, the line width at half maximum ∆V and the integrated line intensity∫ Tmbdv. Velocities are in the LSR frame. In several cases where the profiles were non-Gaussian, these values were read off from the spectra. In some cases the spectra were smoothed to improve the signal to noise ratio. In this paper, only the autocorrelation spectra are analyzed. 4. Results The basic parameters of molecular transitions derived from Gaussian analysis are assembled in Table A.1, while the spec- tra are shown in Fig. A.1. The sensitivity achieved and detection rate for each transition are summarized in Fig. 1. The histogram counts as detected only those lines with Tmb > 3σ. 13CO(2−1), C18O(2−1), CS(2−1) and CS(3−2) lines were detected in all sources. HCO+(1−0) and H13CO+(1−0) lines were detected in all but one source. The detection rates in C34S(2−1) and C34S(3−2) transitions were 1 http://www.iram.es/IRAMES/telescope/telescopeSummary/telesco- pe summary.html M. Szymczak et al.: Molecular line study of high-mass protostars 3 Table 1. List of targets. Name α(J2000) δ(J2000) σα σδ Vp Sp (arcsec) (arcsec) (km s−1) (Jy) 21.407−0.254 18 31 06.3403 −10 21 37.305 0.28 0.80 +89.0 2.0 22.335−0.155 18 32 29.4109 −09 29 29.435 0.27 1.10 +35.7 2.8 22.357+0.0661 18 31 44.144 −09 22 12.45 +80. 23.707−0.1983 18 35 12.3625 −08 17 39.409 0.06 0.40 +79.0 3.2 23.966−0.1093 18 35 22.2167 −08 01 22.395 0.35 1.60 +71.0 4.3 24.147−0.0093 18 35 20.9501 −07 48 57.470 0.03 0.19 +17.9 6.4 24.541+0.3123 18 34 55.7212 −07 19 06.630 0.90 0.90 +105.5 4.4 24.635−0.323 18 37 22.7932 −07 31 37.950 0.50 1.20 +35.5 1.0 25.410+0.1052 18 37 16.9 −06 38 30.4 +97. 26.598−0.024 18 39 55.9268 −05 38 44.490 0.03 0.18 +23.0 2.0 27.221+0.136 18 40 30.5446 −05 01 05.450 0.03 0.18 +119.0 3.0 28.817+0.365 18 42 37.3470 −03 29 41.100 0.02 0.18 +91.0 1.0 30.316+0.069 18 46 25.0411 −02 17 45.160 0.03 0.16 +35.5 1.3 30.398−0.297 18 47 52.2623 −02 23 23.660 0.02 0.14 +98.2 1.5 31.056+0.361 18 46 43.8558 −01 30 15.690 0.05 0.28 +81.0 1.0 31.156+0.045 18 48 02.3471 −01 33 35.095 0.10 0.90 +41.0 0.8 31.585+0.080 18 48 41.8975 −01 09 43.085 0.50 0.70 +95.8 0.8 32.966+0.0412 18 51 24.5 +00 04 33.7 +92. 33.648−0.2243 18 53 32.5508 +00 32 06.525 0.50 1.0 +62.6 20.0 33.980−0.019 18 53 25.0184 +00 55 27.260 0.05 0.50 +59.0 1.0 34.753−0.092 18 55 05.2410 +01 34 44.315 0.08 0.50 +53.0 1.6 35.791−0.1753 18 57 16.9108 +02 27 52.900 0.04 0.17 +60.8 5.6 36.115+0.5523 18 55 16.8144 +03 05 03.720 0.02 0.23 +74.2 7.2 36.704+0.096 18 57 59.1149 +03 24 01.395 0.08 0.17 +53.0 1.9 37.030−0.039 18 59 03.6435 +03 37 45.140 0.14 0.50 +79.0 1.2 37.479−0.105 19 00 07.1457 +03 59 53.245 0.07 0.36 +62.8 1.8 37.600+0.426 18 58 26.8225 +04 20 51.770 0.03 0.70 +91.2 2.0 39.100+0.4913 19 00 58.0394 +05 42 43.860 0.34 0.17 +15.2 2.8 1 Position is from Walsh et al. 1998, 2 Position is from Beuther et al. 2002a, 3 This source was reported in Niezurawska et al. 2005 Table 2. Observing parameters Transition Frequency Ref. HPBW Res. Tsys (GHz) (′′) (km s−1) (K) HCO+(1−0) 89.188518 2 27 0.13 200 H13CO+(1−0) 86.754330 1 27 0.13 200 13CO(2−1) 220.398686 2 11 0.11 1200 C18O(2−1) 219.560328 2 11 0.11 1340 CS(2−1) 97.980953 1 25 0.12 260 CS(3−2) 146.969049 2 17 0.16 690 CS(5−4) 244.935560 1 10 0.10 1830 C34S(2−1) 96.412961 1 25 0.12 290 C34S(3−2) 144.617147 2 17 0.16 560 C34S(5−4) 241.016113 1 10 0.10 2100 The references for the line frequencies are 1 - Lovas (2003), 2 - Brand et al. (2001) about half of those in CS(2−1) and CS(3−2) lines. Because the sensitivities achieved for these four lines were comparable, these detection rate differences reflect a real drop in the number of sources exhibiting emission at the same level in the C34S(2−1) and C34S(3−2) lines. In contrast, the lower detection rates in the CS(5−4) and C34S(5−4) transitions appear to reflect the drop in sensitivity. 4.1. Systemic velocities Five of the observed lines (C18O(2−1), H13CO+(1−0) and the J=2−1, 3−2 and 5−4 transitions of C34S) are expected to be optically thin (Plume et al. 1997; Brand et al. 2001; Purcell et 1/2520/2716/2819/1914/2525/2517/1827/2825/25 C34S (5-4) (5-4) C34S (3-2) (3-2) 13CO (2-1) C18O (2-1) HCO+ (1-0) H13CO+ (1-0) (2-1) C34S (2-1) 28/28 Fig. 1. The average sensitivity achieved for each transition (top) and the detection rate (bottom). The ratio of the number of de- tected to observed objects is shown in each of the bars. al. 2006). These lines can be used to determine source systemic velocities. In order to test whether these species trace the same or similar kinematic regimes we compare their line parameters. The C34S(5−4) line is excluded from the following analysis due to very low number of detections. The average intensities of the H13CO+(1−0) and C34S lines are very similar and are a factor of 8 weaker than the average 4 M. Szymczak et al.: Molecular line study of high-mass protostars intensity of the C18O(2−1) line. This implies that the parameters of the latter line, especially Vp, are determined most accurately. We note that some line rest frequencies adopted from Brand et al. (2001) differ slightly from those recommended by Lovas (2003). In the extreme case of C34S(3−2) this results in the ve- locity difference of 0.07 km s−1. Moreover, the uncertainties in the line rest frequencies introduce a maximum uncertainty of ±0.17 km s−1 to the derived line velocity for the H13CO+(1−0). We assume that the above uncertainties affect the velocity esti- mates by up to 0.24 km s−1. Comparison of the velocities of the four optically thin lines in our sample reveals no significant aver- age differences higher than 0.30 km s−1. This suggests the same kinematic behaviour of these low-density gas tracers. At 100 K the thermal linewidths of C18O(2−1) and H13CO+(1−0) are 0.24 km s−1 whereas those of C34S(2−1) and C34S(3−2) are 0.20 km s−1. The observed linewidths are much broader, suggesting that turbulence or bulk gas motions play a significant role in the line broadening. The mean linewidth ratios of the optically thin lines are 5−10% higher than unity. This bias is relatively small and suggests that the lines trace the same molecular gas in the beam. The systemic velocities are listed in Table 4. They are primarily the C34S(2−1) and C34S(3−2) line peak velocities. If emission in these lines is absent or weak the other optically thin lines are used. In two sources, 37.030−0.039 and 37.600+0.426, the systemic veloc- ities are derived from CS(5−4) and HCO+(1−0) profiles, re- spectively. We conclude that in most cases the observed opti- cally thin lines are well fitted by single Gaussian profiles (devi- ations are discussed in Sect. 4.2.2) and their peak velocities are within ±0.4 km s−1 of each other for almost all sources in the sample. Therefore, these lines provide reliable estimates of sys- temic velocity of sufficient accuracy to allow comparison with the methanol maser velocities listed in Szymczak et al. (2002). 4.2. Shape of profiles We analyse the molecular line profiles in order to search for specific signatures of ordered motions such as infall, outflow or rotation. Inward motion can be signposted by blue asymmetric profiles (Myers et al. 1996; Fuller et al. 2005) if the molecular spectral lines trace sufficiently dense gas. Signatures of outflow or rotation are generally manifested in the line wings. 4.2.1. Asymmetry We analysed line asymmetry quantitatively using the asymmetry parameter (Mardones et al. 1997), δv=(vthick − vthin)/∆Vthin, where vthick and vthin are the peak velocities of optically thick and optically thin lines, respectively and ∆Vthin is the line width at half maximum of the optically thin line. We used C34S(2−1) as the optically thin line and the best available mea- sure of the systemic velocity of MYSOs. Figure 2 shows his- tograms of the distribution of δv for the optically thick lines 13CO(2−1), HCO+(1−0), CS(2−1), CS(3−2) and CS(5−4). There are approximately equal numbers of blue and red asym- metric profiles in our sample. Specifically, we note that there is no evidence for an excess of blue-shifted emission in the opti- cally thick lines. Such an excess is postulated as the signature of inward motion of the gas (Myers et al. 1996). We suggest that motions other than infall, i.e. turbulence, rotation and out- flow, are the dominant factor producing asymmetric profiles for most of the sources in our sample. It is possible that infall sig- natures could be masked by the relatively low resolution (typi- -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 13CO(2-1) HCO+(1-0) CS(2-1) CS(3-2) CS(5-4) Fig. 2. Histograms of the distribution of the asymmetry parame- ter δv for the five transitions. The range of |δv| < 0.25 marked by the dotted lines corresponds to the spectra with no asymme- cally ≥ 0.2 pc, i.e. at a distance of 5 kpc and spatial resolution of 10′′) of our observations, since even the near kinematic dis- tances are >3 kpc for ∼80% of the sources (the average Dnear is 5.2± 2.5 kpc for the whole sample). We therefore examined separately the 5 closest (Dnear < 2.8 kpc) objects with well-determined asymmetry parameters. Two of these, 26.598−0.024 and 30.316+0.069, consistently show negative values of δv, i.e. blue asymmetry, in the 13CO(2−1), HCO+(1−0), CS(2−1) and CS(3−2) line pro- files (Fig. A.1). The corresponding values of δv are −0.38, −0.82, −0.15 and −0.18 for source 26.598−0.024 and −0.43, −0.60, −0.33 and −0.37 for source 30.316+0.069. Their asym- metry parameters are smaller for the optically thin lines (i.e. C18O(2−1) and H13CO+(1−0)), in the range from −0.20 to 0.0. Such a dependence of the amount of blue asymmetry on the op- tical depth of the transition is typical in molecular cores experi- encing infall (Narayanan et al. 1998). We suggest that these two sources are the clearest infall candidates although source com- plexity or a combination of outflow and rotation could contribute to asymmetries in optically thick profiles. 4.2.2. Wings Wing emission is identified by the presence of residuals after Gaussian fitting and by comparing the same transitions of op- tically thick and thin isopotomers. A single Gaussian function provides a good fit to most of the optically thin lines analyzed in Section 4.1, but in a few cases the residuals are at a level ≥ 3σ, assumed to be wing emission. We cannot exclude the possibility that they are weak separate component(s), given the limitations of our signal to noise ratio and spectral resolution, but we note that the blue and/or red residuals are non-Gaussian in most cases. 4 out of 25 sources detected in the C18O(2−1) line show weak (3-4σ) wing emission of width 4.5−8 km s−1 (Table 3) which mostly is seen from the red or blue sides of the profiles. In the H13CO+(1−0) line the wing emission is seen in 2 out of 17 ob- jects detected (Table 3). 25.410+0.105 is a peculiar source show- ing broad (7−8 km s−1) and symmetric wings in both lines (Fig. A.1). In contrast, the optically thick lines show more frequent ab- sorption dips, multiple components and wings. In several cases identification of the wings is difficult. The 13CO(2−1) profiles are especially complex; commonly they are fit by 2-5 Gaussians. These profiles can be interpreted as multiple emitting regions M. Szymczak et al.: Molecular line study of high-mass protostars 5 Table 3. Statistics of wing occurrence. Entries marked Y or N indicate symmetric wings and no wings respectively, Yb or Yr indicate wing emission seen from the blue and red sides of the profiles, respectively. An interrogation point (?) indicates a ten- tative wing and the absence of entry indicates no observation. Source C18O H13CO+ 13CO HCO+ CS CS (2-1) (1-0) (2-1) (1-0) (2-1) (3-2) 21.407−0.254 N N N 22.335−0.155 N N N Y N N 22.357+0.066 Yb N N N N N 23.707−0.198 N N N N N N 23.966−0.109 N Yr N Y Y Y 24.147−0.009 N N N Y N N 24.541+0.312 Y N N Y N N 24.635−0.323 Yr N Y Y Y Y 25.410+0.105 N Y Y Y Y Y 26.598−0.024 N N N N N N 27.221+0.136 N N N N 28.817+0.365 N N Y Y 30.316+0.069 N N N Y N N 30.398−0.297 Yb N ? Y Y N 31.056+0.361 N N N N 31.156+0.045 N N ? Y Y Y 31.585+0.080 N N N N 32.966+0.041 N N ? N N N 33.648−0.224 N N N N N 33.980−0.019 N N N Y Y Y 34.753−0.092 N ? Y N 35.791−0.175 N N ? Y Y Y 36.115+0.552 N N N Y N N 36.704+0.096 N N Y N N 37.030−0.039 N N 37.479−0.105 ? N 37.600+0.426 Y Y 39.100+0.491 N N N Y Y Y along the same line of sight. The 13CO(2−1) lines show evi- dence of wings in only 3 objects (Table 3). The HCO+(1−0) lines are also complex, often exhibiting two or more components or broad line wings (Fig. A.1). They appear to consist of the superposition of several emitters seen along the line of sight or of (self)absorption by cooler gas on the near side of the source. Wings are identified in 17 out of 27 detections (Table 4). The wing full width ranges from 6 to 20 km s−1 with a mean value of 10.3±3.3 km s−1. Evidence for wings is seen in the CS(2−1) transition for 9 out of 25 sources and in the CS(3−2) transition for 7 out of 19 sources (Table 3). Their full widths are from 8 to 19 km s−1. We conclude that 64% (18/28) of the sources show residual line wings at least in one line when a Gaussian profile is used to fit the CO, HCO+ and CS molecular lines. Detection of the wings may indicate molecular outflows from the MYSOs iden- tified by methanol masers but we caution that such detections based on our data alone are only tentative. 5. Derivation of physical parameters 5.1. Column densities In order to estimate the column density of H13CO+ from the ob- served HCO+(1−0) and H13CO+(1−0) line parameters, we fol- low the procedure outlined in Purcell et al. (2006) and references therein. Briefly, the main assumptions made are: (i) HCO+(1−0) is optically thick and H13CO+(1−0) is optically thin. (ii) Both lines form in the same gas and share the same excitation temper- ature. (iii) The excitation temperature is equal to the rotational temperature. (iv) The gas is in local thermodynamic equilibrium. (v) The beam filling factor is one for both lines. The derived H13CO+ column density, N (H13CO+), (Table 4) ranges from 1.3 − 5.1 × 1012 cm−2 and the median value is 2.2 × 1012 cm−2. We derive a value of N (H13CO+) a fac- tor of 4 smaller than the value found by Purcell et al. (2006) for two of the sources common to both samples, 22.357+0.066 and 23.707−0.198. This is probably because Purcell et al. ap- plied corrections for self-absorption, leading to higher estimates of the HCO+(1−0) line intensities and lower optical depths, compared with our study. We adopt an abundance ratio of [H13CO+/H2]=3×10 −11 (Girart et al. 2000), from which we ob- tain the H2 column density from 4.3 − 17.0 × 10 22 cm−2 with the median value of 7.3× 1022 cm−2. We apply the same method to estimate the column density of C18O, N (C18O), from the line parameters of 13CO(2−1) and C18O(2−1), assuming that 13CO(2−1) is optically thick and C18O(2−1) is optically thin. For our sample N (C18O) is 0.9−32.6×1015 cm−2 (Table 4) with the median value of 4.6×1015 cm−2. The temperature varies between 10 and 30 K. The resulting H2 column density ranges from 5.4 × 10 1.9×1023 cm−2 for an abundance ratio [C18O]/[H2]=1.7×10 (Frerking et al. 1982). We conclude that the CO and HCO+ data provide consis- tent estimates of the column density of H2 towards the methanol maser sources. The range of N (H2) derived here is in good agreement with that reported for high-mass protostar candidates associated with methanol masers; 3 × 1022 − 2 × 1023 cm−2 (Codella et al. 2004; Minier et al. 2005; Purcell et al. 2006). However, it is significantly lower than N (H2)≥ 4 × 10 23 cm−2 reported in some earlier works (e.g. Churchwell et al. 1992) for ultra-compact HII regions. This discrepancy is likely due to the temperatures of 10−30 K derived here which is significantly lower than ≥ 100K assumed in Churchwell et al. (1992). We notice that a dispersion of the N (C18O) is a factor of 7 larger than that of the N (H13CO+) (Table 4). In two sources 22.357+0.066 and 26.598−0.024 the N (C18O) is extremely large (> 1.9×1016 cm−2). In consequence, the values of N (H2) derived from the C18O is a factor of 1.5 and 3.1, respectively, higher that those derived from the H13CO+. This discrepancy suggests that the methanol masers in these sources probe re- gions with the abundance ratio of 13CO/C18O significantly lower than a typical ratio of 6.5−7 (Frerking et al. 1982; Beuther et al. 2000). A decrease of 13CO/C18O ratio is predicted in the PDR model in a clumpy cloud; in small clumps the C18O molecule is nearly completely photodissociated whilst it is protected from photodissociation in large clumps (Beuther et al. 2000 and ref- erences therein). Object 26.598−0.024 with the highest value of N (C18O) is also a candidate infall object (Sect. 4.2.1) and one can speculate that it is the youngest methanol maser in our sam- ple; the maser emission forms in large clumps at nearly systemic velocity. Another explanation of low 13CO/C18O intensity ratio can be that our 11′′ beam probes the methanol maser sites where the C18O cores did not coincide with the13CO cores. This obser- vational fact is well documented in Brand et al. (2001) at least for their sources Mol 98 and Mol 136 (see their Fig. 5). Furthermore, the C18O emission is less extended than the 13CO emission; by a factor of ∼ 3 − 5 for common source 35.791−0.175. This ex- planation seems to be less plausible as a similar effect can be observed for HCO+ and H13CO+ lines. 6 M. Szymczak et al.: Molecular line study of high-mass protostars Table 4. Derived properties 30K 60K Source Vsys dnear dfar N (H 13CO+) N (C18O) lognH2 logN (CS) lognH2 logN (CS) (km s−1) (kpc) (kpc) (1012cm−2) (1015cm−2) (cm−3) (cm−2) (cm−3) (cm−2) 21.407−0.254 90.7 6.0 10.4 - 3.8 - - - - 22.335−0.155 30.9 2.4 14.7 2.1 3.7 6.15±0.15 14.52±0.10 5.91±0.12 14.68±0.06 22.357+0.066 84.2 5.2 10.6 2.2 19.1 5.48±0.09 14.70±0.11 5.27±0.13 14.52±0.21 23.707−0.198 68.9 5.1 10.5 3.2 13.2 5.93±0.08 13.74±0.14 5.57±0.08 13.49±0.07 23.966−0.109 72.7 4.2 11.6 5.1 9.0 >6.7 14.73±0.57 >6.5 15.16±0.52 24.147−0.009 23.1 2.0 14.5 1.4 2.1 5.61±0.08 14.51±0.18 5.42±0.12 14.60±0.56 24.541+0.312 107.8 7.0 9.5 1.5 4.6 - - - - 24.635−0.323 42.7 3.7 13.1 4.7 7.6 >6.7 14.61±0.38 6.39±0.12 14.53±0.23 25.410+0.105 96.0 - 9.5 3.4 7.0 6.42±0.11 14.40±0.08 6.22±0.07 14.53±0.20 26.598−0.024 23.3 1.8 13.4 1.8 32.6 >6.9 14.54±0.42 >6.5 14.86±0.28 27.221+0.136 112.6 - 8.0 - 9.4 - - - - 28.817+0.365 87.0 5.5 9.4 - 5.3 - - - - 30.316+0.069 45.3 2.8 12.2 2.2 3.3 >6.9 14.77±0.18 6.28±0.19 14.59±0.09 30.398−0.297 102.4 6.0 8.5 1.6 3.7 6.12±0.10 14.83±0.08 - - 31.056+0.361 77.6 - 9.6 - 2.9 - - - - 31.156+0.045 38.9 2.7 11.9 2.2 4.8 6.06±0.04 14.11±0.06 5.74±0.06 14.64±0.13 31.585+0.080 96.0 5.4 8.1 - 11.8 - - - - 32.966+0.041 83.4 5.4 8.9 1.3 4.2 - - 4.39±0.11 15.73±0.16 33.648−0.224 61.5 - 10.4 - 2.1 - - - - 33.980−0.019 61.1 3.5 10.6 2.5 4.7 - - 4.52±0.21 15.76±0.13 34.753−0.092 51.1 3.1 11.0 - 1.4 - - - - 35.791−0.175 61.9 4.6 10.3 2.4 3.0 - - - - 36.115+0.552 76.0 4.9 9.0 1.9 8.1 - - >6.890 15.60±0.1990 36.704+0.096 59.8 4.6 10.4 - 0.9 - - - - 37.030−0.039 80.1 5.0 8.3 - - - - - - 37.479−0.105 59.1 - 9.5 - - - - - - 37.600+0.426 90.0 6.5 7.5 - - - - - - 39.100+0.491 23.1 1.0 14.7 2.0 2.9 - - 6.58±0.0890 14.81±0.0890 90 values for kinetic temperature 90 K 5.2. Gas density and temperature We used the escape-probability modelling code RADEX on- line2 to estimate the density and temperature of the gas required for the observed line temperature ratios of CS and C34S. Because these parameters cannot be derived independently for diatomic molecules (Schilke et al. 2001) we calculate the models for 30, 60 and 90 K with gas number densities of 104 − 107 cm−3, CS column densities of 1012−1017 cm−2 and linewidth of 1 km s−1. We performed the calculations for the 16 sources for which all three CS lines were detected and we assumed that beam dilu- tion is comparable for all these transitions. We used a χ2 mini- mization procedure to fit the models to the observed line ratios. The derived parameters are listed in Table 4. We found equally reasonable fits for 10 sources using models at kinetic tempera- tures of both 30 and 60 K. Five sources have good fits only for a single kinetic temperature. We could not find a satisfactory fit for the source 35.791−0.175 as its CS(2−1) and CS(3−2) lines are strongly self-absorbed (Fig. A.1) and thus its line ratios are poorly constrained. Using a temperature of 60 K the average logarithmic num- ber density is 5.7±0.7 and the average logarithmic column den- sity of CS is 14.7±0.6 for the sample. These values are consis- tent with 5.9 and 14.4, respectively, reported for a large sam- ple of massive star formation sites selected by the presence of H2O masers (Plume et al. 1997). Our estimates are also in good agreement with those based on the nine-point CS maps of high- mass protostellar candidates (Beuther et al. 2002a; Ossenkopf et al. 2001) and calculated with more sophisticated models. Taking 2 http://www.strw.leidenuniv.nl/moldata/radex.php the CS fractional abundance as ∼ 8×10−9 (Beuther et al. 2002a) our estimate of the CS column density implies a mean N (H2) of 6.3× 1022 cm−2 which is in very good agreement with the esti- mates based on CO and HCO+ data (Sect. 5.1). Our C34S data are less useful to estimate the gas density and temperature because the line ratios are poorly constrained for most of the targets. 26.598−0.024 is the only source for which we are able to determine C34S line ratios but the results are in- consistent with those obtained from the CS data. This indicates that the escape probability model provides only a crude esti- mate to the physical parameters and the assumption of homo- geneous parameters across the cloud is not fulfilled (Ossenkopf et al. 2001). 6. Discussion 6.1. Kinematics The present survey reveals new information regarding the kine- matics of molecular gas surrounding massive forming stars. In the following we attempt to answer the question of whether the 6.7 GHz methanol maser and the thermal molecular lines arise from similar or different kinematic regimes. The velocity ranges of 6.7 GHz methanol masers, 13CO and HCO+ line wings are plotted in Fig. 3. This plot clearly shows that the systemic velocity derived in this study (Table 4) is in good agreement with the methanol maser central veloci- ties, Vm, derived from Szymczak et al. (2002). We note that in many sources Vm does not coincide with the peak maser veloc- ity Vp. The average value of Vm − Vsys is 0.04±0.60 km s The difference is less than 3 km s−1 for 23 sources (82%). Vm M. Szymczak et al.: Molecular line study of high-mass protostars 7 -20 -10 0 10 20 39.100+0.491 37.600+0.426 37.479−0.105 37.030−0.039 36.704+0.096 36.115+0.552 35.791−0.175 34.753−0.092 33.980−0.019 33.648−0.224 32.966+0.041 31.585+0.080 31.156+0.045 31.056+0.361 30.398−0.297 30.316+0.069 28.817+0.365 27.221+0.136 26.598−0.024 25.410+0.105 24.635−0.323 24.541+0.312 24.147−0.009 23.966−0.109 23.707−0.198 22.357+0.066 22.335−0.155 21.407−0.254 Velocity (km s-1) Fig. 3. Comparison between the velocity ranges of 6.7 GHz methanol maser (thick bars)(Szymczak et al. 2002) and 13CO (dotted bars) and HCO+ (dashed bars) line wings. The dotted vertical line marks the systemic velocity. is offset by >4 and ≤8.1 km s−1 with respect to Vsys in 5 sources (18%), 23.707−0.198, 23.966−0.109, 24.147−0.009, 30.316+0.069 and 32.966+0.041 (Figs. 3 and A.1). This does not necessarily imply that the different species arise from sep- arate regions along the same line of sight. Two of the sources, 24.147−0.009 and 32.966+0.041, have ranges of maser emis- sion ∆Vm ≤4 km s −1 which is a factor of two narrower than the mean value of 8.3±0.9 km s−1 for the sample but this could be simply an effect of inhomogeneous conditions in molecular clumps; the maser emission is sustained in one or a few clumps of sizes a few×1015 cm (Minier et al. 2000). The effect of clumping is clearly seen even in regular structures (Bartkiewicz et al. 2005). The other three sources exhibit maser emission at velocities which differ from the systemic velocity by less than 4 km s−1. In source 30.316+0.069 the maser spectrum is double (Szymczak et al. 2000) and one of the peaks near 49 km s−1 is close to the systemic velocity of 45.3 km s−1, so that the maser emission related to the thermal molecular lines has a width of about 6 km s−1. We conclude, Vm is a reliable estimator of the systemic velocity, with an accuracy better than 3 km s−1, for most of the sources in our sample. The overlap between the velocity ranges of the methanol masers and the 13CO/HCO+ line wings is remarkable. Figure 4 shows a histogram of the ratio of methanol maser velocity spread, ∆Vm, to HCO + line wings spread. This ratio ranges from 0.2−6.7 and the median value is 1.3. Similar trends are ob- served in the ratio of ∆Vm to 13CO line wings spread. In 12 out of 23 sources where we detected 13CO/HCO+ wings, ∆Vm falls entirely within the wing velocity ranges and in 9 sources there is an overshoot of ≤4 km s−1. The 13CO/HCO+ line wings appear to provide a good indication of the presence of outflow and their widths can serve as an approximate measure of outflow veloci- ties. The present observations used beamwidths of 11′′ and 27′′ for 13CO and HCO+ lines, respectively, which samples a small fraction of the molecular cloud, centred on the methanol maser position. The outflow velocity can be reliably estimated from these data only for the fortunate case when the axis of out- flow lies along the line of sight. One source in our sam- ple, 25.410+0.105, has been mapped in the 12CO(2−1) line by Beuther et al. (2002b) who measured a wing velocity range of 14 km s−1, which is comparable with our estimate. In this object the maser emission, with velocity width of 5 km s−1, is closely centered on the systemic velocity. The velocity ranges of the 13CO and HCO+ wings are 11 and 18 km s−1, respectively. This indicates that the maser emission traces a small portion of the kinematic regime of the 13CO and HCO+ lines or it is completely unrelated. Fig. 3 indi- cates that sources 21.407−0.254, 26.598−0.024, 31.156+0.045 and 35.791−0.175 share similar properties with 25.410+0.105. VLBI observations of 35.791−0.175 (Bartkiewicz et al. 2004) support the above interpretation. In this object the 6.7 GHz methanol maser emission appears to come from part of a cir- cumstellar disc. Our sample contains 4 objects (23.707−0.198, 24.147−0.009, 32.966+0.041, 36.115+0.552) for which the velocity range of the maser emission is very similar to or slightly overshoots that of the 13CO/HCO+ line wings. If we assume that the width of 13CO and HCO+ line wings is a measure of the outflow velocity, in these objects the 6.7 GHz methanol masers arise in outflows. This scenario appears to be supported by VLBI observations of 36.115+0.552 (Bartkiewicz et al. 2004); the maser emission comes from two well separated regions which probably represent a bipolar outflow. In this case the methanol maser traces the same or a very similar kinematic regime as that of the 13CO and HCO+ lines. Sources 22.355−0.155 and 27.221+0.136 appear to posses complex kinematics in the regions where the methanol masers operate. A close inspection of their 6.7 GHz spectra (Szymczak et al. 2002) suggests that some spectral features arise from the inner parts of the molecular cloud whilst other features form in outflows. VLBI studies of maser emission and detailed measure- ments of the kinematic properties of the molecular emission are needed to verify this suggestion. 6.2. Implications for the evolutionary status One of the important findings of our observations is the de- tection of considerable number of sources with line wings. We identified residual line wings in 18 out of 28 sources when a Gaussian profile was used to fit the CO, HCO+ and CS molecu- lar lines. The line wings appear to be the best indicators of out- flow motions in most cases. The presence of line wings in about 64% of sources in the sample suggests a close association of the methanol masers with the evolutionary phase when outflows oc- cur. This result is consistent with that reported by Zhang et al. (2005). They mapped the CO(2−1) line in a sample of 69 lu- minous IRAS point sources and found that about 60% of them were associated with outflows. However, with the present data 8 M. Szymczak et al.: Molecular line study of high-mass protostars 0 1 2 3 4 5 6 7 8 Ratio Fig. 4. Histogram of the ratio of methanol maser velocity spread to HCO+ line wings spread. we cannot resolve whether the methanol maser sites and the out- flows have a common origin. Because of clustering in high mass star formation (e.g. Beuther et al. 2002a) it is possible that some masers in the sample are not actually associated with outflowing sources. Codella et al. (2004) proposed an evolutionary sequence for UCHII regions in which the earliest phase is marked by maser emission and molecular outflows not yet large enough to be de- tected with single-dish observations. The present data suggest that our sources are slightly more evolved because several of them show evidence of outflows. Their age therefore seems to be less than a few 104 yr (Codella et al. 2004) which is consis- tent with a statistical estimate of 3− 5× 104 yr for the methanol maser lifetime (van der Walt 2005). 6.3. Constraints on maser models The present study allows us to refine the range of physical con- ditions required to produce strong methanol masers at 6.7 GHz. Theoretical modelling by Cragg et al. (2002) demonstrated that a maser line of 1 km s−1 width attains a peak brightness temper- ature of ∼1011 K for a dust temperature >100 K and a methanol column density > 5 × 1015 cm−2. They found that methanol masers can be produced under a wide range of the physical conditions. In fact, for a methanol fractional abundance from 3× 10−8 to 10−5, masing is predicted for the gas density range 5 − 2 × 108 cm−3 and the methanol column density range 5 × 1015 − 2 × 1018 cm−2 (Cragg et al. 2002). The gas density inferred from our observations is between 105 and 107 cm−3; higher values (> 107 cm−3) are less probable. The hydrogen column density from 1022 to 2×1023 cm−2, inferred here, trans- lates well into the above range of methanol column densities for methanol fractional abundances of 5 × 10−7 − 10−5. This sug- gests that 6.7 GHz maser emission is less probable in environ- ments with a lower methanol fractional abundance of the order of 10−8. We conclude that our study well refines a range of the input parameters of Cragg et al.’s maser model. Specifically, a high methanol fractional abundance of > 5 × 10−7 is required whilst a gas density < 107 cm−3 is sufficient for the production of methanol masers. 7. Conclusions We have observed 10 transitions of HCO+, CO and CS iso- topomers at millimetre wavelengths in order to characterize the physical conditions in a sample of 28 MYSOs identified by the presence of methanol masers. No other preconditions were in- volved in the sample selection. The observations were centred at maser positions known with a sub-arcsecond accuracy. The main conclusions of the paper are summarized as follows: (1) The systemic velocity determined from the optically thin lines C18O(2−1), H13CO+(1−0), C34(2−1) and C34(3−2) agrees within ±3 km s−1 with the central velocity of the methanol maser emission for almost all the sources. (2) 18 out of 28 sources show residual line wings at least in one line when a Gaussian function was used to fit the CO, HCO+ and CS lines. Detection of the line wing emission sug- gests the presence of molecular outflows in these sources. Their occurrence needs to be confirmed by mapping observations. (3) Comparison between the kinematics of the methanol masers and of the thermal molecular lines reveals that they trace a wide range of molecular cloud conditions. In some objects the maser emission occurs in a narrow velocity range centered at the systemic velocity, which may indicate that the innermost parts of a molecular cloud or a circumstellar disc is the site of maser emission. In other objects the velocities of maser features are very similar to, or slightly overshoot, the velocity ranges of the thermal molecular line wings, suggesting that the masers arise in outflows. There are also objects where the maser emission re- veals more complex kinematics. (4) The column density of H2 derived from the CO and HCO+ lines are between 1022 and 2 × 1023 cm−2. We use our measurements of the intensity ratios of the CS lines to in- fer that methanol masers arise from regions with a gas den- sity of 105 − 107 cm−3, a kinetic temperature of 30 − 100K and a methanol fractional abundance of 5 × 10−7 − 10−5. This represents a significant refinement to the input parameters of methanol maser models. Acknowledgements. We like to thank the staff of the IRAM 30 m telescope for help with the observations and the unknown referee for helpful comments. This work has been supported by the Polish MNiI grant 1P03D02729. References Bartkiewicz, A., Szymczak, M., & van Langevelde, H.J. 2004, in Proceedings of the 7th European VLBI Network Symposium, 187 (astro-ph/0412002) Bartkiewicz, A., Szymczak, M., & van Langevelde, H.J. 2005, A&A, 442, L61 Beuther, H., Kramer, C., Deiss, B., & Stutzki, J. 2000, A&A, 362, 1109 Beuther, H., Schilke, P., Menten, K.M., et al. 2002a, ApJ, 566, 945 Beuther, H., Schilke, P., Sridharan, T.K., et al. 2002b, ApJ, 383, 892 Brand, J., & Blitz, L. 1993, A&A, 275, 67 Brand, J., Cesaroni R., Palla, F., & Molinari S. 2001, A&A, 370, 230 Caswell, J.L., Vaile, R.A., Ellingsen, S.P., Whiteoak, J.B., & Norris, R. 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Szymczak et al.: Molecular line study of high-mass protostars 9 Minier, V., Booth, R.S., & Conway, J.E. 2000, A&A, 362, 1093 Minier, V., Burton, M.G., Hill, T., et al. 2005, A&A, 429, 945 Myers, P.C., Mardones, D., Tafalla, M., Williams, J.P., & Wilner, D.J. 1996, ApJL, 465, 133 Narayanan, G., Walker, C.K., & Buckley, H.D. 1998, ApJ, 496, 292 Niezurawska, A., Szymczak, M., Richards, A.M.S., & Cohen R.J. 2005, BaltA, 14, 429 Norris, R.P., Byleveld, S.E., Diamond, P.J., et al. 1998, ApJ, 508, 275 Ossenkopf, Trojan, C., & Stutzki J. 2001. A&A, 378, 608 Plume, R., Jaffe, D.T., Evans, N.J.II, Martin-Pintado, J.,& Gomez-Gonzalez, J. 1997, ApJ, 476, 730 Purcell, C.R., Balasubramanyam, R. Burton, M.G., et al. 2006, MNRAS, 367, Schilke, P., Pineau de Forets, G., Walmsley, C.M., & Martin-Pintado, J. 2001, A&A, 372, 291 Schutte, A.J., van der Walt, D.J., Gaylard, M.J., & MacLeod, G.C. 1993, MNRAS, 261, 783 Szymczak, M., Hrynek, G., & Kus, A.J. 2000, A&AS, 143, 269 Szymczak, M., Kus, A.J., Hrynek, G., Kepa, A., & Pazderski, E. 2002, A&A, 392, 277 Thompson, A.R., Moran, J.M., & Swenson, J.R. 1991, Interferometry and syn- thesis in radio astronomy, van der Walt, J. 2005, MNRAS, 360, 153 van der Walt, D.J., Gaylard, M.J., & MacLeod, G.C. 1995, A&AS, 110, 81 Walsh, A.J., Burton, M.G., Hyland, A.R., & Robinson, G. 1998, MNRAS, 301, Walsh, A.J., Hyland, A.R., Robinson, G., & Burton, M.G. 1997, MNRAS, 291, Zhang, Q., Hunter, T.R., Brand, J., et al. 2005, ApJ, 625, 864 Introduction The sample Astrometric positions Distances Observations and data reduction Results Systemic velocities Shape of profiles Asymmetry Wings Derivation of physical parameters Column densities Gas density and temperature Discussion Kinematics Implications for the evolutionary status Constraints on maser models Conclusions

0704.1221

Dynamics of the Tippe Top via Routhian Reduction

Dynamics of the Tippe Top via Routhian Reduction M.C. Ciocci1 and B. Langerock2 Department of Mathematics, Imperial College London, London SW7 2AZ, UK Sint-Lucas school of Architecture, Hogeschool voor Wetenschap & Kunst, B-9000 Ghent, Belgium October 22, 2018 Abstract We consider a tippe top modeled as an eccentric sphere, spinning on a horizontal table and subject to a sliding friction. Ignoring translational effects, we show that the system is reducible using a Routhian reduction technique. The reduced system is a two dimensional system of second order differential equations, that allows an elegant and compact way to retrieve the classification of tippe tops in six groups according to the existence and stability type of the steady states. 1 Introduction The tippe top is a spinning top, consisting of a section of a sphere fitted with a short, cylindrical rod (the stem). One typically sets off the top by making it spin with the stem upwards, which we call, from now on the initial spin of the top. When the top is spun on a table, it will turn the stem down towards the table. When the stem touches the table, the top overturns and starts spinning on the stem. The overturning motion as we shall see is a transition from an unstable (relative) equilibrium to a stable one. Experimentally, it is known that such a transition only occurs when the spin speed exceeds a certain critical value. Let us make things more precise and first describe in some detail the model used throughout this paper. The tippe top is assumed to be a spherical rigid body with its center of mass ǫ off the geometrical center of the sphere with radius R. In addition, we assume that this eccentric sphere has a mass distribution which is axially symmetric about the axis through the center of mass O and the geometrical center C of the sphere. The tippe top is subject to a holonomic constraint since we only consider motions of the tippe top on a horizontal plane. An important observation is that the system has an integral of motion, regardless of the model of the friction force that is used. This integral is called the Jellet J and is proportional to the initial spin n0 (equation (2.7)). For the sake of completeness we mention here that a nonholonomic model (rolling without slipping) for the tippe top is not appropriate, since the equations of motion then allow two more integrals of motion (the energy and the Routh-integral) that prohibit the typical turning motion, cf. eg. [1, 5]. We will choose the friction force to be linearly dependent on the slip velocity of the contact point Q of the top and the horizontal plane. In earlier works [7, 15] conditions of stability for asymptotic states of the tippe top were retrieved by the Lyapunov function method starting with the Newton equations of motion. We take a different approach to the problem. Our goal is to describe the behaviour of the tippe top in terms of Lagrangian variables and thereby classify its asymptotic motions as function of non-dimensional (physical) parameters http://arxiv.org/abs/0704.1221v3 A/C, the inertia ratio and ǫ/R, the eccentricity of the sphere and of the value the Jellet J . As main novelty, we show that by ignoring the translational velocity of the center of mass in the friction law, the Lagrangian formulation of the dynamics of the tippe top allows a restriction to a system on SO(3) which is amenable to Routhian reduction [22]. The reduced system allows a rather simple stability analysis of the relative equilibria based on the relationship of the value of the Jellet’s integral J and the tumbling angle θ, see below. Through the Routhian reduction we retrieve a simple stability criterium which leads to the same conclusions as in [7, 15]. The hypothesis of neglecting translational effects is similar to the one made by Bou-Rabee et al. [16]. They motivated it by reasoning that for all possible asymptotic states (the relative equilibria of the system) the velocity of the center of mass is zero [7], and, for this reason, in a neighborhood of these solutions the translational friction can be neglected. In [15] the dynamics of the spherical tippe top with small friction has been studied without such an approximation and a full analysis of the asymptotic long term dynamics of the system is given in terms of the Jellet and the eccentricity of the sphere and inertia ratio. Remarkable is the fact that the stability results obtained by means of the Routhian reduction procedure fully coincides with the results of [15]. This justifies a-posteriori the approximation assumption in the friction force and shows that it accurately describes the behavior of the tippe top. The main result of this paper is summarized in the following theorem, also compare with [1, 15]. Theorem 1 In the approximation of negligible translational effects, a spinning ec- centric sphere on an horizontal (perfectly hard) surface subject to a sliding friction is reducible with a Routhian reduction procedure [22]. The relative equilibria of the reduced system are precisely the steady states of the original system. They are purely rolling solutions and except for the trivial state of rest, they are of three types: (i) (non-inverted) vertically spinning top with center of mass straight below the geometric center; (ii) (inverted) vertically spinning top with center of mass straight above the geo- metric center; (iii) intermediate spinning top, the top precesses about a vertical while spinning about its axle and rolling over the plane without gliding. The existence and stability type of these relative equilibria only depend on the inertia ratio A , the eccentricity of the sphere ǫ and the Jellet invariant J . In particular, six regimes are identified in terms of the Jellet invariant and only three exhibit the ‘tipping’ behavior. It turns out that the vertical states always exist, and intermediate states may branch off from them. Qualitative bifurcation diagrams corresponding to the possible dif- ferent regimes are sketched in Fig. 1. The reader is referred to Sec. 3 – Sec. 4 for the details and specification of the parameter ranges. There are three main groups Group I, II and III determined by existence properties of intermediate equilibrium states (similar to the subdivision from [20]). Tippe tops of Group I may admit intermediate states for θ > θc, where θ denotes the angle between the vertical and the symmetry axis of the top. Two subgroups are distinguished according to change in stability type of the intermediate states. Relevant is that for tippe tops belonging to this class the non-inverted position is always stable, so they never flip however large the initial spin when launched under an angle θ close to 0. Tippe tops of Group II may admit intermediate states for all θ, and they show complete inversion when the initial spin is large Group Ia Group Ib Group IIa Group IIb Group IIc Group III Figure 1: Bifurcation diagrams of relative equilibria in function of the Jellet invariant J . Solid black branches correspond to stable relative equilibria, while dashed black branches correspond to unstable ones. The vertical states θ = 0 and θ = π always exist, from which intermediate states (with 0 < θ < π) may branch off. enough. Tops of Group III tend to flip over up to a certain angle θc < π when spun rapidly enough. Since stability results are often in the literature expressed using the ‘initial’ spin n0 of the tippe top, one can read the J 2 in the figures as n20. We anticipate to further results by noting that the instability inequalities are independent of the friction coefficient unless it is zero. The structure of the paper is as follows. In Section 2 our model is described and the equation of motion are given according to the Lagrangian formalism. After introducing the Jellet integral of motion, the Routhian reduction is performed in Section 2.2. The steady states of the system are then calculated and their sta- bility type is determined, yielding a tippe top classification in six groups which is summarized in Section 4.1. 2 Equations of motion As is mentioned in the introduction, we consider the eccentric sphere model of such a top, see Fig. 2. That is, we consider a sphere with radius R whose mass distribution is axially symmetric but not spherically symmetric, so that the center of mass and the geometric center do not coincide. The line joining the center of mass and the geometrical center is an axis of inertial symmetry, that is, in the plane perpendicular to this axis the inertia tensor of the sphere has two equal principal moments of inertia A = B. The inertia moment along the axis of symmetry is denoted by C and the total mass of the sphere is m. The eccentricity ǫ is the distance between the center of mass O and the geometric center C of the sphere, with 0 < ǫ < R. The point Q is the point of contact with the plane of support. We assume that an inertial (laboratory) frame Mxyz is chosen, where M is some point on the table and the z-axis is the vertical. Let us denote the unit vectors along the axis of the reference frame Ox, Oy, Oz fixed to the body by respectively ex, ey, ez. The coordinates of the center of mass are denoted by PSfrag replacements h(θ) = R − ǫ cos(θ) Figure 2: Eccentric sphere version of the tippe top. R is the radius of the sphere, the center of mass O is off center by ǫ. The top spins on a horizontal table with point of contact Q. The axis of symmetry is Oz and the vertical axis is Oz, they define a plane Π (containing ~OQ) which precesses about Oz with angular velocity ϕ̇. The height of O above the table is h(θ). rO = (x, y, z)Mxyz . A second reference frame is denoted by Oxyz, and is defined in such a way that its third axis is precisely the symmetry axis of the top and the y-axis is perpendicular to the plane Π through the z- and z-axes (see Fig. 2). Again we denote the unit vectors along the axis of this reference frame by ex, ey, ez Let (θ, ϕ, ψ) be the Euler angles of the body with respect to the inertial frame, Fig. 2, chosen in such a way that (i) the vertical plane Π is inclined at ϕ to the fixed vertical plane xz and precesses with angular velocity ϕ̇ around the vertical Oz; (ii) the angle θ is the angle between the vertical Oz and the axle Oz of the top; θ̇ causes the nodding (nutation) of the axle in the vertical plane Π; and (iii) the angle ψ orients the body with respect to the fixed-body frame, ψ̇ is the spin about the axle. As it was pointed out before, the tippe top is constrained to move on a horizontal plane. This holonomic constraint is expressed by z = R − ǫ cos θ = h(θ). We assume throughout the paper that the only forces acting on the sphere are gravity G = −mgez and a friction force F exerted at the point of contact Q of the sphere with the plane. It is now immediate to write down the Lagrangian for the tippe L = 1 m(ẋ2 + ẏ2) + (ǫ2m sin2 θ +A)θ̇2 +A sin2 θϕ̇2 + C(ψ̇ + ϕ̇ cos θ)2 −mg(R− ǫ cos θ), (2.1) where g is earth acceleration. This function is defined on the tangent space of the configuration manifold M = IR2 × SO(3). In order to obtain the equations of motion, it only remains to define a suitable friction force F. The Lagrangian equations of motion for the tippe top then read: = QFi , (2.2) where qi represents one of the coordinates (x, y, ϕ, θ, ψ), and QF = QFi dq i is a one- form on M representing the generalized force moment of the friction force at the point of contact. It is defined by, with F = Rf+Rnez the orthogonal decomposition of F: QF = Rf · exdx+Rf · eydy + (q×Rf ) · (eydθ + ezdϕ+ ezdψ). 1In [16] the origin of the reference system attached to the body is in the center of the sphere C, and not in the center of mass O. Modeling the friction force One typically models the friction force F = Rf + Rnez to be proportional to the slip velocity of the point of contact vQ. We denote by Rn = Rnez the normal reaction of the floor at Q, which is of order mg, and Rf = FXex+FY ey is the (sliding) friction which opposes the slipping motion of the body. The fact that the sliding friction opposes the slipping motion is expressed by Rf · vQ ≤ 0. We adopt a viscous friction law [1, 8, 9, 20] and assume that Rf = −µRnvQ. (2.3) Here µ is a coefficient of friction with the dimension of (velocity)−1. It now takes a few tedious computations to arrive to the coordinate expressions for the force moments of the friction force. The coordinates of the point of contact ~OQ := q are Q = (xQ, 0, zQ)Oxyz = (R sin θ, 0, ǫ − R cos θ)Oxyz . The velocity of the point of contact Q equals vQ = vO + ω × q, (2.4) vO = (ẋ, ẏ, h ′(θ)θ̇)Mxyz is the velocity of the center of mass. A coordinate expression for the angular velocity is given by ω = −ϕ̇ sin(θ)ex + θ̇ey + nez, where n := ψ̇ + ϕ̇ cos(θ) (2.5) n is the spin (that is, the component of ω about Oz). The generalized force moments now read: Qx = −µRn(ẋ− sinϕ θ̇(R − ǫ cos θ) + cosϕ sin θ(Rψ̇ + ǫϕ̇)) Qy = −µRn(ẏ + cosϕ θ̇(R− ǫ cos θ) + sinϕ sin θ(Rψ̇ + ǫϕ̇)) Qθ = −µRn(R − ǫ cos θ)(cosϕẏ − sinϕẋ+ (R − ǫ cos θ)θ̇) Qϕ = −µRnǫ sin θ(cosϕẋ+ sinϕẏ + sin θ(ǫϕ̇+Rψ̇)) Qψ = −µRnR sin θ(cosϕẋ+ sinϕẏ + sin θ(Rψ̇ + ǫϕ̇)). For the sake of completeness we write an explicit expression for the normal compo- nent of the reaction force Rn, which can be determined from Newton’s law for the center of mass of the sphere: Rn(θ, ϕ, θ̇, ϕ̇, ψ̇, ẋ, ẏ) = g + θ̇2h′′ + h′ϕ̇ sin(θ)(ϕ̇ cos(θ)− C(ψ̇ + ϕ̇ cos θ)/A) 1/m+ h′/A[−hµ(sinϕẋ− cosϕẏ − θ̇h) + h′] (2.6) To conclude this section, we briefly discuss other models for the tippe top. The eccentric sphere model does not accurately model the contact effects of the tippe top stem. However, it does describe the fundamental phenomenon of the over- turning. For the sake of completeness we remark that as soon as the top rises to spin on its stem, it behaves as a ‘normal’ spinning top with rounded peg, we refer to [12, Chapter 6] for a satisfactory introduction to this topic. Note that our model also does not describe the peculiar ‘Hycaro’ tippe tops by Prof. T. Tokieda [18], which need a non-axisymmetric asymmetric mass distribution. These tippe tops have a ‘preferred direction’ meaning that the top would flip over only if spun in the preferred direction with a certain initial spin, and, no matter what the initial spin is, it would just continue rotating around the rest position when spun the other way round. We refer to [4] for a detailed analysis when elasticity properties of the horizontal surface and tippe top are taken into account. Their model allows for jumps of the tippe top on the horizontal surface (that we assumed to be rigid). Finally, we mention that we do not debate over the issue of whether transitions sliding-rolling and rolling-sliding occur in the motion of an eccentric sphere on a flat surface. We chose to concentrate on the sliding model only, because we were interested in capturing the ‘overturning’ phenomenon which cannot occur under the non-holonomic constraint of pure rolling. We refer the interested reader to [1, 5, 8] for a discussion of the topic and to [3, 13, 19] for an analysis of the motion of the rolling eccentric sphere also called the Routh’s sphere. 2.1 Constants of motion: the Jellet invariant It was first shown by Jellet [6] by an approximate argument, and later proved by Routh [17] that the system, even if dissipative, has a conserved quantity: J = −L · q = const, (2.7) where L is the angular momentum of the tippe top about the center of mass. We prove this by using Euler equations which govern the evolution of the angular momentum L̇ = q× F. The total time derivative of J then becomes: J̇ = −L̇ · q− L · q̇ = 0− (Aω − (A− C)(ω · ez)ez) · (ǫω × ez) = 0 . Straightforward calculations show that Jellet’s constant can be written as J = Cn(R cos(θ)− ǫ) +Aϕ̇R sin2(θ) . (2.8) We emphasize once more that the Jellet’s constant is an exact constant of motion for the tippe top whether or not there is slipping and independent of the expression for F. As we will explain later, it is this constant that to some extent controls the motion of the spinning top. Indeed, it allows a Routhian reduction procedure (see Sec. 2.2), resulting in relatively simple reduced equations from which we are able to recover in full detail the stability properties of the steady states. In the specific case that θ = 0, the Jellet is proportional to n0 = n|θ=0, the spin about the z-axis. Since one typically sets off the tippe top at an angle θ ≈ 0, one can say that the Jellet is proportional to the initial spin n0. Note that the spin at θ = π has an opposite sign to the initial spin n0, meaning that, relative to a body fixed frame, the spin is reversed when the tippe top fulfills a complete inversion. There is a rotational symmetry for which the Jellet is the associated first integral. The action of S1 on IR2 × SO(3) can be defined as a simultaneous rotation about ê3 over the angle Rξ and about k̂ over the angle −ǫξ, where ξ ∈ S1 (see also [16]). Noether’s theorem is applicable in this situation since the work of the friction force at the point of contact vanishes under this action. Note that the total energy of the spinning top is E = T + V is in general not conserved. Here T is the kinetic energy with its rotational and translational part, V = mgh(θ) is the potential energy. The orbital derivative of E is E = vQ ·Rf ≤ 0, (2.9) which is negative semi-definite and vanishes if and only if vQ vanishes. Observe that E(t) decreases monotonically and hence is a suitable Lyapunov function2, see [7]. From (2.9) it follows that dissipation is due to friction. 2E(t) is analytical, therefore it is either strictly monotone or a constant. The energy E is constant only if vQ = 0. Note that E being Lyapunov [7] implies that the limiting solutions for t → ∞ are solutions of constant energy. 2.2 Routhian reduction It turns out that, if we consider an approximation of the friction law, the resulting generalized force moments assume a form that allows us to apply a Routhian re- duction procedure, see [22] and appendix A. In turn, using the reduced equations we are able to study in full detail the stability properties of the tippe top which confirm the results obtained in [15], and also recover those of [16, 20]. We now ignore translational effects in the friction force, i.e. we assume that all terms in Qθ, Qϕ, Qψ containing ẋ and ẏ are neglected. Typically this approximation is justified by noting that for all steady states the velocity of the center of mass is zero, and that in a neighborhood of the steady states it can be neglected. In our situation, it allows to restrict ourselves to a system on SO(3) which is reducible using Routh’s procedure. It is easily seen that within this approximation, if we study the Lagrangian system on SO(3) determined by L′ = 1 (ǫ2m sin2 θ +A)θ̇2 +A sin2 θϕ̇2 + C(ψ̇ + ϕ̇ cos θ)2 −mg(R− ǫ cos θ), Q′ = −µR′n(R− ǫ cos θ)2θ̇dθ − µR′nǫ sin2 θ(ǫϕ̇+Rψ̇)dϕ −µR′nR sin2 θ(Rψ̇ + ǫϕ̇)dψ, with R′n(θ, θ̇, ϕ̇, ψ̇) = Rn(θ, ϕ, θ̇, ϕ̇, ψ̇, ẋ = 0, ẏ = 0), then we essentially study the en- tire approximated system. Indeed, any solution (ϕ(t), θ(t), ψ(t)) to this Lagrangian system will determine the remaining unknowns (x(t), y(t)) as solutions to the fol- lowing system of time-dependent second order differential equations: mẍ = Qx(ẋ, θ(t), ϕ(t), ψ(t)), mÿ = Qy(ẏ, θ(t), ϕ(t), ψ(t)). (2.10) Our next step in the reduction procedure is to consider the Lagrangian system L′ on SO(3) with generalized force form Q′ and perform a simple coordinate transfor- mation, determined by (θ, ϕ, ψ) 7→ (θ, ϕ = ǫϕ+Rψ, c = Rϕ− ǫψ). The Lagrangian L′ and the force Q′ then become L′ = 1 (ǫ2m sin2 θ +A)θ̇2 + A sin (ǫ2+R2)2 (ǫϕ̇+Rċ)2+ (ǫ2+R2)2 (R+ ǫ cos θ)ϕ̇+ (R cos θ − ǫ)ċ −mg(R− ǫ cos θ) Q′ = −µR′n (R − ǫ cos θ)2θ̇dθ + sin2 θϕ̇dϕ where it is understood that R′n is a function of θ, θ̇, ϕ̇ and ċ. The main reason for writing L′ and Q′ in this form is the fact that c is a cyclic coordinate and that Q′c = 0. Indeed, recall that Jellet integral was associated to the symmetry determined by the vector field R∂ϕ − ǫ∂ψ, or in the above introduced coordinate system by the vector field ∂c. In particular, we have made the symmetry generator into a coordinate vector field, and this ensures that we can apply the Routhian reduction procedure ([22] and appendix A), provided the coefficients of Q′ do not depend on c. This is the case since we neglected the terms in the velocity of the center of mass. The conserved quantity associated with the cyclic coordinate is precisely the Jellet integral: = RA sin (ǫ2+R2)2 (ǫϕ̇+Rċ) + C(R cos θ−ǫ) (ǫ2+R2)2 (R+ ǫ cos θ)ϕ̇+ (R cos θ − ǫ)ċ = J/(ǫ2 +R2). The latter equality implies that J(ǫ2 +R2)− (RAǫ sin2 θ + C(R cos θ − ǫ)(R+ ǫ cos θ))ϕ̇ R2A sin2 θ + C(R cos θ − ǫ)2 (2.11) The Routhian reduction procedure defines a Lagrangian system R = L′ − Jċ/(ǫ2 +R2) with two degrees of freedom (θ, ϕ) (here every instance of ċ in R is replaced us- ing (2.11)). The reduced equations of motion are then given by = Q′θ = −µR′n(R− ǫ cos θ)2θ̇ (2.12) = Q′ϕ = −µR′n sin2 θϕ̇, (2.13) where it is understood that (2.11) is used to eliminate ċ in R′n. It takes rather tedious but straightforward computations to show that R can be written as R = T2 + T1 −W, where T2 = Tθθ(θ)θ̇ 2 + Tϕϕ(θ)ϕ̇ (ǫ2m sin2 θ +A)θ̇2 + AC sin2 θ (R2A sin2 θ + C(R cos θ − ǫ)2) (R2+ǫ2) (RǫA sin2 θ+C(R+ǫ cos θ)(R cos θ−ǫ)) R2A sin2 θ+C(R cos θ−ǫ)2 ϕ̇, W = 1 R2A sin2 θ+C(R cos θ−ǫ)2 −mgǫ cos θ. The function W is also called the effective potential. Remark 1 The above defined Routhian function R is not globally defined on the sphere. In order to provide a globally defined system of differential equations for the reduced system, we need to extract the term T1 from the Routhian and consider it as a gyroscopic force (see e.g. [21]). Remark 2 Observe that the effective potential W , obtained through reduction, co- incides with the effective energy on a Jellet’s level surface as it has been used in [15], Sec. 3. Note that R and Q′ are independent of ϕ. This residual symmetry does not lead to a conserved quantity (the friction does not vanish for ∂ϕ). This symmetry is due to the approximation we carried out in the previous section; it is not present in the original system (L,Q) or (2.2). It leads however to a zero eigenvalue of the linearized system at equilibrium points, see Section 4. 3 Steady states The equilibria of the reduced Routhian system (2.12) and (2.13) are determined by θ̇ = 0, ϕ̇ = 0 and ∂W/∂θ = 0, (3.1) i.e. they are the critical points of the effective potential (note that if θ̇ = ϕ̇ = 0 the components of the force vanish). Equation (3.1) is satisfied if (i) sin θ = 0 or, if (ii) f(J2, cos θ) = J mgCR2ǫ cos θ + ǫ sin2 θ + cos θ − ǫ (3.2) Solutions to (i) are θ = 0, π and give the vertical spinning states. Solutions to (ii) only occur if cos(θ) + > 0. (3.3) If this condition is satisfied, solutions to (ii) are the so-called intermediate states. The existence condition only depends on A/C and ǫ/R, and will determine in our classification the three main groups I, II and III, as it was proposed in [20]. The values for θ for which (ii) is satisfied will depend on the Jellet, A/C and ǫ/R. The vertical spinning states correspond to the periodic motion of the tippe top spinning about its axle (which is in vertical position) either in the non-inverted position (θ = 0) or inverted position (θ = π). The intermediate states correspond to those relative equilibria in which the tippe top shows in general quasi-periodic motion precessing about a vertical while spinning about its inclined axle rolling over the plane without gliding (observe that the intermediate states correspond to the tumbling solution of [7]). The condition (3.3) for existence of intermediate states leads to a first classifi- cation of tippe tops into three groups. - Group I [(A/C − 1) < −ǫ/R]: the tippe tops belonging to this group do not ad- mit intermediate states in an interval of the form [0, θc[, where θc is determined by (A/C − 1) cos θc + ǫ/R = 0. - Group II [−ǫ/R < (A/C−1) < ǫ/R]: intermediate states may exist for all θ ∈]0, π[. - Group III [(A/C − 1) > ǫ/R]: tippe tops belonging to this group do not admit intermediate state in an interval of the form ]θc, π[, where cos θc = (ǫ/R)/(1−A/C). In the following section we refine this first classification taking into account the stability type of the steady states and their bifurcations, explaining and giving the details of the J2 versus θ diagrams in Fig. 1 from the introduction. We anticipate that the subdivision in subgroups according to a change in sta- bility type of the intermediate states is based on the simple observation that they lie on a curve f(J2, cos θ) = 0 in the (J2, θ)-plane, and, denoting by ∂2f the partial derivative to the second argument of f , a bifurcation point for intermediate states is characterized as the point where f(J2, cos θ) = 0, ∂2f(J 2, cos θ) = 0. Note that the relation ∂2f(J 2, cos θ) = 0 is essentially the same basic relation (4.27) in [15], however its derivation is different. As a consequence we expect the stability results for intermediate states to confirm earlier known facts. 4 Stability analysis via the reduced equations Determining the (linear) stability of the steady states as given above is an extremely simple task in the reduced setting. Indeed, let (θ0, ϕ0) be an equilibrium. The linearized equations of motion at this relative equilibrium read as Tθθ(θ0)θ̈ = ∂ϕ̇∂θ (θ0)ϕ̇ − (θ0)(θ − θ0)− µmg(R− ǫ cos θ0)2θ̇ Tϕϕ(θ0)ϕ̈ = − ∂ϕ̇∂θ (θ0)θ̇ − µmg sin2 θ0ϕ̇, where we used that R′n equals mg (2.6) at the relative equilibria. It is not hard to show that the characteristic polynomial of this system is p(λ) = λ λ3 + µmg (R− ǫ cos θ0)2 Tθθ(θ0) sin2 θ0 Tϕϕ(θ0) µmg(R− ǫ cos θ0) sin θ0 ∂ϕ̇∂θ Tθθ(θ0)Tϕϕ(θ0) Tθθ(θ0) µmg sin2 θ0 Tθθ(θ0)Tϕϕ(θ0) Due to the translational symmetry in ϕ, one eigenvalue is zero. In Appendix B we show that all remaining eigenvalues have a negative real part, if and only if ∂2W/∂θ2(θ0) > 0, or if mgǫ cos θ0 > J2R2C (R2A sin2 θ0+C(R cos θ0−ǫ)2)2 − (A/C − 1) sin2 θ0 cos θ0 − 4 B sin2 θ0 (A/C) sin2 θ0+(cos θ0−ǫ/R)2 , (4.1) with B given by B := (A−C) cos(θ0)+C ǫR .We will further manipulate this equation to retrieve the stability results, compare also with [20]. We will retrieve six groups depending on how the inertia ratio A/C relates to the eccentricity ǫ/R. Since in the literature results have been expressed in terms of the spin of an initial condition at a vertical state, we introduce n0 := C(R−ǫ) , which is the value of the spin n at θ = 0 for a given Jellet J . Similarly, nπ := − JC(R+ǫ) is the spin of the solution with Jellet J at θ = π. Note that for a fixed J these spins are related by n0 = −nπ R+ǫR−ǫ . Vertical spinning state: θ = 0. For the vertical spinning state θ = 0, the relation (4.1) yields − (1 − ǫ . (4.2) It follows that in Group I, the vertical state θ = 0 is always stable, while for Group II and Group III stability requires that |n0| < n1 := −(1− ǫ . (4.3) Vertical spinning state: θ = π. For the vertical spinning state θ = π, the relation (4.1) yields . (4.4) This condition is never satisfied for Group III, so θ = π is unstable; in the case of Group I and II, when A < (1 + ǫ ), stability requires |nπ| > n2 =: C[(1+ ǫ 1 + ǫ . (4.5) Note that for tippe tops of Group I and II n22 ≥ n2∗, with n∗ := 2 . The equality holds when A 1 + ǫ Intermediate states. Recall that intermediate states are determined by (3.2): f(J2, cos θ0) = 0. Using this condition, the requirement (4.1) becomes 0 < g(cos θ0) where we set g(cos θ0) := −1) cos θ+ ǫ sin2 θ+(cos θ− ǫ . (4.6) We now prove that g(cos θ0) is strictly increasing and changes sign at a bifur- cation point for intermediate states. As we already mentioned, a bifurcation point along the curve in the (J2, cos θ)-plane of intermediate states is determined by the conditions f(J2, cos θ) = 0, ∂2f(J 2, cos θ) = J mgǫR2C − 1) + − 1) cos θ + ǫ (1 − cos2 θ) + (cos θ − ǫ )2) = 0. An elementary substitution of the first equation into the second shows that the function ∂2f(J 2, cos θ) along the intermediate states can also be written as ∂2f(J 2, cos θ) = g(cos θ) mgǫR2C Hence, bifurcation points are given by those θ such that g(cos θ0) = 0. Solving for cos θ gives the two solutions: 1−A/C 1−A/C 1−A/C − (ǫ/R)2. (4.7) Note that 1−A/C − (ǫ/R)2 > 0 only for tippe tops in Groups I and II. Moreover, the solution cos θb = 1−A/C + . . . leads to a contradiction since, for tippe tops of Group I, it is incident with the interval ]0, θc[ where no intermediate states exist and for tippe tops in Group II satisfying 1−A/C − (ǫ/R)2 > 0, the number ǫ/R 1−A/C is greater than 1, implying that the + solution of (4.7) can not equal a cosine. We denote the − solution by xb, i.e. xb := 1−A/C − 1−A/C 1−A/C − (ǫ/R)2. We conclude that a bifurcation point for intermediate states exists if 1 − A/C − (ǫ/R)2 > 0 and |xb| < 1. Before studying in further detail these two conditions, we first show that the function g(cos θ) is strictly increasing for increasing θ. This result implies that, if a bifurcation exists (i.e. a point θ with g(cos θ) = 0) then stability will change. On the other hand, if no bifurcation occurs, the entire branch of intermediate states is either stable or unstable. Let us assume that x = cos θ. If we consider ∂2f(J 2, x) as a function on the submanifold f(J2, x) = 0 and if we compute its derivative w.r.t x, i.e. ∂2,2f − (∂1,2f)(∂2f/∂1f), then after some tedious computations we may conclude that the sign of this derivative is opposite to the sign of 8((A/C − 1)x+ ǫ/R)2 + (A/C − 1)2 (A/C(1−x 2)+(x−ǫ/R)2)2 ((A/C−1)x+ǫ/R)2 > 0 Hence, if there is a bifurcation at a certain x = cos(θb) in the set of intermediate states, then we know that the intermediate states for which θ > θb are stable, while the other branch is unstable. It now remains to study the conditions for the bifurcation point to exist. The first condition says that 1 − A/C − (ǫ/R)2 > 0, implying that we only have to consider Groups I and II. We start with Group I. Group I From (A/C − 1) < −ǫ/R, it follows that 1 − A/C − (ǫ/R)2 > 0 and xb < 1. We have to distinguish between two subgroups: xb < −1 (Group Ia) and xb > −1 (Group Ib). From the previous paragraph it should be clear that if xb < −1 then the value of the function g(x) on the intermediate states −1 < x < 1 is negative, i.e. the entire branch is unstable. Group II We define three subgroups: Group IIa is the group for which 1−A/C− (ǫ/R)2 > 0 and |xb| < 1, Group IIc is defined by 1−A/C− (ǫ/R)2 > 0 and xb < −1 and thirdly Group IIb as the group containing the remaining tippe tops in II. Again from the previous, we immediately conclude that the entire branch of intermediate states is unstable in Group IIc. Group IIb can alternatively be defined as the group containing all tippe tops for which the branch of intermediate state is entirely stable. To show this, we first remark that the remaining tippe tops in II are characterized by 1−A/C−(ǫ/R)2 > 0 and xb > 1 or 1−A/C−(ǫ/R)2 ≤ 0. If 1−A/C−(ǫ/R)2 > 0 and xb > 1 then the intermediate branch is entirely stable. If 1−A/C− (ǫ/R)2 = 0 then xb > 1 and the intermediate branch is stable. The remaining tippe tops we have to consider are characterized by the condition 1−A/C− (ǫ/R)2 < 0. To show that g(x) > 0, we consider two subcases: (i) if 1−A/C < 0 then from (4.6) it is clear that g(x) > 0, (ii) if (ǫ/R)2 > 1−A/C > 0 then it suffices to compute g(0) (g(x) will not change sign since there is no bifurcation point xb): from 1−A/C− (ǫ/R)2 < 0, A/C < 1 and (ǫ/R)2 < 1 we find g(0) = (A/C − 1) + 4 (ǫ/R) A/C+(ǫ/R)2 > (ǫ/R)2 > 0. The above argument also proves stability for intermediate states in Group III. Note that tippe tops in Group II are real ‘tippe tops’ since they admit tipping from a position near θ = 0 to the inverted state near θ = π. Tipping never occur for tops of Group III, though they may rise up to a (stable) intermediate state. Tippe tops of group I never flip over since the position θ = 0 is always stable. 4.1 Tippe Top Classification The following schematic classification summarizes the previous analysis, and pre- sented in Fig. 1. Compare also with Fig. 3 in [15]. Group I: A/C − 1 < −ǫ/R - The non-inverted vertical position θ = 0 is stable for any value of J . - The inverted vertical position θ = π is stable for |nπ| > n2, unstable otherwise, with n2 given by (4.5). - Intermediate states do not exist for all values of θ, but only for θ > θc = arccos(( ǫ )/(1− A Group Ia: 1−A/C − 1−A/C 1−A/C − (ǫ/R)2 < −1 . The entire branch of intermediate states is unstable. Group Ib: if −1 < ǫ/R 1−A/C − 1−A/C 1−A/C − (ǫ/R)2 := cos θb . There is a bifurcation: intermediate state are stable if θ > θb and unstable if θ < θb. Group II: −ǫ/R < (A/C − 1) < ǫ/R. - If |n0| > n1 the equilibria θ = 0 become unstable, with n1 as in (4.3). - If |nπ| > n2 the equilibria θ = π become stable. - There are intermediate states for any θ. We distinguish the following three sub- groups. Group IIa: (A/C − 1) < −(ǫ/R)2 and 1−A/C − 1−A/C 1−A/C − (ǫ/R)2 < 1 . There is a bifurcation of intermediate states. Group IIb: either (A/C − 1) ≥ −(ǫ/R)2 or 1−A/C − 1−A/C 1−A/C − (ǫ/R)2 > 1 . The branch of intermediate states is entirely stable. Group IIc: (A/C − 1) < −(ǫ/R)2 and 1−A/C − 1−A/C 1−A/C − (ǫ/R)2 < −1 . The branch of intermediate states is entirely unstable. Group III: (A/C − 1) > ǫ/R. - The equilibria with θ = 0 become unstable for |n0| > n1. - The equilibria with θ = π are always unstable. - For these tippe tops intermediate states do not exist for θ ∈]θc, π[. Bifurcations in intermediate states do not occur and the intermediate states are stable. Tippe tops in Group II exhibit ‘tipping’ behavior: if the initial spin satisfies |n0| > max(n1, n2 R−ǫ ) 3, then tipping is possible from θ = 0 to θ = π. 5 Remarks and Conclusions We would like to remark that the results on the linear stability for the tippe top presented here are equivalent to the stability properties for the model of the tippe top without the assumption that the translational friction terms can be neglected. In fact, the function g(cos θ) which fully determines the stability of the intermediate states can be retrieved in the expressions for the eigenvalues of the linearized equa- tions of motion about the relative equilibria of the full system. It is also remarkable that the presented stability analysis does not depend on the friction coefficient µ, which suggests that the above stability analysis is valid for a rather large class of possible dissipative friction forces at the point of contact as was pointed out in [15]. Up until now, a linear stability analysis for the intermediate states was not available upon our knowledge in the literature. Most recently Ueda et al. [20] analyzed the motion of the tippe top under the gyroscopic balance condition (gbc) and approached the stability problem by per- turbing the system around a steady state and obtained under linear approximation a first order ode for the perturbation of the variable θ. They derive stability crite- ria in terms of the initial spin n given at the non-inverted position θ = 0. Possible 3About the relation between n1 and n2: in Group IIb n > 1 holds and in Group IIc intermediate states for tippe tops of Group I were not considered. Our approach is based on a complete different technique, namely Routhian reduction and we obtain a more refined and exhaustive classification of tippe tops in six groups (instead of three). Moreover, our analysis does not exclude the possibility of launching the top with its stem down (i.e. θ near π). Acknowledgments The results presented here were obtained while the first author had financial support by the European Community’s 6th Framework Programme, Marie Curie Intraeuro- pean Fellowship EC contract Ref. MEIF-CT-2005-515291, award Nr. MATH P00286 and the second author was Postdoctoral Fellow of the Research Foundation – Flan- ders (FWO) at the Department of Mathematical Physics and Astronomy, Ghent University, Belgium. The authors wish to thank Dr. B. Malengier, Prof. F. Cantrijn, and Prof. J. Lamb for stimulating discussions and the anonymous referee for pointing out reference [15]. References [1] M.C. Ciocci, J.S.W. Lamb, B. Langerock and B. Malengier. Dynamics of the spherical tippe top with small friction. preprint [2] C.M. Cohen. The tippe top revisited. Am. J. Phys. 45, (1977) 12–17. [3] R. Cushman. The Routh’s sphere. Reports on Mathematical Physics 42, (1998) 47–70. [4] C. Friedl. Der Stehaufkreisel, Zulassungsarbeit zum 1. Staatsexamen, Univer- sität Augsburg, http://www.physik.uni-augsburg.de/∼wobsta/tippetop/index.shtml.en [5] C.G. Gray and B.G. Nickel. Constants of the motion for nonslipping tippe tops and other tops with round pegs. Am. J. Phys. 68 (9), (2000) 821–828. [6] J.H. Jellet. A treatise on the theory of friction. MacMillan, London, (1872). [7] S. Ebenfeld and F. Scheck. A new analysis of the tippe top: Asymptotic States and Lyapunov Stability. Annals of Physics 243, (1995) 195–217. [8] T.R. Kane and D. Levinson. A realistic solution of the symmetric top problem. J. Appl. Mech. 45, (1978) 903–909. [9] H.K. Moffatt, Y. Shimomura and M. Branicki. Dynamics of axisymmetric body spinning on a horizontal surface. I. Stability and the gyroscopic approximation. Proc. R. Soc. Lond. A 460, (2004) 3643–3672 [10] I.J. Nagrath and M. Gopal. Control Systems Engineering. New Age Interna- tional Publishers, 4th edition, (2006). [11] H.K. Moffatt and Y. Shimomura. Spinning eggs - a paradox resolved. Nature 416, (2006) 385–386. [12] V.D. Barger and M.G. Olsson. Classical machanics. A modern perspective. McGraw-Hill Book Company, (1973). [13] A.V. Borisov and I.S. Mamaev. Rolling of a rigid body on plane and sphere. Hierarchy of dynamics. Regular and Chaotic Dyn. 7 (2), (2002) 177–200. http://www.physik.uni-augsburg.de/~wobsta/tippetop/index.shtml.en [14] Or. The Dynamics of a tippe top. SIAM J. on A. Math. 54 (3), (1994) 597–609. [15] S. Rauch-Wojciechowski, M. Sköldstam and T. Glad. Mathematical analysis of the Tippe Top. Regul. Chaotic Dyn. 10, (2005) 333–362. [16] N.M. Bou-Rabee, J.E. Marsden and L.A. Romero. Tippe top inversion as a dissipation induced instability. SIAM J. A. Dyn. Sys. 3, (2004) 352–377. [17] E.J. Routh. Dynamics of a system of rigid bodies. MacMillan, NY, (1905). [18] T. Tokieda. Private Communications. The Hycaro Tipe Top was presented for the first time during the meeting Geometric Mechanics and its Applications (MASIE), 12–16/07/2004, EPF Lausanne, CH. [19] S. Torkel Glad, D. Petersson and S. Rauch-Wojciechowski. Phase Space of Rolling Solutions of the Tippe Top. SIGMA 3, (2007) Contribution to the Vadim Kuznetsov Memorial Issue. [20] T. Ueda, K. Sasaki and S. Watanabe. Motion of the Tippe Top: Gyroscopic balance condition and Stability. SIADS 4 (4), (2005) 1159–1194, Society for Industrial and Applied Mathematics. [21] J.E. Marsden, T.S. Ratiu and J. Scheurle. Reduction theory and the Lagrange- Routh equations. Journal of Math. Phys. 41 (6), (2000) 3379–3429. [22] L.A. Pars. A Treatise on Analytical Dynamics. Heinemann Educational Books, (1965). A Routhian reduction for dissipative systems. Assume that a Lagrangian L is given, defined on the tangent space of a manifold M on which a local coordinate system (q1, . . . , qn) is chosen. The system is not conservative, in the sense that there is a one form on M , denoted by Q = Qidq representing the generalized force moments of the non-conservative forces. The equations of motion read: = Qi for i = 1, . . . , n . (A.1) Theorem 2 (cf. [22]) Assume that there exists a coordinate, say q1 such that (i) ∂L = 0, (ii) Qi(q 2, . . . , qn) is independent of q1 for all i = 1, . . . , n, (iii) 2, . . . , qn) = 0; then β = ∂L/∂q̇1 is a constant of the motion. Assume that the latter equation is invertible and allows us to write q̇1 = f(q2, . . . , qn, q̇2, . . . , q̇n, β). The system (A.1) is equivalent to the system = Qi for i = 2, . . . , n , (A.2) where the new Lagrangian R is defined by R = L− q̇1β and such that all occurrences of q̇1 in R and Qi are replaced by f . By equivalent systems we mean that any solution to (A.1) with fixed momentum ∂L/∂q̇1 = β is a solution to (A.2) and vice versa. B Roots of the characteristic polynomial We show that the roots of the polynomial λ3 + µmg (R− ǫ cos θ0)2 Tθθ(θ0) sin2 θ0 Tϕϕ(θ0) µmg(R− ǫ cos θ0) sin θ0 ∂ϕ̇∂θ Tθθ(θ0)Tϕϕ(θ0) Tθθ(θ0) µmg sin2 θ0 Tθθ(θ0)Tϕϕ(θ0) all have negative real parts if and only if ∂ (θ0) > 0. For our convenience we use the following shorthand notations µmg(R− ǫ cos θ0)2 Tθθ(θ0) > 0, β := ∂ϕ̇∂θ Tθθ(θ0)Tϕϕ(θ0) µmg sin2 θ0 Tϕϕ(θ0) > 0 and δ := Tθθ(θ0) and the polynomial under consideration becomes λ3 + (α+ γ)λ2 + (αγ + β2 + δ)λ+ δγ. Note that if sin θ0 = 0 the system (2.13) is singular and a coordinate change is necessary. The characteristic polynomial written above is not singular. Indeed, we have that Tϕϕ(θ0) is proportional to sin 2 θ0 or that γ is well-defined in the case sin θ0 = 0. Similarly, we have that ∂ϕ̇∂θ (θ0) is proportional to sin θ0, and β is well- defined. If we now write the necessary and sufficient conditions for this polynomial to be Hurwitz [10], we obtain the conditions (α+ γ) > 0, (α+ γ)(αγ + β2 + δ)− δγ > 0 and δγ > 0. The first condition is trivially satisfied. We now show that the third condition δ > 0 implies the second condition. We can rewrite the second condition as (1 + α/γ)(αγ + β2 + δ) > δ, and this is valid if δ > 0 or if ∂ (θ0) > 0. 1 Introduction 2 Equations of motion 2.1 Constants of motion: the Jellet invariant 2.2 Routhian reduction 3 Steady states 4 Stability analysis via the reduced equations 4.1 Tippe Top Classification 5 Remarks and Conclusions A Routhian reduction for dissipative systems. B Roots of the characteristic polynomial

0704.1222

Plasmaneutrino spectrum

EPJ manuscript No. (will be inserted by the editor) Plasmaneutrino spectrum A. Odrzywo lek M. Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krakow, Poland November 15, 2018 Abstract. Spectrum of the neutrinos produced in the massive photon and longitudal plasmon decay process has been computed with four levels of approximation for the dispersion relations. Some analytical formulae in limiting cases are derived. Interesting conclusions related to previous calculations of the energy loss in stars are presented. High energy tail of the neutrino spectrum is shown to be proportional to exp(-E/kT), where E is the neutrino energy and kT is the temperature of the plasma. 1 Introduction & Motivation Thermal neutrino loses from plasma are very important for stellar astrophysics [1,2]. Plasmon decay is one of the three main reactions. Extensive calculations for these pro- cesses were done by group of Itoh [3,4,5,6,7,8,9,10,11]. Other influential article include [12,13,14,15,16,17,18,19, 20]. Meanwhile, our abilities to detect neutrinos has grown by many orders of magnitude, beginning with 1.4 tonne experiment of Reines&Cowan [21] up to the biggest exist- ing now 50 kt Super-Kamiokande detector [22]. Recently, ”GADZOOKS!” upgrade to Super-Kamiokande proposed by Beacom&Vagins [23] attract attention of both experi- mental and theoretical physicists. At last one new source of the astrophysical antineutrinos is guaranteed with this upgrade, namely Diffuse Supernova Neutrino Background [25,24]. Pre-supernova stars will be available to observa- tions out to ∼2 kiloparsecs [25]. This technique is the only extensible to megaton scale [25]. Memphys, Hyper- Kamiokande and UNO (Mt-scale water Cherenkov detec- tors cf. e.g. [26]) proposals now seriously consider to add GdCl3 to the one of the tanks with typical three-tank design [27]. Recently, the discussion on the geoneutrino detection [28], increased attention to the deep underwa- ter neutrino observatories [29] with target mass 5-10 Mt [25] and even bigger [30]. It seems that (anti)neutrino as- tronomy is on our doorstep, but numerous astrophysical sources of the ν’s still are not analyzed from the detection point of view. Detection of the solar [31,32,33,34] and supernova neu- trinos [35,36,37,38] was accompanied and followed with extensive set of detailed calculations (see e.g. [39,40,41, 42,43,44] and references therein as a representatives of this broad subject) of the neutrino spectrum. On the con- trary, very little is known about spectral neutrino emission from other astrophysical objects. Usually, some analytical representation of the spectrum is used, based on earlier experience and numerical simulations, cf. e.g. [45]. While this approach is justified for supernovae, where neutrinos are trapped, other astrophysical objects are transparent to neutrinos, and spectrum can be computed with an ar- bitrary precision. Our goal is to compute neutrino spec- tra as exact as possible and fill this gap. Plasmaneutrino process dominates dense, degenerate objects like red giant cores [46], cooling white dwarfs [47] including Ia supernova progenitors before so-called ,,smoldering” phase [48]. It is also important secondary cooling process in e.g. neutron star crusts [49] and massive stars [50]. Unfortunately, ther- mal neutrino loses usually are calculated using methods completely erasing almost any information related to the neutrino energy Eν and directionality as well. This infor- mation is not required to compute total energy Q radiated as neutrinos per unit volume and time. From experimen- tal point of view, however, it is extremely important if given amount of energy is radiated as e.g. numerous keV neutrinos or one 10 MeV neutrino. In the first case we are unable to detect (using available techniques) any tran- sient neutrino source regardless of the total luminosity and proximity of the object. In the second case we can detect astrophysical neutrino sources if they are strong and not too far away using advanced detector which is big enough. Few of the research articles in this area attempt to es- timate average neutrino energy [16,17,52,53] computing additionally reaction rate R. Strangely, they presented fig- ures and formulae for Q/R instead of 1 Q/R. This gives false picture of real situation, as former expression gives 〈Eν + Eν̄〉. Obviously, we detect neutrinos not ν-ν̄ pairs. Q/R do not give average neutrino energy, as in general neutrino and antineutrino spectra are different. As we will see only for longitudal plasmon decay neutrinos energies of neutrinos and antineutrinos are equal. However, dif- ference in all situations where thermal neutrino loses are important is numerically small and formula: 〈Eν〉 ≃ is still a ”working” estimate. http://arxiv.org/abs/0704.1222v1 2 A. Odrzywo lek: Plasmaneutrino spectrum Mean neutrino energy is useful in the purpose of quali- tative discussion of the detection prospects/methods. Quan- titative discussion require knowledge of spectrum shape (differential emissivity dR/dEν). High energy tail is par- ticularly important from an experimental detection point of view. Detection of the lowest energy neutrinos is ex- tremely challenging due to numerous background signal noise sources e.g. 14C decay for Eν < 200 keV [51]. Rele- vant calculations for the spectrum of the medium energy 〈Eν〉 ∼ 1MeV neutrinos emitted from thermal processes has become available recently [52,53,54]. Purpose of this article is to develop accurate methods and discuss various theoretical and practical (important for detection) aspects of the neutrino spectra from astrophysical plasma process. This could help experimental physicists to discuss possi- ble realistic approach to detect astrophysical sources of the neutrinos in the future. 2 Plasmaneutrino spectrum 2.1 Properties of plasmons Emissivity and the spectrum shape from the plasmon de- cay is strongly affected by the dispersion relation for trans- verse plasmons (massive in-medium photons) and longi- tudal plasmons. In contrast to transverse plasmons, with vacuum dispersion relation ω(k) = k, longitudal plasmons exist only in the plasma. Dispersion relation, by the def- inition is a function ω(k) where ~ω is the energy of the (quasi)particle and ~k is the momentum. Issues related to particular handling of these functions are discussed clearly in the article of Braaten and Segel [15]. We will repeat here the most important features of the plasmons. For both types, plasmon energy for momentum k = 0 is equal to ω0. Value ω0 ≡ ω(0) is refereed to as plasma frequency and can be computed from: ω20 = (f1 + f2) dp (2) where v = p/E, E = p2 + m2e (~ = c = 1 units are used), me ≃ 0.511 MeV and fine structure constant is α = 1/137.036 [55]. Functions f1 and f2 are the Fermi-Dirac distributions for electrons and positrons, respectively: e(E−µ)/kT + 1 , f2 = e(E+µ)/kT + 1 . (3) Quantity µ is the electron chemical potential (including the rest mass). Other important parameters include first relativistic correction ω1: ω21 = v2 − v4 (f1 + f2) dp (4) maximum longitudal plasmon momentum (energy) kmax: k2max ≡ ω max = 1 − v 1 + v (f1 + f2) dp Table 1. Plasma properties for typical massive star during Si burning. All values in MeV. kT µ ω0 ω1 mt ωmax ωA 0.32 1.33 0.074 0.070 0.086 0.133 0.002 and asymptotic transverse plasmon mass mt: m2t = (f1 + f2) dp. (6) Value mt is often referred to as thermal photon mass. We also define parameter v∗: interpreted as typical velocity of the electrons in the plasma [15]. Axial polarization coefficient is: (f1 − f2) dp. (8) Value of the ωA is a measure of the difference between neutrino and antineutrino spectra. Set of numerical values used to display sample result is presented in Table 1. Values ω0, ωmax,mt define sub-area of the ω-k plane where dispersion relations for photons ωt(k) and longitu- dal plasmons ωl(k) are found: max (k, ω0) ≤ ωl(k) ≤ ωmax, 0 ≤ k ≤ kmax (9a) k2 + ω20 ≤ ωt(k) ≤ k2 + m2t , 0 ≤ k ≤ ∞ (9b) Dispersion relations are solution to the equations [15]: k2 = Πl (ωl(k), k) (10a) k2 = ωt(k) 2 −Πt (ωt(k), k) (10b) where longitudal and transverse polarization functions are given as an integrals: ωl + vk ωl − vk ω2l − k ω2l − v (f1+f2) dp. (11a) ω2t − k ωt + vk ωt − vk (f1+f2) dp. (11b) Typical example of the exact plasmon dispersion re- lations (dash-dotted) is presented in Fig. 1. As solving eqns. (10a, 10b) with (11) is computationally intensive, three levels of approximation for dispersion relations are widely used: 1. zero-order analytical approximations 2. first order relativistic corrections 3. Braaten&Segel approximation A. Odrzywo lek: Plasmaneutrino spectrum 3 0.05 0.1 0.15 k @MeVD Longitudal 0.05 0.1 0.15 k @MeVD Transverse Fig. 1. Longitudal and transverse plasmon dispersion relation ωl,t(k) for plasma parameters from Table 1. Exact result (dot- sahed) is very close to the Braaten & Segel approximation (solid). Zero-order (dotted) and first order (dashed) approximations are very poor, especially for londitudal mode (left). 0.05 Ω0 0.1 Ωmax k @MeVD Longitudal plasmon 0.2 0.4 0.6 0.8 1 1.2 k @MeVD In-medium photon Fig. 2. Longitudal and transverse plasmon mass. Dotted lines on the right panel show asymptotic transverse mass. Line dashing the same as in Fig. 1. 2.1.1 Approximations for longitudal plasmons For longitudal plasmons, the simplest zero-order approach used in early calculations of Adams et al. [13] and more recently in [53] for photoneutrino process is to put simply: ω(k) = ω0 (12) where ω0 is the plasma frequency (2). Maximum plasmon energy ωmax = ω0 in this approximation. Zero-order ap- proximation is valid only for non-relativistic regime, and leads to large errors of the total emissivity [12]. First relativistic correction to (12) has been introduced by Beaudet et al. [12]. Dispersion relation ωl(k) is given in an implicit form: ω2l = ω , (13) with maximum plasmon energy equal to: ω(1)max = ω20 + ω21 (14) This approximation, however, do not introduce really se- rious improvement (Figs. 1, 2 (left) & 4). Breaking point was publication of the Braaten&Segel approximation [15]. 4 A. Odrzywo lek: Plasmaneutrino spectrum Using simple analytical equation: k2 = 3 ωl + v∗k ωl − v∗k where v∗ is defined in (7) one is able to get almost exact dispersion relation, cf. Figs. 1 & 2, left panels. Solution to the eq. (15) exist in the range 1 < k < kBSmax, where, in this approximation, maximum longitudal plasmon momentum ωBSmax 1 + v∗ 1 − v∗ what gives value slightly different than exact value (Fig. 2, left), but required for consistency of the approximation. 2.1.2 Approximations for transverse plasmons For photons in vacuum dispersion relation is ωt = k. Zero order approximation for in-medium photons is: ω2t = ω 0 + k 2, k ≪ ω0 (17a) valid for small k and: ω2t = m t + k 2, k ≫ ω0 (17b) valid for very large k. Formulae (17a) and (17b) provide lower and upper limit for realistic ωt(k), respectively (cf. Fig. 1, right panel, dotted). First order relativistic corrections lead to the formula: ω2t = ω 0 + k with asymptotic photon mass: ω20 + ω 1/5 (19) Finally, Braaten&Segel approximation leads to: ω2t = k 2 + ω20 ω2t − v 2ωt v∗ k ωt + v∗k ωt − v∗k Asymptotic photon mass mBSt derived from (20) is: 1 − v2 1 + v∗ 1 − v∗ This is slightly smaller (left panel of Fig. 2, dashed) than exact value (solid line). All four relations are presented in Fig. 1. Differences are clearly visible, but they are much less pronounced for transverse than for longitudal plasmons. Inspection of Fig. 2 reveals however, that in the large momentum regime asymptotic behavior is correct only for exact integral re- lations (10b) and may be easily reproduced using (17b) with mt from (6). Let us recapitulate main conclusions. Braaten&Segel approximation provide reasonable approximation, as non- linear equations (15) and (20) are easily solved using e.g. νe,µ,τ ν̄e,µ,τ Fig. 3. Fenmann diagrams for plasmon decay. bisection method. Zero and first-order approximations (12, 17a, 17b) with limiting values (9) provide starting points and ranges. Approximation has been tested by [56] and is considered as the best available [20]. Errors for part of the kT -µ plane where plasmaneutrino process is not dominant may be as large as 5% [56]. At present, these inaccuracies are irrelevant for any practical application, and Braaten&Segel approximation is recommended for all purposes. 2.2 Plasmon decay rate In the Standard Model of electroweak interactions, mas- sive in-medium photons and longitudal plasmons may de- cay into neutrino-antineutrino pairs: γ∗ → νx + ν̄x. (22) In the first-order calculations two Feynmann diagrams (Fig. 3) contribute to decay rate [15,52]. For the decay of the longitudal plasmon squared ma- trix element is: M2l = ω2l − k 2K ·Q1 K ·Q2 2k · q1 k · q2 −Q1 ·Q2 (23a) where K = (ω,k) is four momentum of the plasmon. Q1 = (E1,q1) and Q2 = (E2,q2) is four-momentum of the neutrino and antineutrino, respectively. Squared matrix element for decay of the massive pho- ton is: M2t = C2V Π t + C E1E2 − k · q1 k · q2 + 2CV CAΠtΠA E1 k · q2 − E2 k · q1 (23b) where Πt is defined in (11b) and axial polarization func- tion ΠA reads: ω2t −k ωt + vk ωt − vk ω2t − k ω2t − v (f1−f2) dp A. Odrzywo lek: Plasmaneutrino spectrum 5 Table 2. Relative weight of the M2t (23b) terms for e and µ, τ neutrinos. Flavor Vector Axial Mixed (CV ω +CAωA) (CV ω +CAωA) 2CV CAω (CV ω +CAωA) electron 0.74 0.02 0.24 mu/tau 0.07 0.39 0.54 Fermi constant is GF /(~c) 3 = 1.16637(1)× 10−5 GeV−2 [55] and, in standard model of electroweak interactions, vector and axial coupling constants are: CeV = + 2 sin2 θW , C V = − + 2 sin2 θW , C A = − for electron and µ, τ neutrinos, respectively. The Weinberg angle is sin2 θW = 0.23122(15) [55]. Terms containing CA (so-called axial contribution) in (23b) are frequently treated separately [52] or removed at all [3]. In calculations concentrated on the total emissiv- ity this is justified as anti-symmetric term multiplied by CV CA do not contribute at all and term C A × . . . is sup- pressed relative to the term beginning with C2V × . . . by four orders of magnitude [3]. However, if one attempts to compute neutrino energy spectrum all three terms should be added together, as mixed V-A ,,channel” alone leads to negative emission probability for some neutrino energy range (Fig. 6), what is physically unacceptable. These terms remains numerically small but only for electron neu- trinos. For µ and τ neutrino spectra axial part contributes at ∼ 1% level due to very small value C V = −0.0376 while still CA = −0.5. ”Mixed” term leads to significant differences between νµ,τ and ν̄µ,τ spectra, cf. Fig. 6. Rel- ative contributions of the three transverse ”channels” for electron and µ, τ are presented in Table 2. In general, all the terms in the squared matrix element (23b) should be added. We have only two different spectra: longitudal and transverse one. Particle production rate from plasma in thermal equi- librium is: (2π)5 Zi fγ∗ δ 4(K−Q1−Q2) M where i = l for longitudal mode and i = t for transverse mode. Bose-Einstein distribution for plasmons fγ∗ is: fγ∗ = eωt,l/kT − 1 . (28) and residue factors Zt,l are expressed by polarization func- tions Πt,l (11b, 11a): Z−1t = 1 − Z−1l = − . (30) For massive photons gt = 2 and for longitudal plasmon gl = 1. Differential rates1 has been derived for the first time in [52]. Here, we present result in the form valid for both types of plasmons, ready for calculations using any avail- able form of dispersion relation: dE1 dE2 i fγ∗ Ji S (31) where i = l or i = t. Product S of the unit step functions Θ in (31) restrict result to the kinematically allowed area: S = Θ(4E1E2−m i )Θ(E1+E2−ω0)Θ(ωmax−E1−E2) (32) Four-momenta in the squared matrix element are: Q1 = (E1, 0, 0, E1) Q1 = (E2, E2 sin θ, 0, E2 cos θ) K = (E1 + E2, E2 sin θ, 0, E1 + E2 cos θ) m2i = K ·K = (E1 + E2) 2 − k′2 cos θ = − E21 − E 2E1E2 k′ = ω−1l,t (E1 + E2) ωi = E1 + E2 where ω−1i denotes function inverse to the dispersion re- lation. Jacobian Ji arising from Dirac delta integration in (27) is: J−1i = . (33) Residue factors Zi are given in (30) and (29). Maximum energy ωmax in (32) for longitudal plasmons must be in the agreement with particular approximation used for ωl(k): ω0, (14) or (16) for zero-order (12), first-order (13) or Braaten&Segel (15) approximation, respectively. For trans- verse plasmons ωmax → ∞ and last Θ function in (32) has no effect and may be omitted. 2.3 Longitudal neutrino spectrum 2.3.1 Analytical approximation We begin with general remark on the spectrum. Note, that eq. (31) is symmetric for longitudal mode under change E1,2 → E2,1 because (23a) is symmetric with respect to exchange Q1,2 → Q2,1. Resulting energy spectrum is thus identical for neutrinos and antineutrinos. This is not true for transverse plasmons with axial contribution included, cf. Sect. 2.4. 1 Double differential rate d2Ri/dEd cos θ has an identical form as (31) but now four momenta cannot be given explicitly, unless simple analytical approximation for ωi(k) is used. Ana- lytical approximations for the specrum shape are derived this 6 A. Odrzywo lek: Plasmaneutrino spectrum Using zero-order dispersion relation for longitudal plas- mons (12) we are able to express spectrum by the elemen- tary functions. Longitudal residue factor Zt is now: Z0l = 1, (34) and Jacobian Jl resulting from the integration of the Dirac delta function is: J0l = 1. (35) Now, differential rate d2R/dEd cos θ (cf. (31) and foot- note 1) becomes much more simple and integral over d cos θ can be evaluated analytically. Finally, we get the longitu- dal spectrum: ≡ λ(E) = 1260 π4 α ~3 c9 f(E/ω0) eω0/kT − 1 where normalized spectrum is: f(x) = 4x(x− 1)(8x4 − 16x3 + 2x2 + 6x− 3) + 3(1 − 2x)2 ln(1 − 2x)2 Let us note that f is undefined at x = 1/2; use limit instead: x→1/2 f(x) = 105/32. Function f(x) is symmetric with respect to point x = 1/2, where f has a maximum value (Fig. 4, dotted line). In this limit, correct for non-relativistic, non-degenerate plasma, average neutrino and antineutrino energy is 〈E〉 = ω0/2 and maximum ν energy is ω0. Inspection of Fig. 4 reveals little difference between analytical result (36) and result obtained with first-order relativistic corrections to the dispersion relation (13). 2.3.2 Numerical results Simple formula (36) significantly underestimates flux and the maximum neutrino energy, equal to ωmax rather than ω0. Therefore we have used Braaten & Segel approxima- tion for longitudal plasmon dispersion relation. To derive spectrum we will use form of differential rate (31) provided by [52]. In the Braaten&Segel approxima- tion: ZBSl = ω2l − k 2(ω2l − v 3ω20 − ω l + v JBSl = 1 − βl βBSl = ωl + v∗k ωl − v∗k ω2l v∗ k2(ω2l − v Spectrum is computed as an integral of (31) over dE2. Example result is presented in Fig. 4. Integration of the function in Fig. 4 over neutrino energy gives result in well agreement with both (30) from [15] and (54) from [52]. Ω0�2 Ω0 Ωmax @MeVD Fig. 4. Longitudal plasmon approximate analytical (36) neu- trino spectrum (dotted), with first-order correction used by BPS [12] (dashed), and spectrum computed using [15] disper- sion relation (solid). Plasma properties according to Table 1. 2.4 Transverse plasmon decay spectrum 2.4.1 Analytical approximation Derivation of massive in-medium photon decay spectrum closely follows previous subsection. Semi-analytical for- mula can be derived for dispersion relations (17). For dis- persion relation (17b) transverse residue factor Zt is: Z0t = 1, (38) polarization function Πt is equal to: Π0t = m t , (39) and Jacobian resulting from integration of the Dirac delta function Jt is: J0t = E1 + E2 . (40) Approximate spectrum, neglecting differences between neutrinos and antineutrinos, is given by the following in- tegral: λ(E) = 64 π4α P (cos θ, E/mt) d cos θ 2E(1−cos θ) where rational function P (ct, x) is: 1 + 2(ct− 1)2(2x2 − 1)x2 x(ct− 1)2[1 − 2ct(ct− 1)x2 + 2(ct− 1)2)x4] A. Odrzywo lek: Plasmaneutrino spectrum 7 Ω0 Ωmax kT 0.5 @MeVD Fig. 5. Transverse plasmaneutrino spectrum computed from [15] approximation (solid) with upper (17b) and lower (17a) limits for the dispersion relation (dotted). First-order rela- tivistic correction leads to the spectrum shown as dashed line. Plasma parameters as in Fig. 4. Result presented in Fig. 5 show that spectrum (41) obtained with dispersion relation (17b) agree well in both low and high neutrino energy part with spectrum obtained from Braaten&Segel approximation for dispersion rela- tions. Dispersion relation (17a) produces much larger er- ror, and spectrum nowhere agree with correct result. This fact is not a big surprise: as was pointed out by Braaten [16] dispersion relation is crucial. Therefore, all previous results, including seminal BPS work [12], could be eas- ily improved just by the trivial replacement ω0 → mt. Moreover, closely related photoneutrino process also has been computed [12,3,17,14] with simplified dispersion re- lation (17a) with ω0. One exception is work of Esposito et. al. [57]. It remains unclear however, which result is better, as accurate dispersion relations have never been used within photoneutrino process context. For plasma- neutrino, Eq. (17b) is much better approximation than (17a), especially if one put mt from exact formula (6). High energy tail of the spectrum also will be exact in this case. As formula (41) agree perfectly with the tail of the spectrum, we may use it to derive very useful analyti- cal expression. Leaving only leading terms of the rational function (42) P (ct, x) ∼ x−1(1 − ct)−2 one is able to compute integral (41) analytically: λ(E) ≃ 64 π4α eaκ/2 − 1 where κ = 2x + (2x)−1, x = E/mt, a = mt/kT . Interest- ingly, spectrum (43) is invariant under transformation: E ′E = m2t/4 and all results obtained for high energy tail of the spec- trum immediately may be transformed for low-energy ap- proximation. The asymptotic behavior of (43) for E ≫ kT is of main interest: λ(Eν) = A kT m t exp where for electron neutrinos : = 2.115 × 1030 [MeV−8cm−3s−1] and mt, kT are in MeV. For µ, τ neutrinos just replace A with A (C Formula (44) gives also quite reasonable estimates of the total emissivity Qt and mean neutrino energies 〈Eν〉: Qt = A kT 3m6t (45a) 〈Eν〉 = kT (45b) For a comparison, Braaten & Segel [15] derived exact for- mulae in the high temperature limit kT ≫ ω0: QBSt = V ζ(3) 12π4α kT 3m6t = 0.8A kT 3m6t (46a) 〈EBSν 〉 = 6ζ(3) kT = 0.73 kT (46b) Formulae above agree with ∼25% error in the leading co- efficients. 2.4.2 Numerical results Calculation of the spectrum in the framework of Braaten&Segel approximation requires residue factor, polarization func- tion [15] (transverse&axial) and Jacobian [52]: ZBSt = 2ω2t (ω t − v 3ω20ω t + (ω t + k 2)(ω2t − v k2) − 2ω2t (ω t − k ΠBSt = ω2t − v ωt + v∗k ωt − v∗k , (48) ΠBSA = ωA k ω2t − k ω2t − v 3ω20 − 2 (ω t − k , (49) JBSt = E1 + E2 1 − βBSt βBSt = ωt + v∗k ωt − v∗k Example spectrum, computed as an integral of (31) over dE2 is shown in Fig. 5. 8 A. Odrzywo lek: Plasmaneutrino spectrum Ω0 Ωmax 0.5 @MeVD Fig. 6. Spectrum of the muon neutrinos (dotted) and antineu- trinos (dashed) from transverse plasmon decay. Contributions to the spectra from so-called mixed ,,vector-axial channel” pro- duces significant differences. For electron flavor, contribution from ”mixed channel” lead to unimportant differences. For both flavors contribution from ”axial channel” remains rela- tively small: 10−4 for νe and 10 −2 for νµ. Overall contribution to the total emissivity from µ, τ flavors is suppressed relatively to electron flavor by a factor (C /CeV ) ≃ 3.3 × 10−3. 3 Summary Main new results presented in the article are analytical formulae for neutrino spectra (36, 41) and exact analyti- cal formula (44) for the high energy tail of the transverse spectrum. The latter is of main interest from the detection of astrophysical sources point of view: recently available detection techniques are unable to detect keV plasmaneu- trinos emitted with typical energies 〈Eν〉 ∼ ω0/2 (Fig. 4, 5), where ω0 is the plasma frequency (2). Tail behavior of the transverse spectrum quickly ”decouple” from ω0 dominated maximum area, and becomes dominated by temperature-dependent term exp (−Eν/kT ). Calculation of the events in the detector is then straightforward, as de- tector threshold in the realistic experiment will be above maximum area. This approach is much more reliable com- pared to the typical practice, where an average neutrino energy is used as a parameter in an arbitrary analytical formula. Analytical formulae for the spectrum are shown to be a poor approximation of the realistic situation, especially for longitudal plasmons (Fig. 4). This is in the agreement with general remarks on the dispersion relations presented by Braaten [16]. On the contrary, Braaten & Segel [15] ap- proximation is shown to be a very good approach not only for the total emissivities, but also for the spectrum. Excep- tion is the tail of the massive photon decay neutrino spec- Ω0Ωmax 1.0 10 @MeVD Fig. 7. Typical spectra from the plasma process. Dotted line is a longitudal and dashed transverse spectrum. Only ∼ exp(−Eν/kT ) tail of the transverse spectrum (solid line) contributes to (possibly) detectable signal. Plasma properties according to Table 1. trum: Braaten & Segel [15] formulae lead to underestimate of the thermal photon mass while the formula (44) gives exact result. Numerical difference between mt from (6) and (21) is however small [15]. Calculating of the emissiv- ities by the spectrum integration seems much longer route compared to typical methods, but we are given much more insight into process details. For example, we obtain exact formula for the tail for free this way. 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0704.1223

Quadratic BSDEs with random terminal time and elliptic PDEs in infinite dimension

arXiv:0704.1223v2 [math.PR] 21 Jun 2007 Quadratic BSDEs with random terminal time and elliptic PDEs in infinite dimension. Philippe Briand IRMAR, Université Rennes 1, 35042 Rennes Cedex, FRANCE [email protected] Fulvia Confortola Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca Via R. Cozzi 53 - Edificio U5 - 20125 Milano, Italy [email protected] Mathematics Subject Classification: 60H20; 60H30. Abstract In this paper we study one dimensional backward stochastic differential equations (BSDEs) with random terminal time not necessarily bounded or finite when the generator F (t, Y, Z) has a quadratic growth in Z. We provide existence and uniqueness of a bounded solution of such BSDEs and, in the case of infinite horizon, regular dependence on parameters. The obtained results are then applied to prove existence and uniqueness of a mild solution to elliptic partial differential equations in Hilbert spaces. Finally we show an application to a control problem. 1 Introduction Let τ be a stopping time which is not necessarily bounded or finite. We look for a pair of processes (Yt, Zt)t≥0 progressively measurable which satisfy ∀t ≥ 0,∀T ≥ t Yt∧τ = YT∧τ + ∫ T∧τ F (s, Ys, Zs)− ∫ T∧τ ZsdWs Yτ = ξ on {τ <∞} where W is a cylindrical Wiener process in some infinite dimensional Hilbert space Ξ and the generator F has quadratic growth with respect to the variable z. Moreover the terminal condition ξ is Fτ -measurable and bounded. We limit ourselves to the case in which (Yt)t≥0 is one-dimensional and we look for a solution (Yt, Zt)t≥0 such that (Yt)t≥0 is a bounded process and (Zt)t≥0 is a process with values in the space of the Hilbert-Schmidt operator from Ξ to R such that E (∫ t∧τ <∞,∀t ≥ 0. BSDEs with random terminal time have been treated by several authors (see for instance [21], [6], [3], [23]) when the generator is Lipschitz, or monotone and with suitable growth with respect to y, but Lipschitz with respect to z. Kobylanski [18] deals with a real BSDE with quadratic http://arxiv.org/abs/0704.1223v2 generator with respect to z and with random terminal time. She requires that the stopping time is bounded or P-a.s finite. We generalize in a certain sense the result of Kobylanski, but, to obtain the existence and uniqueness of the solution to (1) for a general stopping time, we have to require stronger assumption on the generator. In particular it has to be strictly monotone with respect to y. We follow the techniques introduced by Briand and Hu in [3], and used successively by Royer [23], based upon an approximation procedure and on Girsanov transform. We can use this strategy even if, under our assumptions, the generator is not Lipschitz with respect to z. The main idea is to exploit the theory of BMO-martingales. It is indeed known that if (Y,Z) solves a quadratic BSDE with bounded (or P-a.s.) finite final time then Zs dWs is a BMO–martingale (see [16]). Then the result on BSDE is exploited to study existence and uniqueness of a mild solution (see Section 5 for the definition) to the following elliptic partial differential equation in Hilbert space H Lu(x) + F (x, u(x),∇u(x)σ) = 0, x ∈ H, (2) where F is a function from H ×R×Ξ∗ to R strictly monotone with respect the second variable and with quadratic growth in the gradient of the solution and L is the second order operator: Lφ(x) = Trace(σσ∗∇2φ(x)) + 〈Ax,∇φ(x)〉 + 〈b(x),∇φ(x)〉. H is an Hilbert space, A is the generator of a strongly continuous semigroup of bounded linear operators (etA)t≥0 in H, b is a function with values in H and σ belongs to L(Ξ,H)- the space of linear bounded operator from Ξ to K satisfying appropriate Lipschitz conditions. Existence and uniqueness of a mild solution of equation (2) in infinite dimensional spaces have been recently studied by several authors employing different techniques (see [5], [14], [9] and [10]). In [13] (following several papers dealing with finite dimensional situations, see, for instance [4], [6] and [20]) the solution of equation (2) is represented using a Markovian forward-backward system of equations dXs = AXsds+ b(Xs)ds + σ(Xs)dWs, s ≥ 0 dYs = −F (Xs, Ys, Zs)ds + ZsdWs, s ≥ 0 X0 = x where F is Lipschitz with respect to y and z and monotone in y, but with monotonicity constant large. A such limitation has then been removed under certain conditions in [17], still assuming F Lipschitz with respect to z, strictly monotone and with arbitrary growth with respect to y. We follow the same approach to deal with mild solution to (2) when the coefficient F is strictly monotone in the second variable (there are not conditions on its monotonicity constant) and has quadratic growth in the gradient of the solution. The main technical point here will be proving differentiability of the bounded solution of the backward equation in system (3) with respect to the initial datum x of the forward equation. To obtain this result we follow [17]. The proof is based on an a-priori bound for suitable approximations of the equations for the gradient of Y with respect to x. We use again classical result on BMO-martingales. In the last part of the paper we apply the above result to an optimal control problem with state equation: dXτ = AXτdτ + b(Xτ )dτ + σr(Xτ , uτ )dτ + σdWτ , X0 = x ∈ H, where u denotes the control process, taking values in a given closed subset U of a Banach space U . The control problem consists of minimizing an infinite horizon cost functional of the form J(x, u) = E e−λσg(Xuσ , uσ)dσ. We suppose that r is a function with values in Ξ∗ with linear growth in u and g is a given real function with quadratic growth in u. λ is any positive number. We assume that neither U nor r is bounded: in this way the Hamiltonian corresponding to the control problem has quadratic growth in the gradient of the solution and consequently the associated BSDE has quadratic growth in the variable Z. The results obtained on equation (2) allows to prove that the value function of the above problem is the unique mild solution of the corresponding Hamilton-Jacobi- Bellman equation (that has the same structure as (2). Moreover the optimal control is expressed in terms of a feedback that involves the gradient of that same solution to the Hamilton-Jacobi- Bellman equation. We stress that the usual application of the Girsanov technique is not allowed (since the Novikov condition is not guaranteed) and we have to use specific arguments both to prove the fundamental relation and to solve the closed loop equation. We adapt some procedure used in [11] to our infinite dimensional framework on infinite horizon. The paper is organized as follows: the next Section is devoted to notations; in Section 3 we deal with quadratic BSDEs with random terminal time; in Section 4 we study the forward backward system on infinite horizon; in Section 5 we show the result about the solution to PDE. The last Section is devoted to the application to the control problem. 2 Notations The norm of an element x of a Banach space E will be denoted |x|E or simply |x|, if no confusion is possible. If F is another Banach space, L(E,F ) denotes the space of bounded linear operators from E to F , endowed with the usual operator norm. The letters Ξ, H, U will always denote Hilbert spaces. Scalar product is denoted 〈·, ·〉, with a subscript to specify the space, if necessary. All Hilbert spaces are assumed to be real and separable. L2(Ξ, U) is the space of Hilbert-Schmidt operators from Ξ to U , endowed with the Hilbert-Schmidt norm, that makes it a separable Hilbert space. We observe that if U = R the space L2(Ξ,R) is the space L(Ξ,R) of bounded linear operators from Ξ to R. By the Riesz isometry the dual space Ξ∗ = L(Ξ,R) can be identified with Ξ. By a cylindrical Wiener process with values in a Hilbert space Ξ, defined on a probability space (Ω,F ,P), we mean a family {Wt, t ≥ 0} of linear mappings from Ξ to L 2(Ω), denoted ξ 7→ 〈ξ,Wt〉, such that (i) for every ξ ∈ Ξ, {〈ξ,Wt〉, t ≥ 0} is a real (continuous) Wiener process; (ii) for every ξ1, ξ2 ∈ Ξ and t ≥ 0, E (〈ξ1,Wt〉 · 〈ξ2,Wt〉) = 〈ξ1, ξ2〉Ξ t. (Ft)t≥0 will denote, the natural filtration of W , augmented with the family of P-null sets. The filtration (Ft) satisfies the usual conditions. All the concepts of measurably for stochastic processes refer to this filtration. By B(Λ) we mean the Borel σ-algebra of any topological space We also recall notations and basic facts on a class of differentiable maps acting among Ba- nach spaces, particularly suitable for our purposes (we refer the reader to [12] for details and properties). We notice that the use of Gâteaux differentiability in place of Fréchet differentia- bility is particularly suitable when dealing with evaluation (Nemitskii) type mappings on spaces of summable functions. Let now X, Z, V denote Banach spaces. We say that a mapping F : X → V belongs to the class G1(X,V ) if it is continuous, Gâteaux differentiable on X, and its Gâteaux derivative ∇F : X → L(X,V ) is strongly continuous. The last requirement is equivalent to the fact that for every h ∈ X the map ∇F (·)h : X → V is continuous. Note that ∇F : X → L(X,V ) is not continuous in general if L(X,V ) is endowed with the norm operator topology; clearly, if this happens then F is Fréchet differentiable on X. It can be proved that if F ∈ G1(X,V ) then (x, h) 7→ ∇F (x)h is continuous from X × X to V ; if, in addition, G is in G1(V,Z) then G(F ) belongs to G1(X,Z) and the chain rule holds: ∇(G(F ))(x) = ∇G(F (x))∇F (x). When F depends on additional arguments, the previous definitions and properties have obvious generalizations. 3 Quadratic BSDEs with random terminal time Let τ be an Ft-stopping time. It is not necessarily bounded or P-a.s. finite. We work with a function F defined on Ω× [0,∞) ×R× Ξ∗ which takes its values in R and such that F (·, y, z) is a progressively measurable process for each (y, z) in R × Ξ∗. We define the following sets of Ft-progressively measurable processes (ψt)t≥0 with values in a Hilbert space K: M2,−2λ(0, τ ;K) = ψ : E e−2λs|ψs| M2loc(0, τ ;K) = ψ : E (∫ t∧τ <∞ ∀t ≥ 0 We want to construct an adapted process (Y,Z)t≥0 which solves the BSDE − dYt = 1t≤τ (F (t, Yt, Zt)dt− ZtdWt), Yτ = ξ on {τ <∞}. (5) We assume that: Assumption A1. There exist C ≥ 0 and α ∈ (0, 1) such that 1. |F (t, y, z)| ≤ C 1 + |y|+ |z|2 2. F (t, ·, ·) is G1,1(R× L2(Ξ,R);R); 3. |∇zF (t, y, z)| ≤ C (1 + |z|); 4. |∇yF (t, y, z)| ≤ C (1 + |z|) Moreover we suppose that there exist two constants K ≥ 0 and λ > 0 such that dP ⊗ dt a.e.: 5. F is monotone in y in the following sense: ∀y, y′ ∈ R, z ∈ Ξ∗, < y − y′, F (t, y, z) − F (t, y′, z) >≤ −λ|y − y′|2; 6. |F (t, 0, 0)| ≤ K; 7. ξ is a Fτ -measurable bounded random variable; we denote by M some real such that |ξ| ≤M P-a.s. We call solution of the equation a pair of progressively measurable processes (Yt, Zt)t≥0 with values in R× Ξ∗ such that 1. Y is a bounded process and Z ∈ M2loc(0, τ ; Ξ 2. On the set {τ <∞}, we have Yτ = ξ and Zt = 0 for t > τ ; 3. ∀T ≥ 0, ∀t ∈ [0, T ] we have Yt∧τ = YT∧τ + ∫ T∧τ F (s, Ys, Zs)ds − ∫ T∧τ ZsdWs. Before giving the main result of this section we prove a lemma which we use in the sequel. The proof involves the Girsanov transform and results of the bounded mean oscillation (BMO, for short) martingales theory. Here we recall a few well-known facts from this theory following the exposition in [15]. Let M be a continuous local (P,F)-martingale satisfying M0 = 0. Let 1 ≤ p <∞. Then M is in the normed linear space BMOp if ||M ||BMOp = sup ∣∣∣E[|MT −Mτ |p|Fτ ]1/p where the supremum is taken over all stopping time τ ≤ T . By Corollary 2.1 in [15], M is a BMOp-martingale if and only if it is a BMOq-martingale for every q ≥ 1. Therefore, it is simply called a BMO-martingale. In particular, M is a BMO-martingale if and only if ||M ||BMO2 = sup ∣∣∣E[〈M〉T − 〈M〉τ |Fτ ]1/2 where the supremum is taken over all stopping time τ ≤ T ; 〈M〉 denotes the quadratic variation of M . This means that local martingales of the form Mt = ξsdWs are BMO-martingales if and only if ||M ||BMO2 = sup ∣∣∣∣∣ ∣∣∣∣∣E ||ξs|| ∣∣∣Fτ ]1/2∣∣∣∣∣ ∣∣∣∣∣ The very important feature of BMO-martingales is the following (see Theorem 2.3 in [15]): the exponential martingale E(M)t = Et = exp 0 ≤ t ≥ T is a uniformly integrable martingale. Lemma 3.1. Let (U, V ), be solutions to Ut = ξ + 1s≤τ [asUs + bsVs + ψs ]ds − Vs dWs (6) where ξ is Fτ–measurable and bounded and as, bs, ψs are processes such that 1) as ≤ −λ for some λ > 0; bsdWs is a BMO-martingale; 3) |ψs| ≤ ρ(s) where ρ is a deterministic function. Moreover we assume that U is bounded. Then we have P-a.s. for all t ∈ [0, T ] |Ut| ≤ e −λ(T−t)‖ξ‖∞ + ρ(s)e−λ(s−t) ds. Proof. Let (U, V ) be a solution of the BSDE (6) such that U is bounded. We fix t ∈ R+ and set for s ≥ t es = e ar dr. By Ito’s formula we have, Ut = eT ξ + 1s≤τesψsds− esVs(dWs − bs). Let QT the probability measure on (Ω,FT ) whose density with respect to P|FT is ET = exp bsdWs − By assumption bsdWs is a BMO-martingale and the probability measures QT and P|FT are mutually absolutely continuous and W t = Wt − br dr for 0 ≤ t ≤ T is a Brownian motion under QT . Taking the conditional expectation with respect to Ft we get |Ut| ≤ E eT |ξ|+ es|ψs|ds ∣∣∣Ft , QT a.s. and thanks to 3) |Ut| ≤ (Et) ET eT |ξ|+ ρ(s)esds ∣∣∣ Ft But from 1) as ≤ −λ and, for all s ≥ t es ≤ e −λ(s−t) P-a.s., from which we get P-a.s. ∀t ∈ [0, T ] |Ut| ≤ e −λ(T−t)||ξ||∞ + ρ(s)e−λ(s−t)ds. Corollary 3.2. Let (Y i, Zi), i = 1, 2, be solutions to Y it = ξ 1s≤τF i(s, Y is , Z s) ds− Zis dWs where ξi is Fτ–measurable and bounded. We assume that Y 1 and Y 2 are bounded and that the Zi are such that ZiddWt are BMO-martingales. Moreover F 1 is −λ-monotone in the following sense: there exists λ > 0 such that ∀y, y′ ∈ R, z ∈ Ξ∗, < y − y′, F 1(t, y, z)− F 1(t, y′, z) >≤ −λ|y − y′|2; and verifies |F 1(t, y, z) − F 1(t, y, z′)| ≤ C |z − z′| 1 + |z|+ |z′| We assume moreover that |F 1(t, Y 2t , Z t )− F 2(t, Y 2t , Z t )| ≤ ρ(t) where ρ is a deterministic function. Then we have P-a.s. for all t ∈ [0, T ] |Y 1t − Y t | ≤ e −λ(T−t)‖ξ1 − ξ2‖∞ + ρ(s)e−λ(s−t) ds. Proof. Let (Y 1, Z1) and (Y 2, Z2) be solutions of the BSDE with data respectively (ξ1, F 1) and (ξ2, F 2) such that Y 1 and Y 2 are bounded. We set Y = Y 1−Y 2 and Z = Z1−Z2. It is enough to write the equation for the difference Y = Y 1 − Y 2 dY t = −1t≤τ [F 1(t, Y 1t , Z t )− F 2(t, Y 2t , Z t )dt+ ZtdWt] dY t = −1t≤τ [(atY t + btZt + ψt)dt+ ZtdWt]. using a linearization procedure by setting F 1(s, Y 1s , Z s )− F 1(s, Y 2s , Z Y 1s − Y , if Y 1s − Y s 6= 0 −λ otherwise F 1(s, Y 2s , Z s )− F 1(s, Y 2s , Z |Z1s − Z (Z1s − Z s ), if Z s − Z s 6= 0 0 otherwise . ψs = F 1(s, Y 2s , Z s )− F 2(s, Y 2s , Z Now we can state the main result of this section, concerning the existence and uniqueness of solutions of BSDE (5). Theorem 3.3. Under assumption A1 there exists a unique solution (Y,Z) to BSDE (5) such that Y is a continuous and bounded process and Z belongs to M2loc(0, τ ; Ξ Proof. Existence. We adopt the same strategy as in [3] and [23], with some significant modi- fications. Denote by (Y n, Zn) the unique solution to the BSDE Y nt = ξ1τ≤n + 1s≤τF (s, Y s , Z s )ds− Zns dWs, 0 ≤ t ≤ n. (7) We know from results of [18] that under A1-1,2,3,4 the BSDE (7) has a unique bounded solution and that ∥∥∥supt∈[0,τ∧n] |Yt|n ≤ (||ξ||∞ + Cn)e and there exists a constant C = Cn, which depends on ∥∥∥supt∈[0,τ∧n] |Y nt | , such that Zns · dWs ≤ Cn. Now we study the convergence of the sequence of processes (Y n, Zn). (i) First of all we prove that, thanks to the assumptions of boundedness and monotonicity A1-5,6, Y n is a process bounded by a constant independent on n. Applying the Corollary 3.2 we have that P-a.s. ∀n ∈ N, ∀t ∈ [0, n] |Y nt | ≤ e −λ(n−t)||ξ1τ≤n||∞ + e−λ(s−t)|F (s, 0, 0)| ds ≤M + . (8) Moreover we can show that for each ǫ > 0 e−ǫs|Zns | 2ds) <∞. (9) To obtain this estimate we take the function ϕ(x) = e2Cx − 2Cx− 1 /(2C2) which has the following properties: ϕ′(x) ≥ 0 if x ≥ 0, ϕ′′(x)−Cϕ′(x) = 1. Thanks to (8) we can say that there exist a constant K0 such that ∀s ∈ [0, T ], Y s + K0 ≥ 0, P-a.s. Now, if we calculate the Ito differential of e−ǫtϕ(Y nt +K0), using the previous properties, we have (9). (ii) Now we prove that the sequence (Y nt )n≥0 converges almost surely. We are going to show that it is an almost definite Cauchy sequence. We define Y n and Zn on the whole time axis by setting Y nt = ξ1τ≤n, Z t = 0, if t > n. Fix t ≤ n ≤ m and set Ŷ = Y m − Y n, Ẑ = Zm − Zn and F̂ (s, y, z) = 1s≤nF (s, y, z). We get, from Ito’s formula Ŷt = Ŷm + 1s≤τ (F (s, Y s , Z s )− F̂ (s, Y s , Z s ))ds − ẐsdWs. We note that |F (s, Y ns , Z s )− F̂ (s, Y s , Z s ))| = |1s>nF (s, ξ1τ≤n, 0)| ≤ C(1 +M)1s>n. Hence, we can apply the Corollary 3.2 with ξ1 = ξ1τ≤m and ξ 2 = ξ1τ≤n, F 1 = F and F 2 = F̂ , ρ(t) = C(1 +M)1s>n and state that ∀n,m ∈ N, with n ≤ m and ∀t ∈ [0, n], P-a.s. |Y mt − Y t | ≤ e −λ(m−t)||ξ1τ≤m − ξ1τ≤n||∞ + C(1 +M)e−λ(s−t) ds ≤ C(1 +M) e−λ(n−t). (10) The previous inequality implies that for each t ≥ 0 the sequence of random variable Y nt is a Cauchy sequence in L∞(Ω), hence converges to a limit, which we denote Yt. If m goes to infinity in the last inequality, it comes that P-a.s., ∀ 0 ≤ t ≤ n |Y nt − Yt| ≤ βe −λ(n−t), where β =M + C(1 +M) . (11) This inequality implies that the sequence of continuous processes (Y n)n∈N converges almost surely to Y uniformly with respect to t on compact sets. The limit process Y is also continuous and from (8) we have that ∀t ∈ R+ |Yt| ≤M + (iii) We show that the sequence (Yn)n also converges in the space M 2,−2λ(0, τ ;R). Indeed we have e−2λt|Y nt − Yt| [∫ n∧τ e−2λt|Y nt − Yt| e−2λt|Y nt − Yt| and using the inequality (11) for the first term, we get that [∫ n∧τ e−2λt|Y nt − Yt| ≤ β2ne−2λn. In addition, from the definition of Y nt on R+, we know that ∀t > n Y t = ξ1τ≤n. Hence, e−2λt|Y nt − Yt| e−2λt|Yt − ξ1n<τ | )2 ∫ τ e−2λtdt e−2λn. Finally we have e−2λt|Y nt − Yt| ≤ e−2λn nβ2 + Hence (Y n) converges to Y in M2,−2λ(0, τ ;R). (iv) To continue, we show that the sequence (Zn)n is a Cauchy sequence in the space M2,−2(λ+ǫ)(0, τ ; Ξ∗). Fix t ≤ n ≤ m and set, as before, Ŷ = Y m−Y n, Ẑ = Zm−Zn and F̂ (s, y, z) = 1s≤nF (s, y, z). We write F (s, Y ms , Z s )− F̂ (s, Y s , Z s ) = a s Ŷs + b s Ẑs + 1s>nF (s, ξ1τ≤n, 0) where an,ms = F (s, Y ms , Z s )− F (s, Y s , Z Y ms − Y , if Y ms − Y s 6= 0 −λ otherwise bn,ms = F (s, Y ns , Z s )− F (s, Y s , Z |Zms − Z (Zms − Z s ), if Z s − Z s 6= 0 0 otherwise . From Ito’s formula we get |Ŷ0| ∫ τ∧m e−2(λ+ǫ)s|Ẑs| 2 ds+ ∫ τ∧m 2e−2(λ+ǫ)sŶsẐsdWs = = e−2(λ+ǫ)τ∧m|Ŷτ∧m| ∫ τ∧m e−2(λ+ǫ)s2(λ + ǫ)|Ŷs| ∫ τ∧m 2e−2(λ+ǫ)sŶs[a s Ŷs + b s Ẑs]ds+ ∫ τ∧m 2e−2(λ+ǫ)sŶsF (s, ξ1τ≤n, 0)ds and taking the expectation we have ∫ τ∧m e−2(λ+ǫ)s|Ẑs| 2ds ≤ Ee−2(λ+ǫ)τ∧m|Ŷτ∧m| 2 + E ∫ τ∧m e−2(λ+ǫ)s2ǫ|Ŷs| ∫ τ∧m 2e−2(λ+ǫ)sŶsb s Ẑsds+ E ∫ τ∧m 2e−2(λ+ǫ)sŶsF (s, ξ1τ≤n, 0)ds. Using the fact that 2e−2(λ+ǫ)sŶsb s Ẑs ≤ 2|Ŷs| 2e−2(λ+ǫ)s|bn,ms | e−2(λ+ǫ)s|Ẑs| we get ∫ τ∧m e−2(λ+ǫ)s|Ẑs| 2 ds ≤ 2Ee−2(λ+ǫ)τ∧m|Ŷτ∧m| 2 + 2E ∫ τ∧m e−2(λ+ǫ)s2ǫ|Ŷs| ∫ τ∧m |Ŷs| 2 e−2(λ+ǫ)s|bn,ms | 2ds+ E ∫ τ∧m 4e−2(λ+ǫ)s|Ŷs||F (s, ξ1τ≤n, 0)|ds ≤ ≤M2e−2(λ+ǫ)n + β2e−2λn 1 + 4 ∫ τ∧m e−2ǫs|bn,ms | 2ds + 4C(1 +M)E ∫ τ∧m e−2λs|Ŷs| We note that |bn,ms | 2 ≤ C(1 + |Zns | 2 + |Zms | and by (9) supn≥1E e−2ǫs|Zns | 2ds <∞. Finally we obtain ∫ τ∧m e−2(λ+ǫ)s|Ẑs| 2 ds ≤ β′(1 + n)e−2λn where β′ depends on M,λ,K. Moreover we have that e−2(λ+ǫ)s|Ẑs| hence e−2(λ+ǫ)s|Ẑs| ≤ β′(1 + n)e−2λn. Hence (Zn) is a Cauchy sequence in M2,−2(λ+ǫ)(0, τ ; Ξ∗) and converges to the process Z in this space. (v) It remains to show that the process (Y,Z) satisfies the BSDE (5). We already know that Y is continuous and bounded and Z belongs to M2,−2(λ+ǫ)(0, τ ; Ξ∗). By definition ∀n ∈ N, ∀T, t such that 0 ≤ t ≤ T ≤ n we have Y nt∧τ − Y T∧τ = ∫ T∧τ F (s, Y ns , Z ∫ T∧τ Zns dWs. (12) Fix t and T . We shall pass to the limit in L1 in the previous equality. The sequence Y nt∧τ converges almost surely to Yt and is bounded by M + uniformly in n. From Lebesgue’s theorem we get that the sequence converges to Yt∧τ in L 1. Moreover, ∫ T∧τ Zns dWs converges in ∫ T∧τ ZsdWs in L 2 since (∫ T∧τ Zns dWs − ∫ T∧τ ZsdWs ≤ e2(λ+ǫ)TE ∫ T∧τ e−2(λ+ǫ)s|Zns − Zs| We can note that ∫ T∧τ F (s, Y ns , Z s )ds converges to ∫ T∧τ F (s, Ys, Zs)ds in L 1. Indeed ∫ T∧τ F (s, Y ns , Z s )ds− ∫ T∧τ F (s, Ys, Zs)ds ∣∣∣∣ ≤ E |F (s, Y ns , Z s )ds− F (s, Ys, Zs)|ds and, by the growth assumption on F , the map (Y,Z) → F (·, Y, Z) is continuous from the space L1(Ω;L1([0, T ];R)) × L2(Ω;L2([0, T ]; Ξ∗) to L1(Ω;L1([0, T ];R)). (By classical result on continuity of evaluation operators, see e.g. [1]). Hence, passing to the limit in the equation (12), we obtain ∀t, T such that t ≤ T Yt∧τ − YT∧τ = ∫ T∧τ F (s, Ys, Zs)− ∫ T∧τ ZsdWs. So to conclude the proof, it only remains to check the terminal condition. Let ω ∈ {τ < ∞}, and n ∈ N such that n ≥ τ(ω). Then |Yτ − ξ1t≤2n|(ω) = |Yn∧τ − ξ1t≤2n|(ω) ≤ |Yn∧τ − Y n∧τ |(ω) + |Y n∧τ − ξ1t≤2n|(ω) ≤ ≤ βeλ(n∧τ)(ω)e−2λn + |Y 2nn∧τ − ξ1t≤2n|(ω) ≤ βe since Y 2nn∧τ = Y τ = Y 2n = ξ1t≤2n Then, Yτ = ξ P-a.s. on the set {τ < ∞}, and the process (Y,Z) is solution for BSDE (5). Uniqueness. Suppose that (Y 1, Z1) and (Y 2, Z2) are both solutions of the BSDE (5) such that Y 1 and Y 2 are continuous and bounded and Z1 and Z2 belong to M2loc(0, τ ; Ξ ∗). It follows directly from the Corollary 3.2 that ∀t ≥ 0 Y 1t − Y t = 0 P-a.s. and then, by continuity, Y 1 = Y 2. Applying Ito’s formula we have that dP⊗ dt-a.e. Z1t = Z 4 The forward-backward system on infinite horizon In this Section we use the previous result to study a forward-backward system on infinite horizon, when the backward equation has quadratic generator. We introduce now some classes of stochastic processes with values in a Hilbert space K which we use in the sequel. • Lp(Ω;L2(0, s;K)) defined for s ∈]0,+∞] and p ∈ [1,∞), denotes the space of equivalence classes of progressively measurable processes ψ : Ω× [0, s[→ K, such that Lp(Ω;L2(0,s;K)) Elements of Lp(Ω;L2(0, s;K)) are identified up to modification. • Lp(Ω;C(0, s;K)), defined for s ∈]0,+∞[ and p ∈ [1,∞[, denotes the space of progressively measurable processes {ψt, t ∈ [0, s]} with continuous paths in K, such that the norm Lp(Ω;C([0,s];K)) = E sup r∈[0,s] is finite. Elements of Lp(Ω;C(0, s;K)) are identified up to indistinguishability. • L2loc(Ω;L 2(0,∞;K)) denotes the space of equivalence classes of progressively measurable processes ψ : Ω× [0,∞) → K such that ∀t > 0 E 2dr <∞. Now we consider the Itô stochastic equation for an unknown process {Xs, s ≥ 0} with values in a Hilbert space H: Xs = e e(s−r)Ab(Xr)dr + e(s−r)AσdWr, s ≥ 0. (13) Our assumptions will be the following: Assumption A2. (i) The operator A is the generator of a strongly continuous semigroup etA, t ≥ 0, in a Hilbert space H. We denote by m and a two constants such that |etA| ≤ meat for t ≥ 0. (ii) b : H → H satisfies, for some constant L > 0, |b(x)− b(y)| ≤ L|x− y|, x, y ∈ H. (iii) σ belongs to L(Ξ,H) such that etAσ ∈ L2(Ξ,H) for every t > 0, and |etAσ|L2(Ξ,H) ≤ Lt −γeat, for some constants L > 0 and γ ∈ [0, 1/2). (iv) We have b(·) ∈ G1(H,H). (v) Operators A+ bx(x) are dissipative (that is 〈Ay, y〉 + 〈bx(x)y, y〉 ≤ 0 for all x ∈ H and y ∈ D(A)). Remark 4.1. We note we need of assumptions (iv) − (v) to obtain a result of regularity of the process X with respect to initial condition x. We start by recalling a well known result on solvability of equation (13) on a bounded interval, see e.g. [12]. Proposition 4.2. Under the assumption A2, for every p ∈ [2,∞) and T > 0 there exists a unique process Xx ∈ Lp(Ω;C(0, T ;H)) solution of (13). Moreover, for all fixed T > 0, the map x→ Xx is continuous from H to Lp(Ω;C(0, T ;H)). E sup r∈[0,T ] p ≤ C(1 + |x|)p, for some constant C depending only on q, γ, T, L, a and m. We need to state a regularity result on the process X. The proof of the following lemma can be found in [17]. Lemma 4.3. Under Assumptions A2 the map x→ Xx is Gâteaux differentiable (that is belongs to G(H,Lp(Ω, C(0, T ;H))). Moreover denoting by ∇xX x the partial Gâteaux derivative, then for every direction h ∈ H, the directional derivative process ∇xX xh, t ∈ R, solves, P− a.s., the equation t h = e eσA∇xF (X σ )∇xX σhdσ, t ∈ R Finally, P-a.s., |∇xX t h| ≤ |h|, for all t > 0. The associated BSDE is: Y xt = Y F (Xxσ , Y σ , Z σ)dσ − ZxσdWσ, 0 ≤ t ≤ T <∞. (14) Here Xx is the unique mild solution to (13) starting from X0 = x. Y is real valued and Z takes values in Ξ∗, F : H × R× Ξ∗ → R is a given measurable function. We assume the following on F : Assumption A3. There exist C ≥ 0 and α ∈ (0, 1) such that 1. |F (x, y, z)| ≤ C 1 + |y|+ |z|2 2. F (·, ·, ·) is G1,1,1(H ×R× Ξ∗;R) ; 3. |∇xF (x, y, z)| ≤ C; 4. |∇zF (x, y, z)| ≤ C (1 + |z|); 5. |∇yF (x, y, z)| ≤ C (1 + |z|) 6. λ > 0 and F is monotone in y in the following sense: x ∈ H, y, y′ R, z ∈ Ξ∗ < y − y′, F (x, y, z) − F (x, y′, z) >≤ −λ|y − y′|2. Applying Theorem 3.3, we obtain: Proposition 4.4. Let us suppose that Assumptions A2 and A3 hold. Then we have: (i) For any x ∈ H, there exists a solution (Y x, Zx) to the BSDE (14) such that Y x is a contin- uous process bounded by K/λ, and Z ∈ L2loc(Ω;L 2(0,∞; Ξ)) with E e−2(λ+ǫ)s|Zs| 2ds < ∞. The solution is unique in the class of processes (Y,Z) such that Y is continuous and bounded, and Z belongs to L2loc(Ω;L 2(0,∞; Ξ)). (ii) For all T > 0 and p ≥ 1, the map x→ (Y x [0,T ] [0,T ] ) is continuous from H to the space Lp(Ω;C(0, T ;R)) × Lp(Ω;L2(0, T ; Ξ)). Proof. Statement (i) is an immediate consequences of Theorem 3.3. Let us prove (ii). Denoting by (Y n,x, Zn,x) the unique solution of the following BSDE (with finite horizon): F (Xxσ , Y σ , Z σ )dσ − Zn,xσ dWσ, (15) then, from Theorem 3.3again, |Y t | ≤ and the following convergence rate holds: t − Y t | ≤ exp{−λ(n− t)}. Now, if x′m → x as m→ +∞ then T − Y T | ≤ |Y T − Y n,x′m T |+ |Y T − Y T |+ |Y n,x′m T − Y exp{−λ(n− T )}+ |Y n,x′m T − Y Moreover for fixed n, Y n,x′m T → Y T in L p(Ω,FT ,P;R) for all p > 1, by Proposition 4.2 in [2] Thus Y T → Y T in L p(Ω,FT ,P;R). Now we can notice that (Y x [0,T ] [0,T ] ) is the unique solution of the following BSDE (with finite horizon): Y xt = Y F (Xxσ , Y σ , Z ZxσdWσ, and the same holds for (Y x [0,T ] [0,T ] ). By similar argument as in [2] we have t∈[0,T ] |Y xt − Y ]1∧1/p [(∫ T |Zxt − Z )p/2]1∧1/p ≤ C E [∣∣∣Y xT − Y [(∫ T ∣∣∣F (s,Xxs , Ys, Zs)− F (s,Xx s , Ys, Zs) ∣∣∣ ds )p+1] 1p+1 and we can conclude that (Y x [0,T ] [0,T ] ) → (Y x [0,T ] [0,T ] ) in Lp(Ω;C(0, T ;R)) × Lp(Ω;L2(0, T ; Ξ)). We need to study the regularity of Y x. More precisely, we would like to show that Y x0 belongs to G1(H,R). We are now in position to prove the main result of this section. Theorem 4.5. Under Assumption the map x → Y x0 belongs to G 1(H,R). Moreover |Y x0 | + 0 | ≤ c, for a suitable constant c. Proof. Fix n ≥ 1, let us consider the solution (Y n,x, Zn,x) of (15). Then, see [2], Proposition 4.2, the map x → (Y n,x(·), Zn,x(·)) is Gâteaux differentiable from H to Lp(Ω, C(0, T ;R)) × Lp(Ω;L2(0, T ; Ξ∗)), ∀p ∈ (1,∞). Denoting by (∇xY n,xh,∇xZ n,xh) the partial Gâteaux deriva- tives with respect to x in the direction h ∈ H, the processes {∇xY t h,∇xZ t h, t ∈ [0, n]} solves the equation, P− a.s., t h = ∇xF (X σ , Y σ , Z σ )∇xX σ hdσ ∇yF (X σ , Y σ , Z σ )∇xY σ hdσ (16) ∇zF (X σ , Y σ , Z σ )∇xZ σ hdσ − σ hdWσ . We note that we can write the generator of the previous equation as φnσ(u, v) = ψ σ + a σu+ b setting ψnσ = ∇xF (X σ , Y σ , Z σ )∇xX anσ = ∇yF (X σ , Y σ , Z σ ) b σ = ∇zF (X σ , Y σ , Z By Assumption A3 and Lemma 4.3, we have that for all x, h ∈ H the following holds P-a.s. for all n ∈ N and all σ ∈ [0, n]: |ψnσ | = ∣∣∣∇xF (Xxσ , Y σ , Z σ )∇xX ∣∣∣ ≤ C|h|, anσ = ∇yF (X σ , Y σ , Z σ ) ≤ −λ ≤ 0, |b ∣∣∣∇zF (Xxσ , Y σ , Z ∣∣∣ ≤ C(1 + |Zn,xσ |). Therefore σ dWσ is a BMO-martingale. Hence bsdWs is also a BMO-martingale and by Lemma 3.1, we obtain: t∈[0,n] t | ≤ C|h|, P− a.s.; and applying Itô’s formula to e−2λt|∇xY 2 and arguing as in the proof of Theorem 3.3, points (iii) and (iv), tanks to the (9), we get: e−2λt(|∇xY 2 + |∇xZ 2)dt ≤ C1|h| Fix x, h ∈ H, there exists a subsequence of {(∇xY n,xh,∇xZ n,xh,∇xY 0 h) : n ∈ N} which we still denote by itself, such that (∇xY n,xh,∇xZ n,xh) converges weakly to (U1(x, h), V 1(x, h)) in M2,−2λ(0,∞;R × Ξ∗) and ∇xY 0 h converges to ξ(x, h) ∈ R. Now we write the equation (16) as follows: t h = ∇xY ∇xF (X σ , Y σ , Z σ )∇xX (∇yF (X σ , Y σ , Z σ ))∇xY σ hdσ (17) ∇zF (X σ , Y σ , Z σ )∇xZ σ hdσ + σ hdWσ and define an other process U2t (x, h) by U2t (x, h) = ξ(x, h) − ∇xF (X σ , Y σ , Z σ )∇xX σ hdσ (∇yF (X σ , Y σ , Z σ(x, h)dσ (18) ∇zF (X σ , Y σ , Z σ (x, h)dσ + V 1σ (x, h)dWσ , where (Y x, Zx) is the unique bounded solution to the backward equation (14), see Proposition 4.4. Passing to the limit in the equation (17) it is easy to show that ∇xY t h converges to U2t (x, h) weakly in L 1(Ω) for all t > 0. Thus U2t (x, h) = U t (x, h), P-a.s. for a.e. t ∈ R + and |U2t (x, h)| ≤ C|h|. Now consider the following equation on infinite horizon U(t, x, h) = U(0, x, h) − ∇xF (X σ , Y σ , Z σ )∇xX (∇yF (X σ , Y σ , Z σ))U(t, x, h)dσ (19) ∇zF (X σ , Y σ , Z σ )V (σ, x, h)dσ + V (σ, x, h)dWσ . We claim that this equation has a solution. For each n ∈ N consider the finite horizon BSDE (with final condition equal to zero): Un(t, x, h) = ∇xF (X σ , Y σ , Z σ )∇xX (∇yF (X σ , Y σ , Z σ ))Un(t, x, h)dσ ∇zF (X σ , Y σ , Z σ)Vn(σ, x, h)dσ − Vn(σ, x, h)dWσ , By the result in [2] we know that this equation has a unique solution (Un(·, x, h), Vn(·, x, h)) ∈ Lp(Ω;C(0, n;R)) × Lp(Ω;L2(0, n; Ξ∗)). The generator of this equation can be rewrite as φt(u, v) = ψt + atu+ btv where ψt = ∇xF (X t , Y t , Z t )∇xX t and |ψt| ≤ C|h|, at = ∇yF (X σ , Y σ , Z σ) ≤ −λ, bt = ∇zF (X σ , Y σ , Z σ ) and |bt| ≤ C(1+ |Z t |). On the interval [0, n] the process Zxs dWs is a BMO- martingale. Hence, from the Lemma 3.1 it follows that P-a.s. ∀n ∈ N, ∀t ∈ [0, n] |Unt | ≤ and as in the proof of existence in the Theorem 3.3, we can conclude that 1. for each t ≥ 0 Un(t, x, h) is a Cauchy sequence in L∞(Ω) which converges to a process U and P-a.s., ∀t ∈ [0, n] |Un(t, x, h)− U(t, x, h)| ≤ |h|e−λ(n−t); 2. V n(·, x, h) is a Cauchy sequence in L2loc(Ω;L 2([0,∞); Ξ∗); 3. The processes limit (U(·, x, h), V (·, x, h) satisfy the BSDE (19). Moreover still from Lemma 3.1 we get that the solution is unique. Coming back to equation (18), we have that (U2(x, h), V 1(x, h)) is solution in R+ of the equation (19). In particular we notice that U(0, x, h) = ξ(x, h) is the limit of ∇xY 0 h (along the cho- sen subsequence). The uniqueness of the solution to (19) implies that in reality U(0, x, h) = limn→∞∇xY 0 h along the original sequence. Now let xm → x. |U(0, x, h) − U(0, xm, h)| ≤ |U(0, x, h) − U n(0, x, h)| + |Un(0, x, h) − Un(0, xm, h)|+ (20) +|Un(0, xm, h) − U(0, xm, h)| ≤ e−λn|h|+ |Un(0, x, h) − Un(0, xm, h)|, where we have used the (1). We now notice that ∇xF , ∇yF , ∇zF are, by assumptions, con- tinuous and |∇xF | ≤ C, |∇yF | ≤ C(1 + |Z|) 2α, |∇zF | ≤ C(1 + |Z|) . Moreover the following statements on continuous dependence on x hold: maps x → Xx, x → ∇xX xh are continuous from H → L P(Ω;C(0, T ;H)) (see [12] Proposition 3.3); the map x→ Y x [0,T ] is continuous from H to L P(Ω;C(0, T ;R)) (see Proposition 4.4 here); the map x→ Zx [0,T ] is continuous from H to L P(Ω;L 2(0, T ; Ξ)) (see Proposition 4.4 here ). We can therefore apply to (20) the continuity result of [12] Proposition 4.3 to obtain in particular that Un(0, x m, h) → Un(0, x, h) for all fixed n as m → ∞. And by (20) we can conclude that U(0, x′m, h) → U(0, x, h) as m→ ∞. Summarizing U(0, x, h) = limn→∞∇xY 0 h exists, moreover it is clearly linear in h and verifies |U(0, x, h)| ≤ C|h|, finally it is continuous in x for every h fixed. Finally, for t > 0, [Y x+th0 − Y 0 ] = lim n,x+th 0 − Y 0 ] = lim n,x+θth 0 hdθ = lim U(0, x+ θth)hdθ = U(0, x)h and the claim is proved. 5 Mild Solution of the elliptic PDE Now we can proceed as in [13]. Let us consider the forward equation Xs = e e(s−r)Ab(Xr)dr + e(s−r)AσdWr, s ≥ 0. (21) Assuming that Assumption A2 holds, we define in the usual way the transition semigroup (Pt)t≥0, associated to the process X: Pt[φ](x) = E φ(X t ), x ∈ H, for every bounded measurable function φ : H → R. Formally, the generator L of (Pt) is the operator Lφ(x) = Trace σσ∗∇2φ(x) + 〈Ax+ b(x),∇φ(x)〉. In this section we address solvability of the non linear stationary Kolmogorov equation: Lv(x) + F (x, v(x),∇v(x)σ) = 0, x ∈ H, (22) when the coefficient F verifies Assumption A3. Note that, for x ∈ H, ∇v(x) belongs to H∗, so that ∇v(x)σ is in Ξ∗. Definition 5.1. We say that a function v : H → R is a mild solution of the non linear stationary Kolmogorov equation (22) if the following conditions hold: (i) v ∈ G1(H,R) and ∃C > 0 such that |v(x)| ≤ C, |∇xv(x)h| ≤ C |h|, for all x, h ∈ H; (ii) the following equality holds, for every x ∈ H and T ≥ 0: v(x) = e−λT PT [v](x) + e−λt Pt ·, v(·),∇v(·)σ + λv(·) (x) dt. (23) where λ is the monotonicity constant in Assumption A3. Together with equation (21) we also consider the backward equation Yt − YT + ZsdWs = F (Xs, Ys, Zs)ds 0 ≤ t ≤ T <∞ (24) where F : H ×R×Ξ∗ → R is the same occurring in the nonlinear stationary Kolmogorov equa- tion. Under the Assumptions A2, A3, Propositions 4.2-4.4 give a unique solution {Xxt , Y t , Z for t ≥ 0, of the forward-backward system (21)-(24). We can now state the following Theorem 5.2. Assume that Assumption A2, Assumption A3 and hold then equation (22) has a unique mild solution given by the formula v(x) = Y x0 . where {Xxt , Y t , Z t , t ≥ 0} is the solution of the forward-backward system (21)-(24). Moreover the following holds: Y xt = v(X t ), Z t = ∇v(X t )σ. Proof. Let us recall that for s ≥ 0, Y xs is measurable with respect to F[0,s] and Fs; it follows that Y x0 is deterministic (see also [7]). Moreover, as a byproduct of Proposition 4.5, the function v defined by the formula v(x) = Y x0 has the regularity properties stated in Definition 5.1. The proof that the equality (23) holds true for v is identical to the proof of Theorem 6.1 in [13]. 6 Application to optimal control We wish to apply the above results to perform the synthesis of the optimal control for a general nonlinear control system on an infinite time horizon. To be able to use non-smooth feedbacks we settle the problem in the framework of weak control problems. Again we follow [13] with slight modifications. We report the argument for reader’s convenience. As above by H, Ξ we denote separable real Hilbert spaces and by U we denote a Banach space. For fixed x0 ∈ H an admissible control system (a.c.s) is given by (Ω,F , (Ft)t≥0,P, {Wt, t ≥ 0}, u) where • (Ω,F ,P) is a complete probability space and (Ft)t≥0 is a filtration on it satisfying the usual conditions. • {Wt : t ≥ 0} is a Ξ-valued cylindrical Wiener process relatively to the filtration (Ft)t≥0 and the probability P. • u : Ω×[0,∞[→ U is a predictable process (relatively to (Ft)t≥0) that satisfies the constraint: ut ∈ U , P-a.s. for a.e. t ≥ 0, where U is a fixed closed subset of U . To each a.c.s. we associate the mild solution X ∈ LrP(Ω;C(0, T ;H)) (for arbitrary T > 0 and arbitrary r ≥ 1) of the state equation: dXτ = (AXτ + b(Xτ ) + σr(Xτ , uτ )) dτ + σ dWτ , τ ≥ 0, X0 = x ∈ H, and the cost: J(x, u) = E e−λtg(Xt, ut) dt, (26) where g : H × U → R. Our purpose is to minimize the functional J over all a.c.s. Notice the occurrence of the operator σ in the control term: this special structure of the state equation is imposed by our techniques. We work under the following assumptions. Assumption A4. 1. The process W is a Wiener process in Ξ, defined on a complete prob- ability space (Ω,F ,P) with respect to a filtration (Ft) satisfying the usual conditions. 2. A, b verify Assumption A2. 3. σ satisfies Assumption A2 (iii) with γ = 0; 4. The set U is a nonempty closed subset of U . 5. The functions r : H × U → Ξ, g : H × U → R are Borel measurable and for all x ∈ H, r(x, ·) and g(x, ·) are continuous functions from U to Ξ and from U to R, respectively. 6. There exists a constant C ≥ 0 such that for every x, x′ ∈ H , u ∈ K it holds that |r(x, u)− r(x′, u)| ≤ C(1 + |u|)|x − x′|, |r(x, u)| ≤ C(1 + |u|), (27) 0 ≤ g(x, u) ≤ C(1 + |u|2), (28) 7. There exist R > 0 and c > 0 such that for every x ∈ H u ∈ U satisfying |u| ≥ R, g(x, u) ≥ c|u|2. (29) We will say that an (Ft)-adapted stochastic process {ut, t ≥ 0} with values in U is an admissible control if it satisfies e−λt|ut| 2dt <∞. (30) This square summability requirement is justified by (29): a control process which is not square summable would have infinite cost. Now we state that for every admissible control the solution to (25) exists. Proposition 6.1. Let u be an admissible control. Then there exists a unique, continuous, (Ft)-adapted process X satisfying E supt∈[0,T ] |Xt| 2 <∞, and P-a.s., t ∈ [0, T ] Xt = e e(t−s)Ab(Xs)ds+ e(t−s)AσdWs + e(t−s)Aσr(Xs, us)ds. Proof. The proof is an immediate extension to the infinite dimensional case of the Proposition 2.3 in [11]. By the previous Proposition and the arbitrariness of T in its statement, the solution is defined for every t ≥ 0. We define in a classical way the Hamiltonian function relative to the above problem: for all x ∈ H, z ∈ Ξ∗, F (x, y, z) = inf{g(x, u) + zr(x, u) : u ∈ U} − λy Γ(x, y, z) = {u ∈ U : g(x, u) + zr(x, u)− λy = F (x, y, z)}. The proof of the following Lemma can be found in [11] Lemma 3.1. Lemma 6.2. The map F is a Borel measurable function from H × Ξ∗ to R. There exists a constant C > 0 such that − C(1 + |z|2)− λy ≤ F (x, y, z) ≤ g(x, u) + C|z|(1 + |u|)− λy ∀u ∈ U . (32) We require moreover that Assumption A5. F satisfies assumption A3 2-3-4. We notice that the cost functional is well defined and J(x, u) <∞ for all x ∈ H and all a.c.s. By Theorem 5.2, the stationary Hamilton-Jacobi-Bellman equation relative to the above stated problem, namely: Lv(x) + F (x, v(x),∇v(x)σ) = 0, x ∈ H, (33) admits a unique mild solution, in the sense of Definition 5.1. 6.0.1 The fundamental relation Proposition 6.3. Let v be the solution of (33). For every admissible control u and for the corresponding trajectory X starting at x we have J(x, u) = v(x)+ − F (Xt,∇v(Xt)σ)− λv(Xt) +∇xv(Xt)σr(Xt, ut) + g(Xt, ut) Proof. We introduce the sequence of stopping times τn = inf{t ∈ [0, T ] : 2ds ≥ n}, with the convention that τn = T if the indicated set is empty. By (30), for P-almost every ω ∈ Ω, there exists an integer N(ω) depending on ω such that n ≥ N(ω) =⇒ τn(ω) = T. (34) Let us fix u0 ∈ K, and for every n, let us define unt = ut1t≤τn + u01t>τn and consider the equation dXnt = b(X t )dt+ σ[dWt + r(X t , u t )dt], 0 ≤ t ≤ T Xn0 = x. Let us define W nt :=Wt + r(Xns , u s )ds 0 ≤ t ≤ T. From the definition of τn and from (27), it follows that |r(Xns , u 2ds ≤ C (1 + |uns |) 2ds ≤ C (1 + |us|) 2ds +C ≤ C + Cn. (36) Therefore defining ρn = exp −r(Xns , u s )dWs − |r(Xns , u the Novikov condition implies that Eρn = 1. Setting dP T = ρndP|FT , by the Girsanov theorem W n is a Wiener process under PnT . Relatively to W n the equation (35) can be written: dXnt = b(X t )dt+ σdW t , 0 ≤ t ≤ T Xn0 = x. Consider the following finite horizon Markovian forward-backward system (with respect to probability PnT and to the filtration generated by {W τ : τ ∈ [0, T ]}). Xnτ (x) = e e(τ−s)Ab(Xns (x)) ds+ e(τ−s)Aσ dW ns , τ ≥ 0, Y nτ (x)− v(X T (x)) + Zns (x)dW F (Xns (x), Y s (x), Z s (x))ds, 0 ≤ τ ≤ T, and let (Xn(x), Y n(x), Zn(x)) be its unique solution with the three processes predictable rela- tively to the filtration generated by {W nτ : τ ∈ [0, T ]} and: E T supt∈[0,T ] |X t (x)| 2 < +∞, Y n(x) bounded and continuous, EnT |Znt (x)| 2dt < +∞. Moreover, Theorem 5.2 and uniqueness of the solution of system (38), yields that Y nt (x) = v(X t (x)), Z t (x) = ∇v(X t (x))G(X t (x)). (39) Applying the Itô formula to e−λtY nt (x), and restoring the original noise W we get e−λτnY nτn(x) = e −λTY nT (x) + λe−λtY ns (x)ds − e−λsZns (x) dWs e−λs [F (Xns (x), Y s (x), Z s (x)) − Z s (x)r(X s , u s )] ds. We note that for every p ∈ [1,∞) we have ρ−pn = exp r(Xns , u s )dW |r(Xns , u · exp p2 − p |r(Xns , u . (41) By (36) the second exponential is bounded by a constant depending on n and p, while the first one has Pn-expectation, equal to 1. So we conclude that Enρ n <∞. It follows that e−2λt|Znt (x)| ρ−2n |Z t (x)| ≤ (Enρ−2n ) |Znt (x)| We conclude that the stochastic integral in (40) has zero expectation. Using the identification in (39) and taking expectation with respect to P, we obtain Ee−λτnY nτn = e E[v(XnT (x))] + E λe−λtY ns (x)ds+ e−λs [F (Xns (x), Y s (x), Z s (x))− Z s (x)r(X s (x), u s )] ds ≤ ≤ e−λTE[v(XnT (x)] + E λe−λsY ns (x)ds + E e−λsg(Xns (x), u s )ds. Now we let n→ ∞. By Proposition 4.4, |Y nt | = sup |v(Xnt )| ≤ ; (43) in particular λe−λsY ns (x)ds ≤ E λe−λs ds ≤ EK(T − τn) and the right-hand side tends to 0 by (34). By the definition of un and (28), g(Xns , u s )ds = E 1s>τng(X s , u0)ds ≤ 1s>τn(1 + |u0| 2)ds ≤ CE(T − τn) (44) and the right-hand side tends to 0 again by (34). Next we note that, again by (34), for n ≥ N(ω) we have τn(ω) = T and v(X T ) = v(X ) = v(Xτn) = v(XT ). We deduce, thanks to (43), that Ev(XnT ) → Ev(XT ), and from (42) we conclude that lim sup Ee−λτnY nτn ≤ e Ev(XT ). On the other hand, for n ≥ N(ω) we have τn(ω) = T and e −λτnY nτn = e −λTY nT = e −λT v(XnT ) = e−λT v(XT ). Since Y n is bounded, by the Fatou lemma, Ee−λT v(XT ) ≤ lim infn→∞ Ee −λτnY nτn . We have thus proved that Ee−λτnY nτn = e Ev(XT ). (45) Now we return to backward equation in the system (38) and write e−λτnY nτn = Y −e−λtF (Xnt , Y t , Z t )dt+ −λe−λtY nt dt+ e−λtZnt dWt+ e−λtZnt r(X t , u t )dt Arguing as before, we conclude that the stochastic integral has zero P-expectation. Moreover, we have Y n0 = v(x), and, for t ≤ τn, we also have u t = ut, X t = Xt, Y t = v(X t ) = v(Xt) and Znt = ∇xv(Xt). Thus, we obtain E[e−λτnY nτn ] = v(x)+ − F (Xt, v(Xt),∇xv(Xt)σ)− λv(Xt) +∇xv(Xt)σr(Xt, ut) dt (46) e−λtg(Xt, ut)dt+ E[e −λτnY nτn ] = v(x)+ − F (Xt, v(Xt),∇xv(Xt)σ) − λv(Xt) +∇xv(Xt)σr(Xt, ut) + g(Xt, ut) dt. (47) Noting that −F (x, y, z)− λy + zr(x, u) + g(x, u) ≥ 0 and recalling that g(x, u) ≥ 0 by (45) and the monotone convergence theorem, we obtain for n→ ∞, e−λtg(Xt, ut)dt+ e Ev(XT ) = v(x)+ − F (Xt,∇xv(Xt)σ) − λv(Xt) +∇xv(Xt)σr(Xt, ut) + g(Xt, ut) dt. (48) Recalling that v is bounded, letting T → ∞, we conclude J(x, u) = v(x)+ e−λt [−F (Xt, v(Xt),∇v(Xt)σ)− λv(Xt) +∇xv(Xt)σr(Xt, ut) + g(Xt, ut)] dt. The above equality is known as the fundamental relation and immediately implies that v(x) ≤ J(x, u) and that the equality holds if and only if the following feedback law holds P-a.s. for almost every t ≥ 0: F (Xt, v(Xt),∇xv(Xt)σ) = ∇xv(Xt)σ + g(Xt, ut)− λv(Xt) where X is the trajectory starting at x and corresponding to control u. 6.0.2 Existence of optimal controls: the closed loop equation. Next we address the problem of finding a weak solution to the so-called closed loop equation. We have to require the following Assumption A6. Γ(x, y, z), defined in 31, is non empty for all x ∈ H and z ∈ Ξ∗. By simple calculation (see [11] Lemma 3.1), we can prove that this infimum is attained in a ball of radius C(1 + |z|), that is, F (x, y, z) = min u∈U ,|u|≤C(1+|z|) [g(x, u) + zr(x, u)]− λy, x ∈ H, y ∈ R, z ∈ Ξ∗, F (x, y, z) < g(x, u) + zr(x, u)− λy if |u| > C(1 + |z|). (49) Moreover, by the Filippov Theorem (see, e.g., [1, Thm. 8.2.10, p. 316]) there exists a measurable selection of Γ, a Borel measurable function γ : H × Ξ∗ → U such that F (x, y, z) = g(x, γ(x, z)) + zr(x, γ(x, z)) − λy, x ∈ H, y ∈ R, z ∈ Ξ∗. (50) By (49), we have |γ(x, z)| ≤ C(1 + |z|). (51) We define u(x) = γ(x,∇xv(Xt)σ) P-a.s. for a.e t ≥ 0. The closed loop equation is dXt = AXtdt+ b(Xt)dt+ σ[dWt+ r(Xt, u(Xt))dt] t ≥ 0 X0 = x By a weak solution we mean a complete probability space (Ω,F ,P) with a filtration (Ft) satisfy- ing the usual conditions, a Wiener process W in Ξ with respect to P and (Ft), and a continuous (Ft)-adapted process Xwith values in H satisfying, P-a.s., e−λt|u(Xt)| 2dt <∞ and such that (52) holds. We note that by (27) it also follows that |r(Xt, u(Xt))| 2dt <∞, P− a.s., so that (52) makes sense. Proposition 6.4. Assume that b, σ, g satisfy Assumption A4, F verifies Assumption A5 and Assumption A6 holds. Then there exists a weak solution of the closed loop equation, satisfying in addition e−λt|u(Xt)| 2dt <∞. (53) Proof. We start by constructing a canonical version of a cylindrical Wiener process in Ξ. An explicit construction is needed to clarify the application of an infinite-dimensional version of the Girsanov theorem that we use below. We choose a larger Hilbert space Ξ ⊃ Ξ in such a way that Ξ is continuously and densely embedded in Ξ with Hilbert-Schmidt inclusion operator J . By Ω we denote the space C([0,∞[,Ξ ) of continuous functions ω : [0,∞[→ Ξ endowed with the usual locally convex topology that makes Ω a Polish space, and by B its Borel σ-field. Since JJ ∗ has finite trace on Ξ , it is well known that there exists a probability P on B such that the canonical processes W t (ω) := ω(t), t ≥ 0, is a Wiener process with continuous paths in Ξ satisfying E[〈W t , ξ ′ ] = 〈J J ∗ξ ′ (t∧ s) for all ξ , t, s ≥ 0. This is called a JJ ∗-Wiener processes in Ξ in [8], to which we refer the reader for preliminary material on Wiener processes on Hilbert spaces. Let us denote by G the P-completion of B and by N the family of sets A ∈ G with P(A) = 0. Let Bt = σ{W s : s ∈ [0, t]} and Ft = σ(Bt,N ), t ≥ 0, where as usual σ(·) denotes the σ-algebra in Ω generated by the indicated collection of sets or random variables. Thus (Ft)t≥0 is the Brownian filtration of W The Ξ-valued cylindrical Wiener process {W t : t ≥ 0, ξ ∈ Ξ} can now be defined as follows. For ξ in the image of J ∗J we take η such that ξ = J ∗J η and define W s = 〈W s,J η〉Ξ′ . Then we notice that E|W 2 = t|J η|2 ′ = t|ξ| Ξ, which shows that the mapping ξ → W s , defined for ξ ∈ J ∗J (Ξ) ⊂ Ξ with values in L2(Ω,F ,P), is an isometry for the norms of Ξ and L2(Ω,F ,P). Consequently, noting that J ∗J (Ξ) is dense in Ξ, it extends to an isometry ξ → L2(ω,F ,P), still denoted ξ → W s . An appropriate modification of {W t : t ≥ 0, ξ ∈ Ξ} gives the required cylindrical Wiener process. We note that the Brownian filtration of W coincides with (Ft)t≥0. Now let X ∈ L (Ω, C(0,+∞;H)) be the mild solution of dXτ = AXτ dτ + b(Xτ ) dτ + σ dWτ X0 = x If together with previous forward equation we also consider the backward equation Yt − YT + ZsdWs = F (Xs, Ys, Zs)ds 0 ≤ t ≤ T <∞ (55) we know that there exists a unique solution {Xxt , Y t , Z t , t ≥ 0} forward-backward system (54)- (55) and by Proposition 5.2, v(x) = Y x0 . is the solution of the of the non linear stationary Kolmogorov equation: Lv(x) + F (x, v(x),∇v(x)σ) = 0, x ∈ H. (56) Moreover the following holds: Yτ (x) = v(Xτ (x)), Zτ (x) = ∇v(Xτ (x))σ (57) We have e−(λ+ǫ)t|Zt| 2dt <∞. (58) and hence 2dt <∞. (59) By (27) we have |r(Xt, u(Xt))| ≤ C(1 + |u(Xt)|), (60) and by (51), |u(Xt)| = |γ(Xt,∇v(Xt(x))σ)| ≤ C(1 + |∇v(Xt(x))σ|) = C(1 + |Zt|). (61) Let us define ∀T > 0 MT = exp 〈r(Xs, u(Xs), dWs〉Ξ − |r(Xs, u(Xs)| . (62) Now, arguing exactly as in the proof of Proposition 5.2 in [11], we can prove that EMT = 1, and M is a P-martingale. Hence there exists a probability P̂T on FT admitting MT as a density with respect to P, and by the Girsanov Theorem we can conclude that {Ŵt, t ∈ [0, T ]} is a Wiener process with respect to P and (Ft). Since Ξ is a Polish space and P̂T+h coincide with P̂T on BT , T, h ≥ 0, by known results (see [22], Chapter VIII, §1, Proposition (1.13)) there exists a probability P̂ on B such that the restriction on BT of P̂T and that of P̂ coincide, T ≥ 0. Let Ĝ be the P̂-completion of B and F̂T be the P̂-completion of BT . Moreover, since for all T > 0, {Ŵt : t ∈ [0, T ]} is a Ξ-valued cylindrical Wiener process under P̂T and the restriction of P̂T and of P̂ coincide on BT modifying {Ŵt : t ≥ 0} in a suitable way on a P̂-null probability set we can conclude that (Ω, Ĝ, {F̂t, t ≥ 0}, P̂, {Ŵt, t ≥ 0}, γ(X,∇v(X)σ(X))) is an admissible control system. The above construction immediately ensures that, if we choose such an admissible control system, then (52) is satisfied. Indeed if we rewrite (54) in terms of {Ŵt : t ≥ 0} we get dXτ = AXτ dτ + b(Xτ ) dτ + σ [r(Xτ , u(Xτ ))dτ + dŴτ ] X0 = x. It remains to prove (53). We define stopping times σn = inf t ≥ 0 : e−λt|Zs| 2ds ≥ n with the convention that σn = ∞ if the indicated set is empty. By (58) for P-a.s. ω ∈ Ω there exists an integer N(ω) depending on ω such that σn(ω) = ∞ for n ≥ N(ω). Applying the Ito formula to e−λtYt, with respect to W , we obtain e−λσnYσn = Y0 − e−λsZs dWs+ e−λs [−F (Xs, Ys, Zs)− λYs(x)ds + Zsr(Xs, u(Xs))] ds. from which we deduce that Ee−λσnYσn + E e−λsg(Xs, u(Xs))ds = Y0+ e−λs [−F (Xs, Ys, Zs)− λYsds + Zsr(Xs, u(Xs)) + g(Xs, u(Xs))] ds = Y0. with the last equality coming from the definition of u. Recalling that Y is bounded, it follows e−λsg(Xs, u(Xs))ds ≤ C for some constant C independent of n. By (29) and by sending n to infinity, we finally prove (53). References [1] A. Ambrosetti, G. Prodi. 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Backward stochastic differential equations: a general introduction. In Back- ward stochastic differential equations (Paris, 1995–1996), volume 364 of Pitman Res. Notes Math. Ser., pages 7–26. Longman, Harlow, 1997. [8] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge, 1992 [9] G. Da Prato and J. Zabczyk. Second Order Partial Differential Equations in Hilbert Spaces. Cambridge University Press, Cambridge, 2002. [10] F. Masiero. Infinite horizon stochastic optimal control problems with degenerate noise and elliptic equations in Hilbert spaces. Preprint, Politecnico di Milano, 2004 (submitted). [11] M. Fuhrman,Y. Hu and G. Tessitore. On a class of stochastic optimal control problems related to BSDEs with quadratic growth. SIAM J. Control Optim. 45 (2006), no. 4, 1279– 1296. [12] M. Fuhrman and G. Tessitore. 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Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, (1999). [23] M. Royer. BSDEs with a random terminal time driven by a monotone generator and their links with PDEs. Stochastics Stochastics Rep. 76 (2004) 281-307.

0704.1224

Non-minimal Wu-Yang wormhole

Non-minimal Wu-Yang wormhole A. B. Balakin,1, ∗ S. V. Sushkov,1, 2, † and A. E. Zayats1, ‡ Department of General Relativity and Gravitation, Kazan State University, Kremlevskaya str. 18, Kazan 420008, Russia Department of Mathematics, Tatar State University of Humanities and Education, Tatarstan str. 2, Kazan 420021, Russia We discuss exact solutions of three-parameter non-minimal Einstein-Yang-Mills model, which describe the wormholes of a new type. These wormholes are considered to be supported by SU(2)- symmetric Yang-Mills field, non-minimally coupled to gravity, the Wu-Yang ansatz for the gauge field being used. We distinguish between regular solutions, describing traversable non-minimal Wu- Yang wormholes, and black wormholes possessing one or two event horizons. The relation between the asymptotic mass of the regular traversable Wu-Yang wormhole and its throat radius is analysed. PACS numbers: 04.20.Jb, 14.80.Hv, 04.20.Gz Keywords: Einstein-Yang-Mills theory, non-minimal coupling, traversable wormhole I. INTRODUCTION Wormholes are topological handles in spacetime linking widely separated regions of a single universe, or “bridges” joining two different spacetimes [1]. Recent interest in these configurations has been initiated by Morris and Thorne [2]. These authors constructed and investigated a class of objects they referred to as “traversable wormholes”. The central feature of wormhole physics is the fact that traversable wormholes are accompanied by an unavoidable violation of the null energy condition, i.e., the matter threading the wormhole’s throat has to be possessed of “exotic” properties [2, 3]. The known classical matter does satisfy the usual energy conditions, hence physical models providing the existence of wormholes must include hypothetical forms of matter. Various models of such kind have been considered in the literature, among them scalar fields [4]; wormhole solutions in semi-classical gravity [5]; solutions in Brans-Dicke theory [6]; wormholes on the brane [7]; wormholes supported by matter with an exotic equation of state, namely, phantom energy [8], the generalized Chaplygin gas [9], tachyon matter [10], etc [11, 12]. The electromagnetic field and the non-Abelian gauge field can also be considered as sources for wormholes when they satisfy the necessary unusual energy conditions. Such possibility can in principle appear if one considers nonlinear electrodynamics [13] or takes into account the non-minimal coupling of gravity with vector-type fields, i.e., with the non-Abelian Yang-Mills field, or Maxwell field. The non-minimal Einstein-Maxwell theory has been elaborated in detail in both linear (see, e.g., [14, 15] for a review) and non-linear (see, [16]) versions. As for the non-minimal Einstein-Yang-Mills theory, two concepts to derive the master equations are known. The first one is a dimensional reduction of the Gauss-Bonnet action [17], this model contains one coupling parameter. The second concept is a non- Abelian generalization of the non-minimal non-linear Einstein-Maxwell theory [16]. We will follow the latter approach and construct a three-parameter non-minimal model being linear in the curvature by analogy with the well-known model proposed by Drummond and Hathrell for linear electrodynamics [18]. Three coupling constants q1, q2, and q3 of the model are shown to introduce a new specific radius associated with the radius a of the wormhole throat. In this work we focus on the example of exact solution of the non-minimal three-parameter EYM model describing the wormhole of a new type, namely, non-minimal wormhole. It can also be indicated as non-minimal Wu-Yang wormhole, since the solution of the non-minimally extended Yang-Mills subsystem of the total self-consistent EYM system of equations is the direct analog of the Wu-Yang monopole [19]. The paper is organized as follows. In Sec. II we briefly describe the formalism of three-parameter non-minimal Einstein-Yang-Mills model. In Sec. III, Subsect. A we adapt this model for the case of static spherically symmetric field configuration, present the exact solution of the Wu-Yang type to the gauge field equations and formulate two key equations for two metric functions σ(r) and N(r). In Subsect. B, C and D we discuss the details of the three- parameter family of exact solutions for the function σ(r). Sec. IV is devoted to the analysis of the solution describing the non-minimal Wu-Yang wormhole. Conclusions are formulated in the last section. ∗Electronic address: [email protected] †Electronic address: sergey˙[email protected]; [email protected] ‡Electronic address: [email protected] http://arxiv.org/abs/0704.1224v2 mailto:[email protected] mailto:[email protected]; [email protected] mailto:[email protected] II. NON-MINIMAL EINSTEIN-YANG-MILLS MODEL The action of the three-parameter non-minimal Einstein-Yang-Mills model has the form1 SNMEYM = ik(a) + χikmnF , (1) where g = det(gik) is the determinant of a metric tensor gik, and R is the Ricci scalar. The Latin indices without parentheses run from 0 to 3, the summation with respect to the repeated group indices (a) is implied. The tensor χikmn, indicated in [16] as non-minimal susceptibility tensor, is defined as follows: χikmn ≡ R (gimgkn − gingkm) + (Rimgkn −Ringkm +Rkngim −Rkmgin) + q3Rikmn . (2) Here Rik and Rikmn are the Ricci and Riemann tensors, respectively, and q1, q2, q3 are the phenomenological param- eters describing the non-minimal coupling of the Yang-Mills and gravitational fields. Following [20], we consider the Yang-Mills field, Fmn, to take the values in the Lie algebra of the gauge group SU(2): Fmn = −iGF (a)mnt(a) , Am = −iGA(a)m t(a) . (3) Here t(a) are Hermitian traceless generators of SU(2) group, G is a constant of gauge interaction, and the group index (a) runs from 1 to 3. The generators t(a) satisfy the commutation relations: [t(a), t(b)] = i ε(a)(b)(c)t(c) , (4) where ε(a)(b)(c) is the completely antisymmetric symbol with ε(1)(2)(3) = 1. The Yang-Mills field potential, Ai, and strength field, Fik, are coupled by the relation Fik = ∂iAk − ∂kAi + [Ai ,Ak] , (5) which guarantees that the equation D̂lFik + D̂kFli + D̂iFkl = 0 (6) turns into identity. Here the symbol D̂k denotes the gauge invariant derivative D̂i ≡ ∇i + [Ai , ] , (7) and ∇m is a covariant spacetime derivative. The variation of the action (1) with respect to Yang-Mills potential A i yields ik ≡ ∇kHik + = 0 , Hik = Fik + χikmnFmn . (8) The tensor Hik is a non-Abelian analog of the induction tensor known in the electrodynamics [21], and thus χikmn can be considered as a non-minimal susceptibility tensor [16]. The variation of the action with respect to the metric gik yields Rik − R gik = 8π T (eff) ik . (9) The effective stress-energy tensor T (eff) ik can be divided into four parts: (eff) ik = T ik + q1T ik + q2T ik + q3T (III) ik . (10) The first term T mn(a) − F (a)in F k , (11) 1 Hereafter we use the units c = G = ~= 1. is a stress-energy tensor of the pure Yang-Mills field. The definitions of other three tensors relate to the corresponding coupling constants q1, q2, q3: ik = RT mn(a) + D̂iD̂k − gikD̂lD̂l F (a)mnF mn(a) , (12) ik = − D̂mD̂l Fmn(a)F l (a)n −RlmFmn(a)F l (a)n − F ln(a) kn +RklF −RmnF (a)im F D̂mD̂m ln(a) + D̂k ln(a) , (13) (III) mnlsF (a)mnF F ls(a) i Rknls + F k Rinls D̂mD̂n k + F . (14) The tensor T (eff) ik satisfies the conservation law ∇kT (eff) ik = 0. The self-consistent system of equations (8) and (9) with (10)-(14) is a direct non-Abelian generalization of the three-parameter non-minimal Einstein-Maxwell model discussed in [16]. This system can also be considered as one of the variants of a non-minimal generalization of the Einstein-Yang-Mills model. III. EXACT SOLUTIONS OF THE STATIC MODEL WITH SPHERICAL SYMMETRY A. Master equations Let us take the metric of a static spherically symmetric spacetime in the form to be especially convenient for studying a wormhole geometry: ds2 = σ2Ndt2 − dr r2 + a2 dθ2 + sin2 θdϕ2 , (15) where the metric functions σ andN depend only on r. The properties of traversable wormholes dictate some additional requirements for the metric (15), which were in great detail discussed in [1, 2]. In particular, we note that (i) the radial coordinate r runs from −∞ to +∞. Two asymptotical regions r = −∞ and r = +∞ are connected by the wormhole’s throat which has the radius a and is located at r = 0. (ii) Since the spacetime of a traversable wormhole has neither singularities nor event horizons, the metric components gtt = σ 2N and −grr = 1/N should be regular and positive everywhere. Note that, in particular, this means that N(r) is positive defined, both σ(r) and N(r) are finite, and neither σ(r) nor N(r) can take zero values. (iii) In addition, one may demand the asymptotical flatness of the wormhole spacetime at r = ±∞. This is guaranteed provided the following boundary conditions for the functions σ and N are satisfied: σ2 (±∞) = 1 , N (±∞) = 1. (16) Below we will search for solutions of the non-minimal Einstein-Yang-Mills model, which satisfy the listed requirements. The non-minimal Yang-Mills equations (8) are satisfied identically, when the gauge field is parameterized as [22, 23] A0 = Ar = 0 , Aθ = itϕ , Aϕ = −iν sin θ tθ , (17) which is known to be the so-called Wu-Yang monopole solution [19]. The parameter ν is a non-vanishing integer, tr, tθ and tϕ are the position-dependent generators of the SU(2) group: tr = cos νϕ sin θ t(1) + sin νϕ sin θ t(2) + cos θ t(3), tθ = ∂θtr, tϕ = ν sin θ ∂ϕtr, (18) which satisfy the following commutation rules [tr, tθ] = i tϕ, [tθ, tϕ] = i tr, [tϕ, tr] = i tθ. (19) The field strength tensor Fik has only one non-vanishing component Fθϕ = iν sin θ tr . (20) Since the effective energy-momentum tensor T (eff) ik is divergence-free, the Einstein equations for the spherical symmetric metric (15) are known to be effectively reduced to the two key equations, say, for equations with i = k = 0 and i = k = r. The components G 00 and G r of the Einstein tensor G i = R i − 12δ i R are G 00 = 1− rN ′ −N r2 + a2 − N a (r2 + a2)2 , (21) G rr = 1− rN ′ −N r2 + a2 − 2rNσ σ(r2 + a2) (r2 + a2)2 . (22) The corresponding components of the effective energy-momentum tensor (see (11)-(14)) take the form T 00 = 2(r2 + a2)2 − 2q1Na (r2 + a2)4 − q1N ′r + q1 + q2 + q3 (r2 + a2)3 (13q1 + 4q2 + q3)Nr (r2 + a2)4 , (23) T rr = 2(r2 + a2)2 (r2 + a2)3 ′r + q1 + q2 + q3 + 2q1Nrσ − (7q1 + 4q2 + q3)Nr (r2 + a2)4 . (24) The difference G 00−G rr=8π T 00−T rr of these equations can be transformed into the decoupled equation for σ(r) only 1− κq1 (r2 + a2)2 (r2 + a2) (r2 + a2)3 (10q1 + 4q2 + q3)r 2 − q1a2 , (25) where κ is a new charge parameter with the dimensionality of area, κ = 8πν2/G2. Solving this equation one can find the function σ(r) in the explicit form for arbitrary values of parameters q1, q2 and q3: σ(r) = r2 + a2 1− κq1 (r2 + a2)2 ] 10q1+4q2+q3 . (26) Excluding the function σ from the Einstein equation with i = k = 0, we obtain the equation for N(r) only: (r2 + a2)2 N ′ +N (r2 + a2) κ(13q1 + 4q2 + q3) (r2 + a2)2 κa2(15q1 + 4q2 + q3) (r2 + a2)3 = 1− κ 2(r2 + a2) κ(q1 + q2 + q3) (r2 + a2)2 . (27) The solution of Eq. (27) can be clearly represented in quadratures. Note that the coefficient Θ(r) ≡ 1− κq1(r2 + a2)−2 in front of the first (highest) derivative in both differential equations (25) and (27) can take, in principle, zero values depending on the sign of the guiding parameter q1. Thus, searching for the solutions of these equations, we have to distinguish three qualitatively different cases q1 < 0, q1 = 0 and q1 > 0. B. The case q1 < 0 For negative q1 the coefficient Θ(r) has only one root, r = 0. In this case, the function σ(r) takes the form σ(r) = r2 + a2 κ|q1| (r2 + a2)2 ] 10|q1|−4q2−q3 4|q1| . (28) Notice that σ(r) given by Eq. (28) turns into zero at r = 0, i.e., σ(0) = 0. This violates the condition (ii), and so this solution cannot describe a traversable wormhole. Note also that −g(0) = 0, where g = −σ2(r2 + a2)2 sin2 θ is the determinant of gik. This means that the chosen coordinate system is ill-defined at r = 0, and well-defined only in the range (0,+∞) (or, equivalently, in (−∞, 0)). C. The case q1 = 0 In this case Θ(r) = r, and so r = 0 is the only root of Θ(r) as in the case q1 < 0. The solution for σ(r) transforms now into σ(r) = r2 + a2 −κ(4q2 + q3) (r2 + a2)2 . (29) The function σ(r) given by Eq. (29) turns into zero at r = 0, i.e., σ(0) = 0. This means that the case q1 = 0 does not admit the existence of traversable wormholes. D. The case q1 > 0 For positive q1 the number of real roots of Θ(r) depends on the value β ≡ (κq1)1/4. In case β > a, Θ(r) has three real roots, namely, r = 0, r = ±r∗, where β2 − a2 . (30) For β = a the roots r = ±r∗ coincide with r = 0, and for β < a one has only one real root r = 0. Below we consider each case separately. I. β < a. Rewrite the solution (26) as follows σ(r) = r2 + a2 (r2 + a2 − β2)(r2 + a2 + β2) (r2 + a2)2 ] 10q1+4q2+q3 . (31) It is clear that due to the condition β < a the expression in square brackets in Eq. (31) is positive for all r. Thus, for all values of the power parameter (10q1 + 4q2 + q3)/4q1 the sign of the function σ(r) inherits the sign of r, and turns into zero at r = 0, i.e., σ(0) = 0. As in previous cases, this means that traversable wormholes do not exist. II. β > a. In this case the expression in square brackets in Eq. (31) vanishes, when r = r∗ ≡ (β2 − a2)1/2. Now, depending on the sign of the power parameter (10q1 + 4q2 + q3)/4q1, the solution σ(r) turns into zero or tends to infinity at r∗. When 10q1+4q2+q3 = 0, one has again σ(0) = 0. Thus, the case β > a also does not admit traversable wormholes. III. β = a (or, equivalently, q1 = a 4/κ). It will be convenient to rewrite the solution (31) for σ(r) in the following form: σ(r) = r2 + a2 r2 + 2a2 r2(r2 + 2a2) (r2 + a2)2 ] 12q1+4q2+q3 . (32) Now the critical point of interest is r = 0. The behavior of σ(r) near r = 0 essentially depends on the sign of the new power parameter, namely, 12q1 + 4q2 + q3. In particular, for 12q1 + 4q2 + q3 > 0 one has σ(0) = 0, and for 12q1 + 4q2 + q3 < 0 one has σ(0) = ∞. Such behavior of σ(r) excludes traversable wormholes. Consider the last particular case, when this parameter vanishes, 12q1 + 4q2 + q3 = 0. Now we obtain σ(r) = r2 + a2 r2 + 2a2 . (33) The function σ(r) given by Eq. (33) is regular and positive in the whole interval (−∞,+∞), moreover, σ(±∞) = 1. Thus, σ(r) given by Eq. (33) satisfies the necessary conditions (i-iii) and the corresponding field configuration can be considered as a candidate in searching for traversable wormholes. In the next section we will complete the solution for σ(r) by the solution for N(r) and discuss the properties of the non-minimal Wu-Yang wormhole solution. IV. NON-MINIMAL WU-YANG WORMHOLE In this section we consider in more details the special case corresponding to the following choice of the non-minimal coupling parameters q1, q2, q3: q1 = a 4/κ, 12q1 + 4q2 + q3 = 0. (34) Then, the equation (27) can be easily integrated in the quadratures to give N(r) = (r2 + a2)3/2 r2 + 2a2 (x2 + a2)3/2 x2 + 2a2 x4 + 2x2 a2 − κ 10a4 + + 3κq2  , (35) where C is a constant of integration. Note that for arbitrary values of a, q2, and C the function N(r) given by Eq. (35) satisfies the boundary condition N(±∞) = 1. Near r = 0 the solution N(r) is, generally speaking, divergent. Such behavior of N(r) is unsuitable for description of traversable wormholes. However, there are special values of parameters q2 and C, namely: C = 0 , q2 = − , (36) for which the solution (35) transforms into N(r) = (r2 + a2)3/2 r2 + 2a2 J(r) , (37) where J(r) = (x2 + a2)3/2 x2 + 2a2 x2 + 2a2 − κ is a function of r and two guiding parameters, a and κ. Note that near r = 0 the function N(r), given by Eq. (37), behaves as N(r) ≃ (3a2)−1(a2 − κ/4) +O(r2) . (39) It is seen that N(r) can be positive, negative, or zero at r = 0 depending on the relation between two parameters: a (the wormhole throat radius) and κ (the charge parameter). It will be convenient further to use a dimensionless parameter α = aκ−1/2. The behavior of N(r) depending on α is illustrated in the Fig.1. Taking into account the relations gtt = σ 2N and −grr = 1/N and using the solutions (33) and (37) for σ(r) and N(r) we finally obtain the following metric, which presents the new exact solution of the non-minimally extended Einstein-Yang-Mills equations: ds2 = (r2 + a2)5/2 r3(r2 + 2a2)3/2 J(r)dt2 − r 3(r2 + 2a2)1/2 (r2 + a2)3/2 − (r2 + a2) (dθ2 + sin2 θdϕ2) . (40) This metric describes a regular (i.e., without singularities) spacetime containing two asymptotically flat regions r = ±∞ connected by a throat located at r = 0. Thus, the metric (40) describes a wormhole, which we will hereafter call as a non-minimal Wu-Yang wormhole. The spacetime structure of the Wu-Yang wormhole essentially depends on the value of the dimensionless parameter α = aκ−1/2. We note that for α > 1/2 the function N(r) is positive defined (see Fig. 1), and so the metric components gtt = σ 2N and −grr = 1/N are finite and positive in the whole region (−∞,+∞). This means that the spacetime has no event horizons, thus in this case the Wu-Yang wormhole is traversable. FIG. 1: Graphs of the function N(r) given for α ≡ aκ−1/2 > 1/2, α = 1/2, and α < 1/2 from up to down, respectively. In case α < 1/2 the function N(r) changes the sign. It is positive for |r| > rh, negative for |r| < rh, and zero at |r| = rh, i.e., N(±rh) = 0 (rh is some parameter, which can be easily found numerically for every α < 1/2). In the vicinity of |r| = rh one has gtt ∼ (r − rh) and grr ∼ (r − rh)−1. This means that the points |r| = rh are nothing but two event horizons of Schwarzschild-like type in the wormhole spacetime, and rh is the radius of horizons. In the accepted nomenclature, the regions |r| > rh with N(r) > 0 and |r| < rh with N(r) < 0 are R- and T-regions, respectively. Thus, in the case α < 1/2 the throat of Wu-Yang wormhole turns out to be hidden in the T-region behind the horizons. Such a wormhole is non-traversable from the point of view of a distant observer. By analogy with black holes one may call such objects as black wormholes. Note that for α = 1/2 two event horizons |r| = rh merge with each other and form an event horizon located at the wormhole’s throat r = 0. Now, in the vicinity of r = 0 one has gtt ∼ r2 and grr ∼ r−2, and this means that r = 0 is an extremal horizon. In this case the T-region is absent, and the event horizon divides two R-regions. Now let us discuss a formula for an asymptotic mass of the Wu-Yang wormhole measured by a distant observer. A mass of a static spherically symmetric configuration is defined as M = 1 limr→±∞ |r| (1− gtt(r)) . Using the metric (40) we can obtain after some algebra the following expression for the mass of the non-minimal Wu-Yang wormhole: ≡ M̃(α) = π , (41) where Γ(z) is gamma function, and α = aκ−1/2 and M̃ = Mκ−1/2 are dimensionless quantities. The graph of M̃(α) is given in Fig. 2. It is worth to note that M̃(α) is positive defined, M̃(α) > 0. Moreover, the function M̃(α) has a minimum M̃min ≈ 0.653 at α = αmin ≈ 0.545. 0.545 0.653 FIG. 2: Wormhole mass fM(α). The shaded region corresponds to α < 1/2. V. CONCLUSIONS In this paper we have considered the non-minimally extended Einstein-Yang-Mills model given by the action (1). The model contains three phenomenological parameters q1, q2 and q3, which determine the non-minimal coupling of the Yang-Mills and gravitational fields. In the framework of this model we have studied static spherically symmetric configurations with the Yang-Mills field possessing the SU(2) symmetry. Basing on the Wu-Yang ansatz for the gauge field we have obtained a three-parameter family of the explicit exact solutions to the non-linear Einstein-Yang-Mills equations. Only one solution from this family is regular and belongs to the class of wormhole spacetimes. We have denoted this solution as a non-minimal Wu-Yang wormhole (see Eq. (40)). Let us emphasize some of its properties. 1. The non-minimal Wu-Yang wormhole corresponds to the specific choice of coupling parameters q1, q2, q3, namely, , q2 = − , q3 = . (42) Thus, the Wu-Yang wormhole geometry turns out to be completely determined by two model parameters: the wormhole throat radius a, and the charge parameter κ = 8πν2/G2, or, equivalently, by a and the dimensionless parameter α ≡ aκ−1/2. Note that in the minimal limit, when q1 = q2 = q3 = 0, the relations (42) yield a = 0, i.e., this wormhole does not exist. In other words, the obtained exact solution is essentially non-minimal. 2. The parameter α can be treated as guiding one. Indeed, in case α > 1/2 the spacetime of Wu-Yang wormhole has no event horizons, and so it is traversable in principle. The condition α > 1/2 equivalently reads a > 1 κ1/2, that is the throat’s radius a of traversable Wu-Yang wormholes is necessary greater than 1 κ1/2. In case α < 1/2 (a < 1 κ1/2) the wormhole spacetime (40) possesses two Schwarzschild-type event horizons at |r| = rh, where rh is an event horizon radius given by the equation σ 2N(rh) = 0. The presence of event horizons means the Wu-Yang wormhole is non-traversable from the point of view of a distant observer. It is worth to note that in this case the wormhole throat located at r = 0 turns out to be hidden behind the horizons. For this reason one can call such objects as black wormholes. For the particular value α = 1/2 (a = 1 κ1/2) two event horizons merge with each other and form a single event horizon at the throat r = 0. Now in the vicinity of r = 0 the metric functions behave as gtt ∼ r2 and grr ∼ r−2, and so the metric (40) behaves near the horizon as the extreme Reissner-Nordström metric. 3. For a distant observer the Wu-Yang wormhole manifests itself through its asymptotical massM . It is determined by the charge parameter κ and expressed through the wormhole throat radius a (see Eq. (41) and Fig. 2). Is it possible for the observer to reconstruct the invisible throat radius using the estimated mass? In principle, yes, but the procedure is ambiguous, since two values of a correspond to one appropriate value of the mass. 4. The important feature is that there exists the lower limit for the mass of the non-minimal Wu-Yang wormhole. In other words, the wormhole mass cannot be less than some minimal value Mmin ≈ 0.653 κ1/2, i.e., M ≥ Mmin. To make estimations we assume that the monopole magnetic charge ν is equal to one, ν = 1, and the square of the constant of gauge interaction is given by G2 = 4παem, where αem = e2/~c ≈ 1/137 is the fine structure constant. Then, in the dimensional units we have Mmin ≈ 10.8Mpl, amin ≈ 9Lpl, where Mpl and Lpl are the Planck mass and the Planck length, respectively. Recently Kirill Bronnikov attracted our attention to the papers [24, 25], where the authors discuss solutions they refer to as regular black holes. He also emphasized that black wormholes obtained in our paper represent the kind of regular black holes. Acknowledgments This work was partially supported by the Deutsche Forschungsgemeinschaft through the project No. 436RUS113/487/0-5 and the Russian Foundation for Basic Research grant No. 05-02-17344. [1] M. Visser, Lorentzian Wormholes: from Einstein to Hawking (AIP Press, New York, 1995). [2] M. S. Morris and K. S. Thorne, Amer. J. Phys. 56, 395 (1988). [3] D. Hochberg and M. Visser, Phys. Rev. D 56, 4745 (1997) [arXiv: gr-qc/9704082]; D. Hochberg and M. Visser, Phys. Rev. D 58, 044021 (1998) [arXiv: gr-qc/9802046]. [4] H. G. Ellis, J. Math. Phys. 14, 104 (1973); K. A. Bronnikov, Acta Phys. Pol. B 4, 251 (1973); C. Barceló and M. Visser, Phys. Lett. B 466, 127 (1999) [arXiv: gr-qc/9908029]; C. Barceló and M. Visser, Class. Quantum Grav. 17, 3843 (2000) [arXiv: gr-qc/0003025]; S. V. Sushkov and S.-W. Kim, Class. Quantum Grav. 19, 4909 (2002) [arXiv: gr-qc/0208069]. [5] S. V. Sushkov, Phys. Lett. A 164, 33 (1992); D. Hochberg, A. Popov, and S. V. Sushkov, Phys. Rev. Lett. 78, 2050 (1997) [arXiv: gr-qc/9701064]. [6] K. K. Nandi, B. Bhattacharjee, S. M. K. Alam, and J. Evans, Phys. Rev. D 57, 823 (1998). [7] L. A. Anchordoqui and S. E. Perez Bergliaffa, Phys. Rev. D 62, 067502 (2000) [arXiv: gr-qc/0001019]; K. A. Bronnikov and S.-W. Kim, Phys. Rev. D 67, 064027 (2003) [arXiv: gr-qc/0212112]; M. La Camera, Phys. Lett. B 573, 27 (2003) [arXiv: gr-qc/0306017]; F. S. N. Lobo, [arXiv: gr-qc/0701133]. [8] S. V. Sushkov, Phys. Rev. D 71, 043520 (2005) [arXiv: gr-qc/0502084]; F. S. N. Lobo, Phys. Rev. D 71, 084011 (2005) [arXiv: gr-qc/0502099]. [9] F. S. N. Lobo, Phys. Rev. D 73, 064028 (2006) [arXiv: gr-qc/0511003]. [10] A. Das and S. Kar, Class. Quantum Grav. 22, 3045 (2005) [arXiv: gr-qc/0505124]. [11] For more details and references dated till 1995 the reader can see the book by Visser [1], and for a more up-to-date recearch in wormhole physics, consult [12] and the references therein. [12] J. P. S. Lemos, F. S. N. Lobo, and S. Q. de Oliveira, Phys. Rev. D 68, 064004 (2003) [arXiv: gr-qc/0302049]. [13] A. V. B. Arellano and F. S. N. Lobo, Class. Quantum Grav. 23, 5811 (2006) [arXiv: gr-qc/0608003]; A. V. B. Arellano and F. S. N. Lobo, Class. Quantum Grav. 23, 7229 (2006) [arXiv: gr-qc/0604095]. [14] V. Faraoni, E. Gunzig, and P. Nardone, Fund. Cosmic Phys. 20, 121 (1999) [arXiv: gr-qc/9811047]. [15] F. W. Hehl and Yu. N. Obukhov, Lect. Notes Phys. 562, 479 (2001) [arXiv: gr-qc/0001010]. [16] A. B. Balakin and J. P. S. Lemos, Class. Quantum Grav. 22, 1867 (2005) [arXiv: gr-qc/0503076]. [17] F. Müller-Hoissen, Class. Quantum Grav. 5, L35 (1988). [18] I. T. Drummond and S. J. Hathrell, Phys. Rev. D 22, 343 (1980). [19] T. T. Wu and C. N. Yang, in Properties of Matter Under Unusual Conditions, edited by H. Mark and S. Fernbach (Interscience, New York, 1969), p. 349. [20] V. Rubakov, Classical Theory of Gauge Fields (Princeton University Press, Princeton and Oxford, 2002). [21] A. C. Eringen and G. A. Maugin, Electrodynamics of continua (Springer-Verlag, New York, 1989). [22] C. Rebbi and P. Rossi, Phys. Rev. D 22, 2010 (1980). [23] A. B. Balakin and A. E. Zayats, Phys. Lett. B 644, 294 (2007) [arXiv: gr-qc/0612019]. [24] K. A. Bronnikov, V. N. Melnikov, and H. Dehnen, Phys. Rev. D 68, 024025 (2003). [25] K. A. Bronnikov, H. Dehnen, and V. N. Melnikov, arXiv: gr-qc/0611022. http://arxiv.org/abs/gr-qc/0611022 Introduction Non-minimal Einstein-Yang-Mills model Exact solutions of the static model with spherical symmetry Master equations The case q1<0 The case q1=0 The case q1>0 Non-minimal Wu-Yang wormhole Conclusions Acknowledgments References

0704.1225

Patterns of dominant flows in the world trade web

Patterns of dominant flows in the world trade web M. Ángeles Serrano,1 Marián Boguñá,2 and Alessandro Vespignani3, 4 1Institute of Theoretical Physics, LBS, FSB, EPFL, BSP 725 - Unil, 1015 Lausanne, Switzerland 2Departament de F́ısica Fonamental, Universitat de Barcelona, Mart́ı i Franquès 1, 08028 Barcelona, Spain 3School of Informatics, Indiana University, Eigenmann Hall, 1900 East Tenth Street, Bloomington, IN 47406, USA 4Complex Network Lagrange Laboratory (CNLL), Institute for Scientific Interchange (ISI), Torino 10133, Italy (Dated: today) The large-scale organization of the world economies is exhibiting increasingly levels of local het- erogeneity and global interdependency. Understanding the relation between local and global features calls for analytical tools able to uncover the global emerging organization of the international trade network. Here we analyze the world network of bilateral trade imbalances and characterize its over- all flux organization, unraveling local and global high-flux pathways that define the backbone of the trade system. We develop a general procedure capable to progressively filter out in a consistent and quantitative way the dominant trade channels. This procedure is completely general and can be applied to any weighted network to detect the underlying structure of transport flows. The trade fluxes properties of the world trade web determines a ranking of trade partnerships that highlights global interdependencies, providing information not accessible by simple local analysis. The present work provides new quantitative tools for a dynamical approach to the propagation of economic crises. I. INTRODUCTION The term “globalization”, when applied to the inter- national economic order, refers to the presence of an in- tricate network of economic partnership among an in- creasing number of countries [1]. In this context, the International trade system, describing the fundamental exchange of goods and services, plays a central role as one of the most important interaction channels between states [2]. For instance, it broadly defines the substrate for the spreading of major economic crises [3, 4, 5], such as the 1997 Asiatic crisis [5, 6, 7] which shows how eco- nomic perturbations originated in a single country can somehow propagate globally in the world. Moreover, commercial trade flows are indeed highly correlated with other types of cross-country economic interactions (flows of services, financial assets, workers) and so stand as a good indicator for more general economic relations [8]. The International trade system as an independent extract of the world economy is therefore still a partial view of the whole system; a complete description would consider the feedback mechanisms that operate between international trade imbalances and other economic variables such as investment, debt, or currency prices. On the other hand, the study of the International trade network in a system’s perspective represents a necessary first step before pro- ceeding with a subsequent more integrative investigation and has proven to be successful in providing insight into some of its global properties. The large size and the entangled connectivity pattern characterizing the international trade organization point out to a complex system whose properties depend on its global structure. In this perspective, it appears natu- ral to analyze the world trade system at a global level, every country being important regardless of its size or wealth and fully considering all the trade relationships. A convenient framework for the analysis of complex in- terconnected systems is network analysis [9, 10]. Within this approach, countries are represented as nodes and trade relationships among them as links. Such visual- izations of bilateral trade relations have been used in recent years to help analyze gravity models [11, 12], of- ten proposed to account for the world trade patterns and their evolution [13]. While the first attempts to study the trade system as a complex network have successfully re- vealed a hierarchical organization [14, 15, 16], these stud- ies focused on topological aspects neglecting fundamental components, such as the heterogeneity in the magnitude of the different bilateral trade relations and their asym- metry. These are essential issues in the understanding of the interplay between the underlying structure and the principles that rule the functional organization of the sys- Here we tackle the quantitative study of the world trade network by implementing the trade flux analysis at a global scale. To this end, we construct the weighted directed network of merchandize trade imbalances be- tween world countries. In this representation, each coun- try appears as a node and a directed link is drawn among any pair whenever a bilateral trade imbalance exists, i.e., whenever bilateral imports does not balance exports. The direction of the arrow follows that of the net flow of money and it is weighted according to the magnitude of the imbalance between the two countries. More precisely, we define the elements Eij that measure the exports of country i to country j and the elements Iij that measure the imports of country i from country j. The trade im- balance matrix is therefore defined as Tij = Eij−Iij and Pajek A B A B A B A B EAB = IBA IAB = EBA TBA = - TAB < 0 FBA EAB = IBA IAB = EBA TAB = - TBA < 0 FAB Totally inhomogeneous Fit kY(k) = k FIG. 1: Measuring local inhomogeneity in fluxes. a, concep- tual representation of the link construction process. b and c, local inhomogeneity for incoming (b) and outgoing (c) con- nections measured by kY (k) as compared to the null model. The diagonal line corresponds to the maximum possible in- homogeneity, with only one connection carrying all the flux. The line kY (k) = 1 is the maximum homogeneity, with all the fluxes equally distributed among the connections. The area depicted in grey corresponds to the average of kY (k) un- der the null model plus two standard deviations. The solid lines are the best fit estimates which give kinY (kin) ∼ k0.6in and koutY (kout) ∼ k0.5out. The inset in (c) sketches a pathway through a country arising from strong local inhomogeneity in incoming and outgoing connections. measures the net money flow from country j to country i due to trade exchanges. Since Eij = Iji and Iij = Eji, T is an antisymmetric matrix with Tij = −Tji, and a di- rected network can be easily constructed by assuming a directed edge pointing to the country with positive bal- ance. The network of the net trade flows is therefore defined in terms of a weighted adjacency matrix F with Fij =| Tij |=| Tji | for all i, j with Tij < 0, and Fij = 0 for all i, j with Tij ≥ 0 (see Fig. 1a for a pictorial de- scription). By using the above procedure we constructed the net- work of trade imbalances by using the data set which reports the annual merchandize trade activity between independent states in the world during the period 1948- 2000, together with the annual values of their Gross Do- mestic Product per capita and population figures (1950– 2000) [17],[25]. The time span of the data set allows us to study the change of trade flow networks with yearly snapshots characterizing the time evolution of the trade system. The most basic topological characterization of each country within the network is given by the number of incoming and outgoing links, kin and kout respectively, which inform us about the number of neighboring coun- tries that contribute positively and negatively to the net trade imbalance of the country under consideration. A precise assessment of the country trade balance cannot however neglect the magnitude of the fluxes carried by each trade relation. This information can be retrieved summing up all the weights of the incoming or outgo- ing links, which give us the total flux of money due to trade entering to or leaving from the country of inter- est. In the network literature, these two variables are called incoming and outgoing strength and are denoted by sinj = i Fij and s i Fji, respectively [18]. The total trade imbalance of a country can then be com- puted as ∆sj = sinj − s j . Depending on ∆sj , countries can be then defined as net consumers and net produc- ers. Net producers export more than they import, the total outcome being a trade surplus which corresponds to ∆sj > 0, whereas net consumers export less than they import, the total outcome being a trade deficit which is indicated by ∆sj < 0. Since one incoming link for a given country is always an outgoing link for another, the sum of all the countries’ trade imbalances in the network must be zero. While the local balance is not conserved, we are therefore dealing with a closed system which is globally balanced (the total flux is conserved). Merchandizes, or equivalently money, flows in the system from country to country with the peculiarity that there is a global flow of money from consumer countries to producer ones. II. LOCAL HETEROGENEITY AND BACKBONE EXTRACTION The obtained networks show a high density of connec- tions and heterogeneity of the respective fluxes among countries. Indeed, as the number of countries increases, so does the average number of trade partners, as well as the total flux of the system, which is seen to grow pro- portional to the aggregated world Gross Domestic Prod- uct [19]. The overall flux organization at the global scale can be characterized by the study of the flux distribution. A first indicator of the system heterogeneity is provided by the probability distribution P (Fij) denoting the prob- ability that any given link is carrying a flux Fij . The observed distribution is heavy-tailed and spans approx- imately four orders of magnitude [19]. Such a feature implies that only a small percentage of all the connec- tions in the network carry most of its total flow F and that there is no characteristic flux in the system, with most of the fluxes below the average and some of them with a much higher value. This is however not totally TABLE I: Sizes of the backbones. Percentage of the original total weight F , number of nodes N and links E in the 1960 and 2000 imbalance networks that remain in the backbone as a function of the significance level α. 1960 2000 α %F %N %E %F %N %E 0.2 88 100 25 92 98 25 0.1 83 100 19 87 98 19 0.05 79 99 15 84 97 15 0.01 69 92 9 75 96 10 unexpected since a large scale heterogeneity is a typical feature of large-scale networks. In addition, the global heterogeneity could just be due to differences in the sizes of the countries, in their population and in their respec- tive Gross Domestic Product. More interesting is there- fore the characterization of the local heterogeneity; i.e. given all the connections associated to each given coun- try, how is the flux distribution for each of them. A local heterogeneity implies that only a few links carry the biggest proportion of the country’s total in-flow or out-flow. Interestingly, such a heterogeneity would de- fine specific pathways within the network that accumu- late most of the total flux. In order to asses the effect of inhomogeneities at the local level, for each country i with k incoming or outgoing trade partners we calcu- late [20, 21] kYi(k) = k p2ij , (1) where k can be either kin or kout in order to discern be- tween inhomogeneities in incoming and outgoing fluxes, and where the normalized fluxes of node i with its neigh- bors are calculated as pij = Fji/sini for incoming con- nections and as pij = Fij/souti for the outgoing ones. The function Yi(k) is extensively used in economics as a standard indicator of market concentration, referred as the Herfindahl-Hirschman Index or HHI [22, 23], and it was also introduced in the complex networks literature as the disparity measure [24]. In all cases, Yi(k) char- acterizes the level of local heterogeneity. If all fluxes emanating from or arriving to a certain country are of the same magnitude, kYi(k) scales as 1 independently of k, whereas this quantity depends linearly on k if the local flux is heterogeneously organized with a few main directions. Increasing deviations from the constant be- havior are therefore indicating heterogeneous situations in which fluxes leaving or entering each country are pro- gressively peaked on a small number of links with the remaining connections carrying just a small fraction of the total trade flow. On the other hand, the deviations from the constant behavior have to be expected for low values of k and it is important to compare the obtained results with the deviations simply produced by statistical fluctuations. To this end, we introduce a null model for the distribution of flows among a given number of neigh- bors in order to assess, in a case by case basis, whether the observed inhomogeneity can just be due to fluctua- tions or it is really significant. The null model with the maximum random homogene- ity corresponds to the process of throwing k − 1 points in a [0, 1] interval, so that the interval ends up divided in k sections of different lengths representing the different values assigned to the k variables pij in the random case. It can be analytically proved that the probability that one of these variables takes a particular value x depends on k and is Prob{x < pij < x+ dx} = (k − 1)(1− x)k−2dx. (2) This probability density function can be used to calculate the statistics of kYNM (k) for the null model. Both the average and the standard deviation are found to depend on k: 〈kYNM (k)〉 = k〈YNM (k)〉 = k + 1 σ2 (kYNM (k)) = k 20 + 4k (k + 1)(k + 2)(k + 3) (k + 1)2 so that each node in the network with a certain in or out degree should be compared to the corresponding null model depending on the appropriate k. In Fig. 1, we show the empirical measures along with the region defined by the average value of the same quan- tity kY (k) plus two standard deviations as given by the null model (shadowed area in grey). For a homogeneously random assignment of weights, this quantity converges to a constant value for large k, which is clearly different from the observed empirical behavior. Most empirical values lie out of the null model domain, which proves that the observed heterogeneity is due to a well definite ordering principle and not to random fluctuations. The direct fit of the data indicates that both in and out fluxes follow the scaling law kYi(k) ∝ kβ with βin = 0.6 for the incoming connections and βout = 0.5 for the out- going ones (see Fig. 1). This scaling represents and in- termediate behavior between the two extreme cases of perfect homogeneity or heterogeneity but clearly points out the existence of strong local inhomogeneities. The emerging picture is therefore consistent with the exis- tence of major pathways of trade flux imbalances (thus money) that enters the country using its major incom- ing links and leaves it through its most inhomogeneous outgoing trade channels (see inset in Fig. 1c). The analysis of the local inhomogeneities in the trade fluxes prompts to the presence of high-flux backbones, sparse subnetworks of connected trade fluxes carrying most of the total flux in the network. This backbone is necessarily encoding a wealth of information being the dominating structure of the trade system. It is also worth remarking that the local heterogeneity is not just encoded Canada Dominican R Mexico El Salvador Nicaragua Costa Rica Panama Venezuela Ecuador Brazil Bolivia Argentina Uruguay United Kingdom Netherlands Belgium Luxemburg France Switzerland Spain German FR German DR Poland Austria Hungary Czechoslovakia Italy Yugoslavia Cyprus Bulgaria Russia Finland SwedenNorwayDenmark Ghana Zaire Ethiopia Malagasy R Morocco Tunisia Turkey Egypt Israel Afghanistan China R of China South Korea Japan India Pakistan Burma R of Vietnam Indonesia Australia New Zealand Canada Haiti Jamaica Trinidad and Tobago Barbados Dominica Grenada St Lucia St Vincent and G Antigua and Barbuda St Kitts-Nevis Mexico Belize Guatemala Honduras Nicaragua Panama Colombia Venezuela Guyana Ecuador Brazil ChileArgentina Uruguay United Kingdom Ireland Netherlands Belgium LuxemburgFrance Switzerland Spain Portugal Germany Poland Austria Hungary Czechoslovakia Slovakia Italy Malta Croatia Yugoslavia Bosnia- Herzegovina Slovenia Greece Cyprus Bulgaria Rumania Russia Estonia Latvia Ukraine Belarus Georgia Finland Sweden Norway Denmark Iceland Cape Verde Guinea-Bissau Equatorial Guinea Gambia Senegal Benin Niger Ivory Coast Liberia Sierra Leone Ghana CamerounNigeria Gabon Central African R Congo Zaire Uganda Kenya Mozambique Zambia Zimbabwe Malawi South Africa Lesotho Botswana Malagasy R Comoros Mauritius Seychelles Morocco Algeria Tunisia TurkeyEgypt Jordan Israel Saudi Arabia Yemen Bahrain Qatar U Arab Emirates China R of China North Korea South Korea Japan India Pakistan Bangladesh Sri Lanka Nepal Thailand Cambodia Laos Malaysia Singapore Philippines Indonesia Australia Papua New Guinea New Zealand Vanuatu Lithuania FIG. 2: Backbone of the world trade system. Snapshots of the α = 10−3 backbone of the world trade imbalance web for the years 1960 and 2000. Notice that the most central economies are depicted at fixed positions to make both graphs more easily comparable. in high flux links in terms of their absolute intensities, but also takes into account the local heterogeneity by com- paring the strength of the fluxes associated to a given country with its total strength. It is then interesting to filter out this special links and provide snapshots of the trade system backbone. This can be achieved by com- paring the link fluxes with the null model used for the calculation of the disparity in a pure random case. The same approach allows us the calculation for each connec- tion of a country i of the probability αij that its normal- ized flux value pij is due to chance. Along these lines, we can identify highly inhomogeneous fluxes as those which satisfy αij = 1− (k − 1) ∫ pij (1− x)k−2dx < α, (5) where α is a fixed significance level. Notice that this ex- pression depends on the number of connections of each country, k. By choosing a global threshold for all coun- tries we obtain a homogeneous criteria that allows us to compare inhomogeneities in countries with different number of connections and filter out links that carry fluxes which can be considered not compatible with a random distribution with an increasing statistical confi- dence. The backbone is then obtained by preserving all the links which beat the threshold for at least one of the two countries at the ends of the link while discounting the rest. By changing the significance level we can fil- ter out the links focusing on progressively more relevant heterogeneities and backbones. An important aspect of this new filtering algorithm is that it does not belittle small countries and then, it offers a systematic procedure to reduce the number of connec- tions without diminishing the number of countries and choosing the backbone according to the amount of trade flow we intend to characterize. It provides a quantitative and consistent way to progressively identify the relevant flow backbone once the level of statistical confidence with respect to the null case is fixed, or instead the total flow we want to represent in the system. Indeed, it is re- markable that when looking at the network of the year 2000 one finds that the α = 0.05 backbone contains only 15% of the original links yet accounting for 84% of the total trade imbalance. Most of the backbones form a giant connected component containing most of the coun- tries in the network, and only for very high values of the confidence level, defining a sort of super-backbones, some disconnected components appear and the number of countries starts to drop. In this respect, the α = 0.01 backbone seems to offer the best trade-off since it keeps nearly all countries, 75% of the total trade imbalances, and one order of magnitude less connections than in the original network (see Table 1). The backbone reduction is extremely effective in sort- ing out the most relevant part of the network and can be conveniently used for visualization purposes. For the sake of space and reproduction clarity, we report the backbones corresponding to α = 10−3, still accounting for approximately 50% of the total flux of the system. Fig. 2 shows two snapshots of such backbones for 1960 and 2000. These high-flux backbones evidence geographi- cal, political and historical relationships among countries which affect the observed trade patterns. For instance, the trade of US with its geographically closer neighbors and also continental neighbors, the case of Russia and the former Soviet republics, or France and its former colonies, the lack of strong trade relations between the two blocks in the cold war, more evident in 1960. In general terms, a recurrent motif present in all years is the star-like struc- ture, formed by a central powerful economy surrounded by small dependent economies. The USA appears as one of those powerful hubs during all this period. However, other countries has gradually lost this role in favor of others. This is the case of the UK, which was the only star-like counterpart of the USA in 1948; since then its direct area of influence has been narrowing. On the con- trary, other countries have arisen for different reasons as new hub economies. This is the case of some European countries, Japan, and most recently, China. III. DIFFUSION ON COMPLEX NETWORKS AND THE DOLLAR EXPERIMENT The picture emerging from our analysis has intriguing similarities with other directed flow networks, such as metabolic networks [21], that transport information, en- ergy or matter. Indeed, the trade imbalances network can be seen as a directed flow network that transport money. In other words, we can imagine that net consumer coun- tries are injecting money in the system. Money flows along the edges of the network to finally reach producer countries. Producer countries, however, do not absorb completely the incoming flux, redistributing part of it through the outgoing links. The network is therefore characterizing a complex dynamical process in which the net balance of incoming and outgoing money is the out- come of a global diffusion process. The realization of such a non-local dynamics in the flow of money due to the trade imbalances spurs the issue of what impact this feature might have on the effect that one economy can have on another. In order to tackle this issue we per- form a simple numerical study, defined as the “dollar ex- periment”. The “experiment” considers running on the networks two symmetric random walk processes. Since we are limited by the yearly frequency of the empirical data, we assume at first approximation that the time scale of the changes in the structure of the underlying trade imbalances network is bigger than the characteris- tic diffusion time of the random walk processes. In the first case we imagine that a consumer country (∆s < 0) is injecting one dollar from its net debit into the system. The dollar at this point travels through the network fol- lowing fluxes chosen with a probability proportional to their intensity, and has as well a certain probability of being trapped in producer countries (∆s > 0) with a probability Pabs = ∆ssin . More precisely, if we consider a consumer country, such as the USA, the traveling dollar goes from country to country always following outgoing fluxes chosen with a probability proportional to their in- tensity. If in its way it finds another source it just crosses it, whereas if it finds a producer country j it has a prob- ability Pabs(j) of being absorbed. Mathematically, this process is a random walk on a directed network with heterogeneous diffusion probability and in the presence of sinks. By repeating this process many times it is possi- ble to obtain the probability eij that the traveling dollar originated in the source i is finally absorbed in the sink j. In other words, for each dollar that a source country i adds to the system, eij represents the fraction of that dol- lar that is retained in country j. The symmetric process considers that each producer country is receiving a dol- lar and the traveler dollar goes from country to country always following incoming links backward chosen with a probability proportional to their intensity. If in its way it finds another sink it just crosses it, whereas if it finds a source j it has a probability Pabs(j) = of remain- ing in that country. The iteration of this process gives the probability gij that yields the fraction originated in the source country j of each dollar that a sink country retains. Consequently, these two quantities are related by the detailed balance condition |∆si|eij = ∆sjgji. (6) The matrices eij and gji are normalized probability dis- tributions and, therefore, they satisfy that j;sink eij = 1 and i;source gji = 1. Using this property in the de- TABLE II: Rankings from the Dollar experiment. Top: effect of two major source countries, USA and Switzerland, on the rest of the world. The first list is a top ten ranking of countries according to eij , where the index i stands for the analyzed source. The second list is the top ten ranking of direct bilateral trade measured as the percentage of flux from the source country, that is, elocalij = Fij/s i . Bottom: major contributors to two major sink countries, Japan and Russia. The first list is a top ten ranking of countries according to gij , i standing for the analyzed sink. The second list is the top ten ranking due to direct trade. In this case, the direct contribution is glocalij = Fji/s i . Countries in boldface have no direct connection with the analyzed country. The values for eij and gij are obtained from the simulation of the dollar experiment described in the text using 106 different realizations for each country, for the year 2000. Net Consumers - Sources USA Switzerland Dollar experiment Bilateral trade Dollar experiment Bilateral trade Japan 19.5% Japan 17.2% France 27.3% France 75.0% Canada 9.9% China 16.7% Germany 10.0% Germany 9.5% China 9.3% Canada 15.6% Russia 9.7% Russia 4.1% Saudi Arabia 6.1% Mexico 5.1% Japan 8.5% Netherlands 2.6% Russia 5.4% Germany 4.8% Ireland 6.9% Ireland 2.3% Germany 4.5% R of China 3.1% Norway 6.0% Belgium 1.7% Indonesia 4.3% Italy 3.0% Saudi Arabia 4.2% Italy 1.2% Malaysia 3.9% Venezuela 2.8% China 3.4% Austria 1.1% Ireland 2.7% South Korea 2.4% Indonesia 2.3% Libya 0.4% South Korea 2.7% Malaysia 2.4% Malaysia 1.9% Nigeria 0.4% Net Producers - Sinks Japan Russia Dollar experiment Bilateral trade Dollar experiment Bilateral trade USA 62.6% USA 40.2% USA 33.3% Germany 9.0% UK 7.3% R of China 9.3% UK 7.2% Italy 8.1% Spain 3.8% Singapore 7.0% Switzerland 7.1% USA 7.7% Switzerland 3.3% South Korea 5.6% Poland 7.0% China 5.9% Singapore 2.4% Germany 5.1% Turkey 6.9% Poland 5.4% Turkey 2.1% UK 4.8% Spain 5.1% Japan 4.4% Panama 2.1% Netherlands 4.8% Greece 3.5% Turkey 4.3% Greece 1.9% China 3.9% Egypt 2.2% Switzerland 4.0% Portugal 1.5% Mexico 2.1% Lithuania 2.0% Netherlands 4.0% Egypt 1.5% Thailand 2.1% Portugal 1.9% UK 3.6% tailed balance condition, we can write ∆sj = i:source eij |∆si| and |∆si| = j:sink gji∆sj . Then, the total trade imbalance of a sink or source coun- try can be written as a linear combination of the trade imbalances of the rest of the source or sink countries, respectively. Therefore, by measuring eij , it is possible to discriminate the effect that one economy has on an- other or, with gij , to find out which consumer country is contributing the most to a producer one, in both cases taking into account the whole topology of the network and the inhomogeneities of the fluxes. The advantage of this approach lies on its simplicity and the lack of tun- able parameters. Indeed, all the information is contained in the network itself, without assuming any kind of mod- eling on the influences among countries. By using this experiment it is possible to evaluate for a consumer country where the money spent is finally go- ing. For each dollar spent we know which percentage is going to any other producer country and we can rank those accordingly. It is important to remark that in this case countries might not be directly connected since the money flows along all possible paths, sometimes through intermediate countries. This kind of ranking is there- fore different from the customarily considered list of the first neighbors ranked by magnitude of fluxes. The anal- ysis indeed shows unexpected results and, as it has been already pointed out in other works [5] applying other methodologies, a country can have a large impact on other countries despite being a minor or undirect trad- ing partner, see Table 2. Similarly, producer countries may have a share of the expenditure of non directly con- nected countries resulting in a very different ranking of their creditors. As an example, for each net dollar that the USA inject into the system, only 9.3% is retained in China although the direct connection imbalance between these countries is 16.7%. Very interestingly, we find that Switzerland spend a large share of his trade imbalance in countries which do not have appreciable trade with it and are therefore not directly connected such as Japan, Indonesia, and Malaysia. The Swiss dollars go to these countries after a long path of trade exchanges mediated by other countries. By focusing on producer countries we find other striking evidence. While the first importer from Russia by looking locally at the ranking of trade im- balances is Germany, the global analysis shows that one third of all the money Russia gains from trade is coming directly or undirectly from the USA. In Table 2, we re- port other interesting anomalies detected by the global analysis. IV. CONCLUSIONS In summary, we have introduced a novel quantitative approach applicable to any dense weighted complex net- work which filters out the dominant backbones while pre- serving most of the nodes in the original connected com- ponent. We have also discussed the behavior of a coupled dynamical process, the dollar experiment, which unveils the global properties of economic and trade partnerships. In a globalized economy, we face ever increasing problems in disentangling the complex set of relations and causality that might lead to crisis or increased stability. Focusing on just the bilateral relations among country economies is a reductionist approach that cannot work in a highly interconnected complex systems. We have proposed the use of the trade network representation and mathemati- cal tools that allow to uncover some basic ordering emerg- ing from the global behavior and the inclusion of non- local effects in the analysis of trade interdependencies. Future work on this grounds might help in the assess- ment of world trade relations and the understanding of the global dynamics underlying major economic crises. Acknowledgments We thank F. Vega-Redondo for useful comments. M. B. acknowledges financial support by DGES grant No. FIS2004-05923-CO2-02 and Generalitat de Catalunya grant No. SGR00889. A.V. is partially supported by the NSF award IIS-0513650. [1] M. A. Centeno, A. Cooke, and S. R. Curran, NetMap Combined Studies, Mapping Globalization Project (Princeton University and University of Wash- ington, 2006). [2] P. R. Krugman, Brookings Papers on Economic Activity 1995, 327 (1995). [3] R. Glick and A. Rose, J. of Intl. Money and Finance 18, 603 (1999). [4] K. Forbes, in In Sebastian Edwards and Jeffrey Frankel (eds.), Preventing Currency Crises in Emerging Markets (University of Chicago Press, Chicago, 2002), pp. 77–124. [5] T. Abeysinghe and K. Forbes, Rev. of Intl. Econ. 13, 356 (2005). [6] M. Goldstein, The Asian Financial Crisis: Causes, Cures and Systemic Implications (International Institute for Economics, Washington, D. C., 1998). [7] M. Noland, L.-G. Liu, S. Robinson, and Z. Wang, Global Economics Effects of the Asian Currency Devalu- ations (International Institute for Economics, Washing- ton, D. C., 1998). [8] P. R. Krugman and M. Obstfeld, International Eco- nomics: Theory and Policy, Seventh Edition (Addison- Wesley, Lebanon, Indiana, U.S.A., 2005). [9] R. Albert and A.-L. Barabási, Reviews of Modern Physics 74, 47 74, 47 (2002). [10] S. N. Dorogovtsev and J. F. F. Mendes, Evolution of networks: From biological nets to the Internet and WWW (Oxford University Press, Oxford, 2003). [11] L. Krempel and T. Plümper, J. of World-Systems Re- search 5, 487 (1999). [12] L. Krempel and T. Plümper, J. of Social Structures 4, 1 (2003). [13] J. H. Bergstrand, The Review of Economics and Statis- tics 67, 474 (1985). [14] M. A. Serrano and M. Boguñá, Phys. Rev. E 68, 015101(R) (2003). [15] D. Garlaschelli and M. I. Loffredo, Phys. Rev. Lett. 93, 188701 (2004). [16] D. Garlaschelli and M. I. Loffredo, Physica A 355, 138144 (2005). [17] K. S. Gleditsch, J. Conflict Resolut. 46, 712724 (2002). [18] A. Barrat, M. Barthélemy, R. Pastor-Satorras, and A. Vespignani, Proc. Natl. Acad. Sci. USA 101, 3747 (2004). [19] M. A. Serrano, J. Stat. Mech. p. L01002 (2007). [20] M. Barthélemy, B. Gondran, and E. Guichard, Physica A 319, 633 (2003). [21] E. Almaas, B. Kovács, T. Vicsek, Z. N. Oltvai, and A.-L. Barabási, Nature 427, 839 (2004). [22] O. C. Herfindahl, Copper Costs and Prices: 1870-1957 (John Hopkins University Press, Baltimore, MD, USA, 1959). [23] A. O. Hirschman, American Economic Review 54, 761 (1964). [24] B. Derrida and H. Flyvbjerg, J. Phys. A 20, 5273 (1987). [25] (version (4.1), http://weber.ucsd.edu/∼kgledits/exptradegdp.html) The following issues should be considered: i) This expanded trade database includes additional estimates for missing values. ii) The definition of state in the international system is as defined by the Correlates of War Project (http://www.correlatesofwar.org/). iii) The http://weber.ucsd.edu/~kgledits/exptradegdp.html http://www.correlatesofwar.org/ figures for trade flows are in millions of current-year US dollars. iv) The import/export values correspond to exchanges of merchandizes. Introduction Local heterogeneity and backbone extraction Diffusion on complex networks and the Dollar experiment Conclusions Acknowledgments References

0704.1226

Hydrogen 2p--2s transition: signals from the epochs of recombination and reionization

Hydrogen 2p–2s transition: signals from the epochs of recombination and reionization Shiv. K. Sethi, Ravi Subrahmanyan, D. Anish Roshi Raman Research Institute , Sadashivanagar, Bangalore 560080, India email: [email protected], [email protected], [email protected] ABSTRACT We propose a method to study the epoch of reionization based on the possible observation of 2p–2s fine structure lines from the neutral hydrogen outside the cosmo- logical H II regions enveloping QSOs and other ionizing sources in the reionization era. We show that for parameters typical of luminous sources observed at z ≃ 6.3 the strength of this signal, which is proportional to the H I fraction, has a brightness temperature ≃ 20µK for a fully neutral medium. The fine structure line from this redshift is observable at ν ≃ 1GHz and we discuss prospects for the detection with several operational and future radio telescopes. We also compute the characteristics of this signal from the epoch of recombination: the peak brightness is expected to be ≃ 100µK; this signal appears in the frequency range 5-10 MHz. The signal from the recombination era is nearly impossible to detect owing to the extreme brightness of the Galactic emission at these frequencies. Subject headings: cosmic microwave background—radio lines:general—line:formation— radiative transfer 1. Introduction Even though the existence of Hydrogen fine structure lines and their explanation using Dirac’s atomic theory has been known for close to a century, a Hydrogen fine structure line has never been detected from an astrophysical object. An interesting Hydrogen fine structure line is the 2p–2s transition. The main difficulty in detecting this line is that the line strength is proportional to the population of either the 2p or 2s states which, being excited states, are not so readily populated in most astrophysical circumstances. Moreover, the line width of the excited 2p state, which is determined by its decay time, is large (99.8 MHz), making the detection of the fine structure line a difficult observation. One astrophysical setting where the feasibility of detecting such a line has been studied is the interstellar H II regions (see, e.g., Dennison, Turner, & Minter 2005 and refer- ences therein; Ershov 1987); in H II regions, the excited levels are populated by recombination. http://arxiv.org/abs/0704.1226v1 – 2 – Here we consider two cosmological settings in which the excited levels are populated by either recombination or pumping by Lyman-α photons from an external source: (a) The Recombination epoch: The Universe makes a transition from a fully ionized to an almost fully neutral medium at z ≃ 1089 (Spergel et al. 2006; for details see, e.g., Peebles 1993 and references therein). During this era, as the density and temperature of the Universe drops, recombination is stalled owing to a high Lyman-α radiation density and progresses either by the depopulation of the 2p state owing to redshifting of the photons out of the line width or the 2-photon decay of the 2s state. This results in a significant 2s and 2p level population during the recombination era. (b) The Reionization epoch: Recent observations suggest that the universe made a transition from nearly fully neutral to fully ionized within the redshift range 6 <∼ z ∼ 15 (Page et al. 2006; White et al. 2003; Fan et al. 2002; Djorgovski et al. 2001; Becker et al. 2001). It is widely believed that this ‘reionization’ was achieved by the percolation of individual H II regions around the sources of reionization. The nature of these sources is not well understood: they might be Pop III stars, active galactic nuclei or star-forming galaxies. During this epoch, a signal from the 2p − 2s fine-structure transition might originate from either within the cosmological H II regions or from the almost fully neutral medium surrounding the H II region. The level population of the first excited state in the former case would be largely determined by recombinations and in the latter case by Lyman-α photons from the central source. We shall show below that for most cases of interest the fine-structure line from within the cosmological H II region might be negligible as compared to the signal from the regions immediately surrounding the H II region. Throughout this work we adopt the currently-favoured ΛCDM model: spatially flat with Ωm = 0.3 and ΩΛ = 0.7 (Spergel et al. 2006; Riess et al. 2004; Perlmutter et al. 1999) with 2 = 0.022 (Spergel et al. 2006; Tytler et al. 2000) and h = 0.7 (Freedman et al. 2001). 2. Fine-structure lines from the reionization epoch Subsequent to the recombination of the primeval baryon gas at redshift z ≃ 1089 (Spergel et al. 2006) and the transformation of the gas to an almost completely neutral state, it is believed that the gas was reionized during epochs corresponding to the redshift range 6 <∼ z ∼ 15. WMAP measurements of cosmic microwave background radiation (CMBR) anisotropy in total intensity and polarization have been used to infer that the baryons were likely neutral at redshifts z >∼ 12 − 15; however, the detection of CMB polarization anisotropy requires substantial ionization by about z ≃ 11 (Page et al. 2006). Observationally, the Gunn-Peterson (GP) test shows that the universe is highly ionized at redshifts lower than z ≃ 5.5; the detection of GP absorption at greater redshifts suggests that the neutral fraction of the intergalactic hydrogen gas rises to at least 10−3 in the redshift range 5.5 <∼ z ∼ 6, and that reionization was not complete till about z ≃ 6 (White et al. – 3 – 2003; Fan et al. 2002; Djorgovski et al. 2001; Becker et al. 2001). However, from the GP test alone it is not possible to infer the neutral fraction of the medium; it only gives a rather weak lower limit of ≃ 10−3 on the neutral fraction of the universe for z >∼ 6. From other considerations it is possible to put more stringent bounds on the neutral fraction; for example, Wyithe & Loeb (2004) obtain a lower limit of 0.1 on the neutral fraction of the universe at z ≃ 6.3 (see also Mesinger & Haiman 2004). Our understanding of the nature of the sources that caused the reionization is far from com- plete. The transition from an almost completely neutral gas to a highly ionized gas during redshifts 6 <∼ z ∼ 15 is a key problem in modern cosmology and considerable theoretical and experimental efforts are currently directed at this unsolved problem. Here we propose a new method, based on the 2p − 2s fine-structure transition, to determine the evolution of the neutral fraction of the intergalactic medium within this epoch. Owing to fine structure splitting, the two possible transitions between the 2s and the 2p states are: 2p1/2–2s1/2 at a frequency ≃ 1058MHz that has an Einstein A coefficient 1.6× 10 −9 s−1 and 2p3/2–2s1/2 at a frequency νps ≃ 9911MHz that has an Einstein A coefficient 8.78 × 10 −7 s−1. The Einstein A coefficient for the latter transition is more than an order of magnitude greater than the former; therefore, in this work we consider only the 2p3/2–2s1/2 transition and hereinafter we refer to this specific transition simply as the 2p–2s transition. The ionizing UV photons from sources in the reionization era create ‘Stromgren spheres’. Whereas the gas in the cosmological H II regions are highly ionized by the photons, the ioniza- tion level of the gas beyond the Stromgren spheres is determined by the history of the gas, the density, and the mean specific intensity of the background ionizing photons, which includes both the photons diffusing out of the Stromgren spheres as well as the background radiation field. In this work we assume that the ionizing sources at these high redshifts are AGNs and in illustrative examples adopt parameters of a few QSOs that have been observed at z ≃ 6. The photons at the Lyman-α transition frequency (here and throughout, unless otherwise specified, we shall con- tinue to refer to frequencies between Lyman-α and Lyman-limit as ’Lyman-α) from a high redshift QSO escape the mostly-ionized Stromgren sphere and are strongly scattered and absorbed in the medium beyond. The population of the 2p level in this region is determined by (a) the intensity of Lyman-α photons from the central source, (b) recombination rate of free electrons, (c) absorption of CMBR photons by electrons in the 2s state (it is assumed here and throughout this work that the only radio source at high redshifts is the CMBR) and (d) collisional transition from the 2s state. The 2s state is populated via (a) the recombination rate of free electrons, (b) collisional transfer of atoms from the 2p state, (c) the spontaneous decay of the 2p state, and (d) transition from the 2p state stimulated by CMBR photons. Additionally, the absorption of photons from the central sources, with energy equal to or in excess of the Lyman-β transition, would result in electronic – 4 – transitions to the second excited state, which could be followed by spontaneous decay to the 2s state. (It might be pointed out here that both the 2s and 2p states could also be populated by atoms cascading from excited states with n > 3. In particular, all photons absorbed from 1s states to any excited state can directly de-excite to the 2s level. However, the rate of transition from 1s to any excited state is roughly ∝ 1/n3 (e.g. Rybicki & Lightman 1979) and, therefore, we include only the most dominant transition in each case.) The population of the ground state is denoted by n1s. We denote the level populations of the the two states—2s1/2 and 2p3/2—by the number densities n2s and n2p; these may be solved for, respectively, from the following two equations of detailed balance: i + cB2p2sn2pnCMBR + Cpsnin2p + A2p2sn2p + cn1sp32 B13,βφ13(ν)nα(ν)dν = A2s1sn2s + Cspnin2s + cB2s2pn2snCMBR, (1) (1− 2f)αBn i + cB2s2pn2snCMBR + Cspnin2s + cn1s B12,αφ12(ν)nα(ν)dν = A2p1sn2p + Cpsnin2p + cB2p2sn2pnCMBR. (2) Here f is the fraction of all the atoms that recombine to the 2s state. In equilibrium f = 1/3 as the n = 2 state splits into three doublets: 2p1/2, 2p3/2 and 2s1/2. αB is the recombination coefficient and ni is the density of the ionized gas. B2p2s = B2s2p = c 2/(8πν3ps)A2p2s is the Einstein B coefficient for the 2p3/2–2s1/2 transition in terms of the corresponding Einstein A coefficient A2p2s (note that the two B coefficients are equal as the two states have the same degeneracy). Cps = Csp = 5.31 × 10 −4 cm3 s−1 is the rate coefficient of transition owing to collisions with electrons. nCMBR is the number density of CMBR photons within the transition line width. nα(ν) is the number density (per unit frequency) of photons with frequency equal to or larger than the Lyman-α frequency (and smaller than the Lyman-limit frequency) at any location; φ13 and φ12 are, respectively, the line profiles corresponding to the Lyman-β and Lyman-α transitions, and p32 is the probability for the electron transition to the 2s state following excitation to n = 3 via absorption of a Lyman-β photon. We have not included the induced Lyman-α transition because the number density of atoms in the 2p state is negligible as compared to that in the 1s state. p32 is the probability that an atom in the third excited state (3p) will decay to the 2s state. A2s1s is the Einstein A coefficient corresponding to the 2-photon decay of the 2s state. Other symbols have their usual meanings. Owing to the fact that the mostly neutral medium in the vicinity of the cosmological Strom- gren spheres is optically thick to Lyman-α scattering, the Lyman-α photons from the decay of the 2p state are strongly scattered by the gas. Therefore, nα(ν) will contain contributions from both the Lyman-α photons from the central source as well as the Lyman-α photons that arise – 5 – from recombinations outside the Stromgren sphere and are multiply-scattered therein: nα(ν) = nsourceα (ν) + n α (ν). We neglect the multiply-scattered Lyman-α photons from the Stromgren sphere that have been reprocessed via recombination within the Stromgren sphere because these would be redshifted redward of the Lyman-α line before encountering the boundary of the Strom- gren sphere. The scattering of recombination photons in an optically thick, expanding medium is a complex problem (Field 1959; Rybicki & Dell’Antonio 1994). One of its first applications was to study the recombination of primeval plasma (Peebles 1968; Zeldovich, Kurt & Sunyaev 1969). In these analyzes it was implicitly assumed that apart from 2-photon decay, in an expanding universe the dominant effect that results in resonant photons ceasing interaction with the gas, and leaving the system, is its redshifting out of the line profile. The effect of scattering off the moving atoms was deemed to be either negligible or at best comparable. This assumption has been borne out by more recent detailed analysis that have taken into account the effect of scattering on the photon escape (Krolik 1990). Taking only the redshift as the main agent of photon escape, it can be shown that the net effect of the scattering of a resonance photon before it drops out of consideration is to reduce the decay time of the 2p state from A21 = 6.2× 10 8 s−1 to A21/τGP (Zeldovich et al. 1969; for more recent work see, e.g., Chluba, Rubino-Martin & Sunyaev 2007, Seager, Sasselov & Scott 1999 and references therein; we give a concise derivation in Appendix A). In this expression, the Gunn-Peterson optical depth τGP = [3/(8πH)]A2p1sλ αn1s. Similar complications exist in computing p32, the probability that an atom in the third excited state will decay spontaneously to the 2s state, in an optically thick medium. In optically thin media, p32 = A32/(A32 + A31); on the other hand, in an optically thick medium, we would need to take into account the ‘trapping’ of the Lyman-β photon owing to resonant scattering. The effect of this scattering in an optically thick medium would be that the fraction of photons that decay directly to the ground state are reabsorbed ‘locally’ to the third excited state and, therefore, all photons absorbed to the third excited state result in an Hα photon and an atom in the 2s state. This means that the appropriate value of p32 is close to unity in an optically thick medium: in this work we assume p32 = 1. The astrophysical setting in which we seek solutions to the algebraic equations above is cos- mological H II regions at high redshift. In particular, we are interested in the signal from the neutral region surrounding the cosmological H II region. For a fully neutral inter-galactic medium (IGM) at z ≃ 6.5, τGP ≃ 6 × 10 5. If we adopt spectral luminosities corresponding to QSOs observed at these high redshifts, it may be shown that the populating of the 2p state via direct recombinations from the free-free state, collisional transfer from the 2s state, and upward transitions from the 2s to 2p state arising from absorption of background CMBR photons may all be neglected. (The rele- vant parameters are: ni ≃ nb ≃ 2.8(1 + z) 3 cm−3 in the H II region surrounding the sources, with ni expected to be much smaller in the neighboring mostly neutral medium; the number density of CMBR photons that might cause a 2p–2s transition is ≃ 5(1 + z) cm−3; the number density – 6 – of Lyman-α photons from the central source, assuming luminosities typical of SDSS quasars at z ≃ 6.5 (more details in §4), is nα ≃ 10 −4 cm−3. First, for these parameters, the dominant process that populates the excited state is the pumping by Lyman-α photons. Second, it may be readily verified that for these plausible values for the parameters the signal expected from the H II region surrounding the central source is negligible as compared to the signal from the surrounding neu- tral region.) Given that the Lyman-α flux from the central QSO is the dominant causative factor for populating the 2p state, the two equations (1 and 2) that determine the level populations is simplified. The number density of atoms in the 2p state is given approximately by: n2p ≃ n1sΓατGP/A2p1s, (3) where n1s = fneunb, with fneu denoting the neutral fraction and nb ≃ 2.7 × 10 −7(1 + z)3 cm−3 is the number density of baryons in the IGM. Γα = B2p1sφ12(ν)n source α (ν) (in units of s −1) is the transition rate to the 2p state owing to the Lyman-α photons from the central source, where nsourceα (ν) is the number density of Lyman-α photons from the central source alone. Similarly, the dominant process that determines the population of the 2s state is the absorption of Lyman-β photons (for details, see the discussion above and §4) and the subsequent decay to the 2s state. The number density of atoms in the 2s state is: n2s ≃ n1sΓβ/A2s1s, (4) where Γβ = B3p1sφ13(ν)n source α (ν) (in units of s −1) is the transition rate to the 2s state owing to the Lyman-β photons from the central source. 3. Fine structure lines from the epoch of recombination The universe made a transition from fully ionized to nearly fully neutral at z ≃ 1089 (Spergel et al. 2006; Peebles 1968; Zeldovich et al. 1969). This transition is mainly accomplished by the 2-photon decay of the 2s state and the slow redshifting of the Lyman-α photons which deplete the 2p state (see Peebles 1993 and references therein for a detailed discussion). The Saha ionization formula, valid for thermodynamic equilibrium conditions between the hydrogen level populations and free electrons, is a poor approximation for studying the epoch of recombination. A good ap- proximation for studying this transition is to assume that all states, excepting 1s, are in equilibrium with the CMBR (matter temperature to a very good approximation remains equal to the CMBR temperature throughout this transition) (Seager et al. 1999; Peebles 1968). However, in this ap- proximation, where the matter temperature and all transitions, excepting the Lyman-α line, are in thermal equilibrium, the 2p–2s signal is unobservable because the excitation temperature for this transition equals the background radiation temperature. Even though this second approximation – 7 – might be useful for studying the evolution in ionization fraction, it will be strictly true only if the dominant mechanisms that determine the level populations of the 2p and the 2s states are either interaction with the CMBR photons or collisions between atoms. As there are a variety of other processes relevant to the determination of the level populations—for example, the free decay of either of the two states—a small deviation from equilibrium is expected in the 2s and 2p level populations and it is our aim here to compute it. One approach to this problem is to simultaneously solve for the level populations of the 2p and the 2s states as well as the change in the ionization fraction. However, assuming thermal equilibrium between these two states is a good approximation for solving the evolution of ioniza- tion. Therefore, the approach we have adopted is to solve for the evolution of ionization using the method of Peebles (1968), and use the resulting ionized/neutral fraction to solve for the populations of the 2s and the 2p states using detailed balance. The resulting equations are: i + cB2p2sn2pnCMBR + Cpsnin2p + A2p2sn2p + n1sA2s1s exp(−(B1 − B2)/(kBTCMBR)) = A2s1sn2s + Cspnin2s + cB2s2pn2snCMBR + βcn2s (5) (1− 2f)αBn i + cB2s2pn2snCMBR + Cspnin2s = A2p1sn2p/τGP + Cpsnin2p + cB2p2sn2pnCMBR + βcn2p,(6) where B1 = 13.6 eV and B2 = 2.4 eV are, respectively, the ionization potentials of the ground and the first excited states, kB is the Boltzmann’s constant, and TCMBR is the temperature of the CMBR. βc is the rate at which the CMBR photons cause a bound-free transition of electrons from the n = 2 state (2s or 2p states). The various other terms in these equations have the same meanings as in the previous case. The main difference is that the CMBR photons and baryonic matter at the time of recombination are hot and dense enough to directly affect the level populations of the excited states by ionizing the excited state, and the two-photon capture to the excited state is not completely negligible (see Peebles 1993 and 1968 for details of the different physical processes that are relevant at this epoch). 4. Expected brightness of the fine structure lines The brightness temperature in the 2p–2s transition is: ∆Tb ≡ Tb − TCMBR = g⋆ n2p(0)hpA2p2sλ e(1 + z) 8πkBH(z) (1− TCMBR/Tex) ≡ τex(Tex − TCMBR) (7) Here n2p(0) = n2p/(1 + z) 3 and λe is the rest wavelength of the 2p–2s transition. Tex = (hpνe/kB)[n2p/(n2s − n2p)] is the excitation temperature corresponding to the transition, where – 8 – hp is the Planck constant. τex is the optical depth of the source in the 2p–2s signal. g⋆ takes into account the selection rules for transitions between the 2p3/2 and 2s1/2 levels. Given the selec- tion rules there are three allowed transitions between these two states; they occur at frequencies ν ≃ 9852, 9875, and 10029 MHz (see, e.g., Ershov 1987). The first two transitions are blended by the natural width of the line, which is approximately 100 MHz, but the third should be observable as a distinct line. This implies that g⋆ = 2/3 if the observing frequency is ≃ 9900/(1 + z) MHz. 4.1. Expectations for signals from cosmological H II regions For computing the strength of this signal in a typical case, we adopt observed parameters of QSO SDSS J1030+0524 (see, e.g., Wyithe & Loeb 2004 and references therein), which is at redshift z = 6.28 and shows no detectable flux beyond the QSO Stromgren sphere: the radius of the Stromgren sphere has been estimated to be R ≃ 4.5Mpc (Mesinger & Haiman (2004) argue that the size of the Stromgren sphere could be roughly 30% higher; this makes no essential difference to our results). It may be noted here that the QSO SDSS J1148+5251 also has similar parameters. We arrive at an estimate of nsourceα (ν) by assuming that Lα photons per second, with wavelengths corresponding to the Lyman-α transition, are emitted by the central source in an effective frequency range ∆νsource, and that the photons are absorbed at a radial distance R. This leads to: nsourceα = Lα/(4πR ∆νsource . (8) We assume typical values: Lα = 10 58 s−1, R = 4.5Mpc and ∆νsource = 30, 000 km s −1. These lead to the following estimate for the transition rate, Γα, in the gas at the boundary of the Stromgren sphere arising due to the photons from the central QSO: Γα ≃ 8× 10 −10 s−1. (9) It should be noted that the mean specific intensity of Lyman-α photons in the IGM would also give a non-zero signal. Assuming a mean specific intensity of ≃ 10−21 erg cm−2 sec−1 Hz−1 sr−1 (this might be needed to couple the HI spin temperature to the matter temperature; see e.g. Madau, Meiksin, & Rees 1997), the expected signal is many orders of magnitude smaller than we have computed from the outskirt of bright sources. If we assume that the central sources are continuum emitters in the frequency range between Lyman-α and Lyman-limit frequencies, we may assume similar parameter values for computing the expectations for Γβ. In that case, Γβ ≃ Γα , (10) – 9 – where fβ and fα are the oscillator strengths of the Lyman-β and Lyman-α transitions, respectively. From Eqs. (3), (4) and (10): A2p1s A2s1sτGP . (11) For a completely neutral medium at z ≃ 6.4, τGP ≃ 6 × 10 5, which may be used to show that n2s ≫ n2p for the parameters of IGM in the redshift range of interest: 6.4 <∼ z ∼ 10. This implies that in the outskirts of the Stromgren sphere surrounding the QSO, the transition is expected to be observable as an absorption feature against the CMBR. Using Eqs. (3) and (4) in Eq. (7), the observable brightness temperature is estimated to be: ∆TB ≃ −20 µK , (12) where fneu is the neutral fraction of hydrogen outside the Stromgren sphere, and might be close to unity. We have adopted parameters typical of QSOs observed at the edge of the reionization epoch in estimating the above temperature decrement. The main uncertainty above is in the estimation of the ‘Lyman-α’ flux from the central source, and as the observed temperature decrement is directly proportional to this flux from the central source, this constitutes a major uncertainty in reliably computing the expectations for the signal. For QSOs that have strong Lyman-α and Lyman-β lines, the blueward side of the lines will be strongly absorbed in the medium just beyond the Stromgren sphere (and this has been observed to happen in many cases), provided that the blueward side photons have not been redshifted to frequencies smaller than the Lyman-α frequency while transversing the Stromgren sphere. In the case of QSOs that have large line fluxes and small Stromgren spheres, the Lyman-α luminosity Lα might be underestimated. 4.2. Expectations for the fine-structure line from recombination Using equations (5) and (6) the level populations of the 2p and the 2s states may be computed. Solving for the level populations and using equation (7), we have computed the expected signal from the recombination epoch; the expected signal is shown in Fig. 1. The fine structure line transition is expected to be an absorption feature, with a maximum temperature decrement of order 100 µK at an observing frequency of about 7 MHz. The width of the decrement in frequency space corresponds to a redshift span ∆z ≃ 200, which is roughly the width of the visibility function at recombination. – 10 – 5. Prospects for the detection of the cosmological fine structure lines As Eq. (12) shows, a detection of the fine structure line in the outer regions of cosmological H II regions is potentially a probe of the cosmological neutral hydrogen density in the vicinity of QSOs in the reionization epoch, and might be a tool for the investigation of the evolution in the neutral fraction with cosmic epoch through the reionization era. Given the cosmological impor- tance of such a measurement, and the fact that there does not exist many reliable methods for the detection of H I at high redshifts (see, e.g., Barkana & Loeb 2001), the detection of the 2p–2s line transition in the cosmological context assumes additional importance. In deriving equation (7) the Hubble expansion was assumed to be the only cause for the velocity width in the observed line. However, an important contribution to the velocity dispersion in the line in this case is the natural width of the fine structure line: owing to the rapid decay of the 2p state, the natural Lorentzian width in the rest frame of the gas is VLor ≃ 100MHz (see, e.g., Dennison et al. 2005). Therefore, the peak brightness in the observed line profile might be suppressed by a factor Vexp/VLor, where Vexp is the line-of-sight peculiar velocity dispersion owing to the Hubble flow across the region being observed. However, in the case Vexp >∼ VLor this suppression has a negligible effect. For QSOs at redshift z ≈ 6.5, the natural Lorentzian width of the fine structure line is equivalent to the peculiar Hubble flow across a proper line-of-sight distance of ≃ 4Mpc. This distance is approximately the size of the Stromgren spheres around QSOs at that redshift; therefore, the natural width of the line does not significantly diminish the expectations, given by equation (7), for the peak brightness temperature. A second inference is that in the case of a QSO that has a smaller Stromgren sphere, the increase in mean brightness temperature is roughly proportion to the inverse of the radius of the Stromgren sphere: 1/R, and not 1/R2. Another assumption that was made while deriving Eq. (7) is that the only radio frequency radiation that needs to be considered for the determination of the level populations and brightness temperature decrement is the CMBR. It may be that the central ionizing source, which may be a QSO, is radio loud. There may also exist radio sources behind the observed Stromgren sphere and within the angular region over which the Lyman-α flux from the QSO is appreciable. Equation (7) may be modified, to account for this, by replacing TCMBR with TCMBR + TB, where TB is the brightness temperature of the radio source at wavelengths corresponding to the rest frequency of the transition, which is ≃ 9GHz. The result would be to enhance the brightness temperature of the line signal by ≃ TB/TCMBR. If the radio source is unresolved, it is appropriate to estimate the expected signal in terms of the optical depth. The optical depth corresponding to equation (12) is ≃ 10−6. We now discuss the feasibility of the detection of the fine structure line towards SDSS J1030+0524. We shall assume that the neutral fraction, fneu, outside the Stromgren sphere of this QSO is unity, – 11 – consistent with the measured GP trough. The redshifted fine structure line would be expected at ≃ 1.36GHz. The observed line ‘width’, considering natural broadening and the Hubble flow across the Stromgren sphere, is expected to be approximately ∆ν ≃ 100MHz/(1 + z); using z = 6.28, we obtain ∆ν = 13.7MHz. The fine structure line would be expected to originate in a shell that is roughly the size of the Stromgren sphere, and fall of as 1/r2 beyond this shell, where r is the distance from the QSO. In the case of SDSS J1030+0524, the angular size of the Stromgren sphere is expected to be 15 arcmin. The frequency of the redshifted line is in the observing bands, and the line width is within the spectral line capabilities, of several currently operational telescopes. However, large-collecting- area arrays like the Giant Metrewave Radio Telescope (GMRT) have large aperture antennas of 45-m diameter and, therefore, poor surface brightness sensitivity for such extended structures. The Australia Telescope Compact Array (ATCA), with 22-m antennas, has reasonable brightness sensitivity for this problem and, additionally, operates with 128-MHz bandwidths in spectral line mode. At ν ≃ 1.3GHz, the ATCA has a system temperature Tsys ≃ 25K and antenna efficient K = 0.1K Jy−1. The five movable 22-m diameter antennas may be configured into an ultra- compact 2-D close-packed H75 (75-m maximum baseline) array, and this would yield a number of baselines sensitive to the 15-arcmin scale fine-structure line signal. The brightness sensitivity of this array, for a 8-arcmin scale structure, is 400 µK in 6 hr integration time. The brightness sensitivity in Fourier synthesis images could be enhanced somewhat, by factors of a few, by ap- propriately weighting the baselines to match the synthetic beam to the expected structure scale; however, the required integration times for a detection are still in the ball park of 103 hr. The detection of the signal from cosmological H II regions, however, might be feasible using facilities under construction or planned for the near future, like the xNTD in Australia or the Square Kilometer Array (SKA). These arrays would have smaller antenna sizes—making the detection of these large-angular-scale structures detectable in interferometers—and significantly more numbers of antennas, giving more numbers of short baselines that would usefully respond to the large- angular-scale fine-structure line. However, the array configuration designs would have to factor in the extraordinary brightness temperature sensitivity requirements for this demanding observation. As an example, the SKA might consist of about 104 12–15 m class reflector antennas, with aperture efficiency of 60%, and the system temperature at 1.4 GHz might be about 20 K. Assuming that the visibilities are optimally weighted so that the synthetic beam of the Fourier synthesis array is matched to the source size of about 15 arcmin FWHM, the line strength would have a peak of about 20 µJy. A 5-σ detection of the fine structure line towards a typical QSO, in a reasonable integration time of about 1 hr, will require that about 250 baselines (just 0.001% of the total) be within about 50 m. The signal from the recombination epoch is observable in the frequency band 6–8 MHz as – 12 – a broad decrement in the brightness temperature of the extragalactic background sky, and would be extremely small compared to the orders of magnitude more intense Galactic non-thermal emis- sion as well as the average low-frequency background brightness temperature arising from the numerous extragalactic radio sources. This decrement may be considered to be a distortion to the CMBR spectrum at long wavelengths, and would be an all-sky cosmological signal. However, owing to the ionosphere, the frequency range in which this feature is expected to appear is too low a radio frequency to be easily accessible using ground-based observatories. Therefore, even though the observation of this signal might be yet another tool to probe the epoch of recombi- nation, new custom-made instruments, which should presumably operate from space and above the ionosphere, will require to be built if a detection of this signal is to be attempted. An addi- tional cause for concern is that the Galactic and Extragalactic background radiations might have low-frequency spectral turnovers at these frequencies, as a result of free-free absorption as well as synchrotron self-absorption, and these would result in significant spectral features in the band that would require a careful modelling in order to detect any CMBR decrement feature arising from fine structure transition absorption. Interference from terrestrial man-made transmitters, as well as from auroral phenomena and solar system objects would also be an issue. 6. Summary and Discussion We have discussed the possibility of detecting the hitherto undetected fine structure line of 2p–2s transition in two cosmological settings: the epoch of recombination at z ≃ 1089 and the epoch of reionization at z ≃6–15. The expectations for the line from the environments of ionization sources in the epoch of reionization are interesting and worthy of attention as a novel tool for the investigation of the reionization process and the cosmological evolution of the gas. The signal is expected to be ob- servable as an extended and weakly absorbing source, which causes a decrement in the brightness of the background CMBR, and may be detected by interferometers as a negative source akin to the Sunyaev-Zeldovich decrements observed along the lines of sight through hot gas in clusters of galaxies. Detection of the 2p–2s line signal from the outskirts of cosmological H II regions at different redshifts within the reionization era may serve to determine the neutral fraction of the medium during the epoch of reionization, which is a quantity of significant interest in modern cos- mology. In particular, we have computed a representative signal strength by adopting parameters typical of a QSO observed at z ≃ 6. These QSOs show GP troughs in their spectra; however, owing to the weakness of the GP test, the spectra have only been useful in setting weak limits on the neutral fraction (constraining the neutral fraction to be >∼ 10 −3) outside the observed H II region. We have shown that for a fully neutral medium the line peak may reach ≃ 20µK, which is – 13 – potentially observable by radio interferometers that are being designed today. Other interesting probes of the reionization epoch include detecting OI line from this epoch (Hernandez-Monteagudo et al. 2006). It is of interest to compare the relative difficulty associated with detecting the neutral hy- drogen in this indirect way, using the redshifted fine structure line, with direct imaging of the redshifted 21-cm line from neutral hydrogen during the epoch of reionization (see, e.g., Sethi 2005; Zaldarriaga, Furlanetto, & Hernquist 2004). The ‘all-sky’ H I signal might be detectable with a peak strength of ≃ 50mK; however, it could be very difficult to detect owing to calibra- tion and foreground contamination issues (e.g., Zaldarriaga et al. 2004; Shaver et al. 1999, and references therein). A better approach might be to attempt to detect the fluctuating component of the sky signal, which could have peak intensities of ≃ a fewmK at observing frequencies of ν ≃ 100–200MHz (Zaldarriga et al. 2004; Shaver et al. 1999). This translates to roughly the same signal strength (specific intensity) as we have obtained in Eq. (12) for the fine structure line. This is not entirely unexpected: even though the level populations of the excited states are much smaller as compared to the ground level population needed in computing the HI signal, the A coef- ficient of the fine structure transition we consider here is roughly 8 orders of magnitude larger than the HI hyperfine transition A coefficient. Currently, there are many ongoing and planned radio interferometer experiments for the detection of the redshifted H I emission/absorption from the epoch of reionization (e.g., Pen, Wu, & Peterson 2004; the LOFAR project at www.lofar.org; the MWA project at www.haystack.mit.edu/ast/arrays/mwa/site/index.html). The detection of the H I signal is firstly more difficult—requiring greater sensitivity—because the typical frequency width of the signal ≃ 0.5MHz (Zaldarriga et al. 2004), which is far smaller than the typical width expected for the fine structure line (≃ 10MHz). Second, the fine structure line is expected at 1.4 GHz, where the sky background temperatures are significantly lower than in the 100–200MHz band, making the telescope system temperatures lower. An additional advantage of the the indirect detection is that the redshifted fine structure line appears at higher frequencies ( >∼ 1GHz), which are relativity free of interference as compared to the low frequency band of 100–200MHz. The main advantage of the direct detection is that unlike the method we suggest here, it is independent of the existence of strong Lyman-α emitters at high redshifts. Another ad- vantage of direct detection is that it may be detected ‘statistically’ and such a detection might be achieved with greater ease than direct ‘imaging’ (e.g Zaldarriaga et al. 2004; Bharadwaj & Sethi 2001); however, the foreground subtraction problem becomes a severe constraint for a statistical detection. To summarize: if strong Lyman-α emitting sources are present at high redshifts, they would facilitate the indirect detection of the neutral hydrogen via enabling the detectability of the fine structure line. The imaging issues and problems associated with the detection of this signal appears to be less of a challenge as comparison to the direct detection of redshifted H I from those epochs. – 14 – Another astrophysical context in which the fine structure line might be detectable is the envi- ronments of high redshift galaxies, which are strong Lyman-α emitters. As equation (12) shows, the observed signal depends on the neutral fraction outside the Stromgren sphere. Therefore, de- tection of this signal would constitute an alternate probe of the neutral fraction of the IGM at large redshifts. The aim of this work has been to examine the detectability of the fine structure line in cos- mological contexts, to point out the cosmological significance of detections, and spawn work that may refine the modelling presented herein and improve the case for appropriate design of future telescopes, which might enable the detection of the fine structure line towards multiple sources in the reionization era. Appendix A: Photon distribution function The evolution of the photon distribution function, neglecting the effect of scattering off mov- ing atoms, is (see e.g. Rybicki & Dell’Antonio 1994): = A2p1sn2pφν − cBνn1sφνnν . (13) Here Bν = 3/(8π)c 2/ν2αA2p1s and H = ȧ/a. The equation may be written as: = −1/τ(nν − n⋆). (14) Here τ = 1/(cBνφνn1s) and f⋆ = A2p1sn2p/(Bνn1sc). The equation above lends itself to a ready interpretation. If the second term on the left hand side (which is owing to the expansion of the Universe) was absent, the distribution function will approach f⋆ on a time scale ≃ τ , where τ ≃ 3 × n1s s ≪ ȧ/a ( n1s here has units cm −3). It may be readily verified that for the recombination epoch and also the epoch of reionization, τ ≪ 1/H , excepting when the neutral fraction of the medium is very small. We work here with the assumption that the neutral fraction is always large enough so that τ ≪ 1/H . In this case the solution to Eq. (14) may be simplified: to leading order the distribution function approaches f⋆ and the first order term (of the order of τH) represents the slow time variation of the distribution function owing to the expansion of the Universe. In this approximation, one may write the solution to Eq. (14) as: nν ≃ n⋆ − τH(t)n⋆. (15) Using these equations we may proceed to prove the contention that the net effect of the ‘trapping’ of Lyman-α photons is to reduce the decay time of the 2p state by a factor τGP . The 1s–2p transition – 15 – rate, which is given by Eq. (6), may be solved using Eq. (15): Bνφ(ν)n(ν)dν ≃ A2p1sn2p − A2p1sn2p/τGP . (16) The first term on the right hand side cancels with the decay term of the 2p state on the right hand side of Eq. (6), and, therefore, the net effect of the scattering of recombination photons is to reduce the decay time of the 2p state by a factor τGP . It may be pointed out that the condition needed to derive the above expression roughly translates to the condition that τGP ≫ 1. For the reionization case, this requires that the neutral fraction >∼ 10 −5. In the case of primordial recombination, it leads to an even weaker condition that the neutral fraction is >∼ 10 Acknowledgment One of us (SKS) would like to thank Jens Chluba for many useful discussions and to Zoltan Haiman for many useful comments on the manuscript. REFERENCES Barkana, R. and Loeb, A. 2001, Phys. Rep., 349, 125 Becker, R. H. et al. 2001, AJ, 122, 2850 Bharadwaj, S. & Sethi, S. K. 2001, J. Astrophys. Astr., 22, 293 Chluba, J., Rubino-Martin, J. A., & Sunyaev, R. A. 2007, MNRAS, 374, 1310 Dennison, B., Turner, B. E., & Minter, A. H. 2005, ApJ, 633, 309 Djorgovski, S. G., Castro, S., Stern, D., & Mahabal, A. A. 2001, ApJL, 560, L5 Ershov, A. A. 1987, Sov. Astron. Lett. 13(2), 115 Fan, X., Narayanan, V. K., Strauss, M. A., White, R. L., Becker, R. H., Pentericci, L., & Rix, H. 2002, AJ, 123, 1247 Field, G. 1959, ApJ, 129, 551 Freedman, W. L. et al. 2001, ApJ, 553, 47 Hernandez-Monteagudo, C., Haiman, Z. 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L., Becker, R. H., Fan, X. & Strauss, M. A. 2003, ApJ, 126, 1R Wyithe, J. S. B. & Loeb, A. 2004, Nature, 432, 194 Zaldarriaga, M., Furlanetto, S. R. & Hernquist, L. 2004, ApJ, 608, 622 Zeldovich, Ya. B., Kurt, V. G. & Sunyaev, R. A. 1969, JETP Lett, 28, 146 This preprint was prepared with the AAS LATEX macros v5.2. http://arxiv.org/abs/astro-ph/0603450 http://arxiv.org/abs/astro-ph/0404083 http://arxiv.org/abs/astro-ph/0603449 – 17 – Fig. 1.— The expected brightness temperature decrement in the background radiation, owing to the fine structure transition in gas at the recombination epoch, is plotted versus the observing frequency. Introduction Fine-structure lines from the reionization epoch Fine structure lines from the epoch of recombination Expected brightness of the fine structure lines Expectations for signals from cosmological H ii regions Expectations for the fine-structure line from recombination Prospects for the detection of the cosmological fine structure lines Summary and Discussion

0704.1227

Superconductor strip in a closed magnetic environment: exact analytic representation of the critical state

Superconductor strip in a closedmagnetic environment: exact analytic representation of the critical state Y.A. Genenko ∗, H. Rauh Institut für Materialwissenschaft, Technische Universität Darmstadt, 64287 Darmstadt, Germany Abstract An exact analytic representation of the critical state of a current-carrying type-II superconductor strip located inside a cylindrical magnetic cavity of high permeability is derived. The obtained results show that, when the cavity radius is small, penetration of magnetic flux fronts is strongly reduced as compared to the situation in an isolated strip. From our generic representation it is possible to establish current profiles in closed cavities of various other geometries too by means of conformal mapping of the basic configuration addressed. Key words: Superconductor strip, Magnetic shielding, Critical state PACS: 74.25.Ha, 74.78.Fk, 74.78.-w, 85.25.Am Relatively high AC losses in superconductor ca- bles and strips present a substantial problem for the implementation of superconductors in high- frequency and low-frequency applications. Recently, a suggestion for improving the current-carrying capability of superconductor strips [1,2,3] and for reducing AC losses in superconductor cables [4] based on the idea of magnetic shielding of applied fields as well as current self-induced fields was put forth. AC losses in superconductor strips caused by the latter type of fields are anticipated to greatly decrease when the strips are exposed to suitably designed magnetic environments [2,3]. Exact ana- lytic representations of sheet current distributions in superconductor strips located between two high- permeability magnets occupying infinite half-spaces were derived before [1,2]; these configurations al- lowed to find the respective current distributions for various other topologically open shielding ge- ometries by application of the method of conformal ∗ Corresponding author. Email address: [email protected] (Y.A. Genenko). mapping. Utilization of the latter tool for analyz- ing sheet current distributions and AC losses in the presence of topologically closed magnetic environ- ments of practical interest requires corresponding reference results. An establishment of such results is the focus of the present communication. We consider an infinitely extended type-II super- conductor strip of width 2w located inside a cylin- drical cavity of radius a in an infinitely extended soft magnet of relative permeability µ, the symme- try axis of this configuration coinciding with the z- axis of a cartesian coordinate system x, y, z. Assum- ing the thickness of the strip to be small compared to its width, variations of the current over the thick- ness of the strip may be ignored and, for mathemat- ical convenience, the state of the strip characterized by the sheet current J alone. When magnetic flux penetrates the superconduc- tor strip in the critical state, the distribution of the sheet current is controlled by the pinning of mag- netic vortices. In conformity with Bean’s hypothe- sis [5], the sheet current adopts its critical value J throughout the flux-penetrated regions of the strip, Preprint submitted to Elsevier 18 August 2021 http://arxiv.org/abs/0704.1227v1 whereas the magnetic field component normal to the strip vanishes in the flux-free regions of the strip. Proceeding in the spirit of previous work [6], a dis- tribution of the sheet current prevails with magnetic flux penetrated from the edges of the strip, but with the central zone −b < x < b of half-width b < w left flux free. In this zone, the distribution of the sheet current is governed by the integral equation [2] dx′J(x′) x− x′ x− a2/x′ = 0 (1) together with the requirement that the total trans- port current equals I. Here, q = (µ − 1)/(µ + 1) means the strength of the image current induced by the magnetic cavity. In the limit µ ≫ 1, i.e. for q → 1, Eq. (1) has the exact analytic solution J(x) = J a2 + b2 πs2(x) s′(x)φ(s(x)), (2) where φ(s) = c2 − s2 arctan (h2 − b2)(c2 − s2) (b2 − s2)(c2 − h2) − arctan h2 − b2 b2 − s2 b2 − s2 arctan b2w2 − b4 a4 − b2w2 with s(x) = x(a2 + b2)/(a2 + x2), c = (a2 + b2)/2a and h = w(a2 + b2)/(a2 + w2). Herein, K and Π denote complete elliptic integrals of the first and, respectively, third kind. Sheet current profiles obtained from Eq. (2) for a range of the geometrical parameters involved, with a fixed value of b, are shown in Fig. 1. This exhibits a flattening of the current profiles together with an increase in the magnitude of the total current up to saturation, when the radius of the magnetic cavity is reduced, precisely as in the case of topologically open magnetic cavities [1,2]. The half-width of the flux-free zone is controlled by the total transport current in the strip and by the geometry of the magnetic environment. An implicit equation for b in the chosen geometry may be found by integrating the sheet current over the width of the strip using Eq. (2) which yields (π/b)(a2 + b2) K2 (2ab/(a2 + b2)) arctan b2(w2 − b2) (a4 − w2b2) . (3) -1,0 -0,5 0,0 0,5 1,0 Fig. 1. Distribution of the sheet current over the flux-free zone of the partly flux-filled strip delineated by b/w = 0.8, with a/w = 1.001, 1.01, 1.1, 2 and infinity (from the upper curve down). The cylindrical magnetic cavity entails a reduc- tion of the depth of penetration of magnetic flux into the strip, ∆(I) = w − b, as compared to the depth in the situation without a magnetic environment, ∆0(I) = w(1− 1− (I/I )2), where I = 2wJ For weak flux penetration, when ∆ ≪ w and hence I ≪ I , the explicit approximate result ∆(I) ≃ a2 − w2 a2 + w2 a2 + w2 is seen to hold. Thus, ∆ strongly decreases with re- spect to ∆0 ≃ (w/2)(I/Ic)2 as a → w. This also means a reduction of AC losses to the same extent which typically scale with ∆2 [6,7]. These losses may be further curtailed by optimization of the shape of the magnetic cavity using, in the limit µ ≫ 1, the method of conformal mapping of the basic cylindri- cal configuration addressed above. References [1] Y.A. Genenko, A. Usoskin, H.C. Freyhardt, Phys. Rev. Lett. 83 (1999) 3045. [2] Y.A. Genenko, A. Snezhko, H.C. Freyhardt, Phys. Rev. B 62 (2000) 3453. [3] Y.A. Genenko, H. Rauh, A. Snezhko, Physica C 372-376 (2002) 1389. [4] M. Majoros, B.A. Glowacki, A.M. Campbell, Physica C 334 (2000) 129. [5] C.P. Bean, Phys. Rev. Lett. 8 (1962) 250. [6] W.T. Norris, J. Phys. D 3 (1970) 489. [7] E.H. Brandt, M.V. Indenbom, Phys. Rev. B 48 (1993) 1289. References

0704.1228

Near- and Far-Infrared Counterparts of Millimeter Dust Cores in the Vela Molecular Ridge Cloud D

Astronomy & Astrophysics manuscript no. 7235-astroph c© ESO 2018 December 8, 2018 Near- and Far-Infrared Counterparts of Millimeter Dust Cores in the Vela Molecular Ridge Cloud D ⋆ M. De Luca1,2, T. Giannini1, D. Lorenzetti1, F. Massi3, D. Elia4, and B. Nisini1 1 INAF - Osservatorio Astronomico di Roma, via Frascati 33, 00040 Monte Porzio Catone (Roma), Italy e-mail: deluca, giannini, lorenzetti, [email protected] 2 Dipartimento di Fisica - Università di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00033 Roma, Italy 3 INAF - Osservatorio Astrofisico di Arcetri, Largo E.Fermi 5, 50125 Firenze, Italy e-mail: [email protected] 4 Dipartimento di Fisica - Università di Lecce, CP 193, 73100 Lecce, Italy Preprint online version: December 8, 2018 ABSTRACT Aims. Identify the young protostellar counterparts associated to dust millimeter cores of the Vela Molecular Ridge Cloud D through new IR observations (H2 narrow-band at 2.12 µm and N broad band at 10.4 µm) along with an investigation performed on the existing IR catalogues. Methods. The association of mm continuum emission with infrared sources from catalogues (IRAS, MSX, 2MASS), JHK data from the literature and new observations, has been established according to spatial coincidence, infrared colours and spectral energy distributions. Results. Only 7 out of 29 resolved mm cores (and 16 out of the 26 unresolved ones) do not exhibit signposts of star formation activity. The other ones are clearly associated with: far-IR sources, H2 jets pointing back to embedded objects not (yet) detected or near-IR objects showing a high intrinsic colour excess. The distribution of the spectral indices pertaining to the associated sources is peaked at values typical of Class I objects, while three objects are signalled as candidates Class 0 sources. Objects with far-IR colours similar to those of T-Tauri and Herbig Ae/Be stars seem to be very few. An additional population of young objects exists associated not with the mm-cores, but with both the diffuse warm dust emission and the gas filaments. We remark the high detection rate (30%) of H2 jets driven by sources located inside the mm-cores. They appear not driven by the most luminous objects in the field, but rather by less luminous objects in young clusters, testifying the co-existence of both low- and intermediate-mass star formation. Conclusions. The presented results reliably describe the young population of VMR-D. However, the statistical evaluation of activity vs inactivity of the investigated cores, even in good agreement with results found for other star forming regions, seems to reflect the limiting sensitivity of the available facilities rather than any property intrinsic to the mm-condensations. Key words. Stars: formation – Infrared: stars – ISM: individual objects: Vela Molecular Ridge – ISM: clouds – ISM: jets and outflows – Catalogs 1. Introduction The association of far-infrared (FIR) sources with the gas and dust emission cores in Giant Molecular Clouds (GMCs) is the starting point of any effort aimed to obser- vationally study high- and intermediate-mass star forma- tion modalities. Indeed the Galactic matter is distributed in such a way that prevents to have a significant num- ber of GMCs located near our Sun. Also low-mass star formation occurs in GMCs, but GMCs are usually found Send offprint requests to: M. De Luca ⋆ Based on observations collected at NTT and 3.6m telescope (ESO - La Silla, Chile). at distances greater than 1-2Kpc, where the sensitivity of the current fore-front instrumentation makes it possible to sample only the most luminous (i.e. the most massive) objects. As a consequence, studying low-mass star forma- tion is limited to nearby dark clouds, leaving GMCs as the privileged targets for high-mass studies. Exceptions exist, which are represented by the Orion and Vela GMCs (usually named Vela Molecular Ridge - VMR) and by the OB association in Scorpius-Centaurus and Perseus, whose distances range between 300 and 700 pc. Orion GMC is by far the most studied star formation site at all wavelengths and both modalities (high- and low-mass) are observation- ally documented to take place in it (e.g. Chen & Tokunaga http://arxiv.org/abs/0704.1228v1 2 M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D 1994). Its location outside the plane of the Galaxy (b≈ - 20◦) has originally sanctioned its leading rôle in star for- mation studies, when extinction and crowding represented insurmountable observational barriers. Also Sco-Cen and Per OB associations are located at about +20◦ and -20◦, respectively. The rapid growth of multi-frequency facil- ities at an increasing spatial resolution, makes it possi- ble now to study, with enough accuracy, even extincted and crowded regions, as the VMR, in the plane of the Galaxy. Thanks to the forthcoming facilities, investigat- ing the plane will be easier and easier and the properties derivable from the VMR will be the most suitable for ob- taining a direct comparison with those of the other galax- ies, whose planes represent the main volume we are able to sample. The VMR is a complex of four GMCs (Murphy & May 1991; Yamaguchi et al. 1999) and it is probably one of the best regions for studying the processes involved in star formation (clustering, isolation, matter outflows). It is lo- cated in the galactic plane (b = ± 3◦) outside the solar cir- cle (ℓ ∼ 260◦ − 275◦) and most of the gas is at a distance of ∼ 700 pc (Liseau et al. 1992). So far we have accumu- lated a large data-base on VMR clouds through ground- based observations from near-IR (NIR) to mm wave- lengths. In particular, our analysis, based on the IRAS point source catalog (IRAS-PSC 1988), unveiled a re- markable concentration of red FIR sources with bolomet- ric luminosities < 104L⊙ (Liseau et al. 1992, Lorenzetti et al. 1993). These are very young intermediate-mass stars and we found (Massi et al. 2000, 2003) that FIR sources with Lbol > 10 3L⊙ coincide with young embedded clus- ters (size ∼ 0.1 pc, ∼ 50-100 members). In particular, we find that the region of the VMR named cloud D (here- inafter VMR-D) hosts a large number of these, revealing a high efficiency in this mode of star formation. At the same time, the presence of IRAS sources having bolomet- ric luminosities of only few solar luminosities, shows that the formation of isolated, low-mass stars is also active in this region. We have searched the region around a complete sample of IRAS sources in VMR-D for protostellar jets, using NIR imaging and spectroscopy (Lorenzetti et al. 2002), and we have discovered a significant number of shock tracers (H2 and [FeII] line emission knots), which signal the pres- ence of protostellar jets in VMR-D. We have also clarified the details of the interactions between jets and circumstel- lar environment as well as the properties of the exciting sources (Caratti o Garatti et al. 2004, Giannini et al. 2001, 2005). Recently we mapped a ∼ 1 deg2 area of the VMR-D in the 1.2mm continuum of dust emission, along with the 12CO(1–0) and 13CO(2–1) transitions (see Fig. 1 in Massi et al. 2007 and Figs. 3-4 in Elia et al. 2007). The aim of the present paper is to correlate dust map and avail- able FIR and NIR catalogues to have a complete census of the VMR-D young population that allows us to assess the pre-main sequence evolutionary stage properties. The correspondence between dust cores and molecular clumps has been already presented in Elia et al. (2007). An increasing number of millimeter surveys towards star formation regions along with a comparison with existing FIR catalogues is now available. For the VMR complex, in particular, an investigation of the Vela-C cloud has been carried out by Moriguchi et al. (2003) and by Baba et al. (2006), and mm-studies of few individual sources belong- ing to VMR-D are reported by Faundez et al. (2004) and Fontani et al. (2005). In the following, we give in Sect. 2 a short summary of the results on the mm-maps (both dust and gas) we have presented elsewhere. The criteria for associating IR coun- terparts to the dust emission cores are discussed in Sect. 3, along with the analysis of a particular selected area, as an example. New IR observations are also presented in this section. The results are then summarized in Sect. 4 and discussed in Sect. 5. In appendix A, the associations for all the regions of dust emission are discussed separately. 2. The investigated region Our observations of the VMR-D carried out in the mil- limeter range with the SEST (ESO - La Silla) telescope are described elsewhere and consist of three maps of about 1 deg2, both in the dust continuum emission at 1.2mm (Massi et al. 2007) and in the molecular transi- tions 12CO(1-0), at 2.6mm, and 13CO(2-1), at 1.3mm (Elia et al. 2007). The 12CO data (resolution: 43”, sampling: 50”) out- line a filamentary distribution of diffuse molecular emis- sion connecting regions of enhanced intensity where there is evidence of clustered, intermediate-mass, star forma- tion in progress (Elia et al. 2007, Massi et al. 2007). The map of 13CO emission, because of its lower abun- dance with respect to 12CO, traces denser regions of the molecular cloud, which present a quite clumpy structure. Summarizing the results from the 13CO map, we have found 49 clumps with mass, size and mean velocity rang- ing from 2 to 140M⊙, from 0.15 to 0.67pc and from 1 to 13 km s−1, respectively (see Tab. 3 in Elia et al. 2007). The dust map (resolution: 24”, corresponding to about 0.1 pc, in fast scanning mode of 80 arcsec s−1) also shows a clumpy structure and we have individuated 29 cores of mass and size in the ranges 0.2 - 80M⊙ and 0.03 - 0.3 pc respectively (see Tab. 1 in Massi et al. 2007), almost all of them nearly coincident with the brightest regions of the velocity integrated CO maps. In addition, other 26 cores, whose size is under the spatial resolution, have been iden- tified, even if their genuine nature remains uncertain. 3. Association with point-sources In this work we aim to correlate the millimeter emis- sion with objects from both new IR images (1-10µm) carried out on selected areas (see Sect. 3.1) and the infrared existing catalogs of point sources in order to find out the sources that dominate the dust cores heat- ing. In particular, the considered catalogs are: (i) the M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D 3 IRAS point source catalog (IRAS-PSC 1988), that pro- vides flux densities at 12, 25, 60 and 100µm; (ii) the MSX (Midcourse Space Experiment) point source cata- log (Price et al. 2001) at 8.28, 12.13, 14.65 and 21.3µm; (iii) the 2MASS point source catalog (Cutri et al. 2003), based on J , H , Ks bands, complemented with litera- ture data (Massi et al. 1999) of deeper IRAC2 images (Moorwood et al. 1992). 3.1. Observations An observational campaign has been carried out by using NIR (narrow-band H2 1-0S(1) at λ=2.13µm) and mid- IR (MIR) (N broad-band) imaging facilities to observe fields selected from the mm-emission maps. All the rec- ognized dust cores (except one) have been imaged in H2, with the aim of searching for protostellar jet evidences. The 10.4µm survey covered those cores associated to the presence of a young embedded cluster (Massi et al. 2003, 2006), aiming, thanks to an adequate spatial resolution, to pick-up those source(s) (if any) that dominate(s) the detected fluxes at FIR wavelengths. 3.1.1. NIR imaging Broadband J , H , Ks and narrow-band images in the H2 1-0S(1) (λ=2.13µm, ∆λ=0.03µm) were obtained in January 2006 with SofI1 (Lidman et al. 2000) at ESO-NTT (La Silla, Chile). The total field of view is 4.9×4.9 arcmin2, which corresponds to a plate scale of 0.29 arcsec/pixel. All the observations were obtained by dithering the telescope around the pointed position and the raw imaging data were reduced by using standard procedures for bad pixel removal, flat fielding, and sky subtraction. 3.1.2. N-band imaging Imaging in the N10.4 broadband filter was carried out in January 2006 with Timmi2 (Saviane & Doublier 2005) at the 3.6m ESO telescope (La Silla, Chile). The adopted plate scale is 0.3 arcsec/pxl, corresponding to a 96′′ × 72′′ field of view. The observations were obtained by chop- ping the signal and by nodding and jittering the tele- scope around the pointed position in the usual ABB′A′ mode. The raw data were reduced by using standard procedures for bad pixel removal and the observed field was flux calibrated by using photometric standard stars (HD29291, HD32887, HD123139). The photometric re- sults are given in Tab. 1, together with the IRAS/MSX results. Although these latter refer to different effective wavelengths and different epochs, ground-based values are significantly lower than IRAS/MSX determinations. These discrepancies have been remarked several times in the literature concerning YSO’s (e.g. Walsh et al. 2001) 1 J and H images were obtained only for those fields con- taining the young clusters. and may be due to the higher environmental contamina- tion suffered by the larger IRAS/MSX beams. 3.2. FIR associations Within molecular clouds the correlation between the po- sitions of dust emission cores and FIR point-like sources represents an important method to obtain a census of both the young stellar population and the different modalities of the star formation. To search the catalogs for sources associated to the dust cores listed in Tab. 1 of Massi et al. (2007), a working definition of the core size has to be firstly provided. Indeed, in that Table the core size is given (col- umn 4), which results from the geometrical mean of the quantities ∆x and ∆y. These latter (see Tab. 2, column 2) are directly provided by the adopted search algorithm (Clumpfind), and indicate the FWHM of the linear profile of the core itself, along its x (right ascension) and y (decli- nation) axes, respectively (Williams et al. 1994). The area covered by a dust core up to the (bidimensional) FWHM flux level can thus be roughly individuated by the ellipse centered at the peak coordinates and having axes ∆x and ∆y. For simplicity, we will call hereinafter this ellipse as dust FWHM-ellipse (see e.g. Fig. 1, where it is represented by the red, inner curve). Analogously, we operatively de- fine as dust 2FWHM-ellipse that with axes 2∆x and 2∆y (red, outer ellipse in Fig. 1). Searching for catalogued sources we use this FWHM value by adopting the following criterion: an IRAS or MSX point source is considered associated to the core if its po- sitional uncertainty ellipse overlaps (or is tangent to) the dust 2FWHM-ellipse. However, to evidence the most com- pelling cases, the association within one FWHM-ellipse are boldfaced in Tab. 1. With respect to the criterion adopted in similar works (e.g. Mookerjea et al. 2004, Beltrán et al. 2006), for which an IRAS/MSX source is associated to a core if it lies inside 90′′/40′′, our criterion both takes into account the dust emission morphology and compensates for the large IRAS/MSX beam. 3.3. NIR associations Both the 2MASS catalog and the IRAC2 catalog reported in Massi et al. (1999) were searched for NIR associations, i.e. the sources that dominates the cores heating. Since the NIR sources positional accuracy is by far larger than the deconvolved core size, there is no need of defining a specific criterion for the association: we simply consider all the NIR sources falling within the FWHM-ellipse of each core that present a valid flux (not an upper limit) at least in a single band (J , H , K). Furthermore, we have tentatively selected the most probable NIR counterpart of the dust core according to the following criteria: 1. Closeness to the peak of dust emission and to the FIR source possibly associated, if any. 2. Intrinsic excess in the two colours (J-H vs H-K) dia- gram (hereafter colour-colour diagram). This criterion 4 M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D makes it possible to pick up the NIR objects whose spectral energy distribution (SED) is typical of a young stellar object (YSO), and does not appear as stellar photosphere reddened by the intervening dust along the line of sight. In this context, to point out their intrinsic colour excess, we will define two loci in the colour-colour diagram (e.g. shaded regions in panel a of Fig. 2): the locus of the red objects (mainly T- Tauri), immediately to the right of the main sequence (reddened) stars and the locus of the very red sources (mainly Class I and Herbig Ae/Be protostars), to the right of the T-Tauri (reddened) stars. 3. Largest spectral index α = d log(λFλ)/d log(λ) among those with α≥ 0. Sources with α < 0, in fact, are gen- erally visible in the optical plates, thus this item cut off at least bright visible stars. 3.4. The core MMS1 Here we describe a typical example of the approach adopted to detect the FIR/NIR counterparts of the mm- core MMS1. Similar considerations have been done for any individual mm-core, and all the results are provided in the Appendix A. As a first step, we overlay the contour map of dust emission (green contours, from 3 σ=3× 15mJy/beam in steps of 3σ) and the H2 narrow-band (gray-scale) image (Fig. 1). The red ellipses centered on the mm-peak repre- sent the FWHM- and the 2FWHM-ellipses within which the association with the IR sources has been searched (see Sections 3.2 and 3.3). The magenta and green ellipses in Fig. 1 individuate the 3σ positional uncertainties of the IRAS2 and MSX point sources, respectively, while the blue asterisks signal the position of the 10.4µm sources observed by Timmi2 in the field of view delimited in fig- ure by the blue line. The 2MASS and IRAC2 NIR sources are labelled 2M ♯ (following an internal numbering) and MGL99 ♯ (following Massi et al. 1999), respectively. The IRAC2 field of view does not cover the whole image and is depicted as the orange rectangle. Both IRS16 (08438 − 4340, corresponding to MSX G263.6200-00.5308 and DGL 4) and G263.5925-00.5364 are located outside the FWHM-ellipse and do not appear directly associated to the core, while the association is more compelling with the sources DGL 1, DGL 2 and G263.5994-00.5236. The IRAS source coincides with a NIR young cluster and an HII region located at the center of the region bordered by the cores MMS1-2-3, almost equidistant from all of them. The NIR cluster has been already investigated in detail by Massi et al. (2003) and we complement those data with the new N -band observation that points out the presence of a very diffuse emission, indicated as DGL 2 Hereinafter we will adopt for the IRAS sources, when avail- able, the shorter names IRS♯ defined in Liseau et al. (1992). Both the original names and these ones, however, are listed in Tab. 1. 4, corroborating the hypothesis that the FIR source can be originated by warm circumstellar matter associated with the most luminous (in the NIR) cluster member, MGL99 Also the MSX source G263.5295-00.5364, that falls at the western border of the 2FWHM-ellipse, does not seem to be related to the dust peak. It has been detected at 8µm only and, presumably, such flux arises from diffuse emis- sion, as an inspection of the MSX image suggests3. To feature the NIR stellar content close to the peak po- sition we give in Fig. 2-a the colour-colour diagram of all the detected sources within the FWHM-ellipse. Here are also drawn: the locus of the main sequence, class V stars (Tokunaga 2000) (dark curve), the locus of the T-Tauri stars (Meyer et al. 1997) (red line) and three reddening lines (blue) (Rieke & Lebofsky 1985), starting from three significant points, with four crosses indicating values of 0, 1, 5 and 10 mag of visual extinction. The separation of the sources in the two regions red and very red (see Sect. 3.3) is also evidenced by a different shading. The arrows on the data points denote constraints on colours derived from upper limits on the NIR photometry. This diagram allows us to select the most interesting objects: in particular, the sources labelled as MGL99 156, MGL99 179 and MGL99 155 show the highest colour excess. We report in Fig. 2-b the Spectral Energy Distribution (SED) of the very red stars within the FWHM-ellipse, to- gether with the Timmi2 measured fluxes and the SED of the MSX object G263.5994-00.5236. The arrows denote again the upper limits. From the SEDs, we see that MGL99 179 and MGL99 156, considering the upper limits, are the sources with the steepest spectral indexes. Moreover, (see Fig. 1), while MGL99 179 has been identified as the counterpart of the Timmi2 source DGL 1, MGL99 156 has lack of detection in N , although its K magnitude is comparable to that of MGL99 179. This allows us to suggest that the main contributor to the millimeter flux should be the object MGL99 179=DGL 1. However, considering the richness of very red sources in the field (Tab. 2), a contribution to the millimeter flux from multiple sources or from objects too much embed- ded to be investigated with the described tools, cannot be ruled out. Moreover, the HII region at the southern border of the core could provide an additional external heating by means of UV photons. 3 It lacks of any NIR suitable counterpart (unfortunately, at that position we have no IRAC2 data) although the nearest NIR significant object, MGL99 156, 27 arcseconds apart, may contribute to the measured flux. M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D 5 Fig. 1. MMS1 field of view (center [J2000]: 08:45:32.8, -43:50:12.3). Grayscale image: H2 emission; green contours: dust continuum (from 3σ, in steps of 3 σ). See text for other details. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 [H-K] 2M-25930 MGL99-155 MGL99-179 MGL99-156 red very red 1 2 5 10 2 -20.5 -20.0 -19.5 -19.0 -18.5 -18.0 -17.5 -17.0 -16.5 -16.0 2M-25930 MGL99-123 MGL99-155 MGL99-179 MGL99-156 G263.5994-00.5236 DGL-2 DGL-1 Fig. 2. a - Colour-colour diagram of the sources falling within the FWHM-ellipse around MMS1. b - Spectral energy distribution of the very red sources within the FWHM-ellipse. See text for more details. 6 M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D 4. Results All the results derived from the association to dust emis- sion cores of sources from both FIR/NIR catalogs and our ground-based new observations are given in three different Tables organized as follows: - Table 1 - Here all the dust cores detected in the mm map (column 1) having an association with a MIR and/or FIR object are listed. The core identification follows that of the dust map (Massi et al. 2007), where the naming MMS# refers to the resolved cores while umms# (second part of the Table) are under-resolved (i.e. size smaller than the SIMBA beam4, see Tab. 2 of Massi et al. 2007). Then, for those cores associ- ated with a FIR source, IRAS and MSX names, flux densities and distances from the peak (d) are given, along with an indicator (CC = correlation coefficient for each band, see IRAS manual) of the point-like na- ture of the IRAS source. For the IRAS objects is also indicated the 1.2mm flux at the IRAS coordinates ob- tained by integrating the dust map within 24 arcsec (= SIMBAHPBW) aperture. Finally, for comparison purposes with IRAS (12µm) and MSX (8.28 and 12.13µm) fluxes, the results of the Timmi2 observa- tions (at 10.4µm, objects coded as DGL ♯) are also reported in the last columns of this Table (again with the distances from the peak). As anticipated, the bold- faced lines correspond to FIR/MIR associations within the corresponding FWHM-ellipse, and, consequently, have to be considered as more robust cases. We remark here that all the cores reported in Tab. 1 are also associated with CO clumps, with the exception of MMS29, umms1-23-24 and 25, which are outside the CO map coverage5 (Elia et al. 2007). - Table 2 - This Table lists the NIR counterparts of all the cores. In column 2 the core size is identified through the FWHM-ellipse axes ∆x and ∆y (see Sect. 3.2). In column 3 the numbers of NIR red or very red sources that are located within the FWHM-ellipse are given. The census of the NIR population is based, whenever it is possible, on the IRAC2 images (Massi et al. 1999, objects coded as MGL99 ♯) that are deeper than the 2MASS frames (K band limit magnitude 18 instead of 14). Otherwise 2MASS images have been exploited (objects coded as 2M ♯). In columns 4 to 6 the loca- tion of the most probable candidate counterpart to the mm emission or to the FIR associated object is given, while the 7th column signals the morphology of the H2 emission as detected in our ground-based observations. 4 The circumstance of having an associated FIR source makes some of the under-resolved cores suitable targets in the next future for high spatial resolution southern facilities in the mm-range (e.g. ALMA). 5 The integrated 12CO and 13CO maps show a noticeable enhancement of the integrated emission towards umms1, al- though this enhancement is not fully mapped. - Table 3 - This table lists all the IRAS sources, falling within the dust map, but not associated to any core, that present at least two valid detections (not upper limits) between 12 and 60µm and a flux increasing with the wavelength, namely F12µm < F25µm < F60µm, or upper limits at 12 and 25µm compatible with this condition. Studying the VMR-D young population, these sources have some relevance. Indeed these red and cold sources are not randomly distributed, as the not-red sources do, but tend to be located along the gas filaments. As in Tab. 3, we also report the 1.2mm flux derived at the IRAS position. Table 1. Dust cores with associated FIR and/or MIR point sources. mm IRAS associated sources a MSX associated sourcesa Timmi2 observationsa core id F12 F25 F60 F100 F1.2mm CC d id F8.3 F12.1 F14.7 F21.3 d id RA Dec F10.4 d (Jy) (”) (Jy) (”) (J2000) (J2000) (Jy) (”) MMS1 08438-4340 13.4 56.0 638.3 1576 0.203 ECDB 49 G263.5994-00.5236 1.6 2.2 2.0 4.1 29 DGL 1 8:45:33 -43:50:04 0.05 17 (IRS16) G263.6200-00.5308 1.8 5.1 6.9 15.6 51 DGL 2 8:45:33 -43:49:48 1.8 33 G263.5925-00.5364 0.5 <0.7 <0.5 <1.6 56 DGL 4 8:45:36 -43:51:02 diffuse ∼49 MMS2 G263.6338-00.5497 0.4 <6.9 <5.4 1.6 25 DGL 3 8:45:34 -43:52:26 0.25 26 MMS3 08438-4340 13.4 56.0 638.3 1576 0.203 ECDB 55 G263.6385-00.5217 0.9 1.3 0.8 1.4 20 DGL 5 8:45:38 -43:51:14 0.08 26 (IRS16) G263.6366-00.5148 0.3 <0.9 <0.7 <1.9 39 DGL 4 8:45:36 -43:51:02 diffuse ∼55 G263.6329-00.5127 0.4 0.7 <0.9 <2.7 46 G263.6200-00.5308 1.8 5.1 6.9 15.6 54 MMS4c 08448-4343 8.7 88.1 326.6 1005 0.984 AABB 7 G263.7759-00.4281 7.5 10.1 13.8 65.5 5 MGL99 57d 8:46:35 -43:54:31 8.79 7 (IRS17) G263.7733-00.4151 1.2 1.3 0.5 2.6 53 MGL99 25d 8:46:34 -43:54:50 0.21 15 G263.7867-00.4437 0.3 0.8 <0.4 <1.2 63 MGL99 40d 8:46:33 -43:54:39 0.14 17 MMS12 08470-4321 44.9 130.1 342.6 406.9 0.331 AAAA 11 G263.7434+00.1161 30.8 52.6 70.3 93.5 3 DGL 7 8:48:49 -43:32:29 18.3 5 (IRS19) MMS18 08472-4326A 0.9 0.9 10.6 <406.9 0.044 BABF 28 G263.8432+00.0945 0.3 <0.7 <0.5 <1.3 16 DGL 8 8:49:03 -43:37:55 0.07 19 MMS20 08474-4323 1.5 1.1 <106.4 125.9 <0.005 BCAD 39 G263.8221+00.1494 0.1 <0.9 <0.6 <1.9 37 DGL 9 8:49:12 -43:35:52 0.08 28 MMS21 08474-4325 <0.3 1.0 <16.4 57.5 0.207 DAHD 7 DGL 9 8:49:12 -43:35:52 0.08 39 MMS22 08476-4306 5.7 44.0 216.3 503.7 0.337 AAAB 10 G263.6177+00.3652 3.9 6.0 7.6 27.7 12 DGL 11 8:49:26 -43:17:12 2.01 12 (IRS20) DGL 10 8:49:26 -43:17:21 0.02 19 MMS24 08476-4306 5.7 44.0 216.3 503.7 0.337 AAAB 48 G263.6280+00.3847 0.3 <0.6 <0.5 <1.3 28 <0.03 (IRS20) MMS25 08477-4359 9.0 26.3 317.0 580.8 0.174 BAAA 20 G264.3225-00.1857 4.7 5.0 2.3 8.0 19 DGL 12 8:49:33 -44:10:60 0.21 30 (IRS21) MMS26 08477-4359 9.0 26.3 317.0 580.8 0.174 BAAA 16 G264.3225-00.1857 4.7 5.0 2.3 8.0 16 DGL 12 8:49:33 -44:10:60 0.21 6 (IRS21) MMS27 DGL 13 8:49:36 -44:11:46 0.03 16 MMS28 08483-4305 1.5 2.5 <39.0 189.2 0.049 EEFD 14 G263.6909+00.4713 0.1 0.6 <0.7 <2.0 11 not obs. MMS29 08483-4305 1.5 2.5 <39.0 189.2 0.049 EEFD 45 not obs. umms1 08446-4331 <1.1 0.5 7.4 42.8 0.160 GBBB 2 not obs. umms8 08458-4332 1.1 2.7 17.8 52.7 0.017 CABA 24 <0.03 umms9 08458-4332 1.1 2.7 17.8 52.7 0.017 CABA 29 <0.03 umms11 G263.7651-00.1572 0.2 <0.5 0.5 <1.2 11 DGL 6 8:47:43 -43:43:48 0.07 10 umms16 08464-4335 <0.3 <0.3 6.6 <47.2 0.160 -JB- 19 <0.03 umms23 G263.5672+00.4036 0.1 <0.9 <0.7 <1.9 17 not obs. umms24 G263.5622+00.4185 0.2 <0.7 <0.5 <1.5 13 not obs. umms25 G263.5622+00.4185 0.2 <0.7 <0.5 <1.5 19 not obs. Notes to the table: the mm sources labeled as umms♯ are not resolved by the SIMBA HPBW (see text and Massi et al. 2007). aBold faced sources are more compelling associations (within the FWHM-ellipse of the core, see text). bPoint source correlation coefficient encoded as alphabetic characters (A=100%, B=99%, ..., N=87%) according to the IRAS Catalogs and Atlases Explanatory Supplement (IRAS-PSC 1988). cThis core has been divided into two components by Giannini et al. (2005). dNames following the numbering convention used in Massi et al. (1999). A description of the Timmi2 observations of these objects is given in Giannini et al. (2005). 8 M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D Table 2. NIR sources and H2 emission associated with dust cores. mm core FWHM ♯ of NIRa Counterparts candidates H2 emission name ∆x-∆y red / very red sourcesb name RA Dec morphologyc (”) within the FWHM-ellipse (J2000) (J2000) MMS1d 59-64 5 / 8 e MGL99 179 08:45:32.86 -43:50:03.80 diffuse, knots MMS2d 59-55 1 / 3 e MGL99 25 08:45:35.93 -43:51:45.60 diffuse, jet-like MMS3d 61-58 0 / 1 e MGL99 36 08:45:39.21 -43:51:34.70 diffuse MMS4d 82-71 19 / 43 f MGL99 57 08:46:34.77 -43:54:30.63 diffuse, jet-like, knots MMS5 40-31 0 / 0 diffuse MMS6 27-23 0 / 0 diffuse MMS7 29-37 1 / 4 - MMS8 24-26 0 / 1 2M 9671 08:48:39.13 -43:31:31.36 - MMS9 32-36 0 / 0 - MMS10 25-28 0 / 0 - MMS11 36-24 0 / 1 2M 14732 08:48:46.54 -43:37:44.02 - MMS12d 57-37 8 / 20 f MGL99 49 08:48:48.51 -43:32:29.08 diffuse, knots MMS13 41-20 2 / 0 f MGL99 2 08:48:50.04 -43:33:19.47 - MMS14 29-38 0 / 0 - MMS15 33-20 0 / 0 - MMS16 23-36 0 / 0 diffuse, jet-like MMS17 67-37 0 / 2 jet-like MMS18d 36-36 1 / 0 diffuse MMS19 26-60 3 / 2 2M 36076 08:49:08.49 -43:35:37.79 - MMS20d 38-32 0 / 0 knots MMS21d 54-57 0 / 3 2M 29953 08:49:13.39 -43:36:29.20 knots MMS22d 35-37 9 / 13 f MGL99 98 08:49:26.23 -43:17:11.11 diffuse, jet-like, knots MMS23 39-22 0 / 0 diffuse MMS24d 35-25 0 / 3 f MGL99 90 08:49:32.27 -43:17:14.43 - MMS25d 52-86 3 / 5 diffuse MMS26d 60-67 1 / 8 diffuse MMS27d 67-48 1 / 2 knots MMS28d 30-31 0 / 0 - MMS29d 40-36 1 / 1 2M 36339 08:50:11.04 -43:17:10.74 knots umms1d <24-24 0 / 0 not observed umms2 <24-24 0 / 0 - umms3 <24-24 0 / 0 - umms4 <24-24 0 / 1 - umms5 <24-24 1 / 0 - umms6 <24-24 1 / 0 - umms7 <24-24 0 / 0 - umms8d <24-24 1 / 1 2M 16128 08:47:37.87 -43:43:42.41 - umms9d <24-24 0 / 0 - umms10 <24-24 0 / 0 - umms11d <24-24 0 / 2 2M 9173 08:47:42.93 -43:43:48.05 - umms12 <24-24 0 / 0 - umms13 <24-24 0 / 0 - umms14 <24-24 0 / 0 - umms15 <24-24 0 / 0 - umms16d <24-24 0 / 0 jet-like umms17 <24-24 0 / 0 jet-like umms18 <24-24 0 / 0 jet-like umms19 <24-24 1 / 0 2M 10742 08:48:33.95 -43:30:47.20 knot umms20 <24-24 0 / 0 knot umms21 <24-24 0 / 3 2M 16489 08:48:37.03 -43:13:53.63 - umms22 <24-24 0 / 0 - umms23d <24-24 0 / 1 2M 11799 08:49:24.56 -43:13:15.49 - umms24d <24-24 0 / 0 - umms25d <24-24 0 / 0 - umms26 <24-24 0 / 0 - aWhere not specified, the reported numbers refer to 2MASS data. bThe terms red and very red refer to different regions of the colour-colour diagram (see text and e.g. Fig. 2-a). cBold-faced if inside the FWHM-ellipse. dCores presenting a MIR or FIR association (see Tab. 1). eCore only partially covered by IRAC2 observations. fCore fully covered by IRAC2 observations. M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D 9 5. Discussion 5.1. IR counterparts of dust cores The starting information is the sample of 29 well re- solved and 26 unresolved dust cores (Massi et al. 2007). As illustrated in the AppendixA, some of them are not isolated, but belong to more complex structures of dust emission that present different cores (e.g. the one composed by MMS7, umms13-14-15, Fig.A.6); other are elongated structures formed by aligned knots (e.g. both MMS5, MMS6, umms6 of Fig.A.5 and umms23-24-25 of Fig.A.38). These multiple core structures often present as- sociated FIR counterpart(s) that tend(s) to be located in a position intermediate between the individual cores. This is the reason why in Tab. 1 the same IRAS/MSX source is sometimes assigned to two different cores. Conversely, the NIR counterparts distribution is more clearly defined because of the increased spatial resolution at such wave- lengths. In Appendix A all the observational details per- taining to each individual dust core are presented; however we can draw here some general remarks. Firstly, all the in- formation given in Tabs. 1 and 2 are statistically summa- rized in Tab. 4, where, among the 29 resolved cores (MMS) only 8 are clearly associable to a MIR/FIR source (with or without a near-IR counterpart); 7 have some MIR/FIR source in their neighbourhood, but additional evidences (i.e. the presence of H2 jets) point back to an embedded object not (yet) detected; 7 additional cores seem to be associated with very red NIR objects; the 7 remaining cores do not present any sign of star formation activity at the current instrumental sensitivity. If the same ap- proach is applied to the sample of the 26 unresolved cores (umms), we obtain that 4 are associable to a MIR/FIR source, 3 to a NIR counterpart, 3 to jet-like structures, and 16 appear as inactive sites. Substantially, both MMS and umms present a similar statistics of associated cat- egories, although the latter sample is more widely dom- inated by objects that could be artifacts, caused by the searching algorithm, or sites harbouring weak IR counter- parts. The total number of resolved cores associable to an IR object (irrespective of being NIR or FIR sources) is 15 (column 6 of Tab. 4) with respect to the 14 unassociated cores (column 7): such percentages are in full agreement with those found in other galactic surveys of star forming regions (e.g. Yonekura et al. 2005, Mookerjea et al. 2004, Beltrán et al. 2006). Such categorization, however, does not necessarily reflect a property intrinsic to the cores themselves, but is likely the product of the limiting instrumental sensitivities of the considered facilities. In fact, recent results of SPITZER MIR surveys have substantially modified the percentage of active vs inactive cores in favour of the first ones (e.g. Young et al. 2004). Such a caveat should be taken into account when drawing general conclusions from our anal- ysis. 5.2. Star formation modalities and evolutionary stages Different modalities of star formation are simultaneously active in VMR-D. Such a co-existence is confirmed by the presence of 8 clusters (see last column of Tab. 4) and by the remaining cases of isolated star formation. This twofold modality, already recognized in Orion (e.g. Chen & Tokunaga 1994) and now in VMR-D as well, seems to be a feature of all the regions where intermediate and high- mass stars form. Indeed, it is likely that the lacking detec- tion of the isolated mode in far and massive star forming regions is only due to limitations on sensitivity and spatial resolution. To investigate whether or not the different cores of VMR- D harbour protostellar objects in different evolutionary stages we have constructed the distribution of the spec- tral slope of the sources associated to the cores (see Fig. 3). For each source the slope α is calculated through the re- lationship α = ∆ log(λiFλi)/∆ log(λi), between the wave- lengths λ1 and λ2, corresponding to about 2 and 10µm, respectively. A certain degree of inhomogeneity is intro- duced by the fact that the flux attributed to the 10µm band corresponds to that detected by different instru- ments (MSX, Timmi2, IRAS) operating at different ef- fective wavelengths (8.28, 10.4 and 12µm, respectively). These differences affect only the details of the slope dis- tribution, but do not alter its significance, as proved by the five sources having a multiple detection. Fig. 3 illustrates how the sample of the sources associated to the dust cores presents a distribution highly peaked at values 0 < α < 3, typical of Class I sources. It is worthwhile noting that the NIR contribution to the slope generally relies on 2MASS data, that provide a K band limiting magnitude of about 14mag; deeper NIR surveys (see e.g. Giannini et al. 2005), could make it possible to find weaker NIR counterparts, increasing the number of sources with high α values. As a result, the presented bar graph (Fig.3) could be shifted to- wards larger α values. As expected, the sources associated to the dust cores are essentially Class I objects, although the distribution presents a significant tail toward the less evolved objects, whose slope is greater than 3. In Tab. 3 we also list all the IRAS sources lying inside the region mapped in the 1.2mm continuum and showing a flux increasing with wavelength but not directly associ- ated to any dust core (see Sect. 4). These sources, with only few exceptions, tend to be distributed all along both the diffuse emission detected by MSX at 8.3µm and the gas filaments. To evaluate the intrinsic nature of all the IRAS sources discussed so far (both associated or not), we present, in Fig. 4, an IRAS two colours diagram, [12-25] vs [25-60], with all those sources. As expected, the objects not asso- ciated to the cores have a greater number of upper limits (especially at 12µm), but tend to occupy the same re- gion of the plot where the associated sources are located. This common region pertains to sources that can be de- scribed as a two dust components system, one at 1000K and the other in between 50 and 100K, with variable rela- 10 M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D Table 3. IRAS point sources with increasing fluxes not associated with any core. id F12 F25 F60 F100 F1.2mm CC 08440-4253 <0.3 0.2 1.2 <30.5 <0.005 -BBD 08461-4314 0.7 0.8 5.9 <48.5 0.003 BCDB 08475-4255 <0.4 0.4 3.3 45.6 0.020 EDDB 08475-4311 <0.5 0.5 9.4 <37.8 0.020 DEC- 08481-4258 <0.4 1.1 7.6 <34.8 0.008 BDC- 08459-4338 <0.2 0.6 6.5 38.3 0.005 -DCB 08442-4328 <1.4 2.2 41.3 113.9 0.020 HDCA 08448-4341 1.3 6.6 <327 <1005 <0.005 DAC- 08468-4330 <0.3 0.4 3.4 <31.6 0.010 FDD- 08491-4310 0.5 0.6 7.6 38.6 <0.005 EBCC 08496-4320 0.6 0.7 7.6 <37.1 <0.005 BCBH 08463-4343 <0.3 0.5 7.4 <43.5 <0.005 NBDG 08478-4403 0.5 0.5 <5.4 <580.8 <0.005 CCJG aSee note b in Tab. 1. Table 4. Statistics about the dust cores population. Cores Associated with MIR/FIR only very red H2 jet no source NIR or no source or embedded NIR object FIR object H2 only cluster 29 resolved 8 7 7 7 15 14 6 26 unresolved 4 3 3 16 7 19 2 tive contributions. The IRAS selected sources have colours definitely redder and colder than those pertaining to pre- main sequence stars (T-Tauri and Herbig Ae/Be; see, e.g., Berrilli et al. 1992), apart from few cases. While the ab- sence of IRAS sources with the colours of T-Tauri stars in our sample is probably due to an observational bias (in fact the IRAS detectability limit is > 0.5L⊙ in Taurus, corresponding to > 10L⊙ in VMR-D), the doubtful pres- ence of Herbig Ae/Be appears to be an intrinsic property of VMR-D. Indeed, Herbig Ae/Be located inside dust cores having masses comparable to those belonging to the VMR- D ones have been already found at distances of 1Kpc or less (e.g. Henning et al. 1998). This difference likely re- flects different star formation histories, being in VMR-D a shorter time (≈ 106 yr) elapsed from the earliest collapse events (Massi et al. 2000) with respect to other studied massive clouds. We have calculated the bolometric luminosity, Lbol, for all the IRAS sources from 12µm to 1.2mm. The bar graph in Fig. 5 shows our result: the sources associated to the dust cores (both resolved and not) are, on average, objects of intermediate luminosity (Lbol ∼ 10 3L⊙), while the unas- sociated FIR sources, even showing similar SEDs, tend to be lower luminosity objects (Lbol ∼ 10 2L⊙). Therefore, we can firmly conclude that massive star formation (Lbol > 104L⊙) does not occur in VMR-D, and that our sample of Class I sources is not contaminated by ultracompact HII regions, which would be indistinguishable based on their FIR colours alone (Wood & Churchwell 1989). An attempt to search for Class 0 objects within our sam- ple of IRAS sources has also been done by applying the criterion proposed by André et al. (1993) for low-mass protostars: Lbol/L1.3mm . 2 × 10 4. None of the selected IRAS sources (Tabs. 1 and 3) strictly satisfies this crite- rion6. It should be said, however, that overestimates of the IRAS fluxes result in overestimates of the Lbol/L1.3mm ra- tio. Thus, leaving any quantitative approach, we point out that three sources of our sample (08446-4331, 08474-4325 and 08464-4335) show a ratio Lbol/L1.3mm one order of magnitude less than the others and thus they are likely the youngest objects in the field. This hypothesis is strongly supported by (i) the tight association of these sources with the mm cores umms1, MMS21 and umms16, respectively; (ii) by the lack of a measured flux at 12µm7, and, but only for the last case, (iii) by the presence of H2 jet-like emis- sion crossing the dust peak (no H2 images are available for the core umms1). 5.3. H2 survey A powerful approach to indirectly probe ongoing star for- mation activity is to investigate over the presence and characteristics of collimated flows, which commonly ex- 6 Hatchell et al. (2007) reduces such limiting ratio to Lbol/L1.3mm . 3×10 3 and underlines how this indicator should vary with envelope mass. 7 For the core MMS21 with the associated source 08474-4325 we report in Tab. 2 a candidate NIR counterpart, but such as- sociation is quite questionable (see Appendix A and Fig. A.18). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D 11 tend far from the most embedded regions where the young protostars are located. For this reason, we have complemented our search for infrared counterparts of mm cores with a survey in the narrow-band filter cen- tered at the H2 2.12µm line, which represents the main NIR cooling channel of shock-excited gas at thousands of Kelvin (e.g. Gredel 1994). Our observations have been also aimed to define the occurrence of the jet phenomenon in intermediate-mass star forming regions and to understand whether jets in clusters are associated with the massive cores or with some closeby low-mass component. A de- tailed description of the H2 emission detected in the var- ious fields is given in Appendix A, while here we briefly comment on some general aspects. Out of 54 investigated fields, we find a positive detection (in the form of jet- like morphologies, knots and diffuse emission) in 23 fields, the large majority (18) being related to resolved cores. We note that the occurrence of jets in proximity of umms cores testifies in favour of the genuine nature of some of these latter. The H2 emission seems to be directly connected with the mm cores in about one third of the investigated fields (namely MMS2-19-22-29, umms16-19 and possibly MMS16), but we are able to identify a possible (NIR) driv- ing source in only two cases (MMS22 and MMS29, see Tab. 2, last column). Given the poor sensitivity of MSX and IRAS observations, Class 0 protostars might already be embedded in the other dust cores, which represent interesting cases to be investigated with SPITZER. All the fields of young clusters (those associated with cores MMS2-4-12-22 and 25) show H2 emission and close to three peaks (MMS2-4-22), all associated with an IRAS source, we have found a sub − parsec scale jet (with ex- tension 0.30, 0.30 and 0.68 pc, respectively). Only in one case the jet seems to be emitted by the most luminous object in the field (i.e. the one that contributes the most to the FIR flux in the IRAS bands), in all the other cases the driving source is a less luminous (and less massive) object in the cluster. A similar result was pointed out by Lorenzetti et al. (2002), who, having surveyed a sample of 12 IRAS protostellar candidates in VMR, have detected H2 emission in 5 fields, clearly coming from low-mass ob- jects clustered around the IRAS source, and not from the IRAS source itself. This feature is likely related to a dif- ferent duration of the jet phase in low and high luminosity (mass) sources: this topic will be discussed in depth in a forthcoming paper (Giannini et al. 2007, in preparation). 6. Conclusions A southern sky area of 1×1deg2 belonging to the star forming region VMR-D, previously surveyed at mmwavelengths to identify the dust cores, has been stud- ied by means of both IR catalogues (IRAS/MSX/2MASS) and a set of new dedicated observations, to identify the young protostellar counterparts associated to dust cores. The motivation for the new IR observations is twofold: (i) to perform a complete survey of all the recognized dust cores in the H2 narrow-band filter (2.12µm) to search ev- idence of protostellar jets; and (ii) to obtain a broad band N (10.4µm) survey of those cores associated to embedded clusters, aiming to pick up the source(s) that mainly con- tribute(s) in the far-IR regime. The main conclusions of this work are summarized here: - In the majority of cases, MIR and/or FIR sources as- sociated with dust cores do not coincide with the mm peaks, although they are located in their vicinity. In those cases of close-by cores, often the IR source is located in between them. - The resolved mm cores (i.e. those larger than the in- strumental beam) are more frequently associated to a NIR or FIR counterpart than the unresolved ones (smaller than the instrumental beam). The existence of signs of star formation activity around these latter in the form of H2 jets, however, attests the genuine nature of lots of them. - The statistics of active vs inactive cores is in good agreement with that found in other star forming re- gions, but should be critically revised in the light of more sensitive observations that will become available in near future (e.g. SPITZER MIPS and IRAC maps). - The SEDs of the associated sources present a slope α = d log(λFλ)/d log(λ) between 2 and 10µm whose distribution is strongly peaked at values typical of Class I sources (0 < α < 3), in some cases even larger. - An attempt has been done to search for Class 0 sources. Ten IRAS sources (with upper limits at 12µm) do not present any NIR reliable counterpart, but no one of them satisfies the criterion Lbol/L1.3mm . 2×10 4 pro- posed by André et al. (1993). However, considered the probable overestimates of the IRAS fluxes (and bolo- metric luminosities), we indicate the three objects with the lowest Lbol/L1.3mm ratios as the youngest IRAS sources of the region. - The sources associated to the dust cores, both resolved and unresolved, have all the same FIR colours, typical of a black-body stratification between 50 and 1000K, with a stronger contribution of the former component. In other words, sources with FIR colours typical of pre-main sequence T-Tauri and Herbig Ae/Be stars seem to be absent, indicating VMR-D as a young (∼ 106 years) region. - Sources associated with unresolved cores are system- atically less luminous (average Lbol ≃ 1.5 × 10 than those related to the resolved ones (average Lbol ≃ 1.5 × 103L⊙), providing evidence that two modalities of star formation, namely low- and intermediate-mass, are simultaneously present in VMR-D. - This occurrence is also confirmed by the existence of a further (low luminosity) population of young objects that have the same colours as the sources associated to the dust cores (resolved or unresolved), but are lo- cated, on the contrary, in the diffuse warm dust and along the gas filaments. - Observing the dust cores in the H2 1-0S(1) line ev- idences a high detection rate (30%) of jets driven 12 M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D by sources located inside the cores. 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In cases of multiple estimates of the 10µm flux (Timmi2, MSX, IRAS) the value obtained with better spatial resolution has been used to determine α. 130 K 150 K 200 K 100 K T-Tauri Herbig Ae/Be -0.5 0.0 0.5 1.0 1.5 2.0 2.5 [25-60] ↑ ↑ ↑ Associated with: MMS (1FWHM-ellipse) MMS (2FWHM-ellipse) umms (1FWHM-ellipse) Not associated Fig. 4. Two colours diagram for all the IRAS sources listed in Tabs. 1 and 3. The mean error bars, the extinction vector corresponding to 5mag of visual extinction and the locus of blackbodies (leftmost line) are indicated (see text for more details). 14 M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D <1.5 1.5-2 2-2.5 2.5-3 3-3.5 3.5-4 >4 Log(Lbol / L ) s Associated Not associated Fig. 5. Luminosity distribution of both associated and not associated IRAS sources (Tabs. 1 and 3). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 1 Online Material M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 2 Appendix A: Counterparts of the dust cores In the following we present a brief discussion of the NIR to FIR associations found for each dust core. Other signs of star formation activity (e.g. H2 knots and jets or young embedded clusters) will be also evidenced. For each core (or group of cores) we will show the SofI H2 (or 2MASS H band) gray scale image supplied with: dust emission contours, position of the most interesting sources, cov- erage of IRAC2 and Timmi2 fields of view and arrows indicating the occurrence of H2 knots (see Sect. 3.4 for more details). Whenever needed for the analysis, the cor- responding colour-colour diagrams and SED plots will be shown as well. – MMS2 (Figs. A.1, A.2): one MSX associated ob- ject inside the FWHM-ellipse, G263.6338-00.5497, with a N band counterpart, DGL 3, having a flux (F10.4µm = 0.25 Jy) smaller than that measured by MSX (F8.3µm = 0.4 Jy). Inside the FWHM-ellipse, many NIR stars (recogniz- able in the SofI H2 image) have not been detected by 2MASS and are only upper limits for IRAC2 in both J and H bands. The object showing the highest colour excess (MGL99 25) and steepest spectral index (Fig. A.2) is not visible in the N band (at 30mJy sensitivity level). We have tentatively indicated that one in Tab. 2 as candidate NIR counterpart. However, an intense H2 line emission is present all over the core, both diffuse and in knots and, remarkably, a well collimated H2 jet crosses the very center of the core. No reliable exciting source has been detected along the jet. Likely, it is heavily embedded near the peak position and contributes significantly (or mainly) to the observed dust emission. – MMS3 (Figs. A.3, A.4): an IRAS source (IRS16) is marginally associated to the peak (see Sect. 3.4). Three MSX detections (one of them inside the FWHM- ellipse) fall in a region of enhanced and diffuse 8µm emission and don’t seem to be point-like (as suggested by the MSX image and by the failed detection in N). A Timmi2 source (DGL 5, identified with MGL99 65) is instead observed in the direction of the IRAS source and NIR cluster. A steep decrease in the number of NIR detections towards the peak suggests a high extinction level. Moreover, the majority of the IRAC2 sources within the FWHM-ellipse presents very red colours. MGL99 36, the one closest to the peak position, could be the main source associated to the millimeter core. – MMS4: a detailed analysis of this core has been already presented in a dedicated paper (Giannini et al. 2005), which the reader is referred to. – MMS5-6, umms6 (Fig. A.5): a dust elongated struc- ture of connected cores pointing towards south-west in the direction of MMS4, the brightest core of the whole dust map; a dust filament which is likely undergoing fragmentation. Neither MIR-FIR associations, nor interesting NIR sources are present (although no IRAC2 data are available); signs of star formation activity are quite hidden and can be only recognized as a faint H2 emission around MMS5 and MMS6. – MMS7, umms13-14-15 (Fig. A.6): resolved core (MMS7) surrounded by three unresolved cores (umms13-14-15). The J , H and Ks photometry of the cluster of about 20 members around the central core will be analyzed in a forthcoming paper. Neither FIR emission nor H2 features are present (no 10µm image has been collected). We just remark here the colour excess of 4 very red stars within the FWHM-ellipse of MMS7. – MMS8-9 (Fig. A.7): these cores are part of a long tail of connected cores (extending for about 1 pc), going from the bright core MMS12 to umms19, whose dynamical behaviour is not clear (Massi et al. 2007). No FIR point sources or clues of H2 emission have been observed. Timmi2 observations, although not covering the whole dust emission, gave negative results as well. Only one very red NIR object, 2M 9671 (no IRAC2 data available), lies within the FWHM-ellipse of MMS8. – MMS10-11 (Fig. A.8): without any MIR-FIR coun- terpart, these are the only two resolved cores included in the 13CO map that lack of an associated CO clump (Elia et al. 2007). The colour-colour diagram of the 2MASS sources (no IRAC2 data available) within the FWHM-ellipse gives only one reddened candidate (2M-14732) for MMS11. – MMS12 (Figs.A.9, A.10): one of the brightest cores, MMS12 (∼ 18M⊙) coincides with a young embedded cluster having in its center the IRAS source 08470-4321 (IRS19), the MSX G263.7434+00.1161 and the Timmi2 DGL 7 object. The new observed flux at 10µm (F10.4µm = 18.3 Jy) is less than a half of the IRAS/MSX measurements at 8-12µm. We ascribe this discrepancy to the presence of a strong diffuse contribution to the flux at these wavelengths. The correspondence of these sources with the NIR object MGL99 49 has been already discussed in Massi et al. (1999) and is confirmed here. Complex and intense H2 emission is also present, especially at the peak position, and at the eastern part of the core; it cannot be clearly distinguished as a single jet-like structure. – MMS13 (Fig.A.11): no MIR-FIR association and no significant H2 emission for this core at the south of MMS12. The two possible NIR counterparts have red colours (MGL99 2, MGL99 7). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 3 – MMS14-15-16 (Fig. A.12): neither MIR-FIR associ- ations nor NIR red sources detected by 2MASS (no IRAC2 data). We note a faint H2 emission aligned with both the peak MMS16 and the H2 knot visible at the east of MMS12 (see also Fig.A.9). We cannot exclude the existence of an embedded exciting source near the MMS16 peak position. – MMS17 (Fig. A.13): a double H2 knot (probably jet- like) in proximity of the peak suggests the existence of star forming activity, but no MIR-FIR objects have been detected and the 2MASS data do not point out any interesting source within the FWHM-ellipse. – MMS18 (Figs.A.14, A.15): one IRAS source (08472- 4326A) is associated within 2FWHM-ellipse, while one MSX source (G263.8432+00.0945) and one Timmi2 object (DGL 8) are inside the FWHM-ellipse. The positional uncertainties of these three objects seem to exclude their coincidence, although the IRAS and MSX fluxes (due to their beam sizes) are surely con- taminated by the Timmi2 source (whose counterpart, 2M 29896, peaks in the H band) and probably by diffuse emission, visible in the H2 filter as well. – MMS19 (Fig. A.16): core connected to MMS20 and MMS21. Neither MIR-FIR sources associated nor H2 emission detected. We signal two very red 2MASS ob- jects (2M-36076, 2M 36192) within the FWHM-ellipse. – MMS20 (Fig. A.17): one IRAS (08474-4323, corre- sponding to MSX G263.8221+00.1494) source turns out to be very marginally associated (positional uncer- tainty tangent to the 2FWHM-ellipse), but the dust emission around its position seems to be negligible. One Timmi2 source (DGL 9) has been detected within 2FWHM-ellipse, in the middle between this core and MMS21 not signalled by IRAS/MSX (SofI images points out at least three objects, partially resolved by 2MASS in two very red sources, 2M 27831 and 2M 37241). – MMS21 (Figs. A.18, A.19): one IRAS source, 08474- 4325, whose fluxes indicate this one as one of the youngest objects of VMR-D, coincides with the peak position. The Timmi2 observation has given no results (for the source DGL 9 see MMS20 description) and no MSX sources are reported in the catalogs. Three 2MASS sources fall close to the IRAS uncertainty ellipse center, two of which (2M-29953, 2M 37193) have very red colours. The SofI H2 image, however, points out the presence of a complex morphology, which cannot be resolved in individual sources with the SofI spatial resolution. – MMS22 (Figs. A.20, A.21): characterized by a power- ful bipolar jet (0.7 pc long, discussed in a forthcoming paper) arising from the IRAS source 08476-4306 (IRS20), located 10′′ at the west of the peak. In cor- respondence with the IRAS source there are a young cluster, a MSX point source (G263.6177+00.3652) and two 10µm objects: DGL 11 (main counterpart of the IRAS/MSX object) and DGL 10 (2σ de- tection, outside the FWHM-ellipse). The DGL 11 flux, F10.4µm = 2.01 Jy, is significantly smaller than those measured by MSX and IRAS: F8.3µm = 3.9 Jy, F12µm = 5.7 Jy, F12.1µm = 6.0 Jy, but the Timmi2 ob- servation points out a diffuse emission which can have contributed to the MIR fluxes measured by IRAS and MSX. The presence of nebular NIR and H2 emission completes the picture of this crowded region. The NIR cluster characteristics have been discussed in detail by Massi et al. (1999) and the conclusions reported in that paper about the candidate NIR counterpart (MGL99 98) of IRS20 are confirmed by our Timmi2 observation. MGL99 98 is also the best candidate as exciting source of the jet, although the complexity of the jet morphology and the difficulty to discriminate between nebular and point-like unresolved emission makes this identification questionable. We also remark one dark strip clearly visible in the H2 image, likely due to an obscuring dust lane crossing the cluster center. – MMS23 (Fig. A.22): isolated, small core with no IR detected sources near the peak (not observed at 10µm). H2 multiple knots are visible to the north of the peak together with a faint emission near the center. We cannot exclude the presence of an embedded, young, low-mass stellar object producing that emission, but more sensitive observations are required to confirm this possibility. – MMS24 (Fig.A.23): this core is connected with the brighter MMS22 and is characterized by diffuse K emission close to a MSX source (within 2FWHM- ellipse) not detected by Timmi2. The NIR source MGL99 90, at the border of the FWHM-ellipse, shows the highest colour excess. – MMS25-26 (Figs. A.24, A.25): this complex re- gion is constituted by two not well resolved cores (also linked to the brighter MMS27) and, between them, in correspondence with a young NIR cluster (Massi et al. 2006), there are the IRAS source 08477- 3459 (IRS21), the MSX source G264.3225-00.1857 and the Timmi2 object DGL 12 (corresponding to the NIR MGL99 27). Moreover, the 10µm emission observed by Timmi2 shows a diffuse emission (in the surroundings of the source MGL99 35) and, probably, another point source corresponding to MGL99 32, although an artifact in the Timmi2 image prevented us to give a reliable estimate of its flux. Remarkable is also a shell-like K emission (clearly visible also in the H2 image) approximately centered near MGL99 63. The colour-colour diagram (Fig. A.25) points out the M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 4 existence of many red and very red sources within the FWHM-ellipse of both the cores, lots of which having upper limits in the J and H bands. The SEDs of the MIR-FIR objects and of the NIR stars with larger colour excess show significant discrepancies among the fluxes of the MIR-FIR detections. No clear evidence has been found for NIR stars (if any) that can be most likely associated to the mm core8. – MMS27 (Fig. A.26): although this is one of the bright- est cores of the whole dust map, no IRAS-MSX point sources have been detected. Only one faint object (DGL 13), counterpart of the NIR very red 2M 53071 (we lack of IRAC2 data for this source), can be seen at 10 µm, inside the FWHM-ellipse, together with three more very red sources. Interesting is the case of 2M 47136, the closest one to the peak: no point-like source can be extracted from the K image which shows instead a very diffuse emis- sion well observable also in the H2 image of Fig.A.26. It is crossed by a dark horizontal strip (quite likely due to obscuring dust). A knot of H2 emission is also visible above this strip and one more on the other side of the core, in the south-west direction, making this core of a peculiar interest for future investigations. – MMS28-29 (Fig. A.27): inside the FWHM of MMS28 there are the IRAS 08483-4305 and MSX G263.6909+00.4713 sources and, remarkably, H2 aligned knots are visible between the two peaks. We lack of both Timmi2 and IRAC2 observations, but the 2MASS data reveal one very red source, 2M 36339, incompatible with the position of the FIR sources, but aligned with the knots, which could be their exciting source. In the following we present the unresolved cores not previously discussed. – umms1 (Fig. A.28): despite the high noise level of the dust map in this position, this unresolved core is one of the most interesting cases: (i) it is by far the most intense core among the unresolved ones, (ii) its coordinates coincide with an IRAS source showing fluxes increasing with wavelength and (iii) the 12CO integrated emission map presents an increase towards this peak, altough it falls immediately outside that map (see note 5). Unfortunately we lack of both IRAC2 and SofI images at this position and only one 2MASS (red) object falls inside the FWHM-ellipse. 8 The lack of any Timmi2 detectable flux in correspondence of MGL99 50, the previously hypothesized NIR counterpart of the IRAS source (Massi et al. 1999), makes that association questionable. Observing its SED, indeed, it is quite unlikely that it could be missed, if point-like, at 10µm. – umms2-3-4-5 (Fig. A.29): region of high noise level of the dust map. These cores are probably artifacts of the reconstruction algorithm and do not present any feature suggesting star formation activity. – umms7-10-12 (Fig. A.30): a small cluster within 2FWHM-ellipse of umms10 (forthcoming dedicated paper) is the only noticeably feature of this field. – umms8-9-11 (Figs. A.31, A.32): dust emission as- sociated with an IRAS source (08458-4332, having flux increasing with λ, associated to umms8-9) and a MSX-Timmi2 object (G263.7651-00.1572-DGL 6, associated with umms11 and the 2MASS objects 2M 9173 and 2M 11131). One very red and one red NIR object (2M-116128 and 2M 18032, respectively) within the FWHM-ellipse of umms8. – umms16 (Fig. A.33): core remarkably crossed by H2 jet-like emission and with an IRAS source (08464- 4335), although the Timmi2 observation gave no results and no red or very red NIR stars have been observed by 2MASS as possible jet exciting source. – umms17-18 (Fig. A.34): also this couple of cores presents an intense H2 jet-like emission, but without any NIR-MIR interesting source. – umms19-20 (Fig. A.35): these cores are part of a chain of connected cores (see description of MMS8-9). No FIR-MIR point sources have been observed and only a knot of H2 emission is visible in the FWHM- ellipse of umms19, close to a red 2MASS source (2M-10742). – umms21 (Figs.A.36): connected to the previous cores. No FIR-MIR point sources or clues of H2 emission have been observed. One very red NIR object (2M-16489) within the FWHM-ellipse. – umms22 (Figs.A.37): No FIR point source or clues of H2 emission have been observed (no Timmi2 data available). Absence also of NIR red objects within the FWHM-ellipse. – umms23-24-25 (Figs.A.38): thin, elongated dust structure with three cores coinciding with a sim- ilar shaped K emission. Two MSX sources, G263.5672+00.4036 and G263.5622+00.4185, are tightly associated to umms23 and umms24, respec- tively. Two very red sources fall within the FWHM of umms23, but they seem to be incompatible with the position of the MSX sources (for which we suspect a high contamination from diffuse emission). – umms26 (Figs. A.39): a small cluster outside 2FWHM-ellipse (forthcoming paper) is the only M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 5 noticeably feature of this field. M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 6 Fig.A.1. MMS2 field of view (center [J2000]: 08:45:34.200, -43:51:54.40). Grayscale image: H2 emission; green con- tours: dust continuum (from 3 σ, in steps of 3σ); red ellipses: FWHM-ellipse and 2FWHM-ellipse of the core (see Sect. 3.2 for an explanation); green and magenta ellipses: MSX and IRAS 3σ positional uncertainty; orange line: IRAC2 field of view. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 [H-K] MGL99-4 MGL99-25 MGL99-14 MGL99-9, MGL99-15 1 2 3 -20.0 -19.5 -19.0 -18.5 -18.0 -17.5 2M-26276 MGL99-25 MGL99-4 MGL99-14 MGL99-9 MGL99-15 Fig.A.2. a) Colour-colour diagram (see text for more details) of the NIR sources within the FWHM-ellipse of MMS2. The Spectral Energy Distributions of the very red ones are shown in panel b). To reduce confusion the wavelength range is limited to the J , H , K bands and the main objects are in red colour. Dashed lines refer to 2MASS sources and arrows denote upper limits. M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 7 Fig.A.3. MMS3 field of view (center [J2000]: 08:45:39.5, -43:51:25.0). 0.0 0.5 1.0 1.5 2.0 2.5 3.0 [H-K] MGL99-65 MGL99-36 1 2 3 -20.2 -20.0 -19.8 -19.6 -19.4 -19.2 -19.0 -18.8 MGL99-36 MGL99-65 Fig.A.4. MMS3: colour-colour and SED diagrams. Fig.A.5. MMS5-6 and umms6 field of view (center [J2000]: 08:46:52.0, -43:53:01.3). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 8 0.0 0.5 1.0 1.5 2.0 [H-K] 2M-16616 2M-10095 2M-18266 2M-8124 Fig.A.6. MMS7 and umms13-14-15 field of view (center [J2000]: 08:47:58.9, -43:39:22.9) and colour-colour diagram. M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 9 0.0 0.5 1.0 1.5 2.0 2.5 3.0 [H-K] 2M-9671 Fig.A.7. MMS8-9 field of view (center [J2000]: 08:48:41.3, -43:31:36.2) and colour-colour diagram. M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 10 0.0 0.5 1.0 1.5 2.0 2.5 [H-K] 2M-14732 Fig.A.8. MMS10-11 field of view (center [J2000]: 08:48:44.2, -43:37:19.1) and colour-colour diagram. M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 11 Fig.A.9. MMS12 field of view (center [J2000]: 08:48:48.5, -43:32:20.8). 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 [H-K] MGL99-49 1 2 5 10 2 5 10 MGL99-49 DGL-7 IRS19 G263.7434+00.1161 Fig.A.10. MMS12 colour-colour and SED diagrams. M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 12 0.0 0.5 1.0 1.5 [H-K] MGL99-2 MGL99-7 Fig.A.11. MMS13 field of view (center [J2000]: 08:48:49.4, -43:33:11.2) and colour-colour diagram. Fig.A.12. MMS14-15-16 field of view (center [J2000]: 08:48:51.6, -43:31:09.9). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 13 Fig.A.13. MMS17 field of view (center [J2000]: 08:48:57.2, -43:38:23.1). Fig.A.14. MMS18 field of view (center [J2000]: 08:49:03.2, -43:38:05.2). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 14 0.0 0.5 1.0 1.5 [H-K] 2M-29896 1 2 5 10 2 5 10 -19.0 -18.5 -18.0 -17.5 -17.0 -16.5 -16.0 -15.5 -15.0 2M-29896 DGL-8 08472-4326A G263.8432+00.0945 Fig.A.15. MMS18 colour-colour and SED diagrams. 0.0 0.5 1.0 1.5 2.0 2.5 [H-K] 2M-36192 2M-36076 Fig.A.16. MMS19 field of view (center [J2000]: 08:49:08.5, -43:35:43.7) and colour-colour diagram. M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 15 0.0 0.5 1.0 1.5 2.0 [H-K] 2M-37241 2M-27831 Fig.A.17. MMS20 field of view (center [J2000]: 08:49:11.2, -43:35:25.9) and colour-colour diagram. Fig.A.18. MMS21 field of view (center [J2000]: 08:49:13.0, -43:36:21.8). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 16 0.0 0.5 1.0 1.5 2.0 2.5 [H-K] 2M-37193 2M-29953 2M-34100 1 2 5 10 2 5 10 -19.0 -18.5 -18.0 -17.5 -17.0 -16.5 -16.0 2M-29953 2M-37193 2M-34100 08474-4325 Fig.A.19. MMS21 colour-colour and SED diagrams. M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 17 Fig.A.20. MMS22 field of view (center [J2000]: 08:49:26.0, -43:17:13.0). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 18 0 1 2 3 4 5 [H-K] MGL99-98 1 2 5 10 2 5 10 MGL99-98 DGL-10 DGL-11 IRS20 G263.7434+00.1161 Fig.A.21. MMS22 colour-colour and SED diagrams. Fig.A.22. MMS23 field of view (center [J2000]: 08:49:30.2, -44:04:10.0). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 19 0.0 0.5 1.0 1.5 2.0 2.5 3.0 [H-K] MGL99-90 Fig.A.23. MMS24 field of view (center [J2000]: 08:49:30.1, -43:17:00.2) and colour-colour diagram. Fig.A.24. MMS25-26 field of view (center [J2000]: 08:49:33.5, -44:10:34.5). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 20 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 [H-K] 1 2 5 10 2 5 10 DGL-12 IRS21 G263.7434+00.1161 Fig.A.25. MMS25-26 colour-colour and SED diagrams. M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 21 0.0 0.5 1.0 1.5 2.0 2.5 3.0 [H-K] ] 2M-47136 2M-53071 Fig.A.26. MMS27 field of view (center [J2000]: 08:49:35.2, -44:11:52.8) and colour-colour diagram. M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 22 0.0 0.5 1.0 1.5 2.0 2.5 [H-K] 2M-36339 Fig.A.27. MMS28-29 field of view (center [J2000]: 08:50:10.0, -43:16:41.3) and colour-colour diagram. Fig.A.28. umms1 field of view (center [J2000]: 08:46:25.7, -43:42:28.3) (2MASS H band image). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 23 Fig.A.29. umms2-3-4-5 field of view (center [J2000]: 08:46:44.0, -43:19:48.2). Fig.A.30. umms7-10-12 field of view (center [J2000]: 08:47:37.4, -43:26:21.7). Fig.A.31. umms8-9-11 field of view (center [J2000]: 08:47:39.6, -43:43:36.1). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 24 0.0 0.5 1.0 1.5 2.0 2.5 [H-K] 2M-9173 2M-11131 2M-18032 2M-16128 1 2 5 10 2 5 10 -19.0 -18.5 -18.0 -17.5 -17.0 -16.5 -16.0 2M-9173 2M-11131 2M-16128 DGL-6 08458-4332 G263.7651-00.1572 Fig.A.32. umms8-9-11 colour-colour and SED diagrams. Fig.A.33. umms16 field of view (center [J2000]: 08:48:15.3, -43:47:06.5). Fig.A.34. umms17-18 field of view (center [J2000]: 08:48:24.6, -43:31:36.7). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 25 Fig.A.35. umms19-20 field of view (center [J2000]: 08:48:33.9, -43:30:46.0). Fig.A.36. umms21 field of view (center [J2000]: 08:48:36.4, -43:31:11.5). Fig.A.37. umms22 field of view (center [J2000]: 08:48:36.3, -43:16:45.9). M. De Luca et al.: IR Counterparts of mm Dust Cores in the VMR-D, Online Material p 26 0.0 0.5 1.0 1.5 2.0 2.5 [H-K] 2M-15995 2M-11799 Fig.A.38. umms23-24-25 field of view (center [J2000]: 08:49:25.9, -43:12:39.2) and colour-colour diagram. Fig.A.39. umms26 field of view (center [J2000]: 08:49:58.9, -43:22:55.1). Introduction The investigated region Association with point-sources Observations NIR imaging N-band imaging FIR associations NIR associations The core MMS1 Results Discussion IR counterparts of dust cores Star formation modalities and evolutionary stages H2 survey Conclusions Counterparts of the dust cores

0704.1229

High-precision covariant one-boson-exchange potentials for np scattering below 350 MeV

JLAB-THY-07-632 High-precision covariant one-boson-exchange potentials for np scattering below 350 MeV Franz Gross Thomas Jefferson National Accelerator Facility, Newport News, VA 23606 College of William and Mary, Williamsburg, VA 23187 Alfred Stadler Centro de F́ısica Nuclear da Universidade de Lisboa, 1649-003 Lisboa, Portugal and Departamento de F́ısica da Universidade de Évora, 7000-671 Évora, Portugal All realistic potential models for the two-nucleon interaction are to some extent based on boson exchange. However, in order to achieve an essentially perfect fit to the scattering data, characterized by a χ2/Ndata ∼ 1, previous potentials have abandoned a pure one boson-exchange mechanism (OBE). Using a covariant theory, we have found a OBE potential that fits the 2006 world np data below 350 MeV with a χ2/Ndata = 1.06 for 3788 data. Our potential has fewer adjustable parameters than previous high-precision potentials, and also reproduces the experimental triton binding energy without introducing additional irreducible three-nucleon forces. A good understanding of the interaction between two nucleons is essential for the study of nuclear structure and nuclear reactions. In the long history of theoretical mod- els of the NN interaction, One-Boson-Exchange (OBE) models played a role of special importance. Yukawa’s [1] insight that a short-range force can be generated through the exchange of particles of finite mass led to the dis- covery of the pion, and later it was found that the ex- change of a pion can quantitatively describe the longer- range part of the NN interaction. Since the range of the force is inversely proportional to the exchanged mass, the exchange of heavier mass bosons generates NN forces of intermediate to short range. It was found that the vector bosons ω and ρ contribute to the observed spin-orbit force and strong repulsion at short internucleon distances [2], and that scalar bosons provide intermediate attraction. Today, with the development of potentials based on chi- ral perturbation theory (ChPT) [3], we understand that these scalar bosons are an approximate representation of the two-pion exchange mechanism [4], which gives strong attraction even if there were no two-pion resonances at masses of around 500 MeV [5]. It is possible, of course, to construct phenomenolog- ical NN potentials that, with a sufficently large num- ber of parameters, give an accurate description of the NN scattering data. However, OBE potentials have sev- eral important advantages. First, they provide a phys- ical mechanism for the interaction between nucleons. This implies that the parameters in these models have a physical meaning, and that, at least in principle, they can be related to, or even be determined through other physical processes. Second, it is possible to construct consistent electroweak currents for systems interacting through OBE, since the underlying microscopic processes are known [6]. With phenomenological potentials this construction is less straightforward because there is no implied microscopic description of the flow of electroweak charges through a nuclear system. Third, when OBE is used in a covariant formalism without time ordering, effective three- and many-body forces are automatically generated from the off-shell couplings of purely two-body OBE [7, 8]. With phenomenological potentials three- body forces must be independently constructed. Finally, OBE models are relatively simple, and depend only on a moderate number of parameters. A quantitatively accu- rate OBE model represents a very economical description of the NN interaction. OBE models also have their limitations. Since they are not fundamental interactions, their validity does not extend to very short distances where QCD should pro- vide the correct description. In potential models, this unknown short-distance part of the interaction is usually parameterized phenomenologically through vertex form factors with adjustable parameters. These form factors also serve to regularize otherwise divergent loop integrals that appear when the kernel is iterated. But parameters that describe the unknown short distance physics cannot be avoided; even more fundamental potential models de- rived from ChPT require subtraction constants to renor- malize and absorb infinities arising from the unknown short range physics. At fourth order, a potential based on ChPT will have at least 24 unknown subtraction con- stants (parameters) [9]. After early phase shift analyses by the VPI group [10], both the VPI [11] and Nijmegen [12] groups obtained op- timal values of χ2/Ndata ≈ 1 after eliminating data sets from their analyses, based on statistical arguments about their incompatibility with other data sets [13]. The Ni- jmegen group also updated their OBE potential (Nijm78) to the new phase shift analysis, but they were unable to get the χ2/Ndata of this 15 parameter model (now called Nijm93) below 1.87 [14]. In order to construct very accu- rate NN potentials they abandoned a pure OBE struc- ture and made several boson parameters dependent on in- http://arxiv.org/abs/0704.1229v2 dividual partial-waves. Similarly, the (almost) pure OBE potentials of the Bonn family, such as Bonn A, B, and C, were superseded by the realistic CD-Bonn, which also incorporates partial-wave dependent boson parameters [15]. The Argonne group also motivated their construc- tion of largely phenomenological potentials like AV18 by the apparent failure of the OBE mechanism (apart from the pion-exchange tail) to allow a perfect fit to the data [16]. The main objective of this letter is to show that within the Covariant Spectator Theory (CST) it is, in fact, pos- sible to derive realistic OBE potentials, and that these require comparatively few parameters. This somewhat surprising finding contradicts the earlier conclusion and common belief that the OBE mechanism is missing some important feature of the NN interaction. Accurate OBE models may provide a useful intermediate step between fundamental physics and experiment. In CST [17, 18], the scattering amplitude M is the so- lution of a covariant integral equation derived from field theory (sometimes referred to as the “Gross equation”). In common with many other equations, it has the form M = V − V GM (1) where V is the irreducible kernel (playing the role of a potential) and G is the intermediate state propagator. As with the Bethe-Salpeter (BS) equation [19], if the kernel is exact and nucleon self energies are included in the propagator, iteration of the CST equation generates the full Feynman series. In cases where this series does not converge (nearly always!) the equation solves the problem nonperturbatively. With the BS equation the four-momenta of all A intermediate particles are sub- ject only to the conservation of total four-momentum pi, so the integration is over 4(A − 1) vari- ables. In the CST equation, all but one of the intermedi- ate particles are restricted to their positive-energy mass shell, constraining A−1 energies (they become functions of the three-momenta) and leaving only 3(A−1) internal variables, the same number of variables as in nonrelativis- tic theory. Since the on-shell constraints are covariant, the resulting equations remain manifestly covariant even though all intermediate loop integrations reduce to three dimensions, which greatly simplifies their numerical so- lution and physical interpretation. This framework has been applied successfully to many problems, in particular also to the two- and three-nucleon system [7, 8, 20]. The specific form of the CST equation for the two- nucleon scattering amplitude M , with particle 1 on-shell in both the initial and final state, is [20] M12(p, p ′;P ) = V 12(p, p ′;P ) (2π)3 V 12(p, k;P )G2(k, P )M12(k, p ′;P ) , (2) where P is the conserved total four-momentum, and p, p′, and k are relative four-momenta related to the momenta of particles 1 and 2 by p1 = P + p, p2 = P − p, and M12(p, p ′;P ) ≡ Mλλ′,ββ′(p, p ′;P ) = ūα(p, λ)Mαα′ ;ββ′(p, p ′;P )uα′(p ′, λ′) (3) is the matrix element of the Feynman scattering ampli- tude M between positive energy Dirac spinors of particle 1. The propagator for the off-shell particle 2 is G2(k, P ) ≡ Gββ′ (k2) = (m+ /k2) m2 − k22 − iǫ h4(k22) (4) with k2 = P − k1, k 1 = m 2, and h the form factor of the off-shell nucleon (related to its self energy), normalized to unity when k22 = m 2. In this paper we use h(p2) = (Λ2N −m −m2)2 + (m2 − p2)2 , (5) where ΛN is an adjustable cutoff parameter. The indices 1 and 2 refer collectively to the two helicity or Dirac indices of particle 1, either {λλ′} or {αα′}, and particle 2, {ββ′}. The covariant kernel V is explicitly antisymmetrized. In its Dirac form it is V αα′;ββ′(p, k;P ) Vαα′;ββ′(p, k;P ) + (−) IVβα′;αβ′(−p, k;P ) , (6) where the isospin indices have been suppressed, so that the factor of (−)I (with I=0 or 1 the isospin of the NN state) insures that the remaining amplitude has the sym- metry (−)I under particle interchange {p1, α} ↔ {p2, β} as required by the generalized Pauli principle. This sym- metry insures that identical results emerge if a different particle is chosen to be on-shell in either the initial or final state. Next we assume that the kernel can be written as a sum of OBE contributions V b12(p, k;P ) = ǫbδ Λb1(p1, k1)⊗ Λ 2(p2, k2) + |q2| f(Λb, q) (7) with b = {s, p, v, a} denoting the boson type, q = p1 − k1 = k2 − p2 = p − k the momentum transfer, mb the boson mass, ǫb a phase, and δ = 1 for isoscalar bosons and δ = τ1 · τ2 = −1 − 2(−) I for isovector bosons. All boson form factors, f , have the simple form f(Λb, q) = + |q2| with Λb the boson form factor mass. The use of the absolute value |q2| amounts to a covariant redefinition of the propagators and form factors in the region q2 > 0. It is a significant new theoretical improvement that removes all singularities and can be justified by a detailed study of the structure of the exchange diagrams. The axial TABLE I: Mathematical forms of the bNN vertex func- tions, with Θ(p) ≡ (m − /p)/2m. The vector propaga- tor is ∆µν = gµν − qµqν/m v with the boson momentum q = p1 − k1 = k2 − p2. JP (b) ǫb Λ1 ⊗ Λ2 Λ(p, k) or Λ µ(p, k) 0+(s) − Λ1Λ2 gs − νs [Θ(p) + Θ(k)] 0−(p) + Λ1Λ2 gpγ −gp(1− λp) Θ(p)γ5 + γ5Θ(k) 1−(v) + Λ Λν2∆µν gv γµ + κv iσµν(p− k)ν +gvνv [Θ(p)γ µ + γµΘ(k)] 1+(a) + Λ Λν2gµν gaγ vector bosons are treated as contact interactions, with the structure as in (7) but with the propagator replaced by a constant, m2a+ |q 2| → m2 with a nucleon mass scale. The explicit forms of the numerator functions Λb1⊗Λ 2 can be inferred from Table I. Note that λp = 0 corresponds to pure pseudovector coupling, and that the definitions of the off-shell coupling parameters λ or ν differ for each boson. In the most general case the kernel is the sum of the exchange of pairs of pseudoscalar, scalar, vector, and ax- ial vector bosons, with one isoscalar and one isovector meson in each pair. If the external particles are all on- shell, it can be shown that these 8 bosons give the most general spin-isospin structure possible (because the vec- tor mesons have both Dirac and Pauli couplings, the re- quired 10 invariants can be expanded in terms of only 8 boson exchanges), explaining why bosons with more com- plicated quantum numbers are not required. By allowing boson masses (except the pion) to vary we let the data fix the best mass for each boson in each exchange channel. Finally, we break charge symmetry by treating charged and neutral pions independently, and by adding a one- photon exchange interaction, simplified by assuming the neutron coupling is purely magnetic, iσµνqν , and that all electromagnetic form factors have the dipole form. To solve the CST NN equation numerically, it was expanded in a basis of partial wave helicity states as described in [20]. Previous models of the kernel, such as models IA, IB, IIA, and IIB of [20] and the updated, ν-dependent ver- sions such as W16 used in [7], had been obtained by fit- ting the potential parameters to the Nijmegen or VPI phase shifts. In a second step the χ2 to the observables was determined. The models presented in this paper were fit directly to the data, using a minimization program that can constrain two of the low-energy parameters (the deuteron binding energy, Ed = −2.2246 MeV, and the 1S0 scattering length, a0 = −23.749 fm, chosen to fit the very precise cross sections at near zero lab energy). This was a significant improvement, both because the best fit to the 1993 phase shifts did not guarantee a best fit to the 2006 data base, and because the low-energy constraints stabilized the fits. After the first fit was found, it would TABLE II: Values of the 27 parameters for WJC-1 with 7 bosons and 2 axial vector contact interactions. All masses and energies are in MeV; other couplings are dimensionless; Gb = g b/(4π). Parameters in bold were varied during the fit; those labeled with an ∗ were constrained to equal the one above. The deuteron D/S ratio is ηD, and the triton binding energy is Et. Experimental values are in parentheses. b I Gb mb λb or νb κv Λb π0 1 14.608 134.9766 0.153 — 4400 π± 1 13.703 139.5702 −0.312 — 4400∗ η 0 10.684 604 0.622 — 4400∗ σ0 0 2.307 429 −6.500 — 1435 σ1 1 0.539 515 0.987 — 1435 ω 0 3.456 657 0.843 0.048 1376 ρ 1 0.327 787 −1.263 6.536 1376∗ h1 0 0.0026 — — — 1376 a1 1 −0.436 — — — 1376 ΛN = 1656; ηD = 0.0256(1) (0.0256(4)); Et = −8.48 (−8.48) TABLE III: Values of the 15 parameters for WJC-2 with 7 bosons. See the caption to Table II for further explanation. b I Gb mb λb or νb κv Λb π0 1 14.038 134.9766 0.0 — 3661 π± 1 14.038∗ 139.5702 0.0 — 3661∗ η 0 4.386 547.51 0.0 — 3661∗ σ0 0 4.486 478 −1.550 — 3661 σ1 1 0.477 454 1.924 — 3661 ω 0 8.711 782.65 0.0 0.0 1591 ρ 1 0.626 775.50 −2.787 5.099 1591∗ ΛN = 1739; ηD = 0.0257(1) (0.0256(4)); Et = −8.50 (−8.48) then be possible to vary the off-shell sigma coupling, νσ, to give essentially a perfect fit to the triton binding en- ergy. However, the binding energies we report here were obtained from the best fit without any adjustment , con- firming the results reported in Fig. 1 of Ref. [7]. TABLE IV: Comparison of precision np models and the 1993 Nijmegen phase shift analysis. Our calculations are in bold face. models χ2/Ndata Reference #a yearb 1993 2000 2007 PWA93[12] 39c 1993 0.99 — — 1.11 1.12 Nijm I[14] 41c 1993 1.03c — — AV18[16] 40c 1995 1.06 — — CD-Bonn[15] 43c 2000 — 1.02 — WJC-1 27 2007 1.03 1.05 1.06 WJC-2 15 2007 1.09 1.11 1.12 aNumber of parameters bIncludes all data prior to this year. cFor a fit to both pp and np data. dOur fitting procedure uses the effective range expansion. The Ni- jmegen 3S1 parameters were taken from Ref. [21], but as no parameters are available we used those of WJC-1. The parameters obtained in the fits are shown in Ta- bles II and III. The χ2/Ndata resulting from the fits are compared with results obtained from earlier fits in Table IV. The data base used in the fits is derived from the previous SAID and Nijmegen analyses with new data af- ter 2000 added. The current data set includes a total of 3788 data, 3336 of which are prior to 2000 and 3010 prior to 1993. For comparison, the PWA93 was fit to 2514, AV18 to 2526, and CD-Bonn to 3058 np data. We re- stored some data sets previously discarded because their χ2 were no longer outside of statistically acceptable lim- its, and this increased the χ2 slightly. Phase shifts and a full discussion of the data and theory will be published elsewhere. In both of our models the high momentum cutoff is provided by the nucleon form factor and not the me- son form factors. Hence the very hard pion form fac- tors merely reflect the fact that the nucleon form fac- tors are sufficient to model the short range physics in the pion exchange channel. The off-shell scalar couplings are perhaps the most uncommon features of these models. They are clearly essential for the accurate prediction of three-body binding energies [7]. It is gratifying to see that the pseudoscalar components of the pion couplings (proportional to λp) remain close to zero, even when un- constrained, and that effective masses of all the bosons remain in the expected range of 400-800 MeV. Aside from this, the parameters of WJC-2 are quite close to values expected from older OBE models of nu- clear forces. A possible exception is the pion coupling constant, somewhat larger than the g2/(4π) = 13.567 found by the Nijmegen group. The high-precision model WJC-1 shows some novel features: (a) gπ0 > gπ± , (b) large gη, and (c) small gω. Why do these OBE models work so well? We are re- minded of the Dirac equation; it automatically includes the p4/(8m3) energy correction that contributes to fine structure, the Darwin term (including the Thomas pre- cession), the spin-orbit interaction, and the anomalous gyromagnetic ratio. Similarly, the CST automatically generates relativistic structures hard to identify, and im- possible to add to a nonrelativistic model without new parameters. We draw the following major conclusions from this work: (1) The reproduction of the np data by the WJC- 1 kernel is essentially as accurate as any other np phase shift analysis or any other model. This surprising result is achieved with only 27 parameters, fewer than used by previous high precision fits to np data. It remains to be seen whether the results will be equally successful once the pp data are included. (2) Model WJC-1 gives us a new phase shift analysis, updated for all data until 2006, which is useful even if one does not work within the CST. (3) The larger number of parameters of WJC- 1 is not necessary unless one wants very high precision; model WJC-2 with only 15 parameters is also excellent and comparable to previously published high precision fits. (4) The OBE concept, at least in the context of the CST where it can be comparatively easily extended to the treatment of electromagnetic interactions [6] and systems with A > 2, can be a very effective description of the nuclear force. Acknowledgements: This work is the conclusion of an effort extending over more than a decade, sup- ported initially by the DOE through grant No. DE-FG02- 97ER41032, and recently supported by Jefferson Science Associates, LLC under U.S. DOE Contract No. DE- AC05-06OR23177. A. S. was supported by FCT un- der grant No. POCTI/ISFL/2/275. We also acknowledge prior work by R. Machleidt and J.W. Van Orden, who wrote some earlier versions of parts of the NN code. The data analysis used parts of the SAID code supplied to us by R. A. Arndt. Helpful conversations with the the Ni- jmegen group (J. J. de Swart, M. C. M. Rentmeester, and R.G.E. Timmermans) and with R. Schiavilla are grate- fully acknowledged. [1] H. Yukawa, Proc. Phys. Math. Soc. Japan 17, 48 (1935). [2] G. Breit, Proc. Nat. Acad. Sci. USA 46 746 (1960); Phys. Rev. 120, 287 (1960). [3] S. Weinberg, Phys. Lett. B 251, 288 (1990). [4] M. C. M. Rentmeester, R. G. E. Timmermans and J. J. de Swart, Phys. Rev. C 67, 044001 (2003). [5] M. T. Peña, F. Gross and Y. Surya, Phys. Rev. C 54, 2235 (1996). [6] F. Gross and D. O. Riska, Phys. Rev. C 36, 1928 (1987). [7] A. Stadler and F. Gross, Phys. Rev. Lett. 78, 26 (1997). [8] A. Stadler, F. Gross and M. Frank, Phys. Rev. C 56, 2396 (1997). [9] D. R. Entem and R. Machleidt, Invited talk at 7th Int. Spring Seminar on Nuclear Physics, Maiori, Italy, May 2001. arXiv:nucl-th/0107057; Phys. Rev. C 68, 041001 (2003). [10] M. H. Mac Gregor, R. A. Arndt and R. M. Wright, Phys. Rev. 182, 1714 (1969). [11] SAID interactive dial-in program, R. A. Arndt, private communication [12] V. G. J. Stoks, R. A. M. Klomp, M. C. M. Rentmeester and J. J. de Swart, Phys. Rev. C 48, 792 (1993). [13] J. R. Bergervoet, P. C. van Campen, W. A. van der Sanden, and J. J. de Swart, Phys. Rev. C 38, 15 (1988). [14] V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, and J. J. de Swart, Phys. Rev. C 49, 2950 (1994). [15] R. Machleidt, Rev. C 63, 024001 (2001). [16] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995). [17] F. Gross, Phys. Rev. 186, 1448 (1969); Phys. Rev. D 10, 223 (1974); Phys. Rev. C 26, 2203 (1982). [18] F. Gross, Phys. Rev. C 26, 2226 (1982). [19] E. E. Salpeter and H. A. Bethe, Phys. Rev. 84, 1232 (1951). [20] F. Gross, J. W. Van Orden, and K. Holinde, Phys. Rev. C 45, 2094 (1992). [21] J. J. de Swart, C. P. F. Terheggen and V. G. J. Stoks, Int. Symp. on the Deuteron, Dubna, Russia, July, 1995, arXiv:nucl-th/9509032. http://arxiv.org/abs/nucl-th/0107057 http://arxiv.org/abs/nucl-th/9509032

0704.1230

Pseudodifferential operators and weighted normed symbol spaces

Pseudodifferential operators and weighted normed symbol spaces J. Sjöstrand Ecole Polytechnique FR 91120 Palaiseau cédex, France [email protected] UMR7640–CNRS Abstract In this work we study some general classes of pseudodifferential operators where the classes of symbols are defined in terms of phase space estimates. Résumé On étudie des classes générales d’opérateurs pseudodifférentiels dont les classes de symboles sont définis en termes d’éstimations dans l’espace de phase. Keywords and Phrases: Pseudodifferential operator, symbol, modulation space. Mathematics Subject Classification 2000: 35S05 Contents 1 Introduction 2 2 Symbol spaces 5 3 Effective kernels and L2-boundedness 7 4 Composition 12 5 More direct approach using Bargmann transforms 18 http://arxiv.org/abs/0704.1230v1 6 Cp classes 21 7 Further generalizations 24 1 Introduction This paper is devoted to pseudodifferential operators with symbols of limited reg- ularity. The author [28] introduced the space of symbols a(x) on the phase space E = Rn × (Rn)∗ with the property that |χ̂γa(x ∗)| ≤ F (x∗), ∀γ ∈ Γ (1.1) for some L1 function F on E∗. Here the hat indicates that we take the Fourier transform, Γ ⊂ E is a lattice and χγ(x) = χ0(x − γ) form a partition of unity, γ∈Γ χγ, χ0 ∈ S(E). A. Boulkhemair [4] noticed that this space is identical to a space that he had defined differently in [3]. It was shown among other things that this space of symbols is an algebra for the ordinary multiplication and that this fact persists after quantization, namely the corresponding pseudodifferential operators (say under Weyl quantization) form a non-commutative algebra: If a1, a2 belong to the class above with corresponding L1 functions F1 and F2 then a 1 ◦ a 2 = a 3 where a3 belongs to the same class and as a correponding function we may take F3 = CNF1 ∗ F2 ∗ 〈·〉 −N for any N > 2n. Here ∗ indicates convolution and aw : S(Rn) → S ′(Rn) is the Weyl quantization of the symbol a, given by awu(x) = (2π)n ei(x−y)·θa( , θ)u(y)dydθ. (1.2) The definition (1.1) is independent of the choice of lattice and the corresponding function χ0. When passing to a different choice, we may have to change the function F to m(x∗) = F ∗ 〈·〉−N0 for any fixed N0 > 2n. We then gain the fact that the weight m is an order function in the sense that m(x∗) ≤ C0〈x ∗ − y∗〉N0m(y∗), x∗, y∗ ∈ E∗. (1.3) (See [11] where this notion is used for developing a fairly simple calculus of semi- classical pseudodifferential operators, basically a special case of Hörmander’s Weyl calculus [26].) The space of functions in (1.1) is a special case of the modulation spaces of H.G. Feichtinger (see [12, 14]), and the relations between these spaces and pseudod- ifferential operators have been developed by many authors; K. Gröchenig [18, 19], Gröchenig, T. Strohmer [22], K. Tachigawa [32], J. Toft [33], A. Holst, J. Toft, P. Wahlberg [25]. Here we could mention that Boulkhemair [5] proved L2-continuity for Fourier integral operators with symbols and phases in the original spaces of the type (1.1), that T. Strohmer [31] has applied the theory to problems in mobile com- munications and that Y. Morimoto and N. Lerner [27] have used the original space to prove a version of the Fefferman-Phong inequality for pseudodifferential operators with symbols of low regularity. This result was recently improved by Boulkhemair Closely related works on pseudodifferential - and Fourier - integral operators with symbols of limited regularity include the works of Boulkhemair [6, 7], and many others also contain a study of when such operators or related Gabor localization operators belong to to Schatten-von Neumann classes: E. Cordero, Gröchenig [9, 10], C. Heil, J. Ramanathan, P. Topiwala [24], Heil [23], J. Toft [34], and M.W. Wong [37]. The present work has been stimulated by these developments and the prospect of using “modulation type weights” to get more flexibility in the calculus of pseu- dodifferential operators with limited regularity. In the back of our head there were also some very stimulating discussions with J.M. Bony and N. Lerner from the time of the writing of [28, 29] and at that time Bony explained to the author a nice very general point of view of A. Unterberger [36] for a direct microlocal analysis of very general classes of operators. Bony used it in his work [1] and showed how his approach could be applied to recover and generalize the space in [28]. However, the aim of the work [1] was to develop a very general theory of Fourier integral operators related to symplectic metrics of Hörmander’s Weyl calculus of pseudodifferential op- erators, and the relation with [28] was explained very briefly. See [2] for even more general classes of Fourier integral operators. In the present paper we make a direct generalization of the spaces of [28]. Instead of using order functions only depending on x∗ we can now allow arbitrary order functions m(x, x∗). See Definition 2.1 below. In Proposition 2.4 we show that this definition gives back the spaces above when the weight m(x∗) is an order function of x∗ only. In Section 3 we consider the quantization of our symbols and show how to define an associated effective kernel on E ×E, E = T ∗Rn, which is O(1)m(γ(x, y)) where γ(x, y) = (x+y , J−1(y − x)) and J : E∗ → E is the natural Hamilton map induced by the symplectic structure. We show that if the effective kernel is the kernel of a bounded operator : L2(E) → L2(E) then our pseudodifferential operator is bounded in L2(Rn). In particular if m = m(x∗) only depends on x∗, we recover the L2-boundedness when m is integrable. This result was obtained previously by Bony [1], but our approach is rather different. In Section 4 we study the composition of pseudodifferential operators in our classes. If aj are symbols associated to the order functions mj, j = 1, 2, then the Weyl composition is a well defined symbol associated to the order function m3(z, z given in (4.11), provided that the integral there converges for at least one value of (z, z∗) (and then automatically for all other values by Proposition 4.1). This statement is equivalent to the corresponding natural one for the effective kernels, namely the composition is well defined if the composition of the majorant kernels , J−1(y − x)) and m2( , J−1(y − x)) is well-defined, see (4.16), (4.17). In Section 5 we simplify the results further (for those readers who are familiar with Bargmann transforms from the FBI - complex Fourier integral operator point of view). In Section 6 we use the same point of view to give a simple sufficient condition on the order function m and the index p ∈ [1,∞], for the quantization aw to belong to the Schatten–von Neumann class Cp for every symbol a belonging to the symbol class with weight m. See [34, 35, 25, 20, 21] for related results and ideas. In Section 7 we finally generalize our results by replacing the underlying space ℓ∞ on certain lattices by more general translation invariant Banach spaces. We believe that this generalization allows to include modulation spaces, but we have contented ourselves by establishing results allowing to go from properties on the level of lattices to the level of pseudodifferential operators. The results could undoubtedly be even further generalized. In this section and the preceding one, we have been inspired by the use of lattices and amalgan spaces in time frequency analysis, in particular by the work of Gröchenig and Strohmer [22] that uses previous results by Fournier–Stewart [15] and Feichtinger [13]. We have chosen to work with the Weyl quantization, but it is clear that the results carry over with the obvious modifications to other quantizations like the Kohn-Nirenberg one, actually for the general symbol-spaces under consideration the results could also have been formulatated directly for classes of integral operators. Similar ideas and results have been obtained in many other works, out of which some are cited above and later in the text. Acknowledgements. We thank J.M. Bony for a very stimulating and helpful re- cent discussion. The author also thanks K. Gröchenig, T. Strohmer, A. Boulkhemair and J. Toft for several helpful comments and references. 2 Symbol spaces Let E be a d-dimensional real vector space. We say that m : E →]0,∞[ is an order function on E if there exist constants C0 > 0, N0 ≥ 1, such that m(ρ) ≤ C0〈ρ− µ〉 N0m(µ), ∀ρ, µ ∈ E. (2.1) Here 〈ρ− µ〉 = (1 + |ρ− µ|2)1/2 and | | is a norm on E. Let E be as above, let E∗ be the dual space and let Γ be a lattice in E ×E∗, so that Γ = Ze1+Ze2+...+Ze2d where e1, ..., e2d is a basis in E×E ∗. Let χ ∈ S(E×E∗) have the property that τγχ = 1, τγχ(ρ) = χ(ρ− γ). (2.2) Let m be an order function on E ×E∗, a ∈ S ′(E). Definition 2.1 We say that a ∈ S̃(m) if there is a constant C > 0 such that ‖χwγ a‖ ≤ Cm(γ), γ ∈ Γ, (2.3) where χγ = τγχ and χ γ denotes the Weyl quantization of χγ . The norm will always be the the one in L2 if nothing else is indicated. To define the L2-norm we need to choose a Lebesgue measure on E, but clearly that can only affect the choice of the constant in (2.3). Proposition 2.2 S̃(m) is a Banach space with ‖a‖eS(m) equal to the smallest possible constant in (2.3). Changing Γ, χ and replacing the L2 norm by the Lp-norm for any p ∈ [1,∞] in the above definition, gives rise to the same space with an equivalent norm. Proof The Banach space property will follow from the other arguments so we do not treat it explicitly. Let m,Γ, a be as in Definition 2.1. Let Γ̃ be another lattice and let χ̃ be another function with the same properties as χ. We have to show that eγ a‖Lp ≤ C̃m(γ̃), γ̃ ∈ Γ̃. Lemma 2.3 ∃ψ ∈ S(E × E∗) such that γ∈Γ ψ γ = 1, where ψγ = τγψ. Proof Let χ̃ ∈ S(E × E∗) be equal to 1 near (0, 0), and put χ̃ǫ(x, ξ) = χ̃(ǫ(x, ξ)). γ∈Γ(1− χ̃ γ)#χγ → 0 in S 0(E×E∗), when ǫ→ 0, so for ǫ > 0 small enough, (χ̃ǫγ) wχwγ = 1− (1− χ̃ǫγ) has a bounded inverse in L(L2, L2). Here S0 is the space of all a ∈ C∞(E × E∗) that are bounded with all their derivatives. By a version of the Beals lemma (see for instance [11]), we then know that the inverse is of the form Ψw where Ψ ∈ S0. Also τγΨ = Ψ, γ ∈ Γ. Put ψ γ = Ψ w ◦ (χ̃ǫγ) w for ǫ small enough and fixed, so that ψγ = τγψ0, ψ0 ∈ S (using for instance the simple pseudodifferential calculus in [11]). γ = 1. ✷ Now, write eγ a = Here (using for instance [11]) ‖χ̃eγψ γ ‖L(L2,Lp) ≤ Cp,N〈γ̃ − γ〉 −N , 1 ≤ p ≤ ∞, N ≥ 0. Hence, if N is large enough, eγ a‖Lp ≤ Cp,N 〈γ̃ − γ〉−N‖χwγ a‖L2 (2.4) ≤ C̃p,N,a 〈γ̃ − γ〉−Nm(γ) ≤ Ĉp,N,a,m( 〈γ̃ − γ〉−N+N0)m(γ̃) ≤ Čm(γ̃). Conversely, if ‖χ̃w eγ a‖Lp ≤ Constm(γ̃), γ̃ ∈ Γ̃, we see that by the same arguments that ‖χwγ a‖L2 ≤ O(1)m(γ), γ ∈ Γ. ✷ Next, we check that this is essentially a generalization of a space introduced by Sjöstrand [28] and independently and in a different way by Boukhemair [3]. It is a special case of more general modulation spaces (see [12, 14]). That follows from the next result if we take an order function m(x, x∗) independent of x. Proposition 2.4 Let m = m(x, x∗) be an order function on E × E∗ and let χ ∈ S(E), j∈J χj = 1, where J ⊂ E is a lattice and χj = τjχ. Then S̃(m) = {a ∈ S ′(E); ∃C > 0, |χ̂ju(x ∗)| ≤ Cm(j, x∗)}. (2.5) Proof Let K ⊂ E∗ be a lattice and choose χ∗ ∈ S(E∗), such that k∈K χ k = 1, where χk = τkχ. If a belongs to the set in the right hand side of (2.5), then by Parseval’s relation, ‖χ∗k(D)(χj(x)u(x))‖L2 ≤ C̃m(j, k). (2.6) Now χ∗k(D) ◦ χj(x) = χ j,k, where χj,k = τj,kχ0,0, χ0,0 ∈ S, (j, k) ∈ J × J ∗, so a ∈ S̃(m). Conversely, if a ∈ S̃(m), we get (2.6). According to Proposition 2.2, we can replace the L2 norm by any Lp norm, and the proof shows that we can equally well replace the L2 norm that of FLp. Taking FL∞, we get ‖χ∗k(x ∗)χ̂ju(x ∗)‖L∞ ≤ Ĉm(j, k), and since m is an order function, we deduce that a belongs to the set in the right hand side of (2.5). ✷ 3 Effective kernels and L2-boundedness A closely related notion for effective kernels in terms of short time Fourier transforms has been introduced by Gröchenig and Heil [20]. We now take E = R2n ≃ T ∗Rn. If a, b ∈ S(E), we let a#b = (e σ(Dx,Dy)a(x)b(y))y=x (3.1) denote the Weyl composition so that (a#b)w = aw ◦ bw. Here σ(Dx,ξ, Dy,η) = Dξ ·Dy −Dx ·Dη where we write (x, ξ), (y, η) instead of x, y whenever convenient. We know that the Weyl composition is still well-defined when a, b belong to various symbol spaces like S(m) = {a ∈ C∞(E); |Dαxa(x)| ≤ Cαm(x)}, (3.2) when m is an order function on E. (See Example 4.3 below for a straight forward generalization.) Let ℓ(x) = x · x∗ be a linear form on E and let a be a symbol. Then, eiℓ#a = e σ(Dx,Dy)(eiℓ(x)a(y))y=x (3.3) = (eiℓ(x)e σ(ℓ′(x),Dy)a(y))y=x = eiℓ(x)(e where Hℓ = ℓ − ℓ′x · (with “x = (x, ξ)”) is the Hamilton field of ℓ. Similarly, a#eiℓ = eiℓ(x)(e− Hℓa). (3.4) From (3.3), (3.4), we get eiℓ#a#e−iℓ = eHℓa, (3.5) where we notice that (eHℓa)(x) = a(x+Hℓ), and 2 #a#ei 2 = eima, (3.6) if m is a second linear form on E. If a ∈ S(E) is fixed, we may consider that a is concentrated near (0, 0) ∈ E×E∗. Then we say that e−Hℓeima is concentrated near (Hℓ, m) ∈ E ×E ∗. Conversely, if b is concentrated near a point (x0, x 0) ∈ E ×E ∗, we let y∗0 ∈ E ∗ be the unique vector with x0 = Hy∗0 and write b = e 0 eix 0a = e−iy 2 #a#ei 2 #eiy 0 , (3.7) where a is concentrated near (0, 0) ∈ E × E∗. To make this more precise, let (as in [30]) Tu = C eiφ(x,y)u(y)dy, C > 0, (3.8) be a generalized Bargmann transform where φ(x, y) is a quadratic form on Cn × Cn with detφ′′xy 6= 0, Imφ yy > 0, and with C > 0 suitably chosen, so that T is unitary L2(Rn) → HΦ(C n) = Hol (Cn) ∩ L2(e−2Φ(x)L(dx)), where L(dx) denotes the Lebesgue measure on Cn and Φ is the strictly plurisubharmonic quadratic form given by Φ(x) = sup −Imφ(x, y). (3.9) We know ([30]) that if ΛΦ = {(x, ); x ∈ Cn}, then ΛΦ = κT (E), (3.10) where κT : C 2n ≃ EC ∋ (y,−φ′y(x, y)) → (x, φ x(x, y)) ∈ C 2n (3.11) is the linear canonical transformation associated to T . Here ∂ ∂Re x following standard conventions in complex analysis. If a ∈ S0(E) we have an exact version of Egorov’s theorem, saying that TawT−1 = ãw, (3.12) where ã ∈ S0(ΛΦ) is given by ã ◦ κT = a. In [30] it is dicussed how to define and estimate the Weyl quantization of symbols on the Bargmann transform side, by means of almost holomorphic extensions and contour deformations. We retain from the proof of Proposition 1.2 in that paper that ãwu(x) = eΦ(x)Keff ea (x, y)u(y)e −Φ(y)L(dy), u ∈ HΦ(C n), (3.13) where the kernel is non-unique but can be chosen to satisfy ea (x, y) = ON (1)〈x− y〉 −N , (3.14) for every N ≥ 0. (This immediately implies the Calderón-Vaillancourt theorem for the class Op (S0(E)).) If a ∈ S(E), then for every N ∈ N |KeffTawT−1(x, y)| ≤ CN(a)〈x〉 −N〈y〉−N , x, y ∈ Cn, (3.15) where CN(a) are seminorms in S. Identifying x ∈ Cn with κ−1T (x, ) ∈ E, we can view Keff TawT−1 as a function Keffaw(x, y) on E × E and (3.15) becomes |Keffaw(x, y)| ≤ CN(a)〈x〉 −N〈y〉−N , x, y ∈ E. (3.16) Now, let b in (3.7) be concentrated near (x0, x 0) = (Jy 0) ∈ E × E ∗ with a ∈ S(E), where we let J : E∗ → E be the map y∗ 7→ Hy∗ (and we shall prefer to write Jy∗ when we do not think of this quantity as a constant coefficient vector field). Then by (3.5)–(3.7), we have b = e−iy 0#eix 0/2#a#eix 0/2#eiy 0 , (3.17) bw = e−i(y ◦ ei(x w/2 ◦ aw ◦ ei(x w/2 ◦ ei(y . (3.18) Now it is wellknown that if z∗ ∈ E∗ then e−i(z ∗)w = (e−iz )w is a unitary oper- ator that can be viewed as a quantization of the phase space translation E ∋ x 7→ x+Hz∗ ∈ E. On the Bargmann transform side these quantizations can be explicitly represented as magnetic translations, i.e. translations made unitary by multiplica- tion by certain weights. In fact, let ℓ(x, ξ) = x∗0 · x+ x0 · ξ be a linear form on C which is real on ΛΦ, so that x∗0 · x+ x0 · (x) ∈ R, ∀x ∈ Cn. (3.19) By essentially the same calculation as in the real setting, we see that (eiℓ)wu(x) = eix 0·(x+ x0)u(x+ x0), u ∈ HΦ, and here we recall from the unitary and metaplectic equivalence with L2(Rn) (via T ) that (eiℓ)w : HΦ → HΦ is unitary, or equivalently that − Φ(x) + Φ(x+ x0) + Re ix∗0 · (x+ = 0, ∀x ∈ Cn. (3.20) (A simple calculation shows more directly the equivalence of (3.19) and (3.20).) Notice also that if we identify u with a function ũ(ρ) on ΛΦ via the natural projection (x, ξ) 7→ x, then u(x+ x0) is identified with ũ(ρ+Hℓ), where the Hamilton field Hℓ is viewed as a real constant vector field on ΛΦ. It follows that bw has a kernel satisfying |Keffbw (x, y)| = |K aw(x+ Jx∗0 − x0, y − Jx∗0 − x0)| and from (3.16) we get |Keffbw (x, y)| ≤ CN(a)〈x− (x0 − Jx∗0)〉 −N〈y − (x0 + Jx∗0)〉 −N , (3.21) so the kernel of bw is concentrated near (x0 − Jx∗0, x0 + Jx∗0). Now, let m be an order function on E × E∗ and let a ∈ S̃(m). Choose a lattice Γ ⊂ E×E∗ and a partition of unity as in (2.2) as well as a function ψ ∈ S(E×E∗) as in Lemma 2.3. Write aγ, aγ = ψ γ ãγ, ãγ = χ γ a, (3.22) where ‖ãγ‖ ≤ Cm(γ). Then, using that ψ 0 is continuous: L 2(E) → S(E), we see that aγ is concentrated near γ in the above sense and more precisely, |Keffaw(x, y)| ≤ CNm(γ)〈x− (γx− Jγx∗)〉 −N〈y− (γx+ Jγx∗)〉 −N , x, y ∈ E, (3.23) where we write γ = (γx, γx∗) ∈ E ×E Let q(x, y) = (x+y , J−1(y − x)) = (qx(x, y), qx∗(x, y)), so that q−1(γ) = (γx − Jγx, γx + Jγx), and hence 〈q(x, y)− γ〉 ≤ O(1)〈x− (γx − Jγx∗)〉〈y − (γx + Jγx∗)〉, so (3.23) implies |Keffawγ (x, y)| ≤ CN(a)m(γ)〈q(x, y)− γ〉 −N (3.24) ≤ C̃N(a)m(q(x, y))〈q(x, y)− γ〉 N0−N , where we used that m is an order function in the last inequality. Choose N with N0 −N < −4n, sum over γ and use (3.22) to get |Keffaw(x, y)| ≤ C(a)m(q(x, y)) = C(a)m( , J−1(y − x)), x, y ∈ E. (3.25) We get Theorem 3.1 Let a ∈ S̃(m), where m is an order function on E × E∗, E = T ∗Rn. Then aw has an effective kernel (rigorously defined after applying a Bargmann transform as above) satisfying (3.25), where C(a) is a S̃(m) norm of a. In particular, if M(x, y) = m(x+y , J−1(y − x)) is the kernel of an L2(E)-bounded operator, then aw is bounded: L2(Rn) → L2(Rn). As mentioned in the introduction, the statement on L2-boundedness here is due to Bony [1], who obtained it in a rather different way. A calculation, similar to the one leading to (3.25), has been given by Gröchenig [18]. Corollary 3.2 If M is the kernel of a Shur class operator i.e. if , J−1(y − x))dy, sup , J−1(y − x))dx <∞, then aw is bounded: L2(Rn) → L2(Rn). Corollary 3.3 Assume m(x, x∗) = m(x∗) is independent of x, for (x, x∗) ∈ E×E∗ and m(x∗) ∈ L1(E∗), then aw is bounded: L2(Rn) → L2(Rn). 4 Composition Let a, b ∈ S(E), E = Rn× (Rn)∗, (x0, x 0), (y0, y 0) ∈ E ×E ∗ and consider the Weyl composition of the two symbols ex·x 0a(x − x0), e x·y∗0b(x − y0) , concentrated near (x0, x 0) and (y0, y 0) respectively: σ(Dx,Dy)(ex·x 0a(x− x0)e y·y∗0b(y − y0))(z, z). (4.1) We work in canonical coordinates x ≃ (x, ξ) and identify E and E∗. Then σ(x∗, y∗) = Jx∗ · y∗, J = , tJ = −J, J2 = −1, and e σ(Dx,Dy) is convolution with k, given by k(x, y) = (2π)2n ei(x·x ∗+y·y∗+ 1 Jx∗·y∗)dx∗dy∗. The phase Φ = x · x∗ + y · y∗ + 1 Jx∗ · y∗ has a unique nondegenerate critical point (x∗, y∗) = (2Jy,−2Jx) and the corresponding critical value is equal to −2σ(x, y) = −2Jx · y. Hence k = Ce−2iσ(x,y) = Ce−2iJx·y for some (known) constant C. The composition (4.1) becomes ei(−2J(z−x)·(z−y)+x·x 0+y·y 0)a(x− x0)b(y − y0)dxdy = (4.2) Ceiz·(x ei(−2Jx·y+x·x 0+y·y 0)a(x+ z − x0)b(y + z − y0)dxdy. The exponent in the last integral can be rewritten as −2Jx · y + x · x∗0 + y · y 0 = −2J(x− J−1y∗0) · (y + J−1x∗0) + Jx∗0 · y and the composition (4.1) takes the form eiz·(x 0)d(z), where d(z) = Ce σ(x∗0 ,y e−2iσ(x,y)a(x+ z − (x0 + Jy∗0))b(y + z − (y0 − Jx∗0))dxdy. Since σ(x, y) is a nondegenerate quadratic form, we have for every N ≥ 0 by inte- gration by parts, |d(z)| ≤ CN 〈(x, y)〉−N〈x+ z − (x0 + Jy∗0)〉 −N〈y + z − (y0 − Jx∗0)〉 −Ndxdy. Hence for every N ≥ 0, |d(z)| ≤ CN〈z − (x0 + Jy∗0)〉 −N〈z − (y0 − Jx∗0)〉 Using the triangle inequality, we get (1 + |z − a|)(1 + |z − b|) ≥ 1 + |z − a|+ |z − b| ≥ 1 + |a− b|+ |z − (1 + |z − a|)(1 + |z − b|) ≥ (1 + |a− b|)1/2(1 + |z − |)1/2 and hence for every N ≥ 0, |d(z)| ≤ CN〈(x0+ Jx∗0)− (y0− Jy∗0)〉 −N〈z− Jx∗0+y0+ Jy∗0)〉 −N . (4.3) Clearly, we have the same estimates for the derivatives of d(z). It follows that the composition (4.1) is equal to eiz·z 0c(z − z0), where z∗0 = x 0 + y 0, z0 = (x0 − Jx∗0 + y0 + Jy∗0), (4.4) and where c ∈ S and for every seminorm p on S and every N , there is a seminorm q on S such that p(c) ≤ 〈(x0 + Jx∗0)− (y0 − Jy∗0)〉 −Nq(a)q(b). (4.5) It follows that : eiz·z 0c(z − z0) ∈ S̃(〈· − (z0, z with corresponding norm bounded by qN,M(a)qN,M(b)〈(x0 + Jx∗0)− (y0 − Jy∗0)〉 for all N,M ≥ 0 where qN,M are suitable seminorms on S. If a1 ∈ S̃(m1), a2 ∈ S̃(m2) then c = a1#a2 is well-defined and belongs to S̃(m provided that the integrals defining m 3 and m3 below converge. Here (replacing summation over lattices by integration) 3 (z, z 〈z∗ − (x∗ + y∗)〉−N〈z − Jx∗ + y + Jy∗)〉−N(4.6) ×〈(x+ Jx∗)− (y − Jy∗)〉−Nm1(x, x ∗)m2(y, y ∗)dxdydx∗dy∗ In order to understand the integral (4.6), we put x̃ = 1 Jx∗, ỹ = 1 Jy∗, z̃ = 1 and study the set Σ(z, z∗) where the arguments inside the three brackets vanish simultaneously:  x̃+ ỹ = z̃, x+ y − x̃+ ỹ = 2z, x− y + x̃+ ỹ = 0, which can be transformed to Σ(z, z∗) : x̃− x = z̃ − z, ỹ + y = z̃ + z, x̃+ ỹ = z̃. (4.7) Now it is clear that for every M > 0 there is an N > 0 such that 3 (z, z ∗) ≤ O(1) dist (x, x∗, y, y∗; Σ(z, z∗))−Mm1(x, x ∗)m2(y, y ∗)dxdydx∗dy∗. (4.8) Since m1, m2 are order functions, we have m1(x, x ∗) ≤ O(1)dist (x, x∗, y, y∗; Σ(z, z∗))N0m1(Π Σ (x, x ∗, y, y∗)) m2(y, y ∗) ≤ O(1)dist (x, x∗, y, y∗; Σ(z, z∗))N0m2(Π Σ (x, x ∗, y, y∗)), where ΠΣ : (E × E ∗)2 → Σ(z, z∗) is the affine orthogonal projection and we write ΠΣ(x, x ∗; y, y∗) = (Π Σ (x, x ∗; y, y∗),Π Σ (x, x ∗; y, y∗)). We conclude that for N large enough, 3 (z, z ∗) ≤ O(1)m3(z, z ∗), (4.9) where m3(z, z Σ(z,z∗) m1(x, x ∗)m2(y, y ∗)dΣ (4.10) or more explicitly, m3(z, z Jx∗−x=1 Jz∗−z Jy∗+y=1 Jz∗+z x∗+y∗=z∗ m1(x, x ∗)m2(y, y ∗))dx. (4.11) Reversing the above estimates, we see that m3(z, z ∗) ≤ O(1)m 3 (z, z ∗), if N > 0 is large enough. Proposition 4.1 If the integral in (4.10) converges for one value of (z, z∗), then it converges for all values and defines an order function m3. Proof Suppose the integral converges for the value (z, z∗) and consider any other value (z + t, z∗ + t∗). We have the measure preserving map Σ(z, z∗) ∋ (x, x∗, y, y∗) 7→ (x+ t, x∗ + t∗, y + Jt∗ + t, y∗) ∈ Σ(z + t, z∗ + t∗), m3(z + t, z ∗ + t∗) = Σ(z,z∗) m1(x+ t, x ∗ + t∗)m2(y + Jt∗ + t, y∗)dx ≤ C〈(t, t∗)〉N0〈t+ t∗〉N0m3(z, z ≤ C̃〈(t, t∗)〉2N0m3(z, z The proposition follows. ✷ From the above discussion, we get Theorem 4.2 Let m1, m2 be order functions on E × E ∗ and define m3 by (4.11). Assume that m3(z, z ∗) is finite for at least one (z, z∗) so that m3 is a well-defined order function by Proposition 4.1. Then the composition map S(E)× S(E) ∋ (a1, a2) 7→ a1#a2 ∈ S(E) (4.12) has a bilinear extension S̃(m1)× S̃(m2) ∋ (a1, a2) 7→ a1#a2 ∈ S̃(m3), (4.13) Moreover, ‖a1#a2‖eS(m3) ≤ O(1)‖a1‖eS(m1)‖a2‖eS(m2). (4.14) We end this section by establishing a connection with the effective kernels of Section 3. Let aj be as in the theorem with a3 = a1#a2. According to Theorem 3.1, we then know that awj has an effective kernel Kj = K (x, y) satisfying Kj(x, y) = O(1)mj(q(x, y)), where q(x, y) = ( , J−1(y − x)). (4.15) Since the composition of the effective kernels of aw1 and a 2 is an effective kernel for aw3 = a 1 ◦ a 2 we expect that m3(q(x̃, ỹ)) = C m1(q(x̃, z̃))m2(q(z̃, ỹ))dz̃, (4.16) or more explicitly, x̃+ ỹ , J−1(ỹ − x̃)) = C x̃+ z̃ , J−1(z̃ − x̃))m2( z̃ + ỹ , J−1(ỹ − z̃))dz̃, (4.17) Writing x̃+ ỹ z∗ = J−1(ỹ − x̃), x̃+ z̃ x∗ = J−1(z̃ − x̃), z̃ + ỹ y∗ = J−1(ỹ − z̃), we check that the integral in (4.17) coincides with the one in (4.11) up to a constant Jacobian factor, so the results of this section fit with the ones of Section 3. Example 4.3 Let aj ∈ S̃(mj), j = 1, 2, where mj are order functions on E×E the form mj(x, x ∗) = m̃j(x)〈x ∗〉−Nj , Nj ∈ R, m̃j(x) ≤ C〈x− y〉 Mjm̃j(y), x, y ∈ E, Mj ≥ 0. Then, the effective kernels K1, K2 of a 1 , a 2 satisfy Kj(x, y) = O(1)mj( , J−1(y − x)) = O(1)m̃j( )〈x− y〉−Nj . Then a1#a2 is well-defined and belongs to S̃(m3), where , J−1(y − x)) = )〈x− z〉−N1〈z − y〉−N2m̃2( z + y provided that the last integral converges for at least one (and then all) value(s) of ((x+ y)/2, J−1(y − x)). If we use that ) ≤ O(1)m̃1( )〈z − y〉M1 z + y ) ≤ O(1)m̃2( )〈x− z〉M2 , we get , J−1(y−x)) ≤ O(1)m̃1( )m̃2( 〈x−z〉−N1+M2〈z−y〉−N2+M1dz. (4.18) Thus m3 and a1#a2 ∈ S̃(m3) are well-defined if − (N1 +N2) +M1 +M2 < −2n. (4.19) The integral I in (4.18) is O(1) in any region where x− y = O(1). For |x− y| ≥ 2, we write I ≤ I1 + I2 + I3, where • I1 is the integral over |x− z| ≤ |x− y|. Here 〈z − y〉 ∽ 〈x− y〉. • I2 is the integral over |z − y| ≤ |x− y|. Here 〈x− z〉 ∽ 〈x− y〉. • I3 is the integral over |x−z|, |z−y| ≥ |x−y|. Here 〈x−z〉 ∽ 〈y−z〉 ≥ 1 〈x−y〉. We get I1 ∽ 〈x− y〉 −N2+M1 ∫ 〈x−y〉 〈r〉−N1+M2+2n−1dr ∽ 〈x− y〉−N2+M1+(−N1+M2+2n)+ , with the convention that we tacitly add a factor ln〈x−y〉 when the expression inside (..)+ is equal to 0. Similarly (with the same convention), I2 ∽ 〈x− y〉 −N1+M2+(−N2+M1+2n)+ . In view of (4.19), we have 〈x−y〉 r−(N1+N2)+M1+M2+2n−1dr ∽ 〈x− y〉−(N1+N2)+M1+M2+2n. it follows that I ∽ 〈x− y〉max(−N2+M1+(−N1+M2+2n)+,−N1+M2+(−N2+M1+2n)+), (4.20) so with the same convention, we have m3(x, x ∗) ≤ O(1)m̃1(x)m̃2(x)〈x ∗〉max(−N2+M1+(−N1+M2+2n)+,−N1+M2+(−N2+M1+2n)+). (4.21) This simplifies to m3(x, x ∗) ≤ O(1)m̃1(x)m̃2(x)〈x ∗〉max(−N2+M1,−N1+M2) (4.22) if we strengthen the assumption (4.19) to: −N1 +M2, −N2 +M1 < −2n. (4.23) 5 More direct approach using Bargmann trans- forms By using Bargmann transforms more systematically (from the point of view of Fourier integral operators with complex phase) the results of Section 3, 4 can be obtained more directly. The price to pay however, is the loss of some aspects that might be helpful in other situations like the ones with variable metrics. Let F be real d-dimensional space as in Section 2 and define T : L2(F ) → C) as in (3.8)–(3.11). Then we have Proposition 5.1 If m is an order function on F × F ∗, then S̃(m) = {u ∈ S ′(F ); e−Φ(x)|Tu(x)| ≤ Cm(κ−1T (x, (x)))}, (5.1) where the best constant C = C(m) is a norm on S̃(m). Proof Assume first that u belongs to S̃(m) and write u = γ∈Γ ψ γ u as in Lemma 2.3. The effective kernel of ψwγ satisfies |Keffψwγ (x, y)| ≤ CN〈x− γ〉 −N〈y − γ〉−N , (5.2) for every N > 0, where throughout the proof we identify FC with F ×F ∗ by means of π ◦κT and work on the latter space. Here π : ΛΦ → F C is the natural projection. Then we see that |e−Φ/hTu(x)| ≤ CN(u) m(γ)〈x− γ〉−N = O(m(x)). Conversely, if e−Φ/hTu = O(m(x)), then since the effective kernel of χwγ also satis- fies (5.2), we see that e−Φ/hTχwγ u = ON(〈x−γ〉 −Nm(γ)), implying ‖e−Φ/hTχwγ u‖L2 = O(m(γ)), and hence ‖χwγ u‖ = O(m(γ)). ✷ With this in mind, we now take a ∈ S̃(Rn × (Rn)∗;m) and look for an explicit choice of effective kernel for aw. Let T : L2(Rn) → HΦ(C n) be a Bargmann trans- form as above. Consider first the map a 7→ Kaw(x, y) ∈ S ′(Rn ×Rn) from a to the distribution kernel of aw, given by Kaw(x, y) = (2π)n ei(x−y)·τa( , τ)dτ (5.3) (2π)2n ei(x−y)·τ+i( −t)·sa(t, τ)dtdsdτ. We view this as a Fourier integral operator B : a 7→ Kaw(x, y) with quadratic phase. The associated linear canonical transformation is given by: κB : (t, τ ; t ∗, τ ∗) = ( , τ ; s, y − x) 7→ (x, τ + ; y,−τ + ) = (x, x∗; y, y∗), which we can write as κB : (t, τ ; t ∗, τ ∗) 7→ (t− , τ + ,−τ + ). (5.4) From the unitarity of T , we know that T ∗T = 1, where T ∗v(y) = C e−iφ(x,y)v(x)e−2Φ(x)L(dx). (5.5) We can therefore define the effective kernel of aw to be Keff(x, y) = e−Φ(x)K(x, y)e−Φ(y), (5.6) where TawT ∗v(x) = K(x, y)v(y)e−2Φ(y)L(dy), v ∈ HΦ(C n), (5.7) K(x, y) = C2 ei(φ(x,t)−φ(y,s))Kaw(t, s)dtds. We write this as K(x, y) = C2 ei(φ(x,t)−φ ∗(y,s))Kaw(t, s)dtds, with φ∗(y, s) = φ(y, s), so K(x, y) = (T ⊗ T̃ )(Kaw)(x, y), (5.8) where (T̃ u)(y) = C ∗(y,s)u(s)ds = (Tu)(y). (5.9) We see that T̃ : L2(Rn) → HΦ∗(C n) is a unitary Bargmann transform, where Φ∗(y) = sup Imφ∗(y, s) = sup Imφ(y, s) = Φ(y). (5.10) The canonical transformation associated to T̃ is κ eT : (s, (y, s)) 7→ (y,− (y, s)). (5.11) ι(s, σ) = (s,−σ), (5.12) we check that κ eT = ικT ι, ι : (x, (x)) 7→ (x, (x)). (5.13) Clearly T⊗ T̃ is a Bargmann transform with associated canonical transformation κT × (ικT ι), so in view of (5.4) the map a 7→ K is also a Bargmann transform with associated canonical transformation (E×E∗)C ∋ (t, τ ; t∗, τ ∗) 7→ (κT ((t, τ)− J(t∗, τ ∗)), ικT (((t, τ)+ J(t∗, τ ∗))), (5.14) where E = Rn × (Rn)∗. The restriction to the real phase space is E ×E∗ ∋ (t, τ ; t∗, τ ∗) 7→ (5.15) (κT ((t, τ)− J(t∗, τ ∗)), ικT (((t, τ) + (t∗, τ ∗))) ∈ ΛΦ × ιΛΦ = ΛΦ × ΛΦ∗ , and this restriction determines our complex linear canonical transformation uniquely. As in Section 3 we may view the effective kernel Keff(x, y) in (5.6) as a function on E×E, by identifying x, y ∈ Cn with κ−1T (x, (x)), κ−1T (y, (y)) ∈ E respec- tively. With this identification and using also the general characterization in (5.1) (with T replaced by T ⊗ T̃ )), we see that if a ∈ S ′(E), then a ∈ S̃(m) iff Keff(t− Jt∗, t+ Jt∗) = O(1)m(t, t∗), (t, t∗) ∈ E × E∗, (5.16) where we shortened the notation by writing t instead of (t, τ) and t∗ instead of (t∗, τ ∗). Theorem 3.1 now follows from (5.16), (5.6), (5.7). Theorem 4.2 also follows from (5.16), (5.6), (5.7) together with the remark that the kernel K(x, y) = Ka(x, y) is the unique kernel which is holomorphic on C n×Cn, such that the corresponding Keffaw given in (5.6) is of temperate growth at infinity and (5.7) is fulfilled. Indeed, then it is clear that Keff(a1#2)w(x, y) = Keffaw1 (x, z)Keffaw2 (z, y)L(dz) (5.17) and the bound (5.16) for a1#a2 withm = m3 follows directly from the corresponding bounds for aj with m = mj. 6 Cp classes In this section we give a simple condition on an order function m on E × E∗ (E = T ∗Rn) and a number p ∈ [1,∞] that implies the property: ∃C > 0 such that: a ∈ S̃(m) ⇒ aw ∈ Cp(L 2, L2) and ‖aw‖Cp ≤ C‖a‖eS(m). (6.1) Here Cp(L 2, L2) is the Schatten–von Neumann class of operators: L2(Rn) → L2(Rn), see for instance [16]. Letm be an order function on E×E∗ and let p ∈ [1,+∞]. Consider the following property, where q is given in (4.15) and Γ ⊂ E is a lattice, ∃C > 0 such that if |aα,β| ≤ m(q(α, β)), α, β ∈ Γ, (6.2) then (aα,β)α,β∈Γ ∈ Cp(ℓ 2(Γ), ℓ2(Γ)) and ‖(aα,β)‖Cp ≤ C. Notice that if (6.2) holds and if we fix some number N0 ∈ N ∗, then if (Aα,β)α,β∈Γ is a block matrix where every Aα,β is an N0 ×N0 matrix then same as (6.2) with aα,β replaced by Aα,β and | · | by ‖ · ‖L(CN0 ,CN0 ). (6.3) Proposition 6.1 The property (6.2) only depends on m, p but not on the choice of Proof Let m, p,Γ satisfy (6.2) and let Γ̃ be a second lattice in E. Let (a eα,eβ ) be a Γ̃ × Γ̃ matrix satisfying |a eα,eβ | ≤ m(q(α̃, β̃)). Let π(α̃) ∈ Γ be a point that realizes the distance from α̃ to Γ, so that |π(α̃) − α̃| ≤ C0 for some constant C0 > 0. Let N0 = max#π −1(α) and choose an enumeration π−1(α) = {α̃1, ..., α̃N(α)}, N(α) ≤ N0, for every α ∈ Γ. Then we can identify (aeα,eβ)eΓ×eΓ with the matrix (Aα,β)α,β∈Γ×Γ where Aα,β is the N0 ×N0 matrix with the entries (Aα,β)j,k = eαj ,eβk , if 1 ≤ j ≤ N(α), 1 ≤ k ≤ N(β), 0, otherwise. Then ‖Aα,β‖ ≤ Cm(q(α, β)) and we can apply (6.3) to conclude. ✷ Theorem 6.2 Let m be an order function and p ∈ [1,∞]. If (6.2) holds, then we have (6.1). Proof Assume that (6.2) holds and let a ∈ S̃(m). Define K(x, y) as in (5.7). It suf- fices to estimate the Cp norm of the operator A : L 2(e−2ΦL(dx)) → L2(e−2ΦL(dx)), given by Au(x) = K(x, y)u(y)e−2Φ(y)L(dy), or equivalently the one of Aeff : L 2(Cn) → L2(Cn), given by Aeffu(x) = Keff(x, y)u(y)L(dy), (6.4) with Keff given in (5.6). Recall that Keff(x, y) = O(1)m(q(x, y)) (identifying Cn with T ∗Rn via πx ◦ κT ), so K(x, y) = O(1)m(q(x, y))e Φ(x)+Φ(y). For α, β ∈ Γ we have (identifying Γ with a lattice in Cn) K(x, y) = eFα(x−α)K̃α,β(x, y)e Fβ(y−β), (6.5) where Fα(x− α) = Φ(α) + 2 (α) · (x− α) (6.6) is holomorphic with Re Fα(x− α) = Φ(x) +Rα(x− α), Rα(x− α) = O(|x− α| 2), (6.7) |∇kx∇ yK̃α,β(x, y)| ≤ C̃k,ℓm(q(α, β)), |x− α|, |y − β| ≤ C0. (6.8) Here we identify α, β ∈ E with their images πxκT (α), πxκT (β) ∈ C n respectively. In fact, the case k = ℓ = 0 is clear and we get the extension to arbitrary k, ℓ from the Cauchy inequalities, since K̃α,β is holomorphic. We can also write Keff(x, y) = eiGα(x−α)Kα,β(x, y)e −iGβ(y−β), (6.9) where Gα(x− α) = ImFα(x− α), Kα,β = e Rα(x−α)K̃α,β(x, y)e Rβ(y−β), |∇kx∇ yKα,β(x, y)| ≤ Ck,ℓm(q(α, β)), |x− α|, |y − β| ≤ C0. (6.10) Consider a partition of unity χα(x), χα(x) = χ0(x− α), χ0 ∈ C 0 (Ω0;R), (6.11) where Ω0 is open with smooth boundary. Let Ωα = Ω0 + α, so that (6.10) holds for (x, y) ∈ Ωα × Ωβ. Let W : L2(Cn) → β∈Γ L 2(Ωβ) be defined by (e−iGβ(x−β)u(x))|Ωβ so that the adjoint of W is given by W ∗v = eiGα(x−α)vα(x)1Ωα(x), v = (vα)α∈Γ ∈ L2(Ωα). Then W and its adjoint are bounded operators and Aeff =W ∗AW, (6.12) where A = (Aα,β)α,β∈Γ and Aeff : L 2(Cn) → L2(Cn), Aα,β : L 2(Ωβ) → L 2(Ωα) are given by the kernels Keff(x, y) and χα(x)Kα,β(x, y)χβ(y) respectively. It now suffices to show that L2(Ωβ) → L2(Ωβ) belongs to Cp with a norm that is bounded by a constant times the S̃(m)-norm of Let e0, e1, .. ∈ L 2(Ω0) be an orthonormal basis of eigenfunctions of minus the Dirichlet Laplacian in Ω0, arranged so that the corresponding eigenvalues form an increasing sequence. Then eα,j := ταej , j = 0, 1, ... form an orthonormal basis of eigenfunctions of the corresponding operator in L2(Ωα). From (6.10) it follows that the matrix elements Kα,j;β,k of Aα,β with respect to the bases (eα,·) and (eβ,·) satisfy |Kα,j;β,k| ≤ CNm(q(α, β))〈j〉 −N〈k〉−N , (6.13) for every N ∈ N. We notice that (Kα,j;β,k)(α,j),(β,k)∈Γ×N is the matrix of A with respect to the orthonormal basis (eα,j)(α,j)∈Γ×N. We can represent this matrix as a block matrix (Kj,k)j,k∈N, where K j,k : ℓ2(Γ) → ℓ2(Γ) has the matrix (Kα,j;β,k)α,β∈Γ. Since (6.2) holds and a ∈ S̃(m), we deduce from (6.13) that ‖Kj,k‖Cp ≤ C̃N〈j〉 −N〈k〉−N . (6.14) Choosing N > 2n, we get ‖A‖Cp ≤ ‖Kj,k‖Cp <∞. (6.15) Hence aw ∈ Cp and the uniform bound ‖a w‖Cp ≤ ‖a‖eS(m) also follows from the proof. ✷ Example 6.3 Assume that ‖m(·, x∗)‖Lp(E)dx ∗ <∞. (6.16) Then ( m(q(α, β)) α,β∈Γ α + β , J−1(β − α)) α,β∈Γ (6.17) is a matrix where each translated diagonal {(α, β) ∈ Γ × Γ; α − β = δ} has an ℓp norm which is summable with respect to δ ∈ Γ. Now a matrix with non-vanishing elements in only one translated diagonal has a Cp norm equal to the ℓ p norm of that diagonal, so we conclude that the Cp norm of the matrix in (6.17) is bounded by , δ)‖ℓp <∞. We clearly have the same conclusion for every matrix (aα,β)α,β∈Γ satisfying |aα,β| ≤ m(q(α, β)), so (6.2) holds and hence by Theorem 6.2 we have the property (6.1). 7 Further generalizations Let E be a d-dimensional real vector space and let Γ ⊂ E be a lattice. We shall extend the preceding results by replacing the ℓ∞(Γ)-norm in the definition of the symbol spaces by a more general Banach space norm. Let B be a Banach space of functions u : Γ → C with the following properties: If u ∈ B, γ ∈ Γ, then τγu ∈ B, and ‖τγu‖B = ‖u‖B. (7.1) δγ ∈ B, ∀γ ∈ Γ, (7.2) where τγu(α) = u(α − γ), δγ(α) = δγ,α, α ∈ Γ. (The last assumption will soon be replaced by a stronger one.) If u = γ∈Γ u(γ)δγ ∈ B, we get ‖u‖B ≤ |u(γ)|‖δγ‖B = C‖u‖ℓ1, where C = ‖δγ‖B (is independent of γ). Thus ℓ1(Γ) ⊂ B. (7.3) We need to strengthen (7.2) to the following assumption: If u ∈ B and v : Γ → C satisfies |v(γ)| ≤ |u(γ)|, ∀γ ∈ Γ, (7.4) then v ∈ B and ‖v‖B ≤ C‖u‖B, where C is independent of u, v. It follows that ‖u(γ)δγ‖B ≤ C‖u‖B, for all u ∈ B, γ ∈ Γ, or equivalently that |u(γ)| ≤ ‖δγ‖B ‖u‖B = C̃‖u‖B, B ⊂ ℓ∞(Γ), and ‖u‖ℓ∞ ≤ C̃‖u‖B, ∀u ∈ B. (7.5) If f ∈ ℓ1(Γ) then using only the translation invariance (7.1), we get u ∈ B ⇒ f ∗ u ∈ B, ‖f ∗ u‖B ≤ ‖f‖ℓ1‖u‖B. (7.6) Using also (7.4) we get the following partial strengthening: Let k : Γ × Γ → Γ satisfy |k(α, β)| ≤ f(α− β) where f ∈ ℓ1(Γ). Then u ∈ B ⇒ v(α) := k(α, β)u(β) ∈ B and ‖v‖B ≤ C‖f‖ℓ1‖u‖B, (7.7) where C is independent of k, u. In fact, u ∈ B ⇒ |u| ∈ B ⇒ f ∗ |u| ∈ B, and v in (7.7) satisfies |v| ≤ f ∗ |u| pointwise. Let Γ̃ ⊂ E be a second lattice and let B̃ ⊂ ℓ∞(Γ̃) satisfy (7.1), (7.4). We say that B ≺ B̃ if the following property holds for some N > d: If u ∈ B and ũ : Γ̃ → C satisfies |ũ(γ̃)| ≤ 〈γ̃ − γ〉−N |u(γ)|, γ̃ ∈ Γ̃, (7.8) then ũ ∈ B̃ and ‖ũ‖ eB ≤ C‖u‖B, where C is independent of u, ũ. If (7.8) holds for one N > d and M > d then it also holds with N replaced by M . This is obvious whenM ≥ N and if d < M < N , it follows from the observation 〈γ̃ − γ〉−M ≤ CN,M eβ∈eΓ 〈γ̃ − β̃〉−M〈β̃ − γ〉−N (cf. (4.20), where I is the integral in (4.18), 2n is replaced by d, and we take M1 = M2 = 0), which allows us to write 〈γ̃ − γ〉−M |u(γ)| ≤ CN,M〈·〉 −M ∗ v, where v(β) := γ〈β̃ − γ〉 −N |u(γ)| and v belongs to B̃ since (7.8) holds. Definition 7.1 Let Γ, Γ̃ be two lattices in E and let B, B̃ be Banach spaces of functions on Γ and Γ̃ respectively, satisfying (7.1), (7.4). Then we say that B ≡ B̃, if B ≺ B̃ and B̃ ≺ B. Notice that this is an equivalence relation. We can now introduce our generalized symbol spaces. With E ≃ Rd as above, let Γ ⊂ E ×E∗ be a lattice and B ⊂ ℓ∞ a Banach space satisfying (7.1), (7.4). Let a ∈ S ′(E). Definition 7.2 We say that a ∈ S̃(m,B) if the function Γ ∋ γ 7→ ‖χwγ a‖ belongs to B. Here χγ is the partiction of unity (2.2). Proposition 2.2 extends to Proposition 7.3 S̃(m,B) is a Banach space with the natural norm. If we replace Γ, χ, B by Γ̃, χ̃, B̃, having the same properties, and with B̃ ⊂ ℓ∞(Γ̃) equivalent to B, and if we further replace the L2 norm by the Lp norm for any p ∈ [1,∞], we get the same space, equipped with an equivalent norm. Proof It suffices to follow the proof of Proposition 2.2: From the estimate (2.4) we get for any N ≥ 0, m(γ̃) eγ a‖Lp ≤ Cp,N 〈γ̃ − γ〉−n ‖χwγ a‖L2 , where we also used that m is an order function. Hence, since B, B̃ are equivalent, ‖χ̃wa · ‖Lp‖ eB ≤ ‖ ‖χw· a‖L2‖B. The reverse estimate is obtained the same way. ✷ As a preparation for the use of Bargmann transforms, we next develop a “con- tinuous” version of B-spaces; a kind of amalgam spaces in the sense of [22, 13, 15]. Let Γ be a lattice in a d-dimensional real vector space E and let B ⊂ ℓ∞(Γ) satisfy (7.1), (7.4). Let 0 ≤ χ ∈ C∞0 (E) satisfy γ∈Γ τγχ > 0. Definition 7.4 We say that the locally bounded measurable function u : E → C is of class [B], if there exists v ∈ B such that |u(x)| ≤ v(γ)τγχ(x). (7.9) The space of such functions is a Banach space that we shall denote by [B], equipped with the norm ‖u‖[B] = inf{‖v‖B; (7.9) holds }. (7.10) This space does not depend on the choice of χ and we may actually characterize it as the space of all locally bounded measurable functions u on E such that |u(x)| ≤ w(γ)〈x− γ〉−N , for some w ∈ B, (7.11) where N > d is any fixed number. Clearly (7.8) implies (7.11). Conversely, if u satisfies (7.11) and χ is as in Definition 7.4, then 〈x〉−N ≤ C 〈α〉−Nταχ(x), so if (7.11) holds, we have, |u(x)| ≤ C 〈α〉−Nχ(x− (γ + α)) (〈·〉−N ∗ w)(β)χ(x− β), and 〈·〉−N ∗ w ∈ B. Similarly, the definition does not change if we replace B ⊂ ℓ∞(Γ) by an equivalent space B̃ ⊂ ℓ∞(Γ̃). Let m1, m2, m3 be order functions on E1 × E2, E2 × E3, E1 × E3 respectively, where Ej is a real vectorspace of dimension dj . Let Γj ⊂ Ej be lattices and let B1 ⊂ ℓ ∞(Γ1 × Γ2), B2 ⊂ ℓ ∞(Γ2 × Γ3), B3 ⊂ ℓ ∞(Γ1 × Γ3) be Banach spaces satisfying (7.1), (7.4). Introduce the Assumption 7.5 If kj ∈ mjBj , j = 1, 2, then k3(α, β) := k1(α, γ)k2(γ, β) converges absolutely for every (α, β) ∈ Γ1 × Γ3. Moreover, k3 ∈ m3B3 and ‖k3/m3‖B3 ≤ C‖k1/m1‖B1‖k2/m2‖B2 where C is independent of k1, k2. Again, it is an easy exercise to check that the assumption is invariant under changes of the lattices Γj and the passage to corresponding equivalent B-spaces. Proposition 7.6 We make the Assumption 7.5, where Bj satisfy (7.1), (7.4). Let Kj ∈ mj[Bj ] for j = 1, 2 in the sense that Kj/mj ∈ [Bj]. Then the integral K3(x, y) := K1(x, z)K2(z, y)dz, (x, y) ∈ E1 ×E3, converges absolutely and defines a function K3 ∈ m3[B3]. Moreover, ‖K3/m3‖[B3] ≤ C‖K1/m1‖[B1]‖K2/m2‖[B2], where C is independent of K1, K2. Proof Write |K1(x, z)| ≤ Γ1×Γ2 k1(α, γ)χ (1)(x− α, z − γ) |K2(z, y)| ≤ Γ2×Γ3 k2(γ, β)χ (2)(z − γ, y − β), with χ(1) ∈ C∞0 (E1 × E2), χ (2) ∈ C∞0 (E2 × E3) as in Definition 7.4 and with kj ∈ mjBj . Then |K3(x, y) ≤ |K1(x, z)||K2(z, y)|dz (α,β)∈Γ1×Γ3 γ,γ′∈Γ2 k1(α, γ)k2(γ ′, β)F (x− α, y − β; γ − γ′), where F (x, y; γ − γ′) = χ(1)(x, z − γ)χ(2)(z − γ′, y)dz χ(1)(x, z − (γ − γ′))χ(2)(z, y)dz. We notice that 0 ≤ F (x, y; γ) ∈ C∞0 (E1 × E3) and that F (x, y; γ) 6≡ 0 only for finitely many γ ∈ Γ. Hence for some R0 > 0, |K3(x, y)| ≤ |γ|≤R0 (α,β)∈Γ1×Γ3 k1(α, γ ′ + γ)k2(γ ′, β) F (x− α, y − β; γ). Since m1(·, ··) k1(·, · ·+γ) ∈ B1, for every fixed γ, and k2/m2 ∈ B2, the assumption 7.5 implies that k3(α, β; γ) := k1(α, γ ′ + γ)k2(γ ′, β) ∈ m3B3, for every γ ∈ Γ. The proposition follows. ✷ We next generalize (5.1). Let F = Rd and define T : L2(F ) → HΦ(F C) as in (3.8)–(3.11). Let m be an order function on F × F ∗, let Γ ⊂ F × F ∗ be a lattice and let B ⊂ ℓ∞(Γ) satisfy (7.1), (7.4). Then we get Proposition 7.7 we have S̃(m,B) = {u ∈ S ′(F ); (e−ΦTu) ◦ π ◦ κT ∈ [B]}, (7.12) where π : ΛΦ ∋ (x, ξ) 7→ x ∈ F C is the natural projection. Proof This will be a simple extension of the proof of (5.1). As there, we identify FC with F ×F ∗ by means of π ◦κT and work on the latter space. Assume first that u ∈ S̃(m,B) and write u = γ∈Γ ψ γ u as in Lemma 2.3, so that (‖χ γ u‖)γ∈Γ ∈ mB. Using (5.2), we see that |e−Φ/hTu(x)| ≤ CN ‖χwγ u‖〈x− γ〉 and hence e−ΦTu ∈ m[B], i.e. u belongs to the right hand side of (7.12) (with the identification π ◦ κT ). Conversely, if e−ΦTu ∈ m[B], then since the effective kernel of χwγ satisfies (5.2), we see that |e−ΦTχwγ u(x)| ≤ CN 〈x− γ〉−N〈y − γ〉−N 〈y − α〉−Naαdy, where (aα) ∈ mB. It follows that |e−ΦTχwγ u(x)| ≤ C̃N〈x− γ〉 〈γ − α〉−Naα = C̃N〈x− γ〉 −Nbγ , where (bγ)γ∈Γ ∈ mB, and hence ‖χ γ u‖ ≤ ĈNbγ, so u ∈ S̃(m,B). ✷ From this, we deduce as in (5.16) that if a ∈ S ′(E), E = F×F ∗, then a ∈ S̃(m,B) Keffaw (t− Jt∗, t+ Jt∗) ∈ m[B], (7.13) where Keffaw is the effective kernel of a w in (5.6), (5.7) after identification of Cd = FC with E via the map π ◦ κT = E → F C. We recall the identity (5.17) for the composition of two symbols. (7.13) can also be written Keffaw(x, y) ∈ m̃[B̃], where m̃ = m ◦ q, [B̃] = [B] ◦ q, (7.14) where q is given in (4.15). The following generalization of Theorem 4.2 now follows from Proposition 7.6. Theorem 7.8 For j = 1, 2, 3, let mj be an order function E × E ∗, where E = Rn × (Rn)∗, let Γj ⊂ E × E ∗ be a lattice and let Bj ⊂ ℓ ∞(Γj) satisfy (7.1), (7.4). Let m̃j = mj ◦ q, Γ̃j = q −1(Γj), ℓ ∞(Γ̃j) ⊃ B̃j = Bj ◦ q. Assuming (as we may without loss of generality) that Γ̃j = Γ × Γ where Γ ⊂ E is a lattice, we make the Assumption 7.5 for m̃jB̃j. Then if aj ∈ S̃(mj , Bj), j = 1, 2, the composition a3 = a1#a2 is well defined and belongs to S̃(m3, B3), in the sense that the corresponding composition of effective kernels in (5.17) is given by an absolutely convergent integral and Keffaw3 ∈ m̃3[B̃3]. We next consider the action of pseudodifferential operators on generalized symbol spaces. Our result will be essentially a special case of the preceding theorem. We start by “contracting” Assumption 7.5 to the case when E3 = 0. Let m1, m2, m3 be order functions on E1×E2, E2, E1 respectively. Let Γj ⊂ Ej, j = 1, 2 be lattices and let B1 ⊂ ℓ ∞(Γ1 × Γ2), B2 ⊂ ℓ ∞(Γ2), B3 ⊂ ℓ ∞(Γ1) be Banach spaces satisfying (7.1), (7.4). Assumption 7.5 becomes Assumption 7.9 If kj ∈ mjBj , j = 1, 2, then k3(α) = k1(α, β)k2(β) converges absolutely for every α ∈ Γ1, and we have k3 ∈ m3B3. Moreover, ‖k3/m3‖B3 ≤ C‖k1/m1‖B1‖k2/m2‖B2 where C is independent of k1, k2. The corresponding “contraction” of Proposition 7.6 becomes Proposition 7.10 Let Assumption 7.9 hold, where Bj satisfy (7.1), (7.4). Let Kj ∈ mj[Bj ] for j = 1, 2. Then the integral K3(x) := K1(x, z)K2(z)dz, x ∈ E1, converges absolutely and defines a function K3 ∈ m3[B3]. Moreover, ‖K3/m3‖[B3] ≤ C‖K1/m1‖[B1]‖K2/m2‖[B2], where C is independent of K1, K2. We get the following result for the action of pseudodifferential operators on generalized symbol spaces. Theorem 7.11 Let m2, m3 be order functions on E = R n× (Rn)∗ and let m1 be an order function on E×E∗. Let Γ̂ ⊂ E×E∗ be a lattice such that Γ̃ := q−1(Γ̂) = Γ×Γ where Γ ⊂ E is a lattice. Let B̂1 ⊂ ℓ ∞(Γ̂), B2, B3 ⊂ ℓ ∞(Γ) satisfy (7.1), (7.4). We make the Assumption 7.9 with Γ1,Γ2 = Γ and with m1, B1 replaced with m̃1 = m1◦q, B̃1 = B1 ◦ q, where q is given in (4.15). Then, if a1 ∈ S̃(m1, B1), u ∈ S̃(m2, B2), the distribution v = a 1 (u) is well- defined in S̃(m3, B3) in the sense that e−Φ(x)Tv(x) = Keffaw1 (x, y)e−Φ(y)Tu(y)L(dy), with Keffaw1 (x, y) as in (5.6), converges absolutely for every x ∈ Cn and ((e−ΦTv) ◦ π ◦ κT ) ∈ [B3], as in (7.12). We shall finally generalize Theorem 6.2. Theorem 7.12 Let p ∈ [1,∞] and let m be an order function on E × E∗ where E = Rn × (Rn)∗. Let Γ ⊂ E be a lattice and B ⊂ ℓ∞(q(Γ × Γ)) a Banach space satisfying (7.1), (7.4). Assume that if (aα,β)α,β∈Γ ∈ (m ◦ q)B ◦ q, then (aα,β) ∈ Cp(ℓ 2(Γ), ℓ2(Γ)) (7.15) and ‖(aα,β)‖Cp ≤ C‖(aα,β)‖(m◦q)B◦q , where q is given in (4.15) and C > 0 is independent of (aα,β). Then there is a (new) constant C > 0 such that If a ∈ S̃(m,B), then aw ∈ Cp(L 2, L2) and ‖aw‖Cp ≤ C‖a‖eS(m,B). (7.16) The proof of Proposition 6.1 shows that the property (7.15) is invariant under changes (Γ, B) 7→ (Γ̃, B̃) with B̃ ⊂ ℓ∞(q(Γ̃× Γ̃)) equivalent to B. Proof We follow the proof of Theorem 6.2. Assume that (7.15) holds and let a ∈ S̃(m,B) be of norm ≤ 1. It suffices to show that Aeff : L 2(Cn) → L2(Cn) is in Cp with norm ≤ C, where Aeff is given in (6.4) and K eff there belongs to m◦q[B ◦q], provided that we identify Cn with E via π ◦ κT . We see that we still have (6.9) where (6.10) should be replaced by |∇kx∇ yKα,β(x, y)| ≤ Ck,ℓaα,β, |x− α|, |y − β| ≤ C0, (7.17) (aα,β)α,β∈Γ ∈ (m ◦ q)B ◦ q, α, β ∈ Γ. Write Aeff =W ∗AW as in (6.12), L2(Ωβ) → L2(Ωβ), A = (Aα,β). The matrix elements Kα,j;β;k of Aα,β now obey the estimate (cf. (6.13)): |Kα,j;β,k| ≤ CN〈j〉 −N〈k〉−Naα,β (7.18) with aα,β as in (7.18). 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0704.1231

Entwining Structures in Monoidal Catrgories

ENTWINING STRUCTURES IN MONOIDAL CATEGORIES B. MESABLISHVILI Abstract. Interpreting entwining structures as special instances of J. Beck’s distribu- tive law, the concept of entwining module can be generalized for the setting of arbitrary monoidal category. In this paper, we use the distributive law formalism to extend in this setting basic properties of entwining modules. 1. Introduction The important notion of entwining structures has been introduced by T. Brzeziński and S. Majid in [4]. An entwining structure (over a commutative ring K) consists of a K-algebra A, a K-coalgebra C and a certain K-homomorphism λ : C ⊗K A → A ⊗K C satisfying some axioms. Associated to λ there is the category MCA(λ) of entwining modules whose objects are at the same time A-modules and C-comodules, with compatibility relation given by λ. The algebra A can be identified with the monad T = −⊗K A : ModK → ModK whose Eilenberg-Moore category of algebras, (ModK) T , is (isomorphic to) the category of right A- modules. Similarly, C can be identified with the comonad G = −⊗K C : ModK → ModK , and the corresponding Eilenberg-Moore category of coalgebras with the category of C- comodules. It turns out that to give an entwining structure C ⊗K A → A⊗K C is to give a mixed distributive law TG → GT from the monad T to the comonad G in the sense of J. Beck [2], which are in bijective corresondence with liftings (or extensions) G of the comonad G to the category (ModK) T ; or, equivalently, liftings T of the monad T to the category (ModK)G. Moreover, the categories M A(λ) , ((ModK) T )G and ((ModK)G) T are isomorphic. Thus, the (mixed) distributive law formalism can be used to study entwining structures and the corresponding category of modules. In this article -based on this formalism- we extend in the context of monoidal categories some of basic results on entwining structures that appear in the literature (see, for example, [5], [6], [11]). The paper is organized as follows. After recalling the notion of Beck’s mixed distribu- tive law and the basic facts about it, we define in Section 3 an entwining structure in any monoidal category. In Section 4, we prove some categorical results that are needed in the next section, but may also be of independent interest. Finally, in the last section we present our main results. We refer to M. Barr and C. Wells [1], S. MacLane [9] and F. Borceux [3] for terminology Supported by the research project ”Algebraic and Topological Structures in Homotopical and Cate- gorical Algebra, K-theory and Cyclic Homology“, with financial support of the grant GNSF/ST06/3-004. 2000 Mathematics Subject Classification: 16W30, 18D10, 18 D35. Key words and phrases: Entwining module, (braided) monoidal category, Hopf algebra. c© B. Mesablishvili, . Permission to copy for private use granted. http://arxiv.org/abs/0704.1231v1 and general results on (co)monads, and to T. Brzeziński and R. Wisbauer [5] for coring and comodule theory. 2. Mixed distributive laws Let T = (T, η, µ) be a monad and G = (G, ε, δ) a comonad on a category A. A mixed distributive law from T = (T, η, µ) to G = (G, ε, δ) is a natural transformation λ : TG → GT for which the diagrams || Gη yy Tε // GT , GT // T, Tδ // TG2 λG // GTG and T 2G Tλ // TGT λT // GTT // GGT TG // GT commute. Given a monad T = (T, η, µ) on A, write AT for the Eilenberg-Moore category of T- algebras, and write FT ⊣ UT : AT → A for the corresponding forgetful-free adjunction. Dually, if G = (G, ε, δ) is comonad on A, then write AG for the category of G-coalgebras, and write FG ⊣ UG : AG → A for the corresponding forgetful-cofree adjunction. 2.1. Theorem. ( see [12] ) Let T = (T, η, µ) be a monad and G = (G, ε, δ) a comonad on a category A. Then the following structures are in bijective correspondences: • mixed distributive laws λ : TG → GT; • comonads Ḡ = (Ḡ, ε̄, δ̄) on AT that extend G in the sense that UT Ḡ = GUT , UT ε̄ = εUT and UT δ̄ = δUT ; • monads T̄ = (T̄ , η̄, µ̄) on AG that extend T in the sense that UGT̄ = TUG, UGη̄ = ηUG and UGµ̄ = µUG. These correspondences are constructed as follows: • Given a mixed distributive law λ : TG → GT, then Ḡ(a, ξa) = (G(a), G(ξa) ·λa), ε̄(a,ξa) = εa, δ̄(a,ξa) = δa, for any (a, ξa) ∈ A T; and T̄ (a, νa) = (T (a), λa · T (νa)), η̄(a,νa) = ηa, µ̄(a,νa) = µa for any (a, νa) ∈ AG. • If Ḡ = (Ḡ, ε̄, δ̄) is a comonad on AT extending the comonad G = (G, ε, δ), then the corresponding distributive law λ : TG → GT is given by TGη // TGT = UTF TGUTF T = UTF TUT ḠF T UT εT ḠFT // UT ḠF T = GUTF T = GT, where εT : F TUT → 1 is the counit of the adjunction F T ⊣ UT . • If T̄ = (T̄ , η̄, µ̄) is a monad on AG extending T = (T, η, µ), then the corresponding mixed distributive law is given by TG = TUGFG = UGT̄FG UGηGT̄ FG // UGFGUGT̄ FG = UGFGTUGFG = GTG GTε // GT , where ηG : 1 → FGUG is the unit of the adjunction UG ⊣ FG. It follows from this theorem that if λ : TG → GT is a mixed distributive law, then (AG) T̄ = (AT)Ḡ. We write (A )(λ) for this category. An object of this category is a three-tuple (a, ξa, νa), where (a, ξa) ∈ A T, (a, νa) ∈ AG, for which G(ξa) · λa · T (νa) = νa · ξa. A morphism f : (a, ξa, νa) → (a ′, ξ′a, ν a) in (A is a morphism f : a → a′ in A such that ξ′a · T (f) = f · ξa and ν a · f = G(f) · νa. 3. Entwining structures in monoidal categories Let V = (V,⊗, I) be a monoidal category with coequalizers such that the tensor product preserves the coequalizer in both variables. Then for all algebras A = (A, eA, mA) and B = (B, eB, mB) and allM ∈ VA, N ∈ AVB and P ∈ BV, the tensor productM⊗AN exists and the canonical morphism (M⊗AN)⊗BP → M⊗A (M⊗BP ) is an isomorphism. Using MacLane’s coherence theorem (see, [9], XI.5), we may assume without loss of generality that V is strict. It is well known that every algebra A = (A, eA, mA) in V defines a monad TA on V by • TA(X) = X ⊗ A, • (ηTA)X = X ⊗ eA : X → X ⊗A, • (µTA)X = X ⊗mA : X ⊗A⊗A → X ⊗ A, and that VTA is (isomorphic to) the category VA of right A-modules. Dually, if C = (C, εC, δC , ) is a coalgebra (=comonoid) in V, then one defines a comonad GC on V by • GC(X) = X ⊗ C, • (εGC)X = X ⊗ εC : X ⊗ C → X, • (δGC)X = X ⊗ δC : X ⊗ C → X ⊗ C ⊗ C, and VGC is (isomorphic to) the category V C of right C-comodules. Quite obviously, if λ is a mixed distributive law from TA to GC, then the morphism λ′ = λI : C ⊗ A → A⊗ C makes the following diagrams commutative: C ⊗ A // A⊗ C , A⊗ C // A, C ⊗ A δC⊗A // C ⊗ C ⊗A C⊗λ′ // C ⊗ A⊗ C C ⊗A⊗A λ′⊗A // A⊗ C ⊗A A⊗λ′ // A⊗A⊗ C // A⊗ C ⊗ C, C ⊗ A // A⊗ C . Conversely, if λ′ : C ⊗A → A⊗C is a morphism for which the above diagrams commute, then the natural transformation −⊗ λ′ : TAGC(−) = −⊗ C ⊗A → −⊗ A⊗ C = GCTA(−) is a mixed distributive law from the monad TA to the comonad GC. It is easy to see that λ′ = (−⊗ λ′)I . When I is a regular generator in V and the tensor product preserves all colimits in both variables, it is not hard to show that λ ≃ −⊗ λI . When this is the case, then the correspondences λ → λI and λ ′ → −⊗ λ′ are inverses of each other. 3.1. Definition. An entwining structure (C,A, λ) consists of an algebra A = (A, eA, mA) and a coalgebra C = (C, εC, δC) in V and a morphism λ : C ⊗ A → A⊗ C such that the natural transformation −⊗ λ : TAGC(−) = −⊗ C ⊗ A → −⊗ A⊗ C = GCTA(−) is a mixed distributive law from the monad TA to the comonad GC. Let be (C,A, λ) be an entwining structure and let Ḡ = (Ḡ, ε̄, δ̄) be the comonad on VA that extends G = GC. Then we know that, for any (V, ξV ) ∈ VA, Ḡ(V, ξV ) = (V ⊗ C, V ⊗ C ⊗A V⊗λ // V ⊗ A⊗ C ξV ⊗C // V ⊗ C). In particular, since (A,mA) ∈ VA, A⊗ C is a right A-module with right action ξA⊗C : A⊗ C ⊗ A A⊗λ // A⊗ A⊗ C ma⊗C// A⊗ C. 3.2. Lemma. View A ⊗ C as a left A-module through ξ̄A⊗C = mA ⊗ C. Then (A ⊗ C, ξ̄A⊗C, ξA⊗C) is an A-A-bimodule. Proof. Clearly (A ⊗ C, ξ̄A⊗C) ∈ AV. Moreover, since (A ⊗ λ) · (mA ⊗ C ⊗ A) = (mA ⊗ A⊗ C) · (A⊗ A⊗ λ), it follows from the associativity of mA that the diagram A⊗ A⊗ C ⊗A A⊗A⊗λ// mA⊗C⊗A A⊗A⊗A⊗ C A⊗mA⊗C A⊗ C ⊗ A A⊗ A⊗ C A⊗A⊗ C // A⊗ C is commutative, which just means that (A⊗ C, ξ̄A⊗C, ξA⊗C) is an A-A-bimodule. Since ε̄(A,mA) : Ḡ(A,mA) → (A,mA) and δ̄(A,mA) : Ḡ(A,mA) → Ḡ 2(A,mA) are morphisms of right A-modules, and since UA(ε̄(A,mA)) = εA = (A ⊗ C −→ A) and UA(δ̄(A,mA)) = δC =(A ⊗ C −→ A ⊗ C ⊗ C), it follows that A ⊗ C −→ A and −→ A⊗C⊗C are both morphisms of right A-modules. Clearly they are also mor- phisms of left A-modules with the obvious left A-module structures arising from the multi- plication mA : A⊗A → A, and hence morphisms of A-A-bimodules. Since C = (C, εC, δC) is a coalgebra in V, it follows that the triple (A⊗ C)λ = (A⊗ C, ε(A⊗C)λ , δ(A⊗C)λ), where ε(A⊗C)λ = A ⊗ C −→ A and δ(A⊗C)λ = A ⊗ C −→ A ⊗ C ⊗ C, is an A-coring. Since, for any V ∈ VA, V ⊗A (A⊗ C) ≃ V ⊗ C, the comonad Ḡ is isomorphic to the comonad G(A⊗C)λ . Thus, any entwining structure (C,A, λ) defines a right A-module structure ξA⊗C on A ⊗ C such that (A ⊗ C, ξ̄A⊗C = mA ⊗ C, ξA⊗C) is an A-A-bimodule and the triple (A⊗ C)λ = (A⊗C, ε(A⊗C)λ , δ(A⊗C)λ) is an A-coring. Moreover, when this is the case, the comonad G(A⊗C)λ on VA extends the comonad GC. It follows that V (A⊗C)λ Conversely, let A = (A, eA, mA) be an algebra and C = (C, εC, δC) a coalgebra in V, and suppose that A⊗ C has the structure ξA⊗C of a right A-module such that the triple A⊗ C = ((A⊗ C,mA ⊗ C, ξA⊗C), A⊗ C A⊗εC // A, A⊗ C A⊗δC // A⊗ C ⊗ C) (1) is an A-coring. Then it is easy to see that the comonad GA⊗C on VA extends the comonad GC on V, and thus defines an entwining structure λA⊗C : C ⊗A → A⊗ C. Summarising, we have 3.3. Theorem. Let A = (A, eA, mA) be an algebra and C = (C, εC, δC) a coalgebra in V. Then there exists a bijection between right A-module structures ξA⊗C making (A⊗C,mA⊗ C, ξA⊗C) an A-bimodule for which the triple (1) is an A-coring and entwining structures (C,A, λ), given by: ξA⊗C // (λA⊗C : C ⊗A eA⊗C⊗A // A⊗ C ⊗A ξA⊗C // A⊗ C) with inverse given by λ // (ξA⊗C : A⊗ C ⊗A A⊗λ // A⊗A⊗ C mA⊗C // A⊗ C) Under this equivalence V (A⊗C)λ 4. Some categorical results Let G = (G, ε, δ) be a comonad on a category A, and let UG : AG → A be the forgetful functor. Fix a functor F : B → A, and consider a functor F̄ : B → AG making the diagram F̄ // F ��? UG}}|| commutative. Then F̄ (b) = (F (b), αF (b)) for some αF (b) : F (b) → GF (b). Consider the natural transformation ᾱF : F → GF, (3) whose b-component is αF (b). It is proved in [7] that: 4.1. Theorem. Suppose that F has a right adjoint R : A → B with unit η : 1 → FU and counit ε : FU → 1. Then the composite tF̄ : FU // GFU Gε // G. is a morphism from the comonad G′ = (FU, ε, FηU) generated by the adjunction η, ε : F ⊣ U : B → A to the comonad G. Moreover, the assignment F̄ −→ tF̄ yields a one to one correspondence between functors F̄ : B → AG making the diagram (2) commutative and morphisms of comonads tF̄ : G ′ → G. Write βU for the composite U ηU // UFU UtF̄ // UG . 4.2. Proposition. The equalizer Ū , if it exists, of the following diagram UUGηG // // UGUG = UUGFGUG, where ηG : 1 → FGUG is the unit of the adjunction UG ⊣ FG, is right adjoint to F . Proof. See [3] or [7]. Let F̄ : B → AG be a functor making (2) commutative and let tF̄ : G ′ → G be the corresponding morphism of comonads. Consider the following composition G′ // AG′ F̄ // AG, where • KG′ : B → AG′, KG′(b) = (F (b), F (ηb)) is the Eilenberg-Moore comparison functor for the comonad G′. • AtF̄ is the functor ((a, θa) ∈ A ) −→ ((a, (tF̄ )a · θa) ∈ AG) induced by the morphism of comonads tF̄ : G ′ → G. 4.3. Lemma. The diagram KG′ // F̄ B B AG′ is commutative. Proof. Let b ∈ B. Then KG′(b) = (F (b), F (ηb)) and AtF̄ (F (b), F (ηb)) = (F (b), (tF̄ )F (b) · F (ηb)). Since (tF̄ )F (b) is the composite FUF (b) (ᾱF )UF (b) // GFUF (b) GεF (b)// GF (b), and since by naturality of ᾱF , the diagram F (b) (ᾱ)b // F (ηb) GF (b) GF (ηb) FUF (b) (ᾱ)UF (b) // GFUF (b) commutes, we have (tF̄ )F (b) · F (ηb) = G(εF (b)) · (ᾱF )UF (b) · F (ηb) = G(εF (b)) ·GF (ηb) · (ᾱF )b = (ᾱF )b = αF (b). (AtF̄ ·KG′)(b) = AtF̄ (KG′(b)) = AtF̄ (F (b), F (ηb)) = (F (b), (tF̄ )F (b) ·F (ηb)) = (F (b), αF (b)), which just means that AtF̄ ·KG′ = F̄ . We are now ready to prove the following 4.4. Theorem. Let G be a comonad on a category A, η, ε : F ⊣ U : B → A an adjunction and F̄ : B → AG a functor with UG · F̄ = F . Then the following are equivalent: (i) The functor F̄ is an equivalence. (ii) The functor F is comonadic and the morphism of comonads tF̄ : G ′ = (FU, ε, FηU) → G is an isomorphism. Proof. Suppose that F̄ is an equivalence of categories. Then F is isomorphic to the comonadic functor UG and thus is comonadic. Hence the comparison functor KG′ : B → AG′ is an equivalence and it follows from the commutative diagram (4) that AtF̄ is also an equivalence, and since the diagram F̄ // UG′ !!C UG~~|| is commutative, tF̄ is an isomorphism of comonads. So (i) =⇒ (ii). Suppose now that tF̄ : G ′ → G is an isomorphism of comonads and F is comonadic. • KG′ is an equivalence, since F is comonadic. • AtF̄ is an equivalence, since tF̄ is an isomorphism. And it now follows from the commutative diagram (4) that F̄ is also an equivalence. Thus (ii) =⇒ (i). This completes the proof of the theorem. 4.5. Remark. In [8], J. Gómez-Torrecillas has proved that F̄ is an equivalence of cate- gories iff tF̄ is an isomorphism of comonads, F is conservative, and for any (X, x) ∈ AG, F preserves the equalizer of the pair of parallel morphisms ηU(X) //UG′(X) U((tF̄ )X) //UG(X) . (5) When tF̄ is an isomorphism of comonads, to say that F preserves the equalizer of the pair of morphisms (5) is to say that F preserves the equalizer of the pair of morphisms ηU(X) // )X )·U(x) // UG′(X), which we can rewrite as ηU(X) // )X ·x) // UG′(X) = UFU(X). (6) Since tF̄ is an isomorphism of comonds, AtF̄ is an equivalence of categories, and thus each object (X, x′) ∈ AG′ is isomorphic to theG ′-coalgebra (X, (t−1 )X · x), where (X, x) ∈ AG. It follows that when tF̄ is an isomorphism of comonds, to say that F preserves the equalizer of (5) for each (X, x) ∈ AG is to say that F preserves the equalizer of (6) for each (X, x′) ∈ AG′ . Thus, when tF̄ is an isomorphism of comonds, F̄ is an equivalence of categories iff F is conservative and preserves the equalizer of (6) for each (X, x′) ∈ AG′ , which according to (the dual of) Beck’s theorem (see [9]), is to say that the functor F is comonadic. Hence our theorem 4.4 is equivalent to Theorem 1.7 of [8]. 5. Some applications Let (C,A, λ) be an entwining structure in a monoidal category V = (V,⊗, I), and let g : I → C be a group-like element of C. (Recall that a morphism g : I → C is said to be a group-like element of C if the following diagrams g⊗g ##F I C ⊗ C are commutative.) 5.1. Proposition. If C has a group-like element g : I → C, then A is a right C-comodule through the morphism gA : A g⊗A // C ⊗ A λ // A⊗ C. Proof. Consider the diagram g⊗A // C ⊗ A A A . The triangle is commutative by (1) of the definition of g and the square is commutative by the definition of λ (see the second commutative diagram in the definition of entwining structures). Now, we have to show that the following diagram g⊗A // C ⊗ A λ // A⊗ C C ⊗ A g⊗A⊗C // C ⊗A⊗ C // A⊗ C ⊗ C is also commutative, which it is since (A⊗ δC)λ = (λ⊗ C)(C ⊗ λ)(δC ⊗ A) by the definition of λ and since the diagram (2) of definition of group-like elements is commutative. Suppose now that V admits equalizers. For any (M,αM) ∈ V C, write ((M,αM) C, iM) for the equalizer of the morphisms (M,αM) iM // M αM // // M ⊗ C. 5.2. Proposition. AC = (A, gA) C is an algebra in V and iA : A C → A is an algebra morphism. Proof. Consider the diagram iA //A //C ⊗ A //A⊗ C ��������� Since g ⊗− : 1V = I ⊗− → C ⊗− is a natural transformation, the diagram g // C // C ⊗ A is commutative. Similarly, since eA⊗− : 1V = I⊗− → C⊗− is a natural transformation, the following diagram is also commutative: eA // // A⊗ C . Now we have: λ(g ⊗ A)eA = λ(C ⊗ eA)g = by the definition of λ = (eA ⊗ C)g = (A⊗ g)eA. Thus there exists a unique morphism eA : I → A C for which iA · eAC = eA. Since • the diagram g⊗A⊗A// C ⊗ A⊗A // C ⊗ A is commutative by naturality of g ⊗−; • λ(C ⊗mA) = (mA ⊗ C)(A⊗ λ)(λ⊗ A) by the definition of λ; • λ(g ⊗ A)iA = (A⊗ g)iA, since iA is an equalizer of λ(g ⊗ A) and A⊗ g; • the diagram A⊗A⊗g// A⊗ A⊗ C // A⊗ C is commutative by naturality of mA ⊗−, we have λ(g ⊗A)mA(iA ⊗ iA) = λ(C ⊗mA)(g ⊗ A⊗ A)(iA ⊗ iA) = = (mA ⊗ C)(A⊗ λ)(λ⊗ A)(g ⊗ A⊗ A)(iA ⊗ iA) = = (mA ⊗ C)(A⊗ λ)(A⊗ g ⊗ A)(iA ⊗ iA) = (mA ⊗ C)(A⊗A⊗ g)(iA ⊗ iA) = = (A⊗ g)mA(iA ⊗ iA). Thus the morphism mA · (iA ⊗ iA) equalizes the morphisms λ · (g ⊗ A) and A ⊗ g, and hence there is a unique morphism mAC : A C ⊗AC → AC such that the diagram AC ⊗ AC iA⊗iA // A⊗ A commutes. It is now straightforward to show that the triple (AC, eAC , mAC) is an algebra in V; moreover, the triangle of the diagram (7) and the diagram (8) show that iA is an algebra morphism. 5.3. Proposition. (A,mA, gA) ∈ V Proof. Since (A,mA) ∈ VA and (A, gA) ∈ V C, it only remains to show that the following diagram is commutative: gA⊗A // A⊗ C ⊗ A A⊗λ // A⊗ A⊗ C // A⊗ C. By the definition of gA, we can rewrite it as g⊗A⊗A// C ⊗ A⊗ A λ⊗A // A⊗ C ⊗ A A⊗λ // A⊗ A⊗ C // C ⊗A // A⊗ C. But this diagram is commutative, since • the middle square commutes because of naturality of g ⊗−; • the right square commutes because of the definition of λ. The algebra morphism iA : A C → A makes A an AC-AC-bimodule and thus induces the extension-of-scalars functor FiA : VAC → VA (X, ρX) −→ (X ⊗AC A,X ⊗AC mA), and the forgetful functor UiA : VA → VAC (Y, ̺Y ) −→ (Y, ̺Y · (Y ⊗ iA)), which is right adjoint to FiA. The corresponding comonad on VA makes A⊗AC A into an A-coring with the following counit and comultiplication: ε : A⊗AC A q // A⊗A mA // A, (where q is the canonical morphism) and δ : A⊗AC A = A⊗AC A C ⊗AC A iA⊗ACA// A⊗AC A⊗AC A = (A⊗AC A)A ⊗ (A⊗AC A). We write A⊗AC A for this A-coring. 5.4. Lemma. For any X ∈ VAC, the triple (X ⊗AC A,X ⊗AC mA, X ⊗AC gA) is an object of the category VC Proof. Clearly (X ⊗AC A,X ⊗AC mA) ∈ VA and ((X ⊗AC A,X ⊗AC gA) ∈ V C. Moreover, by (9), the following diagram X ⊗AC X ⊗AC A⊗ A X ⊗AC A⊗ C ⊗A // X ⊗AC A⊗A⊗ C X ⊗AC A X⊗ // X ⊗AC A⊗ C is commutative. Thus, (X ⊗AC A,X ⊗AC mA, X ⊗AC gA) ∈ V The lemma shows that the assignment X −→ (X ⊗AC A,X ⊗AC mA, X ⊗AC gA) yields a functor F̄ : VA → V (λ) = V (A⊗C)λ It is clear that U(A⊗C)λ ·F̄ = FiA, where U(A⊗C)λ : V (A⊗C)λ → VA is the underlying functor. It now follows from Theorem 3.1 that the composite A⊗AC A A⊗gA // A⊗A⊗ C mA⊗C // A⊗ C is a morphism of A-corings A⊗AC A → (A⊗ C)λ. We write can for this morphism. We say that A is (C, g)-Galois if can is an isomorphism of A-corings. Applying Theorem 4.4 the commutative diagram F̄ // FiA=−⊗ACA $$J (A⊗C)λ U(A⊗C)λ we get: 5.5. Theorem. Let (C,A, λ) be an entwining structure, and let g : I → C be a group-like element of C. Then the functor F̄ : VAC → V is an equivalence if and only if A is (C, g)-Galois and the functor F is comonadic. Let A = (A, eA, mA) and B = (B, eB, mB) be algebras in V and let M ∈ AVB. We call AM (resp. MB) • flat, if the functor − ⊗AM : VA → VB (resp. M ⊗B − : BV → AV) preserves equalizers; • faithfully flat, if the functor − ⊗AM : VA → VB (resp. M⊗B − : BV → AV) is conservative and flat (equivalently, preserves and reflects equalizers); 5.6. Theorem. Let (C,A, λ) be an entwining structure, and let g : I → C be a group-like element of C. If C is flat, then the following are equivalent (i) The functor F̄ : VAC → V (λ) = VA (A⊗C)λ is an equivalence of categories. (ii) A is (C, g)-Galois and ACA is faithfully flat. Proof. Since any left adjoint functor that is conservative and preserves equalizers is comonadic by a simple and well-known application (of the dual of) Beck’s theorem, one direction is clear from Therem 5.5; so suppose that F̄ is an equivalence of categories. Then, by Theorem 4.5, A is (C, g)-Galois and the functor FiA is comonadic. Since any comonadic functor is conservative, FiA is also conservative. Thus, it only remains to show that ACA is flat. Since C is flat by our assumption, A(A⊗ C) is also flat. It follows that the underlying functor of the comonad G(A⊗C)λ on VA preserves equalizers. We recall (for example, from [3]) that if G = (G, εG, δG) is a comonad on a category A, and if A has some type of limits preserved by G, then the category AG has the same type of limits and these are preserved by the underlying functor UG : AG → A. Thus the functor U(A⊗C)λ : VA (A⊗C)λ → VA preserves equalizers, and since F̄ is an equivalence of categories, the functor FiA = −⊗ACA also preserves equalizers, which just means that ACA is flat. This completes the proof. From now on we suppose at all times that our V is a strict braided monoidal category with braiding σX,Y : X ⊗ Y → Y ⊗X . Then the tensor product of two (co)algebras in V is again a (co)algebra; the multiplication mA⊗B and the unit eA⊗B of the tensor product of two algebras A = (A, eA, mA) and B = (B, eB, mB) are given through mA⊗B = (mA ⊗mB)(A⊗ σA,B ⊗B) eA⊗B = eA ⊗ eB. A bialgebra H = (H̄ = (H, eH , mH), H = (H, εH , δH)) in V is an algebra H̄ = (H, eH , mH) and a coalgebra H = (H, εH , δH), where εH and δH are algebra morphisms, or, equivalently, eH and mH are coalgebra morphisms. A Hopf algebra H = (H̄ = (H, eH , mH), H = (H, εH, δH), S) in V is a bialgebra H with a morphism S : H → H , called the antipode of H, such that mH(H ⊗ S)δH = mH(S ⊗H)δH . Recall that for any bialgebra H, the category VH is monoidal: The tensor product (X, δX) ⊗ (Y, δY ) of two right H-comodules (X, δX) and (Y, δY ) is their tensor product X ⊗ Y in V with the coaction δX⊗Y : X ⊗ Y δX⊗δY // X ⊗H ⊗ Y ⊗H X⊗σX,Y ⊗Y // X ⊗ Y ⊗H ⊗H X⊗Y⊗mH // X ⊗ Y ⊗H . The unit object for this tensor product is I with trivial H-comodule structure eH : I → H. 5.7. Proposition. Let H = (H̄ = (H, eH , mH), H = (H, εH , δH)) be a bialgebra in V. For any algebra A = (A, eA, mA) in V, the following conditions are equivalent: • A = (A, eA, mA) is an algebra in the monoidal category V • A = (A, eA, mA) is an H-comodule algebra; that is, A is a right H-comodule and the H-comodule coaction αA : A → A⊗H is a morphism of algebras in V from the algebra A = (A, eA, mA) to the algebra A⊗ H̄ = (A⊗ H̄, eA ⊗ eH , mA⊗H̄). Suppose now that A = (A, eA, mA) is a right H-comodule algebra with H-coaction αA : A → A⊗H . By the previous proposition, A is an algebra in the monoidal category VH , and thus defines a monad TAH = (T H , η H , µ H) on V H as follows: • TAH (X, δX) = (X, δX)⊗ (A, αA); • (ηAH)(X,δX ) = X ⊗ eA; • (µAH)(X,δX) = X ⊗mA. It is easy to see that the monad TAH extends the monad T A; and it follows from Theorem 2.1 that there exists a distributive law λα : T A ·GH → GH ·T A from the monad TA to the comonad GH , and hence an entwining structure (H,A, λ(A,αA)), where λ(A,αA) = (λα)I . Therefore we have: 5.8. Theorem. Every right H-comodule algebra A = ((A, αA), mA, eA) defines an entwin- ing structure (H,A, λ(A,αA) : C ⊗A → A⊗ C). 5.9. Proposition. Let A = ((A, αA), mA, eA) be a right H-comodule algebra. Then the entwining structure λA,αA : H ⊗ A → A⊗H is given by the composite: H ⊗ A H⊗αA// H ⊗ A⊗H σH,A⊗H// A⊗H ⊗H A⊗mH// A⊗H . Proof. Since (A⊗ αA) , (H, δH) ∈ V H , the pair (A⊗H, δA⊗H), where δA⊗H is the com- posite δH⊗αA// H ⊗H ⊗H ⊗A H⊗σH,A⊗H // H ⊗ A⊗H ⊗H H⊗A⊗mH // H ⊗ A⊗H , is also an object of VH , and it follows from Theorem 1.1 that λ(A,αA) is the composite H ⊗ A δA⊗H // H ⊗ A⊗H εH⊗A⊗H // A⊗H. Consider now the following diagram H ⊗A⊗H δH⊗A⊗H // H ⊗H⊗A⊗H H⊗σH,A⊗H // εH⊗H⊗A⊗H H ⊗ A⊗H ⊗H H⊗A⊗mH // εH⊗A⊗H⊗H H ⊗ A⊗H εH⊗A⊗H H ⊗ A⊗H σH,A⊗H // A⊗H ⊗H // A⊗H . Since in this diagram • the triangle commutes because εH is the counit for δH ; • the left square commutes by naturality of σ; • the right square commutes because −⊗− is a bifunctor, it follows that λ(A,αA) = (A⊗mH)(σH,A ⊗H)(H ⊗ αA). Note that the morphism eH : I → H is a group-like element for the coalgebra H = (H, εH , δH). 5.10. Proposition. Let H = (H̄ = (H, eH , mH), H = (H, εH, δH)) be a bialgebra in V, and let A = ((A, αA), eA, mA) be a right H-comodule algebra. Then the right H-comodule structure on A corresponding to the group-like element eH : I → H as in Proposition 4.1 coincides with αA. Proof. We have to show that (A⊗mH)(σH,A ⊗H)(H ⊗ αA)(eH ⊗ A) = αA. But since • clearly (H ⊗ αA)(eH ⊗ A) = (eH ⊗A⊗H) · αA; • (σH,A ⊗H) · (eH ⊗A⊗H) = A⊗ eH ⊗H by naturality of σ; • (A⊗mH) · (A⊗ eH ⊗H) = 1A⊗H since eH is the identity for mH , we have that (A⊗mH)(σH,A ⊗H)(H ⊗ αA)(eH ⊗ A) = = (A⊗mH)(σH,A ⊗H)(eH ⊗A⊗H)αA = = (A⊗mH)(A⊗ eH ⊗H)αA = = 1A⊗H · αA = αA. It now follows from Proposition 5.3 that 5.11. Proposition. A = (A, eA, mA) ∈ V (λA,αA). Recall that for any (X,αX) ∈ V H , the algebra XH = (X,αX) H is the equalizer of the morphisms αX // // X ⊗H. Applying Theorem 5.5 we get 5.12. Theorem. Let H = (H̄ = (H, eH , mH), H = (H, εH , δH)) be a bialgebra in V, let A = ((A, αA), eA, mA) be a right H-comodule algebra, and let λ(A,αA) : H ⊗A → A⊗H be the corresponding entwining structure. Then the functor F̄ : VAH → V (λ(A,αA)) (X, νX) −→ (X ⊗AH A,X ⊗AH mA, X ⊗AH αA) is an equivalence of categories iff the extension-of-scalars functor FiA : VAH → VA (X, νX) −→ (X ⊗AH A,X ⊗AH mA) is comonadic and A is H-Galois (in the sense that the canonical morphism can : A⊗AH A → A⊗H is an isomorphism). Now applying Theorem 5.6 we get 5.13. Theorem. Let H = (H̄ = (H, eH , mH), H = (H, εH , δH)) be a bialgebra in V, let A = ((A, αA), eA, mA) be a right H-comodule algebra, and let λ(A,αA) : H ⊗ A → A ⊗ H be the corresponding entwining structure. Suppose that H is flat. Then the following are equivalent: (i) The functor F̄ : VAH → V (λ(A,αA)) (X, νX) −→ (X ⊗AH A,X ⊗AH mA, X ⊗AH αA) is an equivalence of categories. (ii) A is H-Galois and AHA is faithfully flat. Let H = (H̄ = (H, eH , mH), H = (H, εH , δH)) be a bialgebra in V, and let A = ((A, αA), eA, mA) be a right H-comodule algebra. A right (A,H)-module is a right A- module which is a right H-comodule such that the H-comodule structure morphism is a morphism of right A-modules. Morphisms of right (A,H)-modules are right A-module right H-comodule morphisms. We write VH for this category. Note that the category VH is the category (VH)A of right A-modules in the monoidal category V H , and it follows from Theorem 2.1 that 5.14. Proposition. VH (λ(A,αA)). The following is an immediate consequence of Theorem 5.12. 5.15. Theorem. Let H = (H̄ = (H, eH, mH), H = (H, εH, δH)) be a bialgebra in V, and let A = ((A, αA), eA, mA) be a right H-comodule algebra. Then the functor F̄ : VAH → V is an equivalence of categories iff the extension-of-scalars functor FiA : VAH → VA is comonadic and A is H-Galois. Let H = (H̄ = (H, eH , mH), H = (H, εH , δH), S) be an Hopf algebra in V. Then clearly H̄ = (H, eH , mH) is a right H-comodule algebra. 5.16. Proposition. The composite x : H ⊗H H⊗δH // H ⊗H ⊗H mH⊗H// H ⊗H is an isomorphism. Proof. We will show that the composite y : H ⊗H H⊗δH // H ⊗H ⊗H H⊗S⊗H// H ⊗H ⊗H mH⊗H// H ⊗H is the inverse for x. Indeed, consider the diagram H⊗δH // H ⊗H ⊗H mH⊗H // H⊗H⊗δH H ⊗H ⊗H H⊗δH⊗H // H ⊗H ⊗H ⊗H H⊗H⊗S⊗H mH⊗H⊗H // H ⊗H ⊗H H⊗S⊗H H ⊗H ⊗H ⊗H H⊗mH⊗H mH⊗H⊗H // H ⊗H ⊗H H ⊗H ⊗H // H ⊗H . We have: • Square (1) commutes because of coassociativity of δH ; • Square (2) commutes because of naturality of mH ⊗−; • Square (3) commutes because −⊗− is a bifunctor; • Square (4) commutes because of associativity of mH . yx = (mH ⊗H)(H ⊗ S ⊗H)(H ⊗ δH)(mH ⊗H)(H ⊗ δH) = = (mH ⊗H)(H ⊗mH ⊗H)(H ⊗H ⊗ S ⊗H)(H ⊗ δH ⊗H)(H ⊗ δH), but since mH(H ⊗ S)δH = eH · εH , yx = (mH ⊗H)(H ⊗ eHεH ⊗H)(H ⊗ δH) = = (mH ⊗H)(H ⊗ eH ⊗H)(H ⊗ εH ⊗H)(H ⊗ δH) = = 1H⊗H ⊗ 1H⊗H = 1H⊗H . Thus yx = 1. The equality xy = 1 can be shown in a similar way. 5.17. Proposition. (H, δH) H ≃ (I, eH). Proof. We will first show that the diagram H⊗eH // // H ⊗H δH // // H ⊗H is serially commutative. Indeed, we have: x(H ⊗ eH) = (mH ⊗H)(H ⊗ δH)(H ⊗ eH) = since δH is an algebra morphism = (mH ⊗H)(H ⊗ eH ⊗ eH) = since eH is the unit for mH = H ⊗ eH ; x(eH ⊗H) = (mH ⊗H)(H ⊗ δH)(eH ⊗H) = since eH is a coalgebra morphism = (mH ⊗H)(eH ⊗H)δH = 1HδH = δH . Thus, (H, δH , eH) is isomorphic to the equalizer of the pair (H⊗ eH , eH ⊗H). But since eH : I → H is a split monomorphism in V, the diagram eH // H H⊗eH // // H ⊗H is an equalizer diagram. Hence (H, δH , eH) H ≃ (I, eH). 5.18. Theorem. Let H = (H̄ = (H, eH , mH), H = (H, εH , δH), S) be a Hopf algebra in V. Then the functor V → VH V → V ⊗H is an equivalence of categories. Proof. It follows from Propositions 5.16 and 5.17 that H is H-Galois, and according to Theorem 5.12, the functor V → VH is an equivalence iff the functor − ⊗H : V → VH̄ is comonadic. But since the morphism eH : I → H is a split monomorphism in V, the unit of the adjunction FeH ⊣ UeH is a split monomorphism, and it follows from 3.16 of [10] that FeH is comonadic. This completes the proof. References [1] M. Barr and C. Wells, Toposes, Triples, and Theories, Grundlehren der Math. Wis- senschaften 278, Springer-Verlag, 1985. [2] J. Beck, Distributive laws . Lect. Notes Math. 80, 119-140 (1969). [3] F. Borceux, Handbook of Categorical Algebra. vol. 2, Cambridge University Press, 1994. [4] T. Brzezinski and S, MajidCoalgebra bundles. Comm. Math. Phys. 191, 467-492 (1998). [5] T. Brzezinski and R. Wisbauer, Corings and comodules, London Math. Soc. Lect. Note Ser. 309, Cambridge University Press, Cambridge, 2003. [6] S. Caenepeel, G. Militaru and S. Zhu,Frobenius and separable functors for generalized Hopf modules and nonlinear equations, Lect. Notes Math. 1787, (2003). [7] E. Dubuc, Kan extensions in enriched category theory. Lecture Notes Math. 145 (1970). [8] J. Gómez-Torrecillas, Comonads and Galois corings. Appl. Categ. Struct. 14, No. 5-6, 579-598 (2006). [9] S. MacLane, Categories for the Working Mathematician. Graduate Texts in Mathe- matics Vol. 5, Springer, Berlin-New York, 1971. [10] B. Mesablishvili, Monads of effective descent type and comonadicity. Theory and Applications of Categories 16 (2006), 1–45. [11] R. Wisbauer, Algebras versus coalgebras. Appl. Categ. Struct. (2007) (in press). [12] H. Wolff, V-Localizations and V-mondas. J. of Algebra 24 (1973), 405–438. Introduction Mixed distributive laws Entwining structures in monoidal categories Some categorical results Some applications

0704.1232

CP Violation and Arrows of Time Evolution of a Neutral $K$ or $B$ Meson from an Incoherent to a Coherent State

CP Violation and Arrows of Time: Evolution of a Neutral K or B Meson from an Incoherent to a Coherent State Ch. Berger I. Physikalisches Institut der RWTH, Aachen, Germany L. M. Sehgal∗ Institut für Theoretische Physik (E) der RWTH, Aachen, Germany October 24, 2018 Abstract We study the evolution of a neutral K meson prepared as an in- coherent equal mixture of K0 and K̄0. Denoting the density matrix by ρ(t) = 1 1+ ~ζ(t) · ~σ , the norm of the state N(t) is found to decrease monotonically from one to zero, while the magnitude of the Stokes vector |~ζ(t)| increases monotonically from zero to one. This property qualifies these observables as arrows of time. Requiring mono- tonic behaviour of N(t) for arbitrary values of γL, γS and ∆m yields a bound on the CP-violating overlap δ = 〈KL |KS 〉, which is similar to, but weaker than, the known unitarity bound. A similar requirement on |~ζ(t)| yields a new bound, δ2 < 1 which is particularly effective in limiting the CP-violating overlap in the B0-B̄0 system. We obtain the Stokes parameter ζ3(t) which shows how the average strangeness of the beam evolves from zero to δ. The evolu- tion of the Stokes vector from |~ζ| = 0 to |~ζ| = 1 has a resemblance to an order parameter of a system undergoing spontaneous symmetry breaking. 1 Introduction We examine in this paper the time evolution of a neutral K meson prepared as an equal incoherent mixture of K0 and K̄0. Such a state is easily obtained in a reaction such as e+e− → φ(1020) → K0K̄0, when only one of the kaons in the final state is observed. (Our considerations apply equally to B mesons e-mail: [email protected] http://arxiv.org/abs/0704.1232v2 produced in e+e− → Υ(4s) → B0B̄0.) An incoherent beam of this type is characterized by a density matrix, which we write in the K0-K̄0 basis as ρ(t) = 1+ ~ζ(t) · ~σ The evolution is described by a normalization function N(t), which is the intensity of the beam at time t, and a Stokes vector ~ζ which characterizes the polarization state of the system with respect to strangeness. The beam, which has |~ζ(0)| = 0 at the time of production evolves ultimately into a pure state corresponding to the long-lived K meson KL, with a Stokes vector of unit length: |~ζ(∞)| = 1. In this sense, the system can be regarded as possessing two dynamical func- tions: N(t) which varies from one to zero, and |~ζ(t)| which goes from zero to one. This evolution touches on interesting issues such as the role of CP-violation, and the extent to which the functions N(t) and |~ζ(t)| define arrows of time. The requirement that these functions are monotonic yields constraints on the CP-violating parameter δ = 〈KL |KS 〉. The fact that the incoherent initial state is completely neutral with respect to strangeness and CP quantum numbers is of significance in this regard. In addition the component ζ3(t) of the Stokes vector describes the manner in which the strangeness of the state evolves from zero to final value δ = 3.27× 10−3 and serves as a model for flavour-genesis induced by CP violation in a decaying system. Finally, the evolution of the system from an initial “amorphous” state with ~ζ(0) = 0 to a final “crystalline” state described by a three- dimensional Stokes vector ~ζ(t) with unit length, is suggestive of a phase transition, with ~ζ(t) playing the role of an order parameter of a system undergoing spontaneous symmetry breaking. 2 Density Matrix An arbitrary state of the K meson can be desccribed by a 2 x 2 density matrix which we write, in the K0-K̄0 basis, as [1, 2, 3] ρ(t) = 1+ ~ζ(t) · ~σ Here N(t) is the intensity or norm of the state at time t, calculated from the trace of ρ, N(t) = tr ρ(t) (2) and ~ζ(t) is the Stokes vector, whose components can be expressed as ζi(t) = tr [ρ(t)σi] /trρ(t) (3) An initial state which is a 1:1 incoherent mixture of K0 and K̄0 has the density matrix ρ(t) = |K0 〉〈K0 |+ | K̄0 〉〈 K̄0 | which corresponds to an initial Stokes vector ~ζ(0) = 0. To determine the time evolution, we note that [4] |K0 〉 → |ψ(t) 〉 = |KS 〉e −λSt + |KL 〉e | K̄0 〉 → | ψ̄(t) 〉 = |KS 〉e −λSt − |KL 〉e where we have introduced the eigenstates |KL 〉 = p|K 0 〉 − q| K̄0 〉 |KS 〉 = p|K 0 〉+ q| K̄0 〉 (|p|2 + |q|2 = 1) with eigenvalues λL,S = γL,S + imL,S (7) The overlap of the states |KL 〉 and |KS 〉 is given by the CP-violating parameter δ = 〈KL |KS 〉 = (|p| 2 − |q|2)/(|p|2 + |q|2) = 3.27 × 10−3 (8) The resulting density matrix at time t is ρ(t) = ρ11(t) ρ12(t) ρ21(t) ρ22(t) ρ11(t) = 4(1− δ) e−γSt + e−γLt − 2δe− (γS+γL)t cos∆mt ρ22(t) = 4(1 + δ) e−γSt + e−γLt + 2δe− (γS+γL)t cos∆mt ρ12(t) = e−γSt − e−γLt + i2δe− (γS+γL)t sin∆mt ρ21(t) = ρ 12(t) (10) Using Eqs. (2) and (3), we derive N(t) = 2(1 − δ2) e−γSt + e−γLt − 2δ2e− (γS+γL)t cos∆mt ζ1(t) = Re (pq∗) e−γSt − e−γLt − Im (pq∗) 2δe− (γS+γL)t · sin∆mt e−γSt + e−γLt − 2δ2e− (γS+γL)t cos∆mt ζ2(t) = Im (pq∗) e−γSt − e−γLt +Re (pq∗) 2δe− (γS+γL)t · sin∆mt e−γSt + e−γLt − 2δ2e− (γS+γL)t cos∆mt ζ3(t) = δ e−γSt + e−γLt − 2e− (γS+γL)t cos∆mt e−γS t + e−γLt − 2δ2e− (γS+γL)t cos∆mt . (12) Note that the components ζ1,2(t) involve Re (pq ∗) and Im (pq∗) where p and q are the coefficients in the definition of KL,S in eq.(6). These are convention-dependent, since the relative phase of p and q can be changed by a phase transformation |K0 〉 → eiα|K0 〉, | K̄0 〉 → e−iα| K̄0 〉. A quantity independent of phase convention is ζ21 + ζ 2 = (1− δ e−γSt − e−γLt + 4δ2e−(γS+γL)t sin2 ∆mt e−γSt + e−γLt − 2δ2e− (γS+γL)t cos∆mt Thus the length of the Stokes vector is |~ζ(t)| = ζ21 (t) + ζ 2 (t) + ζ 3 (t) 2N(t)(1 − δ2) e−γSt + e−γLt − 2e− (γS+γL)t cos∆mt + (1− δ2) e−γSt − e−γLt + 4δ2e− (γS+γL)t · sin2∆mt N(t)2 e−(γS+γL)t This equation provides a simple relation between the magnitude of the Stokes vector |~ζ(t)| and the normalization function N(t). In the CP-invariant limit, δ → 0, the density matrix reduces to ρ(t) −→ e−γSt + e−γLt e−γSt − e−γLt e−γSt − e−γLt e−γSt + e−γLt and the limiting form of N(t) and ~ζ(t) is 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 dN/dt Figure 1: Normalization function N(t) (a) and its time derivative (b) as function of time in units of τS for a state prepared as incoherent mixture of K 0 and K̄0. N(t) −→ e−γSt + e−γLt ζ12 ≡ ζ21 (t) + ζ 2 (t) e−γLt − e−γSt e−γSt + e−γLt ζ3(t) −→ 0 (16) The behaviour of N(t) and dN/dt for the K0-K̄0 system is shown in fig.1. The behaviour of the functions ζ212(t), ζ 3 (t) and ζ 2(t) = ζ212(t) + ζ 3 (t) and their time derivatives is shown in fig.2. The function ζ23 is clearly nonmono- tonic, and its derivative has a number of zeros (e.g. t/τS = 4.95, 11.6, 18.2, ...). By comparison the derivative of ζ212 has a distant zero at t/τS = 25.7. As seen in fig.2b these two nonmonotonic functions combine to produce a Stokes vector ζ2(t) which is strictly monotonic, the asymptotic values being ζ212 → (1− δ 2), ζ23 → δ 2, ζ2 → 1. 3 Arrows of time 3.1 The Normalization Arrow N(t) The normalization of the kaon state is given in eq.(11). As seen in fig.1, this function is indeed monotonic for the parameters of the K meson system. This monotonic (unidirectional) property implies that N(t) behaves as an arrow of time. In the absence of CP-violation (δ = 0), the function N(t) is simply the sum of two exponentials (e−γSt + e−γLt)/2, and the monotonic decrease is ensured by the requirement γS, γL > 0 (positivity of the decay matrix). The third term in eq.(11), appearing when δ 6= 0, indicates a KL- KS interference effect. It implies that an incoherent K 0-K̄0 mixture does 2 4 6 8 10 12 14 16 |ζ |2 · 1/2 · 20000 –5e–11 5e–11 1e–10 1.5e–10 2e–10 24 26 28 30 32 34 36 |ζ̇|2 Figure 2: Behaviour of |ζ|2, ζ2 and ζ2 (a) and their time derivatives (b) as function of time. Note the different scale for ζ2 and ζ2 in (a). In (b) only the tail of the time dependence is shown. not evolve like an incoherent KL-KS mixture. Notice however, that the coefficient of the interference term is quadratic in δ, so that the function N(t) is CP-even, remaining unchanged under δ → −δ. Nevertheless the presence of the δ2 term is decisive in determining whether or not N(t) is monotonic, and hence an arrow of time. If we require the function N(t) to be monotonic (dN/dt < 0) then we have from (11) (see also [5]) that 2(1 − δ2) −γSt + γLe −γLt − 2δ2e− (γS+γL)t γS + γL cos∆mt+∆m sin∆mt < 0 (17) from which it follows, as sufficient condition, that (γS + γL) /4 + ∆m2 or δ2 ≤ (1 + r) /4 + µ2 where we have introduced the notation r = γL/γS , µ = ∆m/γS . This con- straint is analogous to, but weaker than, the unitarity constraint derived in [6, 7], which reads δ2unit ≤ (1 + r) /4 + µ2 . (19) 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 dN/dt Figure 3: Behaviour of N(t) (a) and dN/dt (b) for values of parameters µ = µK , r = 0.01, δ = 0.6 violating the monotonicity bound (18). It is clear that the unitarity bound is interesting for a system like K0-K̄0, where r = γL/γS is small, but does not provide a useful constraint for B 0-B̄0, where r is close to 1. To see what happens if the parameters δ, r and µ are allowed to vary, we show in fig.3 the behaviour of N(t) and dN/dt for δ = 0.6, r = 0.01, keep- ing µ at its standard K meson value, µK = 0.47. The function N(t) shows fluctuations, and the derivative dN/dt changes sign. Such a behaviour re- sults from the violation of the bound (18). The fluctuations in N(t) may be regarded as fluctuations in the direction of the time arrow (we call this phenomenon “Zeitzitter”), and can occur when the CP-violating parameter δ2 exceeds the limit (18). This is the manner in which CP violation impacts on the time arrow, even though the function N(t) is CP-even. From the point of view of an observer monitoring the intensity of the kaon beam (for example by measuring the rate of leptonic decays π∓ℓ±ν) the fluc- tuation in N would appear as an inexplicable enhancement or suppression of the beam intensity in certain intervals of time. The effect can be regarded equivalently as violation of unitarity or a flutter in the arrow of time. 3.2 The Stokes Arrow The magnitude of the Stokes vector |~ζ(t)|2 calculated in eq.(14), is a measure of the coherence of the state, and is plotted in fig.2 for the physical K-meson parameters. One sees that the function |~ζ|2 evolves monotonically from 0 to 1, and its derivative remains positive at all times. Thus the Stokes parameter |~ζ(t)| qualifies as an arrow of time. To see how this arrow is affected if the parameters δ, r and µ are allowed to vary, we look at the derivative of the function ζ(t). Writing |~ζ(t)| = e−(γS+γL)t N(t)2 we find that the monotonicity condition d|~ζ(t)|/dt > 0 is equivalent to the condition (γS + γL)N ≤ 0 (21) which implies e∆γt/2 − e−∆γt/2 + 4δ2 sin∆mt ≥ 0 (22) where ∆γ = γS − γL. From this we derive a new upper bound on δ or δ2 < (1− r) This bound is obtained from the requirement that |~ζ(t)| be monotonic (an arrow of time) just as the bound in eq.(18) was derived from the mono- tonicity of N(t). The bound (23) is particularly effective in constraining the value of the overlap parameter in the B0-B̄0 system, in which the decay widths of the two eigenstates are close together, r → 1. In this respect the bound in eq.(23) is complementary to the unitarity bound in eq.(19) which is effective when r → 0. The contrast between the two bounds is highlighted in fig.4. Taking the parameters of the B0-B̄0 system to be r = 0.99, µ = 0.7, we obtain from (23) δB = 〈B L 〉 . 0.0155 . (24) We wish to stress that for a B0-like system the bound in (23) is not just a sufficient condition for monotonic behaviour of |ζ(t)|2, but almost a critical value separating the monotonic and nonmonotonic domains. As an illustra- tion we show in fig.5 the transition in the behaviour of |ζ(t)|2 for a system with parameters µ = 0.7, r = 0.9, as δ is varied from a value 0.1, below the critical value of δcrit = 0.156, to a value 0.2 above δcrit. The fluctuations in |ζ(t)|2, shown in fig.5 are the analog of the fluctuations in N(t), shown in fig.3, which arise when the parameters of the system vi- olate the bound in eq.(18). Whereas the fluctuation in N(t) would reveal itself as an inexplicable Zitter in the beam intensity, the fluctuation in |~ζ(t)| would show up as an unaccountable Zitter in the coherence of the beam. 0.2 0.4 0.6 0.8 1 constraints on δ Figure 4: Constraints on δ in the δ-r plane resulting from unitarity and monotoni- city of |ζ(t)| for a B0-like system with µ = 0.7. The thick line represents the unitarity bound (19) and the thin line our new bound evaluated from (23). The numerical evaluation of (22) (open circles) yields values very close to the approxi- mation given in eq.(23). In both cases, the effect results from a breakdown in the monotonicity of a function, associated with a loss of directionality in an arrow of time. 4 Evolution of Strangeness The component ζ3(t) of the Stokes vector has a special significance: it is the expectation value of σ3, which can be identified with the strangeness operator with eigenvalues +1 for K0 and −1 for K̄0. Thus a measurement of ζ3(t) is simply a measurement of the decay asymmetry into the channels π−ℓ+ν and π+ℓ−ν̄ : ζ3(t) = Γ (π−ℓ+ν; t)− Γ (π+ℓ−ν̄; t) Γ (π−ℓ+ν; t) + Γ (π+ℓ−ν̄; t) 2 4 6 8 10 12 14 16 |ζ(t)|2 Figure 5: Evolution of the Stokes vector |ζ(t)|2 for a B0-like system with para- meters µB = 0.7, r = 0.9. The middle curve corresponds to δcrit = 0.156 obtained from the bound (23). The nonmonotonic upper curve is obtained for δ = 0.2 and the monotonic lower curve for δ = 0.1. Referring to eq.(12), we observe that ζ3(t) is a pure CP-violating observable, since it changes sign under δ → −δ (By contrast, the functions N(t) and |~ζ(t)|, are invariant under δ → −δ). Writing ζ3(t) explicitly as ζ3(t) = δ e−γSt + e−γLt − 2e− (γL+γS)t cos∆mt 2 (1− δ2)N(t) we note that it is a quotient of a function that contains an oscillating term and a monotonic function N(t). The average strangeness ζ3(t) is thus clearly not a monotonic function of time. This is visible in fig.6, where we also show the derivative dζ3(t)/dt. We have here an explicit example of a CP- odd observable emerging from an initial state that has no preferred CP direction. Such observables are not monotonic, and cannot be associated with an arrow of time. 0.001 0.002 0.003 0.004 2 4 6 8 10 12 14 16 –0.0002 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0 2 4 6 8 10 12 14 16 dζ3/dt Figure 6: Evolution of ζ3(t) (a) and its derivative (b) in the K 0-K̄0 system. 5 Summary (1) We have shown that the evolution of an incoherent K0-K̄0 mixture is characterized by two time-dependent functions, the norm N(t) and the magnitude of the Stokes vector |~ζ(t)| both of which evolve mono- tonically and may therefore be associated with microscopic arrows of time. It should be stressed that we are discussing here conditional ar- rows of time, whose existence depends on the degree of CP violation, and not simply on the positivity of the decay widths γL,S. (2) If the parameters γL, γS ,∆m and δ = 〈KL |KS 〉 are allowed to vary, the requirement of monotonic behaviour of N(t) leads to the bound in eq.(18), which is similar to, but weaker than, the unitarity bound (19), derived in [6, 7]. The requirement of monotonicity for |~ζ(t)| leads to a new bound on δ2 given in eq.(23), which is complementary to the unitarity bound (19), and far more restrictive for systems such as B0-B̄0 with r = γL/γS close to unity. (3) A violation of the bounds in eq.(18) and (23) leads to fluctuations in N(t) and ζ(t) associated with fluctuations in the arrow of time (“Zeitzitter”) and a violation of unitarity. (4) It is worth noting that the product N2(1− |~ζ|2) is equal to e−(γL+γS)t and therefore monotonic for all values of δ, r and µ. This product is just four times the determinant of the density matrix ρ(t), (5) The time-dependence of ζ3(t) describes the evolution of strangeness in a beam that is initially an equal mixture of K0 and K̄0. It is an example of flavour-genesis induced by CP violation in a decaying system. (6) The emergence of a non-zero three-dimensional Stokes vector ~ζ(t) from a state that is initially “amorphous” (~ζ(0) = 0), is suggestive of a phase transition. The evolution of the Stokes vector from zero to unit length is reminiscent of an order parameter for a system undergoing spontaneous symmetry breaking. (7) All our considerations have been in the framework of ordinary quan- tum mechanics and CPT invariance. Discussions that involve violation of quantum mechanics and/or CPT symmetry may be found, for ex- ample, in [8]. An early discussion of the arrow of time in connection with K meson decays is given in [9]. Broader issues connected with the arrow of time are discussed, for instance, in [10]. Finally, experimental investigations of discrete symmetries in the decays of K mesons and B mesons produced in e+e− or pp̄ collisions are described in [11]. Acknowledgement: One of us (LMS) wishes to thank Dagmar Bruss (University of Düsseldorf) for a useful correspondence. References [1] U. Fano, Rev. Mod. Phys. 29, 74 (1957) [2] R. G. Sachs, Physics of Time Reversal, University of Chicago Press, [3] L. M. Sehgal Density Matrix Description of Neutral K Meson Decay, Tata Institute report, TIFR - TH- 70-35 (1970) [4] T. D. Lee, R. Oehme and C. N. Yang, Phys. Rev. 105, 1671 (1957) [5] L. M. Sehgal, Decays of Neutral K Mesons Produced in pp̄ Annihilation. A Comment, Aachen preprint, 1973 (unpublished) [6] T. D. Lee and L. Wolfenstein, Phys. Rev. 138, B1490 (1965) [7] J. S. Bell and J. Steinberger, in Proc. Oxford International Conference on Elementary Particles, 1965, pp. 195-222 [8] P. Huet, M. E. Peskin, Nucl. Phys. B 434, 3 (1995) J. Ellis, N. E. Mavromatos and D. V. Nanopoulos, Phys. Lett. B 293, 142 (1992) [9] A Aharony, Ann. Phys. 67, 1 (1971); ibid 68, 163 (1971); A. Aharony and Y. Ne’eman, Int. J. Theor. Phys. 3, 437 (1970) [10] Physical Origins of Time Asymmetry, Eds. Jose Angel Sanchez Asiain, et al, Cambridge University Press (1996) [11] KLOE Collaboration, F. Ambrosino et al, Phys.Lett. B642, 315 (2006) Babar Collaboration (B.Aubert et al), Phys.Rev.Lett. 96, 251802 (2006) Belle Collaboration (E.Nakano et al), Phys.Rev. D73, 112002 (2006) CPLEAR Collaboration, M. Carrol et al, Nucl. Phys. A 626 157c-165c (1997); K. Kleinknecht Uncovering CP Violation, Springer-Verlag, Berlin, Hei- delberg (2003). Introduction Density Matrix Arrows of time The Normalization Arrow N(t) The Stokes Arrow Evolution of Strangeness Summary

0704.1233

Residual entropy in a model for the unfolding of single polymer chains

epl draft Residual entropy in a model for the unfolding of single polymer chains E. Van der Straeten (a)(b) and J. Naudts(c) 1 Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, 2020 Antwerpen, Belgium PACS 82.35.Lr – Physical properties of polymers PACS 05.20.-y – Classical statistical mechanics Abstract. - We study the unfolding of a single polymer chain due to an external force. We use a simplified model which allows to perform all calculations in closed form without assuming a Boltzmann-Gibbs form for the equilibrium distribution. Temperature is then defined by calcula- ting the Legendre transform of the entropy under certain constraints. The application of the model is limited to flexible polymers. It exhibits a gradual transition from compact globule to rod. The boundary line between these two phases shows reentrant behavior. This behavior is explained by the presence of residual entropy. Introduction. – The unfolding of polymers has been studied for many years. Recently, it became possible to investigate the stability of a single polymer against un- folding by applying a mechanical force to the end-point of the molecule. Tools like optical tweezers, atomic force microscopes and soft microneedles are used in this kind of experiments. The unfolding transition of polymers is a single- or a multi-step process, depending on the ex- perimental conditions. For example, with a force-clamp apparatus one is able to study the mechanical unfolding at constant force. One applies a sudden force which is then kept constant with feedback techniques. Numerical simulations [1] show that a certain protein (called ubiqui- tin) unfolds in a single step, while an other protein (called integrin) unfolds in multiple steps. Ubiquitin is studied ex- perimentally in [2]. The findings of [2] support the numer- ical simulations rather well, because in 95% of the cases a clear single-step unfolding process is observed. In [3, 4], force-extension relations of single DNA molecules are ob- tained in the fixed-stretch ensemble. One measures the average applied force while keeping the extension of the DNA molecule constant. Depending on the solvent condi- tions, force plateaus or stick-release patterns are observed. The unfolding transition from compact globule to rod was already theoretically predicted 15 years ago [5], based on heuristic arguments. Nowadays, the theoretical study (a)Research Assistant of the Research Foundation - Flanders (FWO - Vlaanderen) [email protected] [email protected] of this transition is dominated by numerical simulations. The self-avoiding walk (SAW) in two dimensions is in- tensively used to model the unfolding transition of poly- mers [6–8]. The advantage of SAWs is that local interac- tions like monomer-monomer attraction and the excluded volume effect are taken into account. The disadvantage is that one is limited to short walks due to the computa- tional cost (up to chain length 55 [7]). In [6–8], a force- temperature state diagram for flexible polymers in a poor solvent is calculated in the fixed-force ensemble. The most important property is that the force, at which the poly- mer unfolds, goes through a maximum as a function of the temperature. This so called reentrant behavior is ex- plained by the presence of residual entropy [6]. Reentrant behavior has been observed at other occa- sions as well. A realistic, analytical solvable model for the unfolding transition is presented in [9]. The authors ob- tain state diagrams for 4 different molecules. One of these diagrams shows reentrant behavior. The authors of [9] do not comment on this interesting feature. In [10], simpli- fied, analytical solvable, lattice models are considered to describe the unzipping of DNA. The obtained state dia- gram shows reentrant behaviour. This is also caused by the appearance of residual entropy in the model. In [11], the present authors have proposed a simple model to describe single polymers. This model has been used in [12] to compare the outcome of two experiments, which are performed in the fixed-force ensemble and the fixed-extension ensemble. In the present letter, we focus on the fixed-force ensemble, because the model is com- http://arxiv.org/abs/0704.1233v3 E. Van der Straeten et al. 0 0.5 1 1.5 0 0.5 1 1.5 Fig. 1: Plot of the force-extension relation relation at constant temperature, T = 0.1. The value of the parameter h is −1 for the solid line and 0.01 for the dotted line. The value of a is equal to 1 for both lines. pletely solvable in closed form in this ensemble [12]. The more general theory of [14] allows to simplify the calcula- tions of [11, 12]. Of course, this simplified model is of a qualitative nature. More sophisticated models are needed to describe all the experimentally observed features of the unfolding transition. The aim of the present letter is to focus on one of these features, namely the consequences of the appearance of residual entropy. One dimensional random walk with memory. – The model under study is a discrete-time random walk on a one-dimensional lattice. It depends on two parameters ǫ and µ, the probabilities to go straight on, when walk- ing to the right, respectively to the left. This is not a Markov chain since the walk remembers the direction it comes from. The process of the increments is a Markov chain with two states, → and ←. The stationary proba- bility distributions of these two states are [11] p(→) = 1− µ 2− ǫ− µ and p(←) = 1− ǫ 2− ǫ− µ . (1) With these expressions, the average end postion 〈x〉 of the walk and the average number of reversals of direction 〈K〉 of the walk can be written conveniently as [12, 14] = a [p(→)− p(←)] , = 2(1− ǫ)p(→) = 2(1− µ)p(←), (2) with a the lattice parameter and n the total number of steps. Also, an expression for the entropy can be obtained in closed form (neglecting boundary terms) [12, 14] = p(→) [−ǫ ln ǫ− (1 − ǫ) ln(1− ǫ)] +p(←) [−µ lnµ− (1− µ) ln(1− µ)] . (3) 0 0.5 1 1.5 2 2.5 Fig. 2: Plot of the force-entropy relation at constant tempera- ture. The values of the temperature are from top till bottom 1; 0.5; 0.2; 0.1. The value of the parameters h and a are equal to −1 and 1 respectively. We use units in which kB = 1. This one-dimensional random walk can be used as a sim- ple model of a flexible polymer in the fixed-force ensemble. The macroscopic observables are the position of the end point and the number of reversals of direction (kinks) of the walk. The position of the end point measures the effect of an external force applied to the end point. The ground- state of a flexible polymer in a poor solvent is a compact globule. An obvious definition of the Hamiltonian is then H = hK, with h a negative constant with dimensions of energy and K the total number of kinks. A positive value of the parameter h correponds with a polymer is a good solvent. The strength of the solution determines the ab- solute value of h. The contour length of the polymer is equal to na. Thermodynamics. – The Legendre transform of S is the free energy G G = inf E − F 〈x〉 − . (4) The solution of the set of equations ∂G/∂ǫ = 0 and ∂G/∂µ = 0 gives relations for β and F as a function of the model parameters and β = (1− ǫ)(1 − µ) . (5) The most general case when ǫ 6= µ corresponds with a persistent random walk with drift. A persistent random walk [13] without drift is obtained with the choice ǫ = µ. This implies F = 0 but non-vanishing β. Also non- persistent random walk with drift is a special case. This corresponds with ǫ + µ = 1 and implies β = 0 but non- vanishing F . Simple random walk is obtained with ǫ = µ = 1/2. In this case both β and F equal zero. Residual entropy in a model for the unfolding of single polymer chains 0.5 1 1.5 2 Fig. 3: (color online). Plot of the average number of kinks as a function of the temperature and the force. The color code is mentioned to the right. The black solid line, marks the gradual transition from the compact phase to the stretched phases. The black dotted line is an approximation for the solid line, valid at low temperatures only. The value of the parameters h and a are equal to −1 and 1 respectively. The set of equations (5) can be inverted in closed form and has a unique solution for every value of β and F . We also calculated the eigenvalues of the matrix of the sec- ond derivatives of the free energy. These eigenvalues are always non-negative. We conclude that the present model exhibits no phase transition. It is well known that no true phase transition can occur, because of the finite size of single molecules. This poses the problem of defining the different phases of the model. In [6], the sudden change of an appropriate average value is used to obtain the bound- aries between the different phases in the state diagram. We will use the same criterion to define the boundary line between the different phases of the present model. With expressions (2) and (5) one can calculate the force- extension relation at constant temperature. It is shown in figure 1 at low temperature and for two different values of the parameter h. For polymers in a bad solvent (h < 0), one observes a steep increase of the average end-to-end distance at F ≈ 1. For polymers in a good solvent, this steep increase occurs at vanishing force. This in qualita- tive agreement with experimental observations [3, 4]. In the present letter we will focus on polymers in a bad sol- vent. At F ≈ 1 the shape of the polymer changes from compact globule to rod. This becomes a real phase tran- sition (a true force plateau) for T → 0 only. The smallest eigenvalue of the matrix of the second derivatives of the free energy vanishes at this moment. The steep increase of the average end-to-end distance at F ≈ 1 disappears for higher temperatures. Also the force-entropy relation at constant temperature can be obtained. It is shown in figure 2 for different values of the temperature. At low temperatures, the entropy goes through a sharp maximum with increasing force. This maximum disappears at higher 0.5 1 1.5 2 Fig. 4: (color online). Plot of the entropy as a function of the temperature and the force. The color code is mentioned to the right. The black solid line, marks the gradual transition from the compact phase to the stretched phases. The black dotted line is an approximation for the solid line, valid at low temperatures only. The value of the parameters h and a are equal to −1 and 1 respectively. temperatures and the entropy becomes a monotonic de- creasing function of the force. In the present model, the sudden change of the aver- age end-to-end distance can be used to define the gradual transition from compact globule to rod. The boundary line is obtained from the peak value of ∂〈x〉/∂F at constant temperature. Figures 3 and 4 show the average number of kinks and the entropy as a function of force and temper- ature. The black solid line shows the boundary between the compact phase and the stretched phases. The end points of the boundary line are (Tb = 0, Fb = −h/a) and (Tb = −2h/ ln 3, Fb = 0). At low temperatures the bound- ary line is an increasing function of the temperature. At intermediate temperatures the boundary line is a decreas- ing function of the temperature. In the appendix an ap- proximated expression for the boundary line at low tem- peratures is calculated. One obtains F = 1+ 0.35T + . . .. Figures 3 and 4 show the result of the latter expression together with the exact boundary line. The two lines co- incide up to a temperature of approximately 0.5. In the appendix, we show that the positive slope of the boundary line at low temperatures is due to the entropy. This is unexpected, because the entropy is usually not im- portant at low temperatures. However, the entropy of the present model contains two contributions, the usual ther- mal entropy and a configurational entropy, closely related to the zero-temperature phase transition at F = 1. It’s clear from figures 2 and 3, that there are three separate phases at low temperatures. The globular phase (F < 1 and 〈K〉/n ≈ 1) and the stretched phase (F > 1 and 〈K〉 ≈ 0) have a small value of the residual entropy, be- cause only a limited amount of configurations are allowed. The intermediate phase at F ≈ 1 has 〈K〉/n ≈ 1/2. This E. Van der Straeten et al. means there are plenty of allowed configurations. The residual entropy of this phase is very high. The reentrance of the phase boundary is the consequence of a very sub- tle asymmetry in the residual entropy, which prefers the globular phase above the stretched phase. This asymme- try can most clearly be seen in figure 2. Discussion. – To summarize, we solve a simplified model for the unfolding of a polymer in closed form in the fixed-force ensemble. The force-extension relation shows an approximate force plateau at low temperatures. The boundary line between the globular and stretched phases shows reentrant behavior. This reentrant behavior is ex- plained by the presence of residual entropy in the model. We want to stress that the present approach to intro- duce the temperature deviates from the standard way of introducing temperature in statistical mechanics. Usually, one assumes that the equilibrium probability distribution is of the Boltzmann-Gibbs form e−βH , with β the inverse temperature. The present approach is different. We start from a two-parameter model and calculate the average of the macroscopic variables of interest as a function of these two parameters, without assuming the Boltzmann-Gibbs distribution. Then we define the temperature by calculat- ing the Legendre transform (4) of the entropy. Usually, the infimum is taken over all possible probability distribu- tion. This results in the Boltzmann-Gibbs distribution. We take the infimum only over the model parameters. This adds an extra constraint on the equilibrium prob- ability distribution. As a consequence, the resulting dis- tribution is not necessarily of the Boltzmann-Gibbs form, although we started from the Boltzmann-Gibbs definition for the entropy [14]. Indeed, we already pointed out in [11] that the joint probability distribution that after n steps the walk is in x and changed its direction K times is not of the Boltzmann-Gibbs form. The deviations from the Boltzmann-Gibbs form are small and disappear for long chains. The results of the present paper are limited to the fixed- force ensemble, although most experiments are performed in the fixed-stretch ensemble. In [12] we show that it is possible to extend the present model to the fixed-stretch ensemble. However, in this ensemble almost all calcula- tions have to be performed numerically. We also show in [12] that the differences between the two ensembles van- ish in the thermodynamic limit and are negligible for long chains. So we expect only small, finite size corrections to the state diagram after extending the present calculations to the fixed-stretch ensemble. In [3, 4] force plateaus are experimentally observed in stretching experiments in the fixed-stretch ensemble. Following the previous reasoning, this is in agreement with the present calculations up to finite size corrections. To the best of our knowledge, the temperature dependence of this plateau has not yet been studied experimentally. As mentioned in the introduction, SAWs are inten- sively used to study the unfolding transition of polymers. Reentrant behavior is also observed in these models [6–8]. Starting from a phenomenological expression for the free energy near T = 0, one obtains the following boundary line for flexible polymers [6, 15] F = −α+ α√ + ScT, (6) with α a negative model parameter. The second term is a surface correction term. The last term is a contribution due to the residual entropy of the globular phase, with Sc the entropy per monomer. The presence of the residual entropy causes the reentrant behavior. The present model does not contain the surface correction term, because lo- cal interactions are not included. Formula (6) is similar to expression (10) in the thermodynamic limit, because the surface correction term disappears in this limit. A sim- plified version of the SAW is the partially directed walk (PDSAW). This means that steps with negative projection along the x axis are forbidden. Analytical calculations in the thermodynamic limit are possible for this simplified model. The PDSAW is used in [16] as a model for a poly- mer in the fixed-stretch ensemble. It exhibits a true phase transition from a compact phase to a stretched phase. The critical force as a function of the temperature can be ob- tained in closed form in the thermodynamic limit and does not show reentrant behavior. In [15], the PDSAW is stud- ied numerically for finite chains. Reentrant behavior is observed in contrast to the results of [16] in the thermo- dynamic limit. It is argued in [15] that for the PDSAW, the value of Sc is too small to cause reentrant behavior. Reentrant behavior does show up in the numerical sim- ulations because at small temperatures, there is a finite entropy associated with the deformed globule. Together with the surface term this gives rise to the observed reen- trant behavior for finite walks. Our model predicts that the reentrant behavior survives the thermodynamic limit, in contrast to the results of the PDSAW. The reason for this difference is that the restrictions of the PDSAW de- crease the residual entropy. Our model is basicly a two-state model. This kind of model has been used before to model biopolymers, for ex- ample to study the stress-induced transformation from B- DNA to S-DNA. In an attempt to explain the obtained ex- perimental data, the pure two-state model is used in [17]. The application of the two-state model is limited to the region of the transformation from B-DNA to S-DNA. For this reason, the model is combined in [18] with the well known Worm Like Chain model. The combination of the two models results in a reasonable fit to the experimen- tal data. The present work uses a two-state model in an other context, the unfolding transition of single polymers instead of the transformation from B-DNA to S-DNA. To the best of our knowledge, this is the first time that a two-state non-Markovian random walk is used to study the unfolding transition of single polymer chains. We ex- pect that a three-dimensional version of the present model can be used to study the transformation from B-DNA to Residual entropy in a model for the unfolding of single polymer chains S-DNA. In conclusion, our model exhibits a gradual transition from compact globule to rod in qualitative agreement with experimental observations. The boundary line between these two phases shows reentrant behavior in agreement with numerical simulations. Our model predicts that this reentrant behavior survives the thermodynamic limit, in contrast to the results obtained for the partially directed walk. Appendix. – At low temperatures one can replace the average number of kinks by 〈K〉 = n − 〈x〉/a. With this approximation, the free energy becomes G = inf − F 〈x〉 − , (7) with the entropy approximately equal to = − 1 2〈x〉 ln 2〈x〉+ (na− 〈x〉) ln (na− 〈x〉) − (na+ 〈x〉) ln (na+ 〈x〉) . (8) The solution of the equation ∂G/∂〈x〉 = 0 gives the fol- lowing force-extension relation F = −h 4〈x〉2 n2a2 − 〈x〉2 . (9) After inverting this relation, one can calculate the second derivative of the average end-to-end distance with respect to the force at constant temperature. This second deriva- tive equals zero if the following equation holds F = −h T. (10) This is a low temperature approximation for the boundary line between the globular and stretched phases. The factor ln 2/2a is clearly a contribution due to the entropy. REFERENCES [1] P. Szymczak, M. Cieplak, Stretching of proteins in a force- clamp, J. Phys.:Condens. Matter 18, L21 (2006) [2] M. Schlierf, H. Li, J. M. Fernandez, The unfolding kinet- ics of ubiquitin captured with single-molecule force-clamp techniques, PNAS 101, 7299 (2004) [3] C. G. Baumann, V. A. Bloomfield, S. B. Smith, C. Bus- tamante, M. D. Wang, S. M. Block, Stretching of Single Collapsed DNA Molecules, Biophys. J. 78, 1965 (2000) [4] Y. Murayama, Y. Sakamaki, M. Sano, Elastic Response of Single DNA Molecules Exhibits a Reentrant Collapsing Transition, Phys. Rev. Lett. 90, 018102 (2003) [5] A. Halperin, E. B. Zhulina, On the Deformation Be- haviour of Collapsed Polymers, Europhys. Lett. 15, 417 (1991). [6] S. Kumar, D. Giri, Force-induced conformational transi- tion in a system of interacting stiff polymers: Application to unfolding, Phys. Rev. E. 72, 052901 (2005). [7] S. Kumar, I. Jensen, J. L. Jacobsen, A. J. Guttmann, Role of conformational entropy in force-induced biopoly- mer unfolding, Phys. Rev. Lett. 98, 128101 (2007) [8] D. Marenduzzo, A. Maritan, A. Rosa, F. Seno, Stretching of a Polymer below the θ Point, Phys. Rev. Lett. 90, 088301 (2003) [9] A. Imparato, A. Pelizzola, M. Zamparo, Protein mechan- ical unfolding A model with binary variables, J. Chem. Phys. 127, 145105 (2007) [10] D. Marenduzzo, A. Trovato, A. Maritan, Phase diagram of force-induced DNA unzipping in exactly solvable mod- els, Phys. Rev. E 64, 031901 (2001) [11] E. Van der Straeten, J. Naudts, A two-parameter random walk with approximate exponential probability distribu- tion, J. Phys. A: Math. Gen. 39, 7245 (2006). [12] E. Van der Straeten, J. Naudts, A one-dimensional model for theoretical analysis of single molecule experiments, J. Phys. A: Math. Gen. 39, 5715 (2006). [13] I. Claes, C. Van den Broeck, Random walk with persis- tence, J. Stat. Phys. 49, 383 (1987) [14] J. Naudts, E. Van der Straeten, Transition records of sta- tionary Markov chains, Phys. Rev. E. 74, 040103 (2006). [15] S. Kumar, D. Giri, Does changing the pulling direction give better insight into biomolecules?, Phys. Rev. Lett. 98, 048101 (2007) [16] A. Rosa, D. Marenduzzo, A. Maritan, F. Seno, Mechani- cal unfolding of directed polymers in a poor solvent: Crit- ical exponents, Phys. Rev. E 67, 041802 (2003) [17] A. Ahsan, J. Rudnick, R. Bruinsma, Elasticity theory of the B-DNA to S-DNA transition, Biophys. J. 74, 132 (1998) [18] P. Cluzel, A. Lebrun, C. Heller, R. Lavery, J.-L. Viovy, D. Chatenay, F. Caron DNA: An extensible molecule, Sci- ence 271, 792 (1996) Introduction. – One dimensional random walk with memory. – Thermodynamics. – Discussion. – Appendix. –

0704.1234

Generalization of Einstein-Lovelock theory to higher order dilaton gravity

arXiv:0704.1234v2 [hep-th] 19 Oct 2007 IFT–07–1 Generalization of Einstein–Lovelock theory to higher order dilaton gravity D. Konikowska, M. Olechowski Institute of Theoretical Physics, University of Warsaw ul. Hoża 69, PL-00 681 Warsaw, Poland Abstract A higher order theory of dilaton gravity is constructed as a generalization of the Einstein–Lovelock theory of pure gravity. Its Lagrangian contains terms with higher powers of the Riemann tensor and of the first two derivatives of the dilaton. Neverthe- less, the resulting equations of motion are quasi–linear in the second derivatives of the metric and of the dilaton. This property is crucial for the existence of brane solutions in the thin wall limit. At each order in derivatives the contribution to the Lagrangian is unique up to an overall normalization. Relations between symmetries of this theory and the O(d, d) symmetry of the string–inspired models are discussed. http://arxiv.org/abs/0704.1234v2 1 Introduction The equations of motion in the Einstein theory of gravity in 4 space–time dimensions are the most general divergence–free tensor (rank 2) equations bilinear in the first derivatives and linear in the second derivatives of the metric. They can be obtained from the Hilbert–Einstein action which is linear in the Riemann tensor. In more than 4 space–time dimensions, this theory can be generalized to contain higher powers of the Riemann tensor in the action. The corresponding equations of motion involve higher powers of the first derivatives of the metric and are quasi–linear in the second derivatives (all terms are at most linear in the second derivatives, while multiplied by powers of the first derivatives). It has been shown that the contribution to the action of a given order in the Riemann tensor is unique up to an overall normalization. The quadratic contribution is called the Gauss–Bonnet action or the Lanczos action [1]. It has been generalized to higher orders by Lovelock [2]. The quasi–linearity is a very important feature of the Einstein–Lovelock equations of motion. It guarantees that they can be formulated as a Cauchy problem with some constraints on the initial data [3]. On the other hand, it is crucial for the existence of non–singular domain wall solutions in the thin wall limit. This problem for arbitrary order in derivatives was discussed in [4]. Many aspects of the Einstein–Lovelock gravity were discussed in the literature1. Higher derivative corrections to the gravity interactions are present in effective Lagrangians obtained from string theories. The first correction has exactly the form of the Gauss–Bonnet term [9], [10]. The lowest order dilaton interactions were added to the Gauss–Bonnet theory in [11]. However, the α′ expansion in string theories predicts higher derivative corrections not only for the gravitational interactions, such corrections appear also for the dilaton. The effective action for the dilaton gravity with terms up to four derivatives was given in [12], [13]. The effective action with six derivatives was presented in [14], but its gravitational part has a form different from that of the corresponding Einstein–Lovelock action. The dilaton gravity at the field theory level has been investigated by many authors. Some of them included also certain higher order corrections. Yet in most cases such corrections were considered only for gravitational interactions and not for the dilaton. Some higher derivative corrections for both the dilaton and the gravitational interac- tions were considered in [15]–[21] (certain Riemann tensor combinations with dilaton dependent coefficients were analyzed in [22]–[25]). The terms predicted by superstrings up to four derivatives have also been considered in [26]–[28]. The purpose of the present work is to find a generalization of the lowest order dilaton gravity theory to an arbitrary order in derivatives. We start with the Einstein– Lovelock higher order gravity and couple it to the dilaton. There are many ways to do this but we are only interested in the theories where dilaton and gravity interactions 1Quasi–linearity of the Einstein–Gauss–Bonnet theory was reviewed in [5]. A discussion of general quasi–linear differential equations can be found in [6]. For a review on brane–world gravity see eg. [7]. For a discussion of the Lovelock gravity in the context of the equivalence of the Palatini and metric formulations see eg. [8]. are as similar to each other as possible. Equations of motion in such a theory are presented in Section 2. We begin with formulating the conditions which should be fulfilled by such equations. Most of them are simple generalizations of the conditions fulfilled by the Einstein–Lovelock equations of motion. One condition is added in order to eliminate at least some of the possible theories in which the dilaton interactions are not related to the gravitational ones. The equations of motion satisfying all those conditions are constructed in Subsection 2.3. It turns out that at each order those equations are unique up to a numerical normalization. Moreover, they can be obtained by the standard Euler–Lagrange procedure from the Lagrangian presented in Section 3. Section 4 contains the proof that our equations of motion are quasi–linear in the second derivatives of both the metric and the dilaton. The relation between the gravity and the dilaton interactions is discussed in Section 5. We point out that the Lagrangian of our higher order dilaton gravity can be obtained in a simple way from the pure gravity Einstein–Lovelock Lagrangian. We also discuss the relation of the resulting theory to the O(d, d) symmetric theories. We conclude in Section 6. The Appendix contains the explicit formulae for the Lagrangian and the equations of motion up to terms of the sixth order in derivatives. 2 Equations of motion 2.1 Notation Let us start with introducing certain generalizations of the Kronecker delta and the trace operator which will be used later to make the formulae more compact. The generalized Kronecker delta is defined by j1j2···jn i1i2···in = det · · · δj1in · · · · · · · · · · · · · · · δjnin , (1) and should be only employed when the spacetime dimensionality D is sufficient: D ≥ n. Using this definition it is easy to prove some relations among Kronecker deltas of different order. For example: ν j1j2...jn µ i1i2...in = δνµδ j1j2...jn i1i2...in − δνi1δ j1j2...jn µ i2...in − δνi2δ j1j2...jn i1µ ...in − . . .− δνinδ j1j2...jn i1i2...µ . (2) The generalized Kronecker delta can be used to define the following trace–like linear mapping from tenors of rank (n, n) into numbers T (M) = δj1j2···jni1i2···in M i1i2···in j1j2···jn , (3) which reduces to the ordinary trace for n = 1. We will also employ an extension of this operation which maps tensors of rank (n, n) into tensors of rank (1, 1): T νµ (M) = δ ν j1j2···jn µ i1i2···in M i1i2···inj1j2···jn . (4) In the following we will often use T and T evaluated for products of tensors. In order to clearly distinguish between tensors and their contracted counterparts, we will use ∗ indices to indicate the rank of a tensor. For example, R∗∗∗∗ denotes the rank (2, 2) Riemann tensor, and �∗∗φ denotes the rank (1, 1) second derivative of the dilaton, while R is the Ricci scalar and �φ the D’Alembertian acting on the dilaton. Thus, for example, (R∗∗∗∗) (�∗∗φ) = δσ1σ2σ3σ4σ5σ6ρ1ρ2ρ3ρ4ρ5ρ6 R Rρ3ρ3σ3σ4 � φ�ρ6σ6φ , (5) where we used the notation Rρ1ρ2σ1σ2 = R σ1σ2 and � σφ = ∇ρ∂σφ to make the formula more compact. It is easy to see that the sequence of tensors appearing in the product argument of T is not important. Changing such an order is equivalent to interchang- ing the appropriate columns of indices in the generalized Kronecker delta. On the other hand, interchanging two such columns of indices is equivalent to interchanging the corresponding 2 rows and 2 columns in the determinant in Definition (1). Each interchange of two columns (or two rows) changes the sign of the determinant, hence an even number of interchanges leaves the determinant unchanged. 2.2 Conditions Now we want to construct the n–th order dilaton gravity equations of motion. They are to be of the form T (n)µν = 0 , W (n) = 0 , (6) where the tensor T µν and the scalar W (n) satisfy the following conditions (i) They are combinations of terms with exactly 2n derivatives acting on the metric tensor gµν and on the dilaton field φ. There are no derivatives higher than second acting on one object; (ii) Tensor T µν is symmetric in its indices; (iii) The covariant derivative of the tensor is proportional to the scalar: µ = const · (∂µφ)W (n) (the energy–momentum tensor is covariantly con- served if the dilaton equation of motion is fulfilled). It is clear that the above conditions are not sufficient to determine something which could be regarded as an extension of the higher order gravity theory to the dilaton gravity case. For example, all the above conditions are fulfilled by the Einstein–Gauss– Bonnet gravity with only the lowest order terms for the dilaton. We are interested in a theory where the dilaton and the metric are treated in a more symmetric way. It is not obvious how such a symmetry should be defined, because it ought to relate a scalar to a second rank tensor. Or, more precisely, it is supposed to relate the first and second derivatives of the scalar field to the Riemann tensor and its contractions. A simple observation concerning the gravity part is that it contains even–rank tensors only. On the other hand, the first derivative of a scalar is a rank–1 tensor. Hence one can expect that in a gravity–dilaton symmetric theory, the first derivative of the dilaton appears only as a 0–rank tensor: gµν∂µφ∂νφ. However, the feature mentioned above is not invariant under change of variables. Thus, we should specify in which frame it is fulfilled. The theory which relates dilaton to gravity is the string theory so the string frame seems to be a natural choice. Hence our last condition reads: (iv) In the string–like frame, in which the pure gravity term is multiplied by exp(−φ), the first order derivatives of the dilaton appear in the combination (∂µφ)(∂ only. The relation of this condition to the O(d, d) symmetry present in many string–inspired theories will be discussed in Section 5. 2.3 Construction We start our construction with a term in T µ where all 2n derivatives act on the metric tensors. The only pure gravity tensor satisfying Conditions (i)–(iii) (with W (n) = 0) is, up to normalization, equal to the n-th order Lovelock tensor [2]. Because of Condition (iv), it is most natural to work in the frame in which the gravity term is multiplied by exp(−φ). Consequently, the tensor T ν(n)µ starts with T ν (n)µ = −2 −(n+1)e−φδν σ1...σ2nµ ρ1...ρ2nR · · ·Rρ2n−1ρ2nσ2n−1σ2n + . . . (7) The reason for such a normalization will be explained in the next section. Calculating the divergence of (7), we get ∇νT ν (n)µ = 2 −(n+1)e−φ(∂νφ)δ ν σ1...σ2n µρ1...ρ2n Rρ1ρ2σ1σ2 · · ·R ρ2n−1ρ2n σ2n−1σ2n + . . . (8) The above term is produced when the derivative acts on e−φ (derivatives of the Riemann tensor do not contribute due to the Bianchi identity). The first term in T µ shown explicitly in (7) can not be the only one. The reason is that the r.h.s. of (8) is not a product of ∂µφ and a scalar, as Condition (iii) requires. Using Eq. (2) we can rewrite the r.h.s. of (8) as a combination of (2n + 1) terms. The one containing the first term from the r.h.s. of (2) is of the desired form but the remaining 2n terms have different structures of the index contractions. It turns out that similar terms are also present in the following covariant derivative e−φδν σ1...σ2n−1µρ1...ρ2n−1R · · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2� ρ2n−1 σ2n−1 = −e−φ(∂νφ)δν σ1...σ2n−1µ ρ1...ρ2n−1R · · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2� ρ2n−1 σ2n−1 +e−φδν σ1...σ2n−1µρ1...ρ2n−1R · · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2 ∇ν∇ρ2n−1∂σ2n−1φ . (9) The second term on the r.h.s. may be rewritten as e−φδν σ1...σ2n−1µ ρ1...ρ2n−1R · · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2 δσ2nρ2nR ρ2n−1ρ2n σ2n−1ν ∂σ2nφ e−φ(∂νφ)δ δσ1...σ2n−1σ2nρ1...ρ2n−1µ R · · ·Rρ2n−3ρ2n−2σ2n−3σ2n−2R ρ2n−1ρ2n σ2n−1σ2n , (10) where in the last step we interchanged the names of the contracted indices ν and σ2n and rearranged the indices in the generalized Kronecker delta. A term exactly of this structure must be added to (8) in order to obtain an expression proportional to ∂µφ. From Eqs. (2) and (8) it follows that the coefficient should be equal to (−n2−n) instead of the (−1/2) present in (10). This fixes the coefficient of the term in T ν(n)µ which contains (n− 1) Riemann tensors and one second derivative of the dilaton. Now we know the first two terms of the tensor T µ . Using the notation introduced in (3) and (4), they can be written as: T ν (n)µ = −2 −(n+1)e−φT νµ ((R )− 2−(n−1)ne−φT νµ (R∗∗∗∗) (n−1) + . . . (11) Their covariant derivative reads ∇νT ν (n)µ = 2 −(n+1)e−φ(∂µφ)T ((R∗∗∗∗) +2−(n−1)ne−φ(∂νφ)T (R∗∗∗∗) (n−1) + . . . (12) The first term has the structure required by Condition (iii) and determines the first term of the scalar equation of motion2 W (n). However, the second term in (12) is not of the appropriate structure. It means that some additional terms, whose covariant derivatives are products of (n− 1) Riemann tensors with one second derivative of the dilaton, are necessary in T µ . Two such terms are possible: −φT νµ (R∗∗∗∗) (n−2) (�∗∗φ) + c4e −φT νµ (R∗∗∗∗) (n−1) (∂φ)2 . (13) However, it is not enough to have terms with appropriate powers of the Riemann tensor and the dilaton, because their covariant divergences must contain the correct combinations of the generalized Kronecker deltas. To check whether this is possible, we calculate the covariant divergence of (13). When the derivative acts on �∗∗φ in the first term in (13), it gives an additional Riemann tensor multiplied by ∂φ and a pair of new indices. Those new indices are contracted with just one ordinary Kronecker delta and are not under the overall antisymmetrization. Similarly, when the covariant derivative acts on (∂φ)2 in the second term in (13), it gives the second derivative of the dilaton multiplied by ∂φ and a pair of new indices. Those two covariant derivatives should combine with the second term on the r.h.s. of (12) to give an expression proportional to ∂µφ. This fixes the numerical coefficients c3 and c4. The explicit calculation gives c3 = −2(2−n)n(n− 1), c4 = 2−nn. Thus, we have found the first four terms of T T ν (n)µ = −2 −(n+1)e−φ T νµ ((R ) + 4nT νµ (R∗∗∗∗) (n−1) +8n(n− 1)T νµ (R∗∗∗∗) (n−2) (�∗∗φ) −2nT νµ (R∗∗∗∗) (n−1) (∂φ)2 + . . . (14) 2up to an overall normalization. The choice of the relative normalizations of T µν and W (n) shall become clear when the Lagrangian is introduced in Section 3. The covariant divergence of those terms reads ∇νT ν (n)µ = ∂µφ 2−(n+1)e−φ T ((R∗∗∗∗) ) + 4nT (R∗∗∗∗) (n−1) +∂νφ 2 (2−n)n(n− 1)e−φT νµ (R∗∗∗∗) (n−2) (�∗∗φ) −∂νφ 2−nne−φT (R∗∗∗∗) (n−1) (∂φ)2 + . . . (15) The terms in the curly bracket above are the first two terms of the scalar W (n) we are looking for. Equation (15) shows that the procedure of finding T µ and W (n) must be contin- ued. The last two terms on the r.h.s. of (15) do not have the required form, so more terms must be added to T µ . From the steps described so far, it should be clear that each of such new terms must contain exactly 3 (first or second order) derivatives of the dilaton. There are two such terms: −φT νµ (R∗∗∗∗) (n−3) (�∗∗φ) + c6e −φT νµ (R∗∗∗∗) (n−2) (∂φ)2 . (16) The coefficients c5 and c6 can be fixed in the same way as c3 and c4. This procedure can be continued step by step for the terms containing higher and higher powers of the dilaton field with the derivatives acting on it. Eventually, one obtains the term with the maximal number of dilaton fields, namely c e−φδνµ [(∂φ) This is the first term in T µ , the covariant derivative of which need not to be corrected by contributions from any additional terms. This covariant derivative reads e−φδνµ (∂φ)2 = −(∂µφ)e−φ (∂φ)2 + 2ne−φ(∇µ∂σφ)(∂σφ) (∂φ)2 ](n−1) . (17) The second term on the r.h.s. is used to cancel some unwanted part of e−φT νµ (�∗∗φ) [(∂φ)2] (n−1) , which fixes c to be equal to 1 (−1)(n+1). The first term on the r.h.s. of (17) has already the required structure of the product of ∂µφ and a scalar. Thus, the procedure can stop here. The above iterative procedure gives T µν and W (n) satisfying all the four imposed conditions. The resulting gravitational and dilaton equations of motion can be written in the following relatively simple form: T (n)µν = − 2b−an! a!b!(n− a− b)! (R∗∗∗∗) (�∗∗φ) −(∂φ)2 )n−a−b = 0 , (18) W (n) = −e−φ 2b−an! a!b!(n− a− b)! (R∗∗∗∗) (�∗∗φ) −(∂φ)2 )n−a−b = 0 . (19) The existence of T µν and W (n) is a non–trivial result, because in our iterative procedure there are more conditions than available constants. A priori it could happen that there were no solutions other than a trivial one with vanishing T µν and W However, the solution exists and is unique up to an overall normalization. Hence any dilaton gravity equations of motion, satisfying Conditions (i)–(iv), which contain at least one term present in (18) and (19) must also contain all the other terms with uniquely determined coefficients. 3 Lagrangian It is interesting to check whether the equations of motion constructed in Section 2 can be obtained from some D–dimensional action. In such case, T µν and W (n) would satisfy δgµνS (n) = δgµν −gL(n) = −g T (n)µν δg µν , (20) (n) = δφ −gL(n) = −gW (n)δφ . (21) It turns out that indeed the equations of motion (18) and (19) can be obtained from the action with the Lagrangian density given by L(n) = e−φ 2b−an! a!b!(n− a− b)! (R∗∗∗∗) (�∗∗φ) −(∂φ)2 )n−a−b . (22) It is important to underline that for Conditions (i)–(iv) not to be violated, the terms coming from the n-th Lagrangian can appear only in the space–times with dimension- ality D ≥ 2n. Moreover, one should be careful when calculating (20) for D = 2n, as the generalized Kronecker delta (1) can not be employed in (18) for the term of the highest order in the Riemann tensor. The coefficient of that term should be replaced δν σ1σ2...σ2nµ ρ1ρ2...ρ2n −→D=2n δ σ1σ2...σ2n ρ1ρ2...ρ2n − δνρ1δ σ1σ2...σ2n µ ρ2...ρ2n − δνρ2δ σ1σ2...σ2n ρ1µ ...ρ2n − . . .− δνρ2nδ σ1σ2...σ2n ρ1ρ2...µ . (23) Now we can comment on the overall normalization of the tensors T µν . The reason for this particular normalization is that the term e−φRn (with R being the Ricci scalar) appears in the Lagrangian with the coefficient 1. This corresponds to the standard normalizations of the Hilbert–Einstein and Gauss–Bonnet Lagrangians. Proving that the equations of motion derived from the Lagrangian (22) really have the form (6) with T µν and W (n) as given in (18) and (19) is a straightforward but quite tedious calculation. One of the reasons is that apparently several integrations by parts are required. This can be somewhat simplified if one observes that not all those integrations by parts have to be performed explicitly. In case of (21), the reason is as follows. Under the integral (21) there are first (second) derivatives of δφ coming from the variation of the first (second) derivatives of the dilaton. In general, the terms containing second derivatives of δφ should be integrated by parts twice. However, one can notice that the result of a single integration and the terms containing the first derivatives of δφ cancel each other exactly. The situation is a little bit more complicated in case of the gravitational equation of motion. Under the integral (20), there are second derivatives of δgµν coming from the variation of the Riemann tensor and first derivatives of δgµν coming from the variation of the second covariant derivative of the dilaton. Similarly as in the case of the dilaton equation of motion, the terms containing second derivatives of δgµν have to be integrated by parts only once. And although the cancellation of the resulting terms is not complete this time, only some residual integration by parts has to be performed additionally. Of course, the Lagrangian density (22) is not unique. First, one can rewrite L(n) changing the variables gµν and φ. Second, one can add to L(n) any total divergence without changing the resulting equations of motion. However, the form given in Eq. (22) is especially simple and interesting. It is very similar to the form of T µν and W (n). The energy momentum tensor T µν can be obtained from L(n) by replacing the generalized trace T with its tensor extension T µν and multiplying the result by −1/2. In case of the dilaton equation of motion, the analogous relation is even simpler: W (n) = −L(n). We were not able to find any other similarly simple form of the Lagrangian by adding total derivative terms or by changing the variables. For example, we examined the form of the Lagrangian and of the equations of motion in the Einstein–like frame in which the common factor e−φ is absorbed by a suitable Weyl transformation. The results are very complicated and will not be presented here. One of the reasons for such complications is that the Weyl transformation depends on the dimensionality D of the space–time. Thus, many different functions of D appear in the Einstein frame, while there is no explicit dependence on D in our string–like frame. 4 Quasi-linearity It is easy to show that the equations of motion (18)–(19) are quasi–linear in the second derivatives of the metric and the dilaton. Let us introduce in the D–dimensional space– time a (D− 1)–dimensional hypersurface Σ defined by its unit normal vector nµ. The metric induced at this hypersurface is given by hµν = gµν − , (24) where n2 = nρn ρ. The components of the D–dimensional Riemann tensor R∗∗∗∗ corre- sponding to the full metric gµν can be expressed as Rρσµν = R µν −n + 4n[ρD[µK + 4n[µD 4n[µn Kτν] − 4n[µn , (25) where: R is the (D − 1)–dimensional Riemann tensor corresponding to the induced metric hµν ; K is the extrinsic curvature given by Kµν = £nhµν ; (26) Dµ is the covariant derivative with respect to the induced metric hµν ; £n is the Lie derivative along the vector field nµ. Similarly we can write the D–dimensional second covariant (with respect to the metric gµν) derivative of the dilaton ∇µ∇νφ = DµDνφ +n−2 Kµν£nφ+ 2n(µDν)£nφ− 2n(µKτν)Dτφ +n−4 nµnν nφ− (n ρ∇ρnτ )∇τφ . (27) We want to check how the second Lie derivatives of the metric hµν (present in £nKµν) and of the dilaton φ appear in the equations of motion (18) and (19). Such second derivatives are present in (25) and (27) but in both cases they are multiplied by coefficients bilinear in the vector n. After substituting the decompositions (25) and (27) into (18) and (19), one can immediately see that, due to the antisymmetrization present in T µν and W (n), the equations of motion contain terms at most bilinear in n. Thus, the equations of motion (18) and (19) contain terms at most linear in the second Lie derivatives of hµν and φ. We have shown that the equations of motion are quasi–linear in the second Lie derivatives “perpendicular” to the hypersurface Σ. This quasi–linearity has very im- portant consequences. For Σ with a time–like n, this allows us to define a standard Cauchy problem with the initial conditions (values and first Lie derivatives of hµν and φ) given at Σ. For a space–like n, the quasi–linearity is necessary to have non–singular brane solutions even in the thin wall limit. 5 Symmetries The equations of motion presented in Section 2 were obtained assuming some kind of symmetry between the metric and the dilaton. Now we are in a position to investigate such a symmetry in more detail. It is quite amazing that the Lagrangian (22) as well as the equations of motion (18) and (19) can be expressed as functions of n–th perfect “power” of one simple n–independent quantity. Namely: L(n) = −W (n) = e−φT R∗∗∗∗ ⊕ 2� ∗φ ⊕ (−1) (∂φ) , (28) T (n)µν = − e−φT µν R∗∗∗∗ ⊕ 2� ∗φ ⊕ (−1) (∂φ) . (29) One can treat these equations as just a new notation allowing us to rewrite the double sums from (18), (19) and (22) in a compact way3. Yet, on the other hand, it helps to show that the action and the equations of motion depend on some combinations of the dilaton derivatives and tensors obtained from the metric only. In each round parenthesis in Eqs. (28) and (29), there are the rank–4 Riemann tensor, the rank–2 tensor of the second derivatives of the dilaton and the rank–0 tensor built from the first derivatives of the dilaton: R∗∗∗∗ ⊕ 2� ∗φ ⊕ (−1) (∂φ) . (30) All those tensors are under the generalized traces T and T µν . Some of the terms present in these mappings contain traces of the tensors from (30). There are two different rank–2 tensors coming from (30). The first is just �∗∗φ. The second is the Ricci tensor R∗∗ which can be obtained from the Riemann tensor by contraction of its two indices. There are four different ways to contract one pair of indices in R∗∗∗∗, thus in the final result the rank–2 tensors appear always in the combination (2R∗∗ + 2�∗∗φ). There are three different scalars originating from (30): (∂φ) , �φ and the curvature scalar R. There are two different constructions giving R, so the final results depend on a single following scalar combination: R+ 2�φ− (∂φ)2. The above observation allows us to relate our dilaton gravity equations to the corresponding equations in the pure Einstein–Lovelock gravity: L(n) = −W (n) = e−φL(n)E−L R∗∗∗∗ , (R ∗φ) , R+ 2�φ− (∂φ)2 , (31) T (n)µν = e gµν ,R∗∗∗∗ , (R ∗φ) , R+ 2�φ− (∂φ)2 . (32) The recipe for the higher order dilaton gravity can be as follows: Start with the higher order pure gravity Einstein–Lovelock theory. Write the Lagrangian density L(n)E−L and the equations of motion (T E−L)µν in terms of the Riemann tensor, Ricci tensor and the curvature scalar by performing all internal (within a given Riemann tensor) contrac- tions of indices. Then make the substitutions: Rσρ → Rσρ +� , (33) R+ 2�φ− (∂φ)2 . (34) Finally, multiply the result by exp(−φ). The dilaton equation of motion, absent in the pure gravity case, is simply L(n) = 0. It occurs that the form of the Lagrangian and the tensor T µν given in (31) and (32) is very closely related to the string O(d, d) symmetry4. To show this, we consider the 3One could say that Eqs. (28) and (29) make no sense because they contain a sum of tensors of different ranks. To make this mathematically sensible, we should consider a simple sum of spaces of tensors of a given rank. Then the tensors in (28) and (29) should be understood as elements of such a sum space with all but one components set to zero. Finally, the generalized traces T and T µν should be further extended in such a way that when acting on an element of this big space they give the result being the sum of generalized traces of all components. 4For a review on O(d, d) symmetry, see e.g. [29] and the references therein. D–dimensional block–diagonal metric of the form gµν = g̃αβ 0 0 Gmn , (35) where α, β = 1, . . . , (D − d); m,n = (D − d + 1), . . . , D. We assume that the metric components g̃αβ andGmn and the dilaton field φ do not depend on the last d coordinates xm. In such a case, we obtain the following expressions for the second derivatives of the dilaton αφ = �̃ αφ , (36) G−1∂αG ∂αφ , (37) �φ = �̃φ+ (∂α ln detG) ∂ αφ , (38) and for the Ricci tensor and the curvature scalar Rβα = R̃ α ln detG− G−1 (∂αG)G , (39) Rnm = − (∂α ln detG) G−1∂αG G−1�̃G G−1 (∂αG)G −1 (∂αG) , (40) R = R̃ − (∂α ln detG) (∂ α ln detG)− �̃ ln detG G−1 (∂αG)G −1 (∂αG) , (41) where tilde denotes quantities related to the (D−d)–dimensional metric g̃αβ, G should be understood as a d×d matrix (and not its determinant) and Tr and det are the trace and the determinant (acting on d× d matrices). A necessary condition for the O(d, d) symmetry is that the dilaton field φ appear only in the O(d, d) invariant combination Φ = φ− ln detG . (42) Hence any derivative of the dilaton φmust be accompanied by an appropriate derivative of [ln detG]. It is easy to see that there are only three combinations of Eqs. (36)– (41) and the first derivatives of φ which depend on φ and [ln detG] only through the combination Φ: R+ 2�φ− (∂φ)2 = R̃+ 2�̃Φ− ∂αΦ∂αΦ− G−1 (∂αG)G −1 (∂αG) , (43) Rβα +� αφ = R̃ α + �̃ G−1 (∂αG)G , (44) Rnm +� (∂αΦ) G−1∂αG G−1∂αG . (45) These are exactly the combinations which, together with the Riemann tensor with uncontracted indices, appear in the formulation given in Eqs. (31) and (32). Hence the higher derivative contributions to the dilaton gravity theory found in the present paper fulfill the necessary condition for the O(d, d) symmetry formulated before Eq. (42). This does not mean yet that our theory is a part of some O(d, d) symmetric theory. One should check whether all terms depending on G other than [ln detG] form only O(d, d) invariant combinations. Actually, one can calculate that it is really the case for n = 1 and n = 2. The lowest order theory was analyzed from this point of view for the first time in [30]. Our second order Lagrangian L(2) differs from the one found in [31] (for a vanishing tensor field H) by some total derivatives only. Thus, for n = 1, 2 the equations of motion presented in Section 2 are the same as the dilaton and gravity part of the equations obtained as appropriate approximations from the superstring theories. The relation to the O(d, d) symmetry for n > 2 will be discussed elsewhere [32]. The above discussion shows that Condition (iv) from Section 2.2 can be treated as a necessary one for the dilaton gravity model to be part of some O(d, d) symmetric theory. The reason is that there are no O(d, d) invariant expressions containing the first derivatives of the dilaton other than the combination (∂µφ)(∂ 6 Conclusions We have generalized the Einstein–Lovelock theory by adding interactions with the dilaton. The corresponding Einstein and dilaton equations of motion can be written as series in the number of derivatives acting on the fields: Tµν = µν = 0 , (46) (n) = 0 . (47) The n–th contributions T µν and W (n) are sums of terms containing products of the Riemann tensor and the first and second derivatives of the dilaton field. There are 2n derivatives in each such term. We have found the most general equations of motion satisfying Conditions (i)–(iv) given in Section 2.2. The first three conditions are the standard properties of the dilaton gravity theories. The last one was added in order to find the theories in which the dilaton and the metric are treated, as much as possible, on the same footing. Accordingly, we assumed that the rank–1 tensor containing the first derivatives of the dilaton can appear only in the scalar combination (∂µφ)(∂ µφ), as there is no way to build an odd–rank tensor from the metric and the Riemann tensor. It is necessary to specify the frame in which such a condition is to be fulfilled. We have chosen the string frame where the n–th order term from the Einstein–Lovelock theory is multiplied by e−φ. The reason is quite simple: symmetries relating the dilaton and the metric are present in string–motivated theories. We have shown that at each order T µν and W (n) are unique up to a normalization. General expressions for T µν and W (n) for arbitrary n are given in Section 2.3. The explicite formulae for n ≤ 3 are presented in the Appendix. It occurs that the higher order dilaton gravity equations of motion have properties similar to those of the pure Einstein–Lovelock gravity. Namely: • There is an upper limit on the number of terms in (46)–(47) which can be non– zero. For a D–dimensional space–time it is given by the inequality 2n ≤ D (the corresponding limit for pure gravity is 2n < D) • The equations of motion are quasi–linear in the second derivatives. This allows us to treat them as a standard Cauchy initial conditions problem. It is crucial also for the existence of brane–type solutions in the thin wall limit. There is also another very interesting feature of those equations. The form of the scalar and Einstein equations is very similar when written with the help of the generalized Kronecker delta. The tensor T µν can be obtained from the scalar W (n) simply by adding a pair of extra indices µ and ν to each generalized Kronecker delta and dividing by 2. Our dilaton gravity equations of motion can be obtained from an appropriate La- grangian. Of course, such a Lagrangian can be determined only up to some total derivatives. However, we have found that there is one particularly interesting form of L = −W . (48) Moreover, this Lagrangian is related in a simple way to the Einstein–Lovelock one (the same is true also for the gravitational equations of motion). First, one has to write the Einstein–Lovelock Lagrangian as a function of the Riemann tensor, the Ricci tensor and the curvature scalar by performing all internal (within the same Riemann tensor) contractions of indices. Next, one should replace the curvature scalar with the combination R + 2�φ − (∂φ)2, and the Ricci tensor with Rσρ + �σρ . The result is the dilaton gravity Lagrangian. The property that the Lagrangian can be written in terms of only three tensors: one scalar R+2�φ−(∂φ)2, one rank–2 tensor Rσρ+�σρ and the rank–4 Riemann tensor is quite important. We have shown that this is a necessary condition for the dilaton gravity to be a part of any string motivated theory with the O(d, d) symmetry. It turns out that for n = 1, 2 it is also a sufficient one. The contributions L(1) and L(2) to our Lagrangian are, up to total derivatives, the same as those found from demanding the O(d, d) symmetry [30], [31]. It would be interesting to investigate the relation of L(n) to string theories for n > 2 [32]. Most of the interesting features of the Lagrangian and the equations of motion are visible in the string frame only. The theory looks more complicated in other frames. For example, in the most often used Einstein frame there are no simple relations between tensors built from the metric and from the dilaton derivatives and also many coefficients become explicitly D–dependent. The advantages of the string frame should not be surprising. For example, much more explicit solutions in the lowest order dilaton gravity were found in the string frame [33] than in the Einstein one (discussions concerning the relation between the string and the Einstein frames are reviewed in [34]). Our results show that the string frame is the most convenient one to investigate dilaton gravity also at higher orders. Acknowledgments The work of D.K. was partially supported by the EC Project MTKD-CT-2005-029466 “Particle Physics and Cosmology: the Interface” and by the Polish MEiN grant 1 P03B 099 29 for years 2005-2007. M.O. acknowledges partial support from the EU 6th Framework Program MRTN-CT-2004-503369 “Quest for Unification” and from the Polish MNiSW grant N202 176 31/3844 for years 2006-2008. M.O. thanks for hospitality experienced at Institute of Theoretical Physics of Heidelberg University where part of this work has been done. Appendix The dilaton gravity Lagrangian and the corresponding equations of motion can be written as a series in the number of derivatives [D/2] κnL(n) , (A.1) T νµ = [D/2] µ = 0 , (A.2) and the dilaton equation of motion W = −L = 0. The 0–th order terms correspond to the cosmological constant: eφL(0) = 1 , (A.3) eφT ν(0)µ = − δνµ . (A.4) The 1–st order contribution is the standard Einstein gravity interacting with the dila- eφL(1) = R+ 2�φ − (∂φ)2 , (A.5) eφT ν(1)µ = − φL(1) + Rνµ +� . (A.6) The next two orders are given by the following expressions: eφL(2) = eφL(1) Rρ2ρ1 +� Rρ1ρ2 +� +Rρ2ρ4ρ1ρ3R , (A.7) eφT ν(2)µ = − φL(2) + 2 Rνµ +� eφL(1) − 4 Rρµ +� Rνρ +� −4Rνρ2µρ1 Rρ1ρ2 +� + 2Rρ1ρ3µ ρ2R , (A.8) eφL(3) = 3 eφL(2) eφL(1) eφL(1) Rρ2ρ1 +� Rρ3ρ2 +� Rρ1ρ3 +� Rρ2ρ1 +� Rρ4ρ3 +� Rρ1ρ3ρ2ρ4 − 24 Rρ2ρ1 +� Rρ1ρ5ρ3ρ4R −8Rρ2ρ4ρ1ρ3R Rρ3ρ5ρ4ρ6 + 2R Rρ1ρ3ρ5ρ6R , (A.9) eφT ν(3)µ = − φL(3) + 3 Rνµ +� eφL(2) − 12R Rρµ +� Rνρ +� −12RRνρ2µ ρ1 Rρ1ρ2 +� + 6RRρ1ρ3µ ρ2R Rρ1µ +� Rνρ2 +� Rρ2ρ1 +� +24Rνρ2µ ρ1 Rρ3ρ1 +� Rρ2ρ3 +� + 24Rνρ2µ ρ1R Rρ3ρ4 +� +24Rρ1ρ3µ ρ2 Rνρ1 +� Rρ2ρ3 +� − 12Rρ1ρ3µ ρ2R Rνρ4 +� +24Rν ρ3ρ1ρ2 Rρ1µ +� Rρ2ρ3 +� − 12Rν ρ3ρ1ρ2R Rρ4µ +� −24Rρ1ρ3µ ρ2R Rρ4ρ3 +� − 12Rρ1ρ3µ ρ2R Rρ4ρ3 +� −12Rνρ2µ ρ1R Rρ3ρ4ρ2ρ5 + 6R Rν ρ2ρ4ρ5R − 24Rρ1ρ3µ ρ2R Rρ4ρ2ρ1ρ5 . (A.10) References [1] C. Lanczos, Annals Math. 39 (1938) 842. [2] D. Lovelock, J. Math. Phys. 12 (1971) 498. [3] Y. Choquet-Bruhat, J. Math. Phys. 29 (1988) 1891. [4] K.A. Meissner and M. Olechowski, Phys. Rev. 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0704.1235

Fluctuations of the partial filling factors in competitive RSA from binary mixtures

Fluctuations of the partial filling factors in competitive RSA from binary mixtures Arsen V. Subashiev and Serge Luryi Department of Electrical and Computer Engineering, State University of New York at Stony Brook, Stony Brook, NY, 11794-2350 Competitive random sequential adsorption on a line from a binary mix of incident particles is studied using both an analytic recursive approach and Monte Carlo simulations. We find a strong correlation between the small and the large particle distributions so that while both partial con- tributions to the fill factor fluctuate widely, the variance of the total fill factor remains relatively small. The variances of partial contributions themselves are quite different between the smaller and the larger particles, with the larger particle distribution being more correlated. The disparity in fluctuations of partial fill factors increases with the particle size ratio. The additional variance in the partial contribution of smaller particle originates from the fluctuations in the size of gaps between larger particles. We discuss the implications of our results to semiconductor high-energy gamma detectors where the detector energy resolution is controlled by correlations in the cascade energy branching process. PACS numbers: 02.50.Ey, 05.20.-y, 68.43.-h, 07.85.Nc I. INTRODUCTION One dimensional irreversible random sequential ad- sorption (RSA) has been of interest for several decades. Its numerous extensions include RSA with particles ex- panding in the adsorption process [1, 2, 3], two-size par- ticle adsorption [4, 5, 6, 7], and also RSA with an ar- bitrary particle-size distribution function [8]. The inter- est is due to the relevance of this process to a number of physical phenomena in different fields of application, such as information processing [9], particle branching in impact ionization [10] and crack formations in crystals under external stress [11]. The simplest example of RSA is the so-called car parking problem (CPP). In the con- text of CPP, one studies the average number of particles (“cars”) adsorbed on a long line and the variance of this number. Equivalently, one is concerned with the distri- bution function for the size of gaps between the parked cars (see Refs. [12, 13] for the review). The problem of competitive RSA from a binary mix- ture is of special interest because of the non-trivial corre- ∗[email protected] lations in both the particle and gap-size distributions, de- veloped during the deposition. These correlations mani- fest themselves in the final irreversible state correspond- ing to the so-called “jamming limit” — when every gap capable of adsorbing a particle has done so. Numerous studies, reported in the literature for the binary-mixture RSA in the jamming limit, addressed the problem of cor- relations only indirectly, through its manifestation in the fill factor or the gap distribution. Available results in- clude binary mixtures with point-like particles [4, 5] and those with a relatively small particle size ratio, b/a < 2 [6]. Also available are Monte-Carlo studies of the fill factor and the gap-size distribution for a binary-mixture deposition with equal abundance of both particles [7]. The present study is concerned with the correlation between the fluctuations in the number of adsorbed par- ticles of each kind from a two-size binary mixture, as well as with their partial contributions to the fill factor. We present both analytical results and those obtained by Monte-Carlo simulations for a wide range of binary- mixture compositions and size ratios. We are interested in the RSA problem primarily be- cause of its relevance to the propagation of high-energy γ- http://arxiv.org/abs/0704.1235v3 mailto:[email protected] particles through a semiconductor crystal — with parti- cle energy branching (PEB) due to cascade multiplication of secondary electrons and holes [10, 14, 15, 16, 17]. The correlation of energy distribution between secondaries is quite similar to that of the gap distribution in the RSA process [18]. In both cases, the ratio of the variance of the final number of particles to the average particle num- ber in the final (jamming) state can be much less than unity, which is favorable for the detector energy resolu- tion. This ratio (which would be unity if the particle number obeyed a Poisson distribution) is called the Fano factor, Φ [19]. The reported attempts to evaluate Φ employed over- simplified models of the semiconductor band structure. In such models, all crystal properties are characterized by three parameters, namely, the band gap, the phonon frequency, and the ratio of the rate of phonon emission to that of impact ionization. The price of this oversimpli- fication had been that correspondence with experiment could be achieved only by assuming unphysically large rates of phonon losses (about 0.5 eV per created e-h pair). This does not corroborate with the known values for the ratio of the impact ionization and the phonon emission probabilities for high-energy electrons in semiconductors. The model furthermore obscures the role of features in the band structure and the ionization process that are specific to a particular semiconductor. In our earlier work [3], we used an extended RSA model of particles that expand or shrink upon adsorption. The shrinking model is relevant to the PEB problem in that it helps to elucidate such factors as the non-constant den- sity of states in the semiconductor band and the fact that due to momentum conservation the ionization threshold is larger than the actual (bandgap) energy that is lost in impact ionization. The recursive technique employed in Ref. [3] allowed us to assess the accuracy of approximate approaches to the yield and variance calculations (such as, e.g., the average-loss approach of Refs. [15, 16]). In the present work, the RSA model is extended in a different direction — competitive deposition of different- size particles from a binary mixture — that is suitable to simulate the role of multiple channels of pair produc- tion, owing to the multi-valley nature of semiconductor bands. We arrive at a number of qualitative conclusions that should be taken into account in both the interpre- tation of experimental data and the choice of the crystal composition and device structure in gamma detectors op- timized for energy resolution. The paper is organized as follows. Section II presents the basic equations of the recursive approach and the an- alytical results for the fill factor and its variance for the larger particles. In Sect. III, we analyze the results that demonstrate high correlation in the particle distribution. Based on the gained understanding, we formulate in Sec. IV the implications of our results for the Fano factor of semiconductor γ detectors. Our conclusions are summa- rized in Sect. V. Certain analytical results are derived in the Appendix. II. PARTIAL CONTRIBUTIONS TO THE FILL FACTOR AND ITS VARIANCE FOR TWO-SIZE RSA PROBLEM We consider the problem of competitive deposition from a binary mixture of particles with sizes a and b, whose relative contributions to the total flux on the ad- sorbing line are q and p = 1 − q, respectively. We shall use a recursive approach to first study the mean number of particles na(x) and nb(x), adsorbed on a line of length x (in the jamming limit), and then the corresponding variances. Consider a large enough empty length x > a, b. We assume that the adsorption is sequential, i.e. only one particle is adsorbed at a time. The first adsorbed par- ticle will be of size a with the probability of landing at any point q(x − a)/(x − l) or of size b with the landing probability p(x − b)/(x − l). Here l = qa + pb is the “average” particle size in the binary flux. After the first particle is adsorbed, it fills a certain interval [y, y+a] (or [y, y + b]), and leaves two independent segments, whose combined size is either x− a or x− b. The average num- bers of a-particles na(y) and na(x− y− a) (or na(y) and na(x−y−b)) will be subsequently adsorbed in these gaps. Thus, the recursion relation is of the form na(x) = q(x− a) [1 + na(y) + na(x− a− y)] p(x− b) [na(y) + na(x− b− y)] , where the first and the second terms (upper and lower lines) correspond to the cases of the first landed particle being a particle of sort a or b, respectively. These cases must be averaged over all possible landing coordinates y of the first particle in a different way, viz. for a first, < na(y) >a= ∫ x−a na(y)dy , whereas for b first, < na(y) >b= ∫ x−b na(y)dy . Performing the average and using the symmetry between left and right segments we obtain, finally: na(x) = q(x − a) ∫ x−a na(y)dy ∫ x−b na(y)dy. (1) A similar equation holds for the particles of size b: nb(x) = p(x− b) ∫ x−a nb(y)dy ∫ x−b nb(y)dy. (2) With the help of Eqs. (1,2) one can readily derive an equation for the average total covered length f(x), de- fined as f(x) = ana(x) + bnb(x), giving f(x) = xl − qa2 − pb2 ∫ x−a f(y)dy ∫ x−b f(y)dy. (3) Equation (3) agrees with that of Ref. [8] for the total cov- ered length in RSA from a multi-size mixture. However, the advantage of Eqs. (1,2) is that they permit studying the partial contributions to the coverage by each of the two sorts of particles separately. Note that the symmetry between the a- and the b- par- ticles is broken by the initial conditions. To be specific, let b > a. Then, for b-particles the boundary condition at small x is simply nb(x) = 0, 0 ≤ x < b (4) whereas for a-particles we have na(x) = 0, 0 ≤ x < a 1, a < x ≤ min(2a, b) For b > 2a, Eq. (5) should be supplemented with na(x) = 1 + ∫ x−a na(y)dy (6) Eq. (6) accounts for the deposition of smaller particles in small gaps where the larger particle does not fit. Clearly, this process is not influenced by the b-particles and does not involve particle competition. More refined arguments are needed to derive the sec- ond moment of the distribution, i.e. the expected value of the square of the number of particles of a given sort, ua(x) = En a(x). It may not be a priori evident that one can write independent expressions for particles of both sorts, because parameters a and b not only describe the particle size but also designate the sort of a particle. In- deed, we can even have a = b and distinguish the parti- cles by some other parameter, like “color”. Our approach should remain valid in this case too. To be rigorous, we therefore introduce an artificial parameter, the “mass” of a particle, ma and mb, whose value may depend on the particle shape and is simply proportional to the particle length only for a fixed transverse particle size. Hence one can regard ma and mb as independent parameters. Consider a total mass M(x) = mana(x) + mbnb(x) of the particles adsorbed in a line segment x. We first evaluate recursively the mean square of the total mass < M2(x) >= 〈[mana(x)+mbnb(x)] 2〉, and then calculate the second partial derivatives with respect to ma and mb. Using the landing probabilities of particles to perform the averaging, we obtain ua(x) = (x− l) q(x− a) + 2q ∫ x−a ua(y)dy ∫ x−b ua(y)dy + 4q ∫ x−a na(y)dy ∫ x−a na(y)na(x− y − a)dy ∫ x−b na(y)na(x− y − b)dy Similarly, equation for ub(x) reads ub(x) = (x− l) p(x− b) + 2q ∫ x−a ub(y)dy ∫ x−b ub(y)dy + 4p ∫ x−b nb(y)dy ∫ x−a nb(y)nb(x − y − a)dy ∫ x−b nb(y)nb(x − y − b)dy We could have derived Eqs. (1,2) in a similar way, by first evaluating the total average mass M(x) = mana(x) +mbnb(x) recursively, and then calculating the derivatives. For a more general case, when the total mass is a linear functional M(x) = mlnl(x)dl on the mass distribution ml, one would have to use variational deriva- tives δM(x)/δml. For the case of binary mixtures we consider, partial derivatives are sufficient. Similarly, we derive an equation for the correlation function uc(x) = 〈na(x)nb(x)〉 by calculating a mixed derivative of < M2(x) > with respect to ma and mb. For particles uniform in the transverse direction with unit mass density, both the mass and the length of particles are identical, which gives a way to check the equations. An appropriate linear combination of equations for ua, ub, and uc then gives an equation for the variance of the total filled length or, equivalently, for the variance of the wasted length, w(x) = x − f(x). The resulting equation can also be obtained directly, by applying recursion ar- guments to the waste. The identical results obtained can be viewed as an additional proof of Eqs. (7,8). Note the asymmetry in the 4-th terms of Eqs. (7,8) that are proportional, respectively, to 4q and 4p. These terms ensure the correct (linear) asymptotic behavior of the variance at large x. An important feature of Eqs. (1,2,7,8) is that in spite of the competitive character of the deposition of particles of different sorts, the equations for na, nb and the higher moments are independent. This is rooted in the fact that a single deposition step on an empty length x does not depend on the already adsorbed particle distribution. Due to the self-averaging nature of the filling length (and waste length) in the limit x → ∞ the averaged (hence approximate) recursion equations yield exact re- � � � � � � ��� � ������ � � �� �� �� b/a = 2.4 � ��� !"#$%&'() *+,-./012 3456789:; <=>?@ABCD EFGHIJ Adsorbtion length, x FIG. 1: (color online) Average number of adsorbed particles na (solid lines) and nb (dashed lines) as functions of the length x (measured in units of a) of the adsorption interval, assumed initially empty. The results are obtained by iterating Eqs. (1,2) with the assumed ratio of the particle size b/a = 2.4 and the varying fraction q of a-particles in the flux. sults. The recursive technique is in this sense equiva- lent to the alternative “kinetic” approach to RSA that is sometimes regarded as a higher-level theory. In the kinetic approach one considers the rate equation that describes the sequential deposition of particles with the particle distribution on a line characterized by a time- dependent function G(x, t) representing the average den- sity of gaps whose size is between x and x + dx [2, 5]. It has been ascertained for a number of problems that both approaches give the same result for the coverage. Still, each has its own benefits. The kinetic approach allows studying the temporal variation of a state with specified particle distribution. The recursive approach, while simulating a simplified version of the kinetics, al- lows to study more complex effects, such as variance of the adsorbed particles of different size. Evaluation of na(x) and nb(x) is readily done by re- peated iterations of Eqs. (1,2), going from the small to progressively larger lengths x. Results of the numerical recursion are shown in Fig. 1 for a particle size ratio b/a = 2.4 and varying q. The noteworthy features of the functions na(x) and nb(x) are (i) the step-like features at x = a, x = b (which are replicated with ever smaller amplitudes at x = na +mb, where n and m are integers), (ii) the dip in the number of small particles na(x) at x = b, which increases with p, and (iii) the reduction of nb with in- creasing q. We also note that for all q the behavior of both na(x) and nb(x) becomes very close to linear al- ready at x ≈ 7. The asymptotic behavior of na(x) and nb(x) at large x can be obtained by multiplying Eqs. (1,2) by x− l and taking the derivative with respect to x. The resulting differential equations are satisfied by linear functions of the form na = αa(x+ l)− q, nb = αb(x+ l)− p, (9) where αa and αb are arbitrary constants. When correctly chosen (by matching to the recursive solution) these con- stants become the partial filling factors. After the match- ing is done, the total filled length in the asymptotic limit is given by f(x) = θx + (θ − 1)l, where θ = aαa + bαb is the specific coverage. It is worthwhile to stress that the value of the asymptotic solutions (9) consists precisely in that they are asymptotically exact. Hence they provide a sanity check on any solution we could have obtained by a numerical recursion up to moderate values of x. Similarly, Eqs. (7-9) yield the variances at large x, ua − n a = µa(x+ l)− qp[1 + (b − a)αa] 2, (10a) ub − n b = µb(x+ l)− qp[1− (b− a)αb] 2. (10b) Again, these solutions are asymptotically exact; they sat- isfy Eqs. (7,8) with arbitrary values of µa and µb, pro- vided of course that na(x) and nb(x) are in the correct asymptotic form (9) with properly chosen [i.e., satisfying Eqs. (1,2)] coefficients αa and αb. In principle, we could now follow a procedure similar to above, viz. determine µa and µb by matching Eqs. (10) against a numerical recursive solution at some moderate value of x. How- ever, it would be rather difficult to control the numerical accuracy in this procedure, because of the difference of nonlinear functions that enter Eqs. (10), even though that difference itself behaves linearly with x at large x. Fortunately, our model admits of an exact solution based on the use of Laplace transformation (details can be found in [3] and references therein). Below we present an exact evaluation of variance for particles of larger size, while details of similar though lengthier calculations for smaller particles are presented in Appendix. Firstly, we need exact solutions of Eqs. (1,2). To ob- tain these, we substitute x → x + b in Eq. (2) and mul- tiply it by x − l. Taking the Laplace transformation of the resulting equation and using the boundary condition (4), we obtain + b− l ebsNb(s) = qe(b−a)s + p Nb(s) Here Nb(s) is the Laplace transform of nb(x), Nb(s) = e−sxnb(x)dx, (12) Rearranging the terms and multiplying by e−bs, we put Eq. (11) into the form N ′b(s)+ qe−as + pe−bs Nb(s) = − e−bs. (13) For p → ∞, the solution of Eq. (13) is, asymptotically, Nb(s)|s→∞ = s(b− l) e−bs, (14) as follows from the known variation of nb(x) ≈ p(x − b)/(b− l) at small x− b. Hence we have Nb(s) = p exp(−l s) s2β(s) e−q(b−a)tβ(t)dt, (15) where β(s) = exp 1− q exp(−at)− p exp(−bt) To find the asymptotic behavior of nb(x) at large x, it is convenient to use Karamata’s Tauberian theorem for the asymptotic growth rate of steadily growing functions (see e.g. [20], p. 37). According to the theorem, the asymptotics of nb(x) [or na(x) or their variances] can be readily obtained (by taking the inverse Laplace transfor- mation) from the Laurent power series expansion of the Laplace transforms of these functions at small s (see [9] for the mathematical details of this analysis). Function Nb(s) is analytic at all s 6= 0 and at s = 0 it has a second-order pole with the following asymptotic Nb(s) = αb,0l − p +O(s), (17) where αb,0 = p e−q(b−a)sβ(s)ds. (18) To calculate nb(x) at large x, we take the inverse Laplace transformation of (17). This gives nb(x) = αb,0(x + l)− p, (19) with an exponentially small error term, in line with the asymptotics given by Eq. (9). In the limit p = 1, equation (18) duly gives the so- called jamming filling factor R for the standard RSA, αb,0(l = 1) ≡ R = 0.74759 · · · (also called the Renyi constant [21]). In the limit a → 0, Eq. (18) recovers the results of Refs. [4, 5] for the coverage of a line from a binary mixture of finite size particles and point defects. Moreover, Eq. (18) gives the large particle contribution to the total coverage, obtained in [6, 8] for the range a < b < 2a. Here we see that this result remains valid for arbitrary a < b. Next, we perform similar manipulations with Eq. (8) and obtain an equation for the Laplace transform of the variance Ub(s) = L̂[ub(x)], viz. U ′b(s)+ qe−as + pe−bs Ub(s) = − exp(−bs) Rb(s), where Rb(s) = p+ 4psNb(s) + 2s 2N2b (s) qe(b−a)s + p , (21) with Nb(s) defined by Eq. (15). The solution of Eq. (20) can be written in a form similar to Eq. (15), namely Ub(s) = exp(−sl) s2β(s) β(t)e−q(b−a)tRb(t)dt. (22) The integrand in the right-hand side of Eq. (22) is pro- portional to 1/t2 causing the integral to diverge as 1/s for s → 0. This is due to the square-law dependence of u(x) at large x . To separate the regular part needed for the estima- tion of variance, we note that at small t one has Rb(t) ∝ αb,0t −2. Moreover, the series expansion shows that the difference β(t) exp[−q(b− a)t]Rb(t)− 2α −2 is regular at t → 0. Therefore, it is convenient to define an en- tire function κb(t) = β(t) exp[−q(b−a)t]Rb(t)−2α In terms of this function, the solution Ub(s) can be ex- pressed as follows: Ub(s) = exp(−sl) s2β(s) + kb,0 − κb(t)dt , (23) where kb,0 = κb(t)dt. (24) To apply Karamata’s Tauberian theorem, we note that the asymptotic expansion of Ub(s) near its third-order pole is of the form Ub(s) = kb,0 + 2α kb,0l − κb(0)− qp(b− a) 2α2b,0 . (25) Taking the inverse Laplace transformation, we find the asymptotic form of ub(x): ub(x) = α 2 + (kb,0 + 2α b,0l)x +kb,0l − κb(0)− qp(b− a) 2α2b,0, (26) with an exponentially small error term. Using Eq. (19) to subtract n2 (x), we find an equation of the form (10b) with µb = kb,0 + 2pαb,0. The specific variance of the adsorbed number of b-particles is given by (at x → ∞) µb = αb,0(1 + 2p) + 2 β(s)sNb(s)e 2pe−bs +sNb(s) qe−as + pe−bs α2b,0 ds. (27) Integrating by parts the last term and rearranging the result, we finally obtain µb = αb,0(1− 2p) + 4p αb(u) e−bu (1− qe−au −pe−bu du+ 2 β(u)u2 e−luK(u)du, (28) where K(u) = qe−au 2(1− qe−au − pe−bu)− (a+ l)u +pe−bu 2(1− qe−au − pe−bu)− (b+ l)u αb(u) = αb,0 − p e−q(b−a)yβ(y)dy. (30) In the limit of small p → 0, the Fano factor Φ = µb/αb,0 → 1. In this limit, large particles are distributed on the line randomly, without correlations. In the oppo- site limit, p = 1, Eq. (28) reduces to the standard RSA result, first obtained for a lattice RSA model by Macken- zie [22]. The numerical value of the Mackenzie constant, µ0 = 0.0381564 · · · , corresponds to Φ = 0.0510387 · · · , see [9]. Expression (28) for the larger particles has the same structure as the corresponding formula in the stan- dard RSA model (fixed-size CPP). Due to the exponen- tial factors in the integrands of Eq. (28), the dependence of µb on a for a ≪ b is quite weak. The limiting value of the specific variance for a/b → 0 gives the specific vari- ance of the fill factor for the case of finite-size particles (b = 1) mixed with point-size particles, µb,p = αb,p,0(1− 2p) + 4p αb,p(u) e−u (1− e−u) du b,p(u) βp(u)u2 {qe−pu (2− 2e−u − u) +e−(1+p)u [2p (1− e−u)− (1 + p)u] du, (31) where αb,p,0 is the fill factor for this case, αb,p(u) = p e−qyβp(y)dy, αb,p,0 = αb,p(u = 0) βp(u) = exp 1− exp(−t) . (33) It is worth to note that Eqs. (2,8) and their solutions can be readily generalized to the case when particles of the smaller size have an arbitrary distribution in the interval [a1, a2] so long as a2 ≤ b [23]. The above analytic results for the variance of larger particles are essentially exact, as will be confirmed in the next Section by Monte Carlo simulations. For the smaller particles, the calculations are messier and accurate ana- lytical results can be obtained only in a certain range of particle size ratios. Estimations of the variance for small- size particles are further discussed in the Appendix. III. DISCUSSION OF THE RESULTS, COMPARISON WITH MONTE CARLO MODELING Here we present the results of numerical calculations using both the analytical expressions obtained in the pre- ceding section and Monte Carlo simulations. For large- size particles the Monte Carlo results are very close to analytical expressions both for the fill factor and the vari- ance, so we shall not dwell on their comparison. For small-size particles, especially in the range 2 < b/a < 8, analytical calculations are rather unwieldy, so Monte Carlo simulations become indispensable. Larger size ra- tios, b/a > 8, lend themselves to an approximate ana- lytical approach (see Appendix). In this case, we use the Monte Carlo to estimate its accuracy for the small particle contribution. Traditional studies of the generalized RSA via Monte Carlo simulations follow a temporal sequence of events. For the case of adsorption on a line of the length x from a binary mixture, one step of the sequence comprises: (i) selection of a particle from the mixture according to the deposition flux ratio (with the probability q of choosing the small-size (a) particle, and the probability p = 1− q of selecting a particle of larger size b); (ii) random choice of a deposition coordinate of particle center on the line x with formerly deposited particles; (iii) rejection of the particle if it overlaps by any part with formerly deposited particles or with the line bor- ders; otherwise, the particle deposition proceeds with the formation of two new disconnected adsorption lengths. This traditional approach has several drawbacks, that make the modeling very demanding, both in terms of the computer time and memory allocation. Firstly, both the filled length in the jamming limit and the specific fill factor (coverage) depend on the initial length. Due to the self-averaging property of the coverage it tends to a unique exact value in the limit x → ∞. To obtain the accuracy of about 0.1 %, the common strategy has been to use large initial length values (105b -107b) and make additional averaging over a set of about NR =100- 1000 different realizations. Secondly, as time evolves and the jamming limit is ap- proached, the probability of finding a free gap for parti- cle deposition becomes greatly reduced, so that the ad- sorption time tends to infinity. The process is termi- nated when variations of the adsorbed particle number are smaller than those required by the desired accuracy. The recursive analysis of the generalized RSA suggests a revision of the above scheme. Since the deposition is random and sequential, it does not depend on the tem- poral history of the process or the growing number of rejected particles and their coordinates. Therefore one step of the sequence can be chosen as follows: (i) selection of any free deposition length, l1 > a. It is convenient to choose for l1 the outermost free deposition length on the left-hand side. (ii) if l1 < b, then particle of size a is deposited, otherwise the deposited particle is chosen according to the landing probability, given by q(l1 − a)/(l1 − l) for a-particle and p(l1 − b)/(l1 − l) for b-particle, where l = qa+ pb. (iii) random choice of a deposition coordinate (taken as the coordinate of particle’s left end) on the line l1 for a given particle size, i.e. within the interval l1−a for a-size or within l1 − b for b-size particle, with the formation of two new adsorption lengths from the initial length l1. It is readily seen that although the sequence of deposi- tion events is different from the actual temporal sequence of adsorption (the simulated deposition proceeds by se- quentially filling the left-hand lengths), the statistics of divisions is identical and therefore so is the final distri- bution of the gaps, as well as all statistical properties of the jamming state. Our sequential scheme excludes deposition of to-be-rejected particles and therefore is in- comparably faster. Besides, it terminates exactly when the jamming limit (with no gaps larger than unity) is achieved. Direct comparison with the traditional Monte Carlo results, e.g. [5, 7, 8] exhibits total agreement. The difference in the calculation time is especially evident for small (close to zero) q: in the time scale of “real” deposi- tion, the jamming limit will be strongly delayed because of the rarity of events with small particle chosen. In our modified approach, all gaps smaller than b are “rapidly” populated by small-size particles, however small be the value of q. The next step of the revision is to exploit the fact (proven analytically in the preceding section) that in the jamming limit, the linear dependence on the adsorption length of both the average filled length and its variance is exponentially accurate, starting from a reasonably short length, certainly not exceeding x ≈ 10b. Since this lin- ear dependence has only two parameters [actually only one, as the parameter ratio is exactly fixed by analyti- cal considerations, Eqs. (9,10)], both the coverage and the variance can be determined with Monte Carlo simu- lations of short samples. To be sure, in order to achieve the same accuracy as that obtained for long samples, the results should be av- eraged over a sufficient number of realizations NR. This, however, takes little memory or time. Calculations show similar accuracy for different x and NR, so long as their product x×NR is fixed. The results presented below were KLM NOP QRS TUV WXY Z[\]^_`abcde fghijklmn opqrstuvw b/a : Ratio of small particles (a) in the flux, q 2; 5 10; 20 {|} ~�� ��� ��� ��� ��� ��������� �������� �� ¡¢£¤¥¦ total FIG. 2: (color online) Partial contributions of small and large particles to the total coverage depicted as functions of q, for different particle-size ratios b/a in the flow. Open points cor- respond to the limit b/a ≫ 1, as described by the analytical formulae (32) for b-particles and (A13) for a-particles. For the total coverage, the open circles to the wasted length product approximation, Eq. (34). obtained using a sample of size x = 200a for b/a < 10 and x = 400a for b/a = 20, 40, subsequently averaged over 10 000 realizations, which appeared to be sufficient to eliminate any spread of the results in the graphical pre- sentation (producing an accuracy of better than 0.1%). The use of small samples is very effective in reducing the calculation time (with an ordinary PC, high-accuracy results can be obtained in minutes, compared to days in the traditional scheme [8]). Figure 2 shows partial contributions to the coverage as functions of the fraction q of small particles in the binary mixture at different ratios of particle size. As q increases, the coverage with large particles is substituted by that with small particles, producing some decrease in the to- tal coverage. In the regions of corresponding parameters, our results reproduce those of reported analytical calcu- lations (i.e. for b/a < 2 [5, 8] and for a = 0 [6] for the large particle contribution) and those obtained by the Monte Carlo simulations of [5, 7, 8], demonstrating the ª«¬ ­®¯ °±² ³´µ ¶·¸ ¹º»¼½¾¿ÀÁÂà ÄÅÆÇÈÉÊËÌÍÎÏ ÐÑÒÓÔÕÖ×ØÙÚÛ b/a : infinity Ratio of small particles in a flux, q FIG. 3: (color online) Variance of the partial coverage by ad- sorbed b-particles from the binary mixture for different values of the particle size ratio b/a in the flow. validity of our revised approach. It is evident from Fig. 2, that the total coverage in- creases at smaller q, as can be explained by sequential deposition of the two kinds of particles. In the regime of small q, large particles are adsorbed first and their de- position, unobstructed by small particles, is tight. Sub- sequently, the small particle fill the gaps between large particles and this clearly reduces the total wasted length. The effect of increasing the particle size ratio b/a is pronounced only for b/a < 10, then it rapidly saturates. Therefore, for large b/a, say b/a = 20 the coverage by large particles is very close to that obtained for a model mixture of point-like and finite-size particles [by formally letting a = 0 in Eq. (18)]. Such a model, however, has lit- tle relevance to any practical situation, because it simply ignores the partial contribution of small particles to the total coverage. The latter can be described analytically in the limiting case b/a → ∞, Eq. (A13). The partial contribution of small particles steadily grows with the increasing size ratio due to the expand- ing gaps between the large particles. In the limit q → 0, the total coverage can be estimated by observing that the specific wasted length in this case is a simple prod- uct of the specific lengths wasted in initial deposition of large particles and subsequent deposition of small par- ticles, i.e. 1 − θ = (1 − θa)(1 − θb). Since for q = 0 the specific coverage θb = R and since for large size ra- tios (when the gaps between large particles are large) the specific coverage θa = R, we have θ = 1−(1−R) 2=0.936, in agreement with the results reported in the literature [5, 7]. However, the sequential nature of the deposition suggests that the entire q dependence of the total θ can also be approximated by a product of the specific wasted lengths in the competitive deposition of large particles q(1 −R) and subsequent deposition of small particles in the remaining gaps, which gives θ = 1− (1−R)(1− pR) = R[1 + p(1−R)]. (34) This product-waste approximation is shown in Fig. 2 by the open circles. Next, we concentrate on the specific waste variance and the Fano factor. We shall discuss the b- and a-particles separately, since the effects are rather different in na- ture and also since they have been evaluated by differ- ent techniques. Results for large particles are obtained by numerical integration of Eq. (30) and confirmed by Monte Carlo simulations. Results for a particles are ob- tained by Monte Carlo stimulations and are accompanied by analytical expressions in the limit b/a ≫ 1. Variance, µ̃, of the partial contribution of b-particles to the total coverage is shown in Fig. 3 for different par- ticle size ratios. Unlike the particle number variance µ, the variance of coverage, µ̃ = µb, depends only on the size ratio b/a and does not directly scale with b. It is therefore more indicative of the effect of decreasing size àáâ ãäå æçè éêë ìíî ïðñòóô õö÷øùú ûüýþÿ� ����� �� � �� �������� �� � ����� � )*+ , 456789:; <= > ? @ A B C D E F G FIG. 4: (color online) The Fano factor for the number of ad- sorbed b-particles from the binary mixture as functions of q, for different values of the ratio b/a of particle sizes in the flow. Dotted line corresponds to a random particle packing on a lattice with suitable lattice constant (aka monomer ad- sorption). of small particles on the fluctuations of the number of large particles. At q → 0, when the adsorption of large particles is unconstrained by small particles, the vari- ance of large particles is minimal and corresponds to the highly correlated distribution [18] in the standard CPP problem (one-size RSA). The variance rapidly increases with q as the small particle deposition destroys the CPP correlations. The maximum of this effect is shifted to larger q values for larger b/a. For q approaching unity, the variance decreases simply due to the decrease of the average number of adsorbed b-particles. Correlation effects are more adequately characterized by the Fano factor Φb, shown in Fig. 4. With the increas- ing number of competing small particles in the flux, the Fano factor grows from the smallest value Φ = 0.051 · · · , corresponding to the one-size RSA problem, to unity in the limit q → 1. Small coverage by the large particles in the latter limit means that they are distributed randomly on the line, so that Poisson statistics recovers. The most noticeable effect is a rapid decrease of the Fano factor 0.0 0.2 0.4 0.6 0.8 1.0 Small particle contribution to the flux, q b/a: 1.6; 2 5; 10 20; 40 FIG. 5: (color online) The Fano factor for adsorbed a-particles from the binary mixture for different values of the particle size ratio b/a in the flow. Open points show the contribution of fluctuations of the gap sizes between large particles. with 1− q, manifesting a strong enhancement of the cor- relation effects in the large particle distribution. These correlation effects become exhausted only near q ≤ 0.1. The correlation effects increase with b/a but saturate at about b/a = 20. Figure 5 shows the Fano factor for a-particles com- petitively deposited along with large particles. The re- sults are strikingly different at all q 6= 1 (when Φa = Φ, as expected). While the distribution remains correlated (Φa ≤ 1) for small ratios b/a ≤ 5, at larger b/a one has Φa > 1, almost for all q, which means that the number of small particles per unit length is strongly fluctuating. This is due to the widely fluctuating size of the gaps available for small particle deposition between large par- ticles. For large values of b/a and in the entire range of q, the Fano factor Φa can be approximated in terms of the fluctuations of the coverage by the large particles, viz. Φa = (b/a)µb,pR 2/θa, where µb,p is given by Eq. (31) and θa by Eq. (A13). This approximation, which neglects fluctuations of the density of adsorbed a parti- TUV WXY Z[\ ]^_ `ab cde fghij klmno pqrst ��� � ��� � ���� ���� §¨©ª«¬ ­®¯°± FIG. 6: (color online) Variance of the partial number of ad- sorbed a- and b-particles and of the total number of adsorbed particles for b/a = 1.2. Also shown is the fluctuation correla- tion function fcor. cles in the gaps, is shown in Fig. 5 by open points. This contribution is proportional to b/a and for b/a > 10 it is evidently dominant. For the particle energy branching process at small b/a < 2, both the variance of the partial numbers of small and large particles and the total number variance are of importance. We shall illustrate this point in the instance of b/a = 1.2 shown in Fig. 6. We see that at q ≈ 0.5 the fill factor fluctuations are larger for a particles and some- what smaller for b particles, but both are pretty large, compared to the variance of the total number of adsorbed particles. This is due to the strong anti-correlation in their distribution, as evidenced by the specific fluctua- tion correlation function, fcor = x −1〈δnaδnb〉, also plot- ted in Fig. 6. We note that fcor < 0, which means that any excess in the number of a-particles is accompanied by a downward fluctuation in the number of adsorbed b-particles. Importantly, the variance and the Fano fac- tor for the total number of adsorbed particles does not exceed substantially its value for the single-size RSA. Note the asymmetry of the curves for a and b parti- cles, e.g. the variance of large particles goes to zero as q → 1 whereas that of small particles remains finite even as q → 0. This is a feature of our model that allows ”infi- nite” amount of time for the deposition of small particles in the gaps left after the deposition of large particles is completed, but not vice versa. Therefore, the deposition of small particles remains finite even in the limit of q → 0 and the same is true for the a-particle number variance. Another interesting feature of the a-particle num- ber variance, already evident from Fig. 5, is its non- monotonic behavior as function of b/a at small q. This variation is displayed directly in Fig. 7 that shows the dependence of the Fano factor on b/a for q=0.05, 0.1 and 0.2 — where its non-monotonic nature is most pro- nounced. The minimum of the Fano factor is achieved at b/a ≈ 2. Note that the non-monotonic dependence of the Fano factor is accompanied by non-monotonic variations in the dispersion of the gaps between small particles. In Ref. [7] it was found that for q =0.5 the dispersion is noticeably reduced at b/a ≈ 1.55. These effects were interpreted as a manifestation of the so-called “snug fit” events, i.e. particle deposition in gaps that are just barely above the unit length a. In contrast, the Fano factor for b-particles and that for the total number of particles re- main monotonic everywhere. IV. SOME CONSEQUENCES FOR THE ENERGY BRANCHING IN HIGH-ENERGY PARTICLE DETECTORS The model of RSA from binary mixtures is relevant to an important practical problem of particle energy branching (PEB) where high-energy particle propagates in an absorbing medium and multiplies producing sec- ondary electron-hole (e-h) pairs. Multiplication proceeds ²³´ µ¶· ¸¹º »¼½ ¾¿À ÁÂà ÄÅÆ ÇÈÉ q : 0.05 0.1 0.2 Particle size ratio, b/a FIG. 7: (color online) The Fano factor for adsorbed a- and b-particles as functions of the particle size ratio b/a. Also shown is the Fano factor for the total number of adsorbed particles. so long as the particle energy is above the impact ioniza- tion threshold [15]. The energy distribution of secondary particles is random to a good approximation. The affinity between the two problems was fully rec- ognized already in 1965 by van Roosbroek [17] (see also [24]). The PEB process can be considered in terms of a CPP if one identifies the initial particle kinetic energy with an available parking length and the pair creation en- ergy with the car size. Similarly, the kinetic energies of secondary particles can be identified with the new gaps created after deposition of a particle. Full equivalence of PEB to CPP further requires that only one of the secondary particles takes on a significant energy, which corresponds to binary cascades [25]. Otherwise, one has to consider a simultaneous random parking of two cars in one event. To estimate the particle initial energy in PEB, one measures the number N of created electron-hole pairs. Variance of this number, due to the random character of energy branching and also due to random energy losses in phonon emission, limits the accuracy of energy mea- surements. Both the yield N and the e-h pair variance var(N) = (N −N)2 are proportional to the initial en- ergy. The ratio of the e-h pair variance to the yield, i.e. the Fano factor of the PEB process, is a parameter that quantifies the energy resolution of high-energy particle detectors. For semiconductor crystals, the PEB problem has ad- ditional complications due to the energy dependence of phonon losses and the energy dependence of the electron density of states and the impact ionization matrix ele- ment. Full quantitative analysis of the PEB is possible only with detailed numerical calculations, which goes far beyond the scope of the present article. A common feature of the energy branching process in semiconductors is the presence of several pair production channels, associated with the multi-valley energy band structure of the crystal. In Si, Ge and common A3B5 semiconductors, the e-h pair creation produces electrons in one of the ellipsoids near the edge of the Brillouin zone, in 100 (X) or 111 (L) directions. Owing to the difference in the final densities of states and the matrix elements, the impact ionization processes associated with X and L valleys have different but competitive probabilities. Be- cause of its low density of states, the Γ valley is usually not competitive, even when it is the lowest valley. Ultimately, electrons will end up in the lowest energy valley but when the final electron valley is itself degener- ate, as in Ge or Si, the resulting electron states may not be fully equivalent, because of the different collection ki- netics owing to the crystal anisotropy. This effect may have important consequences for the observed variance. For example, in Si diode detectors electrons are created in 6 degenerate energy valleys that represent ellipsoids of revolution elongated along (100) and equivalent direc- tions in k-space. Suppose the diode structure is such that the current flows along the (100) direction, as it is usually the case. Electrons from the two valleys along the cur- rent have a large mass and low mobility. The measured current is hence dominated by electrons from the 4 val- leys elongated perpendicular to the current that have a low mass and high mobility along the current. Since the choice of equivalent valley in the PEB process is fully ran- dom, the number of high-mobility electrons will fluctuate more strongly than the total number of generated carri- ers. These fluctuations will dominate if the inter-valley transition rate is low compared to the inverse collection time. In the opposite limit of high inter-valley transition rates, this effect will average out as the collected current will fluctuate in time. The current fluctuation mecha- nism due to the carrier escape into heavy-mass valleys is a well-known source of noise in multi-valley semiconduc- tors [26]). More detailed account for these effects will be presented elsewhere [23]. Here we shall discuss an opposite situation, that is common to direct-gap semiconductors, such as GaAs or InP. In these materials, the lowest (Γ) electron valley has a very low density of states, compared to that in the satellite (X and L) valleys. Therefore, the probability of electron generation in the Γ-valley can be neglected in first approximation, so that the branching competition occurs only between the satellite valleys of two different kinds. Both the density of states and the threshold en- ergy are different between X and L valleys and we can use the results of the present study to interpret and predict the consequences, at least qualitatively. The binary-mixture RSA model interprets the higher density of states as higher deposition rate and the higher threshold as larger particle size. To make our conclu- 0.0 0.2 0.4 0.6 0.8 1.0 b/a = 1.4 Ratio of the small particles in the flux, q Coverage FIG. 8: (color online) Partial fill factors and the total coverage for b/a = 1.4 as a function of q. Also shown is the total number Ntot of adsorbed particles sions more transparent, let us re-formulate the required results in terms of a random parking problem with cars of two sizes. We are now interested only in the numbers of parked cars and the fluctuations of these numbers. Several qualitative conclusions can be drawn from our results: (i) The total number of parked cars (in the jamming state) will decrease with increasing fraction of larger cars in the flow and with the growth of their size. For b/a = 1.4 the effect is illustrated in Fig. 8 (which can be viewed as an extension of Fig. 2). It follows from the fact that adsorption of a large car excludes larger length for subsequent parking events and thus causes a decrease of the total fill factor. Note that the decrease in the to- tal particle number is accompanied by an increase in the total filled length, as smaller number of cars cover larger area. The next two conclusions (ii) and (iii), illustrated in Fig. 6, are interconnected and will be discussed jointly. (ii) Variance of the total number of parked cars and the Fano factor will both grow with the increasing fraction of larger cars in the flow and with the growth of their size. (iii) Variance of the separate numbers of parked small and large cars and their Fano factors are considerably larger than that of the total number of cars. Therefore, if for some reason one type of cars is neglected or under- counted, the registered variance and the Fano factor can be substantially increased. These conclusions are connected with the nature of the car number fluctuations and the strong anti-correlation between the fluctuations in the number of small and large cars. Fluctuations in the number of parked cars of one kind are strongly enhanced by the presence of more or less randomly distributed cars of the second kind, espe- cially when cars of the second kind dominate. This leads to conclusion (iii). However, the two distributions are anti-correlated (higher number of parked small cars is accompanied by a smaller number of large cars and vice versa). The anti-correlation is particularly strong for a size ratio that is close to unity. One can imagine a case when the two kinds of cars dif- fer only in “color”. In this case, Eqs. (18,28) yield αa,0 = qR, αb,0 = pR, µa = Rqp+q 2µ0 and µb = Rqp+p 2µ0, so that at large x we have < δnaδnb > /x = −(R − µ0)qp. Then, the anti-correlation is almost complete: the fluc- tuations of the total number are much smaller than those of a given color, but still non-zero. Both the individual- color number fluctuations and the anti-correlation are largest at q ≈ 0.5, cf. Fig. 6. The anti-correlation decreases with increasing size ratio, as reflected in our conclusion (ii). To discuss the above conclusions in terms of the PEB problem, we note that estimation of the initial particle energy is equivalent in CPP to a measurement of the unknown length of a parking lot in terms of the total number of cars that were able to fit into it by random parking, assuming that the average fill factor for a given two-size car mixture is known from earlier measurements. The absolute accuracy of such a measurement depends on the variance of the fill factor, and the relative accuracy is determined by the Fano factor. As shown above for a mixture of cars, the larger disparity of car sizes leads to the higher fill-factor variance and therefore reduces the absolute accuracy. A particle detector measures the total number of sec- ondary particles of all sorts (but not their total creation energy, that would be equivalent to the filled length). In any channel, all secondaries that have sufficient energy for further branching will do so. Therefore, only those pair creation energy ratios that leave the channels com- petitive (i.e. b/a < 2) are relevant to the PEB problem — otherwise additional energy branching would be pos- sible. We conclude that the presence of competing channels with different energies [e.g. impact ionization with ex- citation in X and L valleys] will decrease the quantum yield (the number of secondaries per unit energy of the primary particle) and enlarge the Fano factor. The at- tendant loss in energy resolution is not that bad when the ionization energies associated with different valleys are not too disparate. For example, in Ge besides the lowest eight L valleys (EG = 0.66 eV) one has a non- competitive Γ valley (EΓ = 0.8 eV) and six very compet- itive Si-like valleys (EG = 0.85 eV). The downgrading of energy resolution should be more important for crystals with larger (≈ 2) threshold energy ratio. For example, in Si one has besides the 6 lowest valleys (EG = 1.12 eV) in X direction, eight germanium-like L valleys with the gap EL = 2.0 eV. Their effect on the Fano factor in silicon may not be negligible. Finally, reformulating (iii), we stress that any signif- icant disparity in the collection efficiency between dif- ferent equivalent valleys will strongly enhance the Fano factor and downgrade energy resolution. This happens because any collection disparity breaks the symmetry between the equivalent valleys and destroys the anti- correlation, responsible for keeping the total Fano factor low even when the partial particle numbers associated with individual valleys exhibit fully random fluctuations. One possible origin for the asymmetry in the collection efficiency in semiconductors has been discussed above in the case of silicon diodes with the electric field in (100) direction. In germanium diodes all different valleys are equivalent relative to the (100) direction and the sym- metry is not broken. It would be broken, however, if one were to use Ge diodes oriented in (111) direction. This would lead to a situation similar to Si — with a possi- ble degradation in the Fano factor. These effects deserve additional study, both experimental and theoretical. V. CONCLUSIONS We have studied a generalized 1-dimensional competi- tive random sequential adsorption problem from a binary mixture of particles with varying size ratio. Using a re- cursive approach, we obtained independent equations for the number of adsorbed particles of given sort and exact analytical expressions for the partial filling factors and variances for the larger particles. For the smaller parti- cles analytical expressions were obtained in a number of limiting cases. The results have been confirmed by direct Monte Carlo simulations. To do so, we have introduced a modified Monte Carlo procedure that enabled us to explore a wide range of particle size ratios and particle fractions in the flux. A number of qualitative implications have been for- mulated, relevant to the energy branching problem in high-energy particle propagation through a semiconduc- tor crystal. Conclusions made concern the quantum yield and the energy resolution in semiconductor detectors made of crystals with several competing channels of im- pact ionization with different final electronic states. We have found a very strong anti-correlation effects which strongly suppress fluctuations of the total particle number compared to the fluctuations of partial contribu- tions by particles of a given sort. This effect is particu- larly evident when one considers the deposition of similar competing particles, e.g. parking of cars that are differ- ent only in “color”. It may have dramatic consequences for semiconductor γ-radiation detectors, if the symmetry between anti-correlated particles is broken by a biased collection. This leads to an important conclusion that the energy resolution of semiconductor detectors is very sen- sitive to the collection efficiency of competing secondary particles. We have also found a very strong correlation effects that suppress fluctuations of the larger particle number for all particle ratios. As a result, the Fano factor for the larger particles is as a rule considerably smaller than that for the smaller particles. The variance of the cover- age by the smaller particles strongly increases with the growth of the particle size ratio b/a. This effect is due to the fluctuations in the size of gaps between larger par- ticles that serve as receptacles for small-particle deposi- tion. For b/a ≥ 5 the small-particle variance exceeds that for the Poisson distribution in almost the entire range of particle fractions in the flux onto the adsorbing line. Acknowledgement. This work was supported by the New York State Office of Science, Technology and Aca- demic Research (NYSTAR) through the Center for Ad- vanced Sensor Technology (Sensor CAT) at Stony Brook. APPENDIX A: SMALL PARTICLE CONTRIBUTIONS TO COVERAGE AND COVERAGE VARIANCE To calculate the contribution of small particles to the total coverage at large x, we use Eq. (1) with the initial boundary conditions (5). With the substitution x → x+b and using Eq. (6), we rewrite Eq. (1) in the form (x + b− l)na(x + b) = q(b − a)na(b) + qx ∫ x+b−a na(y)dy + 2p na(y)dy (A1) Equation (A1) is valid for all x ≥ b. Taking Laplace transformation of na(x) cut at x < b by a step-function factor, we find that the transform, Ña(s) = e−sxna(x)dx, (A2) satisfies the following equation + b− l ebsÑa(s) = [1 + (b − a)na(b)s] e(b−a)s Ña(s) + J1(s) Ña(s) + J2(s) J1(s) = e−sxna(x)dx, J2(s) = e−sxna(x)dx Rearranging the terms, we rewrite it in form + l + qe−as + pe−bs Ña(s) = − e−bsRa(s) where Ra(s) = q[1+(b−a)na(b)s]+2s qe(b−a)sJ1(s) + pJ2(s) The form of Eq. (A5) is similar to Eq. (20) in which, however Ra should be calculated through J1(s) and J2(s), using Eqs. (5,6). For the case b < 2a we have na(b) = 1, and J1(s) = J2(s), while the explicit expres- sion for J1(s) is easily obtained by substituting na(x) = 1 in Eq. (A4). Solution of Eq. (A5) then enables one to retrieve the result of Ref [5]. To calculate J1(s) and J2(s) for b > 2a, it is necessary to use Eq. (6), which describes RSA of small particles onto a short line x < b. Its ana- lytical solution and therefore the explicit expressions for J1(s) and J2(s) can be obtained for the case b/a < 5 us- ing direct recursion to find na(x) (for one-particle RSA problem!). The result is rather cumbersome but suitable for numerical integration. For the case b/a > 5 one can exploit the exponentially rapid approach of the solution of Eq. (6) to its asymp- totic behavior in the limit x ≫ 1 (see e.g. [27] for the numerical data). This asymptotic solution, na(x) = (x+ 1)− 1, (A7) can be used to calculate J1(s) and then J2(s). To do this, we multiply Eq. (6) by exp(−sx) and integrate between 0 and b− 1. We obtain an equation for J2(s) of the form J ′2(s) + J2(s) = − e−asI(s) +s(b− a)e−bs[na(b)− 1] + 2se −asJ1(s) , (A8) where I(s) = ∫ (b−a)s dyye−y. (A9) Solution of Eq. (A8), satisfying the boundary conditions for na given by Eq. (5), is of the form J2(s) = β̃(s)s2 dtβ̃(t) na(b)(b − a)te −(b−a)t +2tJ1(t)− 1− e−(b−a)t (A10) β̃(t) = exp 1− ev . (A11) The contribution of small particles to the fill factor is then given by β(u)Ra(u)du. (A12) in which β(u) is given by the Eq. (16) and Ra(u) is defined by Eq. (A6). The obtained solution, though rather unwieldy, is suitable for numerical integration and for b/a > 5 it gives the results that agree with Monte Carlo simulations. In the limiting case b/a = b/a ≫ 1 it reduces to a more compact final expression for the contribution to the total coverage from the small particles θa = R due−quβp(u) q(u− 1)− 2pe−u (A13) with βp(u) defined by (33). For q = 1, Eq. (A13) prop- erly gives θa = R, while for q = 0 one has θa = R(1−R). The latter expression corresponds to the coverage by small particles of the gaps between the large particles left after their initial deposition. For arbitrary q, the coverage given by Eq. (A13) is depicted in Fig. 2 by the open squares. Similar approach can be used to calculate the small particle coverage variance. However, for b/a > 2 the equation for the Laplace transform of ua(x) given by Eq. (7), including all contributions to Na(s), becomes rather impractical. In the limiting case b/a ≫ 1, when fluctua- tions of the large particle gaps dominate the variance of small-particle coverage, one gets a more compact result shown in Fig. 5. [1] G. J. Rodgers and Z. Tavassoli, Phys. Lett. A 246, 252 (1998). [2] D. Boyer, J. Talbot, G. Tarjus, P. Van Tassel, and P. Viot, Phys. Rev. E 49, 5525 (1994). [3] A. V. Subashiev and S. Luryi, Phys. Rev. E 75, 011123 (2007). [4] M. C. Bartelt and V. Privman, Phys. Rev. A 44, R2227- R2230 (1991). [5] M. K. Hassan, J. Schmidt, B. Blasius, J. Kurths, Phys. Rev. E 65, 045103(R) (2002). [6] M. K. Hassan and J. Kurths, J. Phys. A 34, 7517 (2001). [7] N. A. M. Araujo, and A. Cadilhe, Phys. Rev. E 73, 051602 (2006). [8] D. J. Burridge and Y. Mao, Phys. Rev. E 69, 037102 (2004). [9] E. G. Coffman, Jr., L. Flatto, P. Jelenkovich, and B. Poonen, Algorithmica 22, 448 (1998). [10] M. Inoue, Phys. Rev. B 25, 3856 (1982). [11] P. Calka, A. Mezin, P. Vallois, Stochastic Processes and their Applications, 115, 983-1016 (2005). [12] J. Talbot, G. Tarjus, P.R Van Tassel, P. Viot, Colloids and surfaces A: Physicochemical and Engineering As- pects B 165, 278 (2000). [13] V. Privman, Colloids Surf A 165 231-240 (2000). [14] R. Devanathan, L. R. Corrales, F. Gao, W. J. Weber, Nuclear Instrum. Methods Phys. Res. A 565, 637-649 (2006). [15] H. Spieler, Semiconductor Detector Systems, Oxford Uni- versity Press, 2005. [16] C. Klein, J. Appl. Phys. 39, 2029 (1968). [17] W. van Roosbroeck, Phys. Rev. 139, A 1702 (1965). [18] This correlation originates from the basic fact that a simple random division of a segment in two parts pro- duces highly correlated pieces: if one is short the other is long and vice versa. Energy branching by impact ion- ization evidently has the similar property, as the sum of secondary-particle energies is fixed by energy conserva- tion. This type of correlations was first pointed out by Ugo Fano in 1947 [19] and bears his name. [19] U. Fano, Phys. Rev, 72, 26 (1947). [20] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regu- lar Variation, Cambridge University Press, Cambridge, 1987. [21] A. Rényi, Publ. Math. Inst. Hung. Acad. Sci. 3 109 (1958); Trans. Math. Stat. Prob. 4, 205 (1963). [22] J. K. Mackenzie, Journ. Chem. Phuys. 37, 723 (1962). [23] S. Luryi and A. V. Subashiev, unpublished. [24] G.D.Alkhazov, A.A. Vorob’ev, A.P Komar, Nucl. Instr. Meth. 48, 1-12 (1967). [25] P. E. Nay, Annals of Mathematical Statistics, 33, 702-718 (1962). [26] Sh. Kogan, Electronic Noise and Fluctuations in Solids, Cambridge University Press, Cambridge, 1996. [27] M. Lal and P. Gillard, Math. Computation 28, 562 (1974).

0704.1236

Sur les repr\'esentations du groupe fondamental d'une vari\'et\'e priv\'ee d'un diviseur \`a croisements normaux simples

Sur les représentations du groupe fondamental d'une variété privée d'un diviseur à roisements normaux simples Niels Borne 27 o tobre 2021 1 Introdu tion 1.1 Une des ription alternative En l'absen e de la ets, la re her he d'une des ription alternative du groupe fondamental étale (dé�ni dans [2℄ en termes de revêtements) est une question lassique, motivée essentiellement par la volonté de déterminer algébriquement des groupes fondamentaux qui ne sont onnus que par voie trans endante. L'étude systématique du lien entre revêtements de, disons, une variété algé- brique proje tive X , et ertain �brés sur X , ommen e ave Weil ([41℄). Celui- i montre qu'un revêtement galoisien non rami�é de surfa es de Riemann Y → X permet d'asso ier à toute représentation V omplexe du groupe de Galois G un �bré sur X : on des end le �bré trivial Y × V sur Y en E = Y × V/G. Weil remarque que ette opération est ompatible ave le produit tensoriel, e qui onfère des propriétés remarquables aux �brés asso iés : ils sont en parti ulier �nis, au sens qu'il existe deux polyn�mes distin ts f, g à oe� ients entiers po- sitifs tels que f(E) ≃ g(E). Il voit dans es �brés la généralisation des �brés en droite de torsion, et ommen e à les ara tériser. Ce travail trouve son aboutissement dans la formulation de Nori ([32℄) : la atégorie des �brés �nis sur X est tannakienne, et le groupe de Tannaka asso iée est le groupe fondamental (pro�ni) de X . Ce i a l'avantage d'être véri�é pour un s héma X propre, réduit, onnexe, sur un orps algébriquement los de ara téristique 0. En ara téristique p, le groupe de Tannaka de la atégorie des �brés essentiellement �nis (le s héma en groupe fondamental de Nori) se surje te dans le groupe fondamental de X . Toutefois, omme le souligne Nori, ette des ription algébrique (les �brés �nis ne dépendant, en fait, que de la topologie de Zariski de X) du groupe fondamental n'a que peu d'utilité, puisque les �brés �nis de rang plus grand que 1 semblent très di� iles à onstruire ex nihilo (i.e. sans utiliser de revêtement). Partant du problème de la détermination algébrique du groupe fondamen- tal, l'étude des ourbes ouvertes (par exemple la droite proje tive moins trois http://arxiv.org/abs/0704.1236v2 points) apparaît plus abordable que elle des ourbes omplètes. En e�et Nori montre dans [33℄ qu'il existe une équivalen e de atégories tannakiennes entre la atégorie des représentations du groupe fondamental de la ourbe ouverte et elles des �brés paraboliques (au sens de Seshadri, [36℄) �nis. Il semble ardu, mais peut-être pas impossible, de onstruire algébriquement de tels �brés. Cependant, Nori ne fait qu'esquisser une preuve de ette équivalen e. Cet arti le répond au sou i d'en donner une démonstration omplète et indépen- dante, su�samment générale pour être valable en toute dimension. Plus pré i- sément, on dé�nit, donné un s héma X propre, normal, onnexe sur un orps k, et D = (Di)i∈I une famille de diviseurs irrédu tibles à roisements normaux simples sur X , la atégorie FPar(X,D) (resp. EFPar(X,D)) des �brés para- boliques (modérés) �nis (resp. essentiellement �nis), et notre résultat prin ipal (théorème 7) s'énon e : Théorème 1. Soit D = ∪i∈IDi, et x ∈ X(k) un point rationnel, x /∈ D. (i) La paire (EFPar(X,D), x∗) est une atégorie tannakienne. (ii) Si k est algébriquement los de ara téristique 0, tout �bré parabolique essentiellement �ni est �ni, et le groupe de Tannaka de (FPar(X,D), x∗) est anoniquement isomorphe au groupe fondamental π1(X −D, x). Le premier point permet de proposer, en ara téristique positive, une dé�- nition du s héma en groupe fondamental modéré πD(X, x) omme groupe de Tannaka de la atégorie (EFPar(X,D), x∗). Ce s héma en groupe est un hybride du s héma en groupe fondamental de Nori ([32℄) et du groupe fondamental mo- déré de Grothendie k-Murre ([21℄). Un fait marquant est l'omniprésen e de ertains hamps de Deligne-Mumford, les hamps des ra ines, tout au long de et arti le. Ils sont onstruits à partir de la paire (X,D) en ajoutant une stru ture d'orbifold le long des diviseurs, et sont en e sens des �s hémas tordus�. Bien qu'il soient absents de l'énon é de notre résultat prin ipal, ils en sont absolument au oeur, leur présen e é lairant d'un jour nouveau d'an iens problèmes. Par exemple le urieux produit dans la atégorie galoisienne des revêtements modérés de [21℄ s'avère être un produit �bré usuel sur un tel hamp des ra ines Pour on lure, on propose une méthode de onstru tion des �brés (parabo- liques) �nis, inspirée de la méthode des petits groupes de Wigner et Ma key en théorie des représentations. 1.2 Organisation de l'arti le Dans la partie 2, on dé�nit les �brés paraboliques sur un s héma X le long d'une famille régulière de diviseurs D = (Di)i∈I , à poids rationnels à dénomi- nateurs dans une famille d'entiers �xée r = (ri)i∈I . Ce i se fait en deux temps : on dé�nit d'abord (partie 2.1.1) les fais eaux paraboliques, puis (partie 2.2) la notion de liberté lo ale pour un tel fais eau. On montre le ara tère ré ursif voir lemme 15 (ii) de ette ondition (proposition 1), qui se simpli�e onsidérablement lorsque la famille est à roisements normaux simples (proposition 2). On rappelle ensuite (partie 2.4.1) la notion de hamp des ra ines asso ié à la donnée de X , D, et r, et la partie se poursuit par le théorème 2 qui identi�e les �brés paraboliques aux �brés usuels sur le hamp des ra ines. Les deux parties suivantes sont de nature plus te hnique. Le résultat essentiel de la partie 3 est la proposition qui donne une inter- prétation du groupe fondamental modéré omme limite proje tive de groupes fondamentaux de hamps des ra ines. C'est une onséquen e à peu près immé- diate du lemme d'Abhyankar. La partie 4 étudie le lien entre groupe fondamental d'un hamp de Deligne- Mumford onvenable et �brés �nis. Ce lien s'exprime par une équivalen e de atégories tannakiennes entre systèmes lo aux k-ve toriels de rang �ni sur le hamp et �brés �nis donné par un fon teur �à la Riemann-Hilbert�, voir le orollaire 7. Le point ru ial (théorème 6) est le fait que les �brés essentiellement �nis sur les hamps des ra ines (et un peu plus généralement sur des s hémas tordus) propres et réduits sur un orps forment une atégorie tannakienne. La partie suivante (partie 5) est une partie de synthèse où l'on assemble les di�érents éléments pour aboutir au théorème 1. En�n la dernière partie (partie 6) est onsa rée à l'appli ation du théorème 1 au al ul expli ite de �brés paraboliques �nis de groupe d'holonomie résoluble. 1.3 Origines et liens ave des travaux existants La dé�nition des �brés paraboliques par rapport à une famille régulière de diviseurs D dans la partie 2 est inspirée de elle de Maruyama-Yokogawa [28℄. Une di�éren e importante ave es auteurs est l'emploi d'indi es multiples (mo- ralement, autant d'indi es que de omposantes irrédu tibles régulières de D) e qui mène naturellement à la dé�nition 1, essentiellement équivalente à elle em- ployée par Iyer et Simpson, voir [23℄. Ce i omplique singulièrement la ondition de liberté lo ale pour un fais eau parabolique : la dé�nition 4 en termes d'ho- mologie de omplexes asso iés à des fa ettes est entièrement originale et semble apporter un é lairage nouveau sur la notion de �bré parabolique �lo alement abélien� employée dans [23℄ (voir la remarque 3). Les hamps des ra ines ont été introduits par Vistoli ([5℄) et Cadman ([14℄). L'identi� ation des �brés paraboliques ave les �brés sur les hamps des ra ines a été initiée dans [10℄ dans la situation à indi e unique, sa généralisation à la situation présente ne pose pas de problème parti ulier. On peut trouver ertains pré urseurs de e résultat, en parti ulier dans le travail de Biswas ([9℄, [8℄), mais es auteurs n'employant que des �brés paraboliques à indi e unique, es résultats ne nous semblent orre ts que dans le as d'un diviseur régulier. La preuve donnée du théorème de Nori parabolique (théorème 7) est omplè- tement indépendante de la preuve de Nori en dimension 1 (Nori n'utilisant pas la dé�nition de Seshadri des �brés paraboliques), mais, grâ e à l'utilisation des hamps des ra ines, suit d'assez près la démonstration de la version lassique (non parabolique) du théorème : en parti ulier la démonstration que les �brés essentiellement �nis sur un �s héma tordu� forment une atégorie tannakienne (théorème 6) est une adaptation dire te de la preuve que Nori donne pour un s héma usuel. Toutefois, l'emploi d'un fon teur de type Riemann-Hilbert (voir �4.4) pour faire le lien entre système lo aux et �brés �nis, bien que naturelle, semble nouvelle dans e ontexte. Les travaux de Grothendie k-Murre ([21℄) sur le groupe fondamental modéré, ainsi que eux de Noohi ([31℄) et Zoonekynd ([43℄) sur le groupe fondamental des hamps de Deligne-Mumford, sont d'autres ingrédients importants de la preuve. Élémentaire, mais à priori un peu surprenante, l'idée de onstruire des �brés �nis par image dire te (voir la proposition 12 et le lemme 20) ne semble pas avoir été déjà employée. En�n, pour on lure sur les insu�san es de et arti le, il serait naturelle- ment souhaitable d'avoir une version du théorème de Nori parabolique pour un diviseur à roisements normaux généraux, ou sur un orps quel onque. 1.4 Remer iements Ce travail doit beau oup à Angelo Vistoli, ses ontours n'étant apparus net- tement qu'à la suite d'une visite à Bologne en janvier 2006. Je l'en remer ie haleureusement. Je tiens également à remer ier Alessandro Chiodo, Mi hel Emsalem, Boas Erez, Madhav Nori, Martin Olsson et Gabriele Vezzosi pour d'intéressantes dis ussions sur le sujet. 2 Fibrés paraboliques le long d'une famille régu- lière de diviseurs Dans ette partie, on notera X un hamp de Deligne-Mumford lo alement noethérien, I un ensemble �ni, D = (Di)i∈I une famille de diviseurs de Cartier e�e tifs sur X , r = (ri)i∈I une famille d'entiers ri ≥ 1. 2.1 Fais eaux paraboliques 2.1.1 Dé�nition Soient d'abord I, C deux atégories monoïdales, I étant supposée stri te. Un fon teur monoïdal F : I → C permet de voir C omme un I-module (relâ hé) sur le monoïde I via l'opération I × C // C (I, C) // F (I)⊗ C On onsidère à présent I = (ZI)op, C1 = (1rZ I)op, où par dé�nition 1 ZI =∏ Z, et C2 = MODX , la atégorie des fais eaux de OX -modules sur X . I et C1 sont vues omme atégories asso iées aux ensembles ordonnés orres- pondants, et munies du produit tensoriel induit par l'addition, quant à C2, elle est munie de sa stru ture monoïdale anonique. En�n, on dispose du fon teur d'in lusion F1 : I → C1 et du fon teur F2 : I // C2 l = (li)i∈I // OX(−lD) = OX(− i∈I liDi) qui permettent de voir C1 et C2 omme des I-monoïdes. Dé�nition 1. On dé�nit la atégorie des fais eaux paraboliques sur X le long de D à poids multiples de 1 omme la atégorie des morphismes de modules sur le monoïde (ZI)op : PAR 1 (X,D) = Hom(ZI )op(( I)op,MODX) Plus en détail, un objet de PAR 1 (X,D) est un ouple (E·, j), où E· : (1rZ I)op → MODX est un fon teur (non né essairement monoïdal !), et j est un isomor- phisme naturel (dit isomorphisme des pseudo-périodes) : (ZI)op × (1 ZI)op (ZI )op×E· ZI)op (ZI)op ×MODX OX(−·D)⊗· 2:llllllllllllllll llllllllllllllll On omettra souvent j pour alléger les notations. Un morphisme (E·, j) → (E ′· , j′) est une transformation naturelle : ZI)op ompatible ave j et j′, en un sens évident. 2.1.2 Opérations élémentaires sur les fais eaux paraboliques Pour l ∈ obj 1 ZI et E· ∈ obj PAR 1 (X,D) on dispose du dé alage dé�ni de la manière usuelle E·[l] : (1rZ +l // (1 ZI)op E· // MODX l'isomorphisme des pseudo-périodes étant induit par elui de E· de la manière évidente. Passons au produit tensoriel des fais eaux paraboliques : donnés E·, E ′· ∈ objPAR 1 (X,D), on dispose, pour tout l ∈ obj 1 ZI , de l'objet El ⊗ E ′· [−l] de PAR 1 (X,D) obtenu omme produit tensoriel externe de El ∈ objMODX par E ′· [−l] ∈ objPAR 1 (X,D). Cette quantité étant dinaturelle en l, on peut dé�nir (E· ⊗ E ′· )· omme la du fon teur de varian e mixte orrespondant dans PAR 1 (X,D) (qui est o omplète vu que MODX l'est) : pour des détails sur la notion de o�n ( oend), voir [27℄, ou bien [10℄, Appendi e B (E· ⊗ E ′· )· = El ⊗ E ′· [−l] Ce produit tensoriel est l'adjoint à gau he (enri hi) du fon teurHom interne naturel de PAR 1 (X,D) (dont nous n'aurons pas usage). Passons au stru tures spé iales : l'in lusion ZI → 1 ZI admet pour adjoint à gau he le fon teur ZI // ZI l // −[−l] où [l] = ([li])i∈I , [·] désignant la partie entière (on espère que ela n'entraî- nera pas de onfusion ave la notation du dé alage). On en déduit que le fon teur d'oubli (ou d'évaluation en zéro) Hom(ZI )op(( ZI)op,MODX)→ Hom(ZI)op((ZI)op,MODX) ≃ MODX admet un adjoint à gau he, qu'on notera E → E ·, dé�ni par E · = E ⊗ OX([−·]D) (1) On appellera E · le fais eau parabolique à stru ture spé iale induit par E . Lorsque D = rE, on dispose d'un fais eau parabolique parti ulier, dé�ni omme fon teur par l→ OX(−lrE) (2) l'isomorphisme des pseudo-périodes étant dé�ni de la manière évidente. On le notera simplement OX(− · rE). En�n il est lair que si f : X ′ → X est un morphisme plat, la paire de fon teurs adjoints (f∗, f∗) entre les atégories MODX et MODX induit une adjon tion similaire entre les atégories PAR 1 (X,D) et PAR 1 (X ′, f∗D). 2.2 Fibrés paraboliques 2.2.1 Fa ette On va dé�nir ertaines appli ations dont le but est Z, vu omme ensemble ordonné. Dé�nition 2. Soit J ⊂ I un sous-ensemble, et (ei)i∈I la base anonique de ZI . On appelle fa ette toute appli ation roissante F : {0 < 1}J → 1 qui est a�ne au sens suivant : il existe une famille d'entiers ǫ = (ǫi)i∈J véri�ant ∀i ∈ J 1 ≤ ǫi ≤ ri telle que : ∀µ = (µi)i∈J ∈ {0 < 1}J F (µ) = F (0) + Par la suite, on identi�era une fa ette entre deux ensembles ordonnés ave le fon teur entre les atégories orrespondantes. La donnée d'une fa ette équivaut bien entendu à elle de J , F (0), et de la famille ǫ. Dans le as parti ulier où F (0) = 0 et ǫ = r|J , on parlera de la fa ette spé iale asso iée à J , et on la notera FJ , il s'agit simplement de �l'in lusion� FJ : {0 < 1}J → 1rZ . Plus généralement, lorsque F (0) = 0, on notera F ǫ fa ette orrespondante. 2.2.2 Complexe asso ié à un fais eau parabolique et une fa ette On ommen er par pré iser les onventions utilisées on ernant les omplexes multiples (essentiellement empruntées à [40℄). Soit J un ensemble �ni. Un om- plexe (de haînes) multiple à valeurs dans une atégorie abélienne A est un fon teur C. = (NJ )op → A tel que si on note (ei)i∈J la base anonique de NJ , alors pour tout multi-indi e α = (αi)i∈J , et pour tout j ∈ J , les morphismes : Cα → Cα−ej véri�ent d A un tel omplexe, on asso ie de la manière son omplexe total Tot(C·)· de la manière usuelle : on ommen e par transformer C· en omplexe anti- ommutatif en �xant un ordre total arbitraire sur l'ensemble J et en modi�ant les di�érentielles de la manière suivante : on pose pour tout multi-indi e α = (αi)i∈J et pour tout j ∈ J : = (−1) L'appellation omplexe anti- ommutatif est justi�ée par le fait qu'alors δjδj −δj′δj , lorsque es expressions ont un sens, pour tout j, j′ dans J . Le omplexe total est alors dé�ni par, pour n ≥ 0 : Tot(C·)n = ⊕α,|α|=nCα où |α| = i∈J αi, et les di�érentielles étant dé�nies par δn = ⊕α,|α|=n i∈J δ Le fait qu'on parte d'un omplexe anti- ommutatif assure qu'on obtient bien ainsi un omplexe (simple). Dé�nition 3. Soient E· ∈ obj PAR 1 (X,D) un fais eau parabolique et F : {0 < 1}J → 1 ZI une fa ette. On appelle omplexe multiple asso ié le prolongement par zéro du fon teur omposé E· ◦ F op à (NJ )op, omme dans le diagramme i-dessous : (NJ )op // MODX ({0 < 1}J)op 66nnnnnnnnnn Par abus, on notera en ore e omplexe multiple E· ◦ F op. Le sens de es dé�nitions apparaît dans le as parti ulier où E· = OX ·, le fais eau stru turel muni de la stru ture parabolique spé iale, et F = FJ , la fa ette spé iale asso iée à J . Lemme 1. Pour i ∈ I on note (OX(−Di) → OX) le omplexe de haînes (simple) on entré en degrés 1 et 0. Pour tout J ⊂ I, il existe un isomorphisme naturel de omplexes multiples : OX · ◦ F J ≃ ⊗i∈J(OX(−Di)→ OX) Démonstration. Cela résulte simplement de l'expression de OX · donnée par (1),�2.1.2. Cet exemple est ru ial pour dé�nir les �brés paraboliques, puisque eux- i seront lo alement somme dire te �nie de fais eaux dé alés du fais eau stru turel, i.e. du type OX ·[l], pour l ∈ obj ZI (voir remarque 3), e qui impose des ontraintes fortes aux omplexes asso iés à haque fa ette. De manière similaire, on a : Lemme 2. Supposons D = rE. Pour tout J ⊂ I, et tout ǫ = (ǫj)j∈J omme dans �2.2.1, il existe un isomorphisme naturel de omplexes multiples : OX(− · rE) ◦ F opǫ J ≃ ⊗i∈J(OX(−ǫiEi)→ OX) Démonstration. Conséquen e dire te de la dé�nition 2. 2.2.3 Dé�nition des �brés paraboliques Lemme 3. Soit E· ∈ objPAR 1 (X,D) un fais eau parabolique, et F : {0 < 1}J → 1 ZI une fa ette. Il existe un morphisme naturel de omplexes multiples EF (0) ⊗ (OX · ◦ F J )→ E· ◦ F qui est un isomorphisme en (multi-)degré 0. Supposons de plus EF (0) lo ale- ment libre de rang �ni, alors si iJ : ∩i∈JDi → X désigne l'immersion fermée anonique, il existe une surje tion naturelle iJ∗i JEF (0) ։ H0(Tot(E· ◦ F op)). On repousse la preuve après la dé�nition suivante. La deuxième assertion montre que, sous les hypothèses du lemme 3, H0(Tot(E· ◦ F op)) est à support dans ∩i∈JDi, e qui donne un sens3 à la : Dé�nition 4. Soit E· ∈ objPAR 1 (X,D) un fais eau parabolique. On dit que 'est un fais eau parabolique lo alement libre ou en ore un �bré parabolique si pour toute fa ette F : {0 < 1}J → 1 ZI , l'homologie Hl(Tot(E· ◦ F op)) du omplexe multiple asso ié est nulle pour l > 0, et est un fais eau lo alement libre de rang �ni sur ∩i∈JDi pour l = 0. On notera Par 1 (X,D) la atégorie des fais eaux paraboliques lo alement libres sur X le long de D à poids multiples de ette dé�nition dépend a priori du hoix d'un ordre total sur I, mais la proposition 1 et la remarque 1 qui s'ensuit montrent qu'il n'en est rien, au moins pour une famille régulière de diviseurs (voir �2.3.1) qui est le seul as que nous onsidérerons Démonstration du lemme 3. La deuxième assertion est une onséquen e de la première, puisque elle- i entraîne l'existen e d'un morphisme de omplexes simples Tot(EF (0)⊗(OX ·◦F J ))→ Tot(E·◦F op) qui est un isomorphisme en de- gré 0, d'où un épimorphisme H0(Tot(EF (0)⊗(OX · ◦F J ))) ։ H0(Tot(E· ◦F op)). L'hypothèse sur EF (0) donne alors H0(Tot(EF (0) ⊗ (OX · ◦ F J ))) ≃ EF (0) ⊗ H0(Tot((OX · ◦F J ))). Le lemme 1 montre que Tot((OX · ◦F J )) est le omplexe de Koszul asso ié à la famille (Di)i∈J , e qui permet de on lure. Reste à montrer la première assertion. L'hypothèse que F est une fa ette montre qu'on a en parti ulier ∀µ = (µi)i∈J ∈ {0 < 1}J F (µ) ≤ F (0) + FJ(µ). Cette inégalité entre appli ations roissantes peut s'interpréter omme l'existen e d'une transformation naturelle ≤ entre les fon teurs orrespondants : {0 < 1}J F (0)+FJ En passant aux atégories opposées et en omposant ave E· ({0 < 1}J)op (F (0)+FJ ) ZI)op E· // MODX on obtient une transformation naturelle : E· ◦ ≤op : E· ◦ (F (0) + FJ )op → E· ◦ F op. Le diagramme suivant ({0 < 1}J)op op,F (0)) (F (0)+FJ ) op,EF (0)) (ZI)op × (1 ZI)op (ZI)op×E· ZI)op (ZI)op ×MODX OX(−·D)⊗· 2:llllllllllllllll llllllllllllllll montre que l'isomorphisme de pseudo-périodes j de E· permet d'identi�er E· ◦ (F (0) + FJ )op ave EF (0)⊗ (OX · ◦F J ), e qui on lut la démonstration du lemme. 2.3 Fibrés paraboliques et revêtements Pour montrer la pertinen e de la dé�nition 4, on doit montrer l'existen e de �brés paraboliques non triviaux, i.e. autre que les sommes dire tes de dé alés de �brés à stru ture spé iale (bien qu'ils soient tous lo alement de e type). La manière la plus dire te de produire de tels �brés paraboliques est d'utiliser des revêtements de Kummer rami�és le long d'une famille régulière de diviseurs. On ommen e par pré iser es notions. 2.3.1 Famille régulière de diviseurs On rappelle le lemme folklorique suivant, ainsi qu'une preuve, repoussée après la dé�nition 5, à laquelle il donne un sens. Lemme 4. Soit X un s héma lo alement noethérien, I un ensemble �ni, D = (Di)i∈I un ensemble de diviseurs de Cartier e�e tifs sur X, et pour tout i ∈ I, si : OX → OX(Di) la se tion anonique. Les onditions suivantes sont équiva- lentes : (i) En tout point x de X, soit une des se tions si est inversible, soit (si)i∈I est une suite régulière (au sens de Serre), (ii) la se tion (si)i∈I : OX → ⊕i∈IOX(Di) est régulière (i.e. le omplexe de Koszul asso ié au morphisme dual n'a pas d'homologie en degré supérieur ou égal à 1), (iii) ∩i∈IDi → X est une immersion fermée régulière, et si II est l'idéal en- gendré par les (si)i∈I , alors en tout point x de ∩i∈IDi, les (si)i∈I forment un système minimal (pour le ardinal) de générateurs de II,x. Dé�nition 5. On dira que D = (Di)i∈I est une famille régulière de diviseurs si pour tout sous-ensemble J ⊂ I, la sous-famille DJ = (Di)i∈J véri�e les onditions équivalentes du lemme 4. Démonstration du lemme 4. (i) =⇒ (ii) : e i résulte du fait que le omplexe de Koszul asso ié soit à un morphisme surje tif, soit à une suite régulière, sont sans homologie en degré plus grand que 1. (ii) =⇒ (iii) : le fait que II soit régulier résulte dire tement de la dé�nition ([3℄, exposé VII, Dé�nition 1.4). De plus en notant EI = ⊕i∈IOX(Di), et iI : ∩i∈IDi → X l'immersion fermée anonique, la régularité de sI = (si)i∈I montre que II/I2I ≃ i∗IE∨I , don en un point x de ∩i∈IDi, II,x/I2I,x est libre de rang #I, par onséquent II,x ne saurait être engendré par moins de #I éléments. (iii) =⇒ (i) ∩i∈IDi → X est une immersion fermée régulière don en parti- ulier quasi-régulière ([3℄, exposé VII, Proposition 1.3). Autrement dit II/I2I est lo alement libre de rang �ni sur ∩i∈IDi et l'homomorphisme surje tif anonique SymOX/II (II/I I ) ։ grII (OX) est un isomorphisme. En un point x de ∩i∈IDi, le lemme de Nakayama montre que le rang de II,x/I2I,x sur OX,x/II,x est le nombre minimal de généra- teurs de II,x sur OX,x, à savoir#I. Don (si)i∈I est une base de II,x/I2I,x et par onséquent les si dé�nissent un isomorphismeOX,x/II,x[(Si)i∈I ] ≃ SymOX,x/II,x(II,x/I I,x), où les Si sont des indéterminées. On en on lut que l'homomorphisme surje tif OX,x/II,x[(Si)i∈I ] ։ grII,x(OX,x) dé�ni par les (si)i∈I est un isomorphisme, autrement dit la famille (si)i∈I est quasi-régulière au sens de [19℄, 0 Dé�nition 15.1.7, don régulière ([19℄, 0 Corollaire 15.1.11). dans un anneau lo al noethérien, la régularité d'une suite ne dépend pas de l'ordre de ses éléments, e qui permet de parler de famille régulière Lemme 5. (i) Toute sous-famille d'une famille régulière l'est également. (ii) Si (li)i∈I est un ensemble d'entiers li ≥ 1, alors la famille (Di)i∈I est régulière si et seulement si la famille (liDi)i∈I l'est. Démonstration. (i) est immédiat. (ii) résulte, par ré urren e, de la ara térisation (i) du lemme 4, puisque qu'un élément est diviseur de zéro (resp. inversible) si et seulement une de ses puissan es l'est. La notion de famille régulière de diviseurs s'étend aux hamps de Deligne- Mumford lo alement noethériens, et le lemme 4 est en ore valide. On a de plus le résultat utile suivant. Proposition 1. Soit X un hamp de Deligne-Mumford lo alement noethé- rien, D une famille régulière de diviseurs. Alors le fais eau parabolique E· ∈ objPAR 1 (X,D) est lo alement libre si et seulement si : (i) ∀l ∈ 1 Z le fais eau El est lo alement libre sur X. (ii) ∀i ∈ I ∀li < l′i ≤ li + 1 ∈ 1riZ coker((El′i)· → (Eli)·) est lo alement libre vu omme objet de PAR( 1 )j 6=i (Di, (Dj ∩Di)j 6=i). Démonstration. Pour démontrer l'équivalen e, on peut supposer que (i) est vrai. Soit J 6= ∅, et i ∈ J . La donnée d'une fa ette F : {0 < 1}J → 1 ZI équivaut à elle d'un triplet (F̃ , li, l i), où F̃ : {0 < 1}J−i → j∈J−i Z est une fa ette, et li < l i ≤ li + 1 dans 1riZ, tels que le diagramme suivant (bi) ommute. {0 < 1}J F // 1 {0 < 1}J−i F̃ // j∈J−i En notant F i0 et F 1 les deux fon teurs {0 < 1}J−i → 1rZ orrespondants, on dispose d'un morphisme de omplexes multiples 1 > 0 : E·◦F i1 op → E·◦F i0 et le fait qu'on suppose (i) vrai montre que e morphisme est inje tif. Il en est don de même pour le morphisme induit Tot(1 > 0) : Tot(E· ◦F i1 )→ Tot(E· ◦F i0 ), et on véri�e que Tot(E· ◦ F op) s'identi�e anoniquement au �ne cone(Tot(1 > 0)) de elui- i. On dispose don d'un morphisme Tot(E· ◦ F op)→ cokerTot(1 > 0) qui est un quasi-isomorphisme ([40℄ 1.5.8), e qui permet de on lure. Remarque 1. En parti ulier, le fait, pour un fais eau parabolique donné, d'être lo alement libre, ne dépend pas de l'ordre hoisi sur l'ensemble d'indi es I. 2.3.2 Fibrés paraboliques relativement à une famille de diviseurs à roisements normaux simples Dé�nition 6. Une famille D = (Di)i∈I de diviseurs de Cartier e�e tifs sur un s héma lo alement noethérien X est dite à roisements normaux simples si pour tout point x de ∪i∈IDi on a : (i) l'anneau lo al OX,x est régulier, (ii) si Ix = {i ∈ I/x ∈ Di}, et si est une équation lo ale de Di en x, alors {si, i ∈ Ix} est une partie d'un système régulier de paramètres. Remarque 2. 1. C'est une légère adaptation de [21℄, De�nition 1.8.2. 2. Il revient au même ([21℄, Lemme 8.1.4) de dire que la famille est à roi- sements normaux et que ha un des Di est régulier. 3. Comme un système régulier de paramètres est ([19℄, Dé�nition 17.1.6) une famille régulière, une famille de diviseurs à roisements normaux simples est en parti ulier une famille régulière au sens de la dé�nition 5. Comme ette dé�nition est invariante par un hangement de base étale, elle a également un sens pour un hamp de Deligne-Mumford. On a de plus : Proposition 2. Soit X un hamp de Deligne-Mumford lo alement noethérien, D une famille de diviseurs à roisements normaux simples. Alors le fais eau parabolique E· ∈ objPAR 1 (X,D) est lo alement libre si et seulement si ∀l ∈ 1 le fais eau El est lo alement libre sur X. Démonstration. Comme, pour tout i ∈ I, la famille (Dj ∩ Di)j 6=i est à roise- ments normaux simples sur Di (voir [19℄, preuve de la Proposition 17.1.7), on peut raisonner par ré urren e sur #I. On on lut à l'aide de la proposition 1 et du lemme suivant (je remer ie Angelo Vistoli pour m'avoir fourni le prin ipe de la preuve) : Lemme 6. Soit R un anneau lo al noethérien régulier, d'idéal maximal m, t ∈ m, t /∈ m2. Soient de plus M et N deux modules libres de rang �ni tel que tM ⊂ N ⊂M . Alors M/N est libre omme R/t-module. Démonstration. De [19℄, Corollaire 17.1.8, on déduit que l'anneau lo al R/t est régulier, et [19℄, Proposition 16.3.7 montre qu'il est de dimension dimR − 1. La formule d'Auslander-Bu hsbaum ([19℄, Proposition 17.3.4) montre que le résultat à démontrer équivaut à profR/tM/N = dimR− 1. Or [19℄, proposition 16.4.8 montre que profR/tM/N = profRM/N . De plus, [19℄, Corollaire 16.4.4 donne pour tout R-module de type �ni P : si k = R/m est le orps résiduel, profR P = inf{m ≥ 0/ExtmR (k, P ) 6= 0}. La suite exa te longue de ohomologie asso iée au fon teur HomR(k, ·) et à la suite exa te ourte de R-modules 0 → N → M → M/N → 0, et une nouvelle appli ation de la formule d'Auslander- Bu hsbaum, permettent de on lure. 2.3.3 Revêtements de Kummer La dé�nition des revêtements de Kummer adoptée i i est elle de [21℄, �1 : donné un ensemble �ni I, r = (ri)i∈I un ensemble d'entiers ri ≥ 1, des s hémas lo alement noethériens X et Y , s = (si)i∈I un ensemble de se tions régulières de OX , un morphisme p : Y → X est dit revêtement de Kummer si Y est muni d'une a tion du s hémas en groupes µ i∈I µri tel qu'il existe un X-isomorphisme µ -équivariant de Y ave Spec(OX [(ti)i∈I ]/(trii − si)i∈I) Alors p : Y → X est l'appli ation naturelle vers le quotient s hématique. Pour i ∈ I, on notera Di (resp. Ei) le diviseur de Cartier asso ié à si (resp. ti), et D = (Di)i∈I (resp. E = (Ei)i∈I) la famille orrespondante. Lemme 7. On suppose que D = (Di)i∈I est une famille régulière de diviseurs sur X. Alors la famille E = (Ei)i∈I est une famille régulière de diviseurs sur Démonstration. Comme p est plat, la famille p∗D = (p∗Di)i∈I est régulière, par exemple d'après le lemme 4 (ii). Or (p∗Di)i∈I = (riDi)i∈I , ave ri ≥ 1, et on peut appliquer le lemme 5 (ii). 2.3.4 Fibrés paraboliques asso iés à un revêtement de Kummer Soit p : Y → X un revêtement de Kummer. On onserve les notations de la partie 2.3.3. On appellera µ -objet (ou objet µ -équivariant) d'un ertain type sur Y tout objet du même type sur le hamp quotient [Y |µ ]. On notera MODY la atégorie des fais eaux µ -équivariants sur Y , p ∗ : µr MODY → MODX l'image dire te le long de [Y |µ ]→ X , et µ PAR 1 (Y, p∗D) la atégorie des µ -fais eaux paraboliques sur Y le long de p∗D à poids multiples de 1 p∗D = (p∗Di)i∈I est onsidérée omme une famille de diviseurs de Cartier e�e tifs µ -équivariants sur Y de la manière anonique). Vu la platitude de [Y |µ ] → X et la fon torialité de PAR 1 (Y, p∗D) en X (�2.1.2), on dispose d'un fon teur anonique ∗ : µr PAR 1 (Y, p∗D)→ PAR 1 (X,D) qu'on peut détailler ainsi : donné un objet (F·, k) de µr PAR 1 (Y, p∗D) (k désignant l'isomorphisme des pseudo-périodes), on lui asso ie (E·, j), où E· = ∗ ◦ F· (en ore noté pµr∗ (F·)), et j est donné par la 2- omposition : (ZI)op × (1 ZI)op (ZI)op×F· ZI)op (ZI)op × µ OY (−·p (ZI )op×p 19kkkkkkkkkkkkkkkkkk kkkkkkkkkkkkkkkkkk (ZI)op ×MODX OX(−·D)⊗· 19kkkkkkkkkkkkkkkkkk kkkkkkkkkkkkkkkkkk la 2-�è he proj étant donnée par la formule de proje tion le long de [Y |µ En omposant e fon teur ave le fon teur µ MODY → µ PAR 1 (Y, p∗D) donné par F → F ⊗OY (− · rE) (voir �2.1.2) on obtient : Dé�nition 7. On notera ·̂ le fon teur µr MODY → PAR 1 (X,D) donné sur les objets par F̂· = pµr∗ (F ⊗OY (− · rE)). Proposition 3. On suppose que D = (Di)i∈I est une famille régulière de di- viseurs sur X, et que F est un µ -fais eau lo alement libre de rang �ni sur Y . Alors le fais eau parabolique sur X asso ié F̂· est un �bré parabolique sur X (au sens de la dé�nition 4). Démonstration. Soit F : {0 < 1}J → 1 ZI une fa ette orrespondant à la donnée de J , F (0), et de la famille ǫ (voir dé�nition 2). Il s'agit de al uler l'homologie du omplexe Tot(F̂·◦F op) ≃ pµr∗ (Tot(F⊗OX(−·rE)◦F op)) ≃ pµr∗ (F⊗Tot(OX(−·rE)◦F op)) vu que le fon teur Tot ommute à tout fon teur onservant les sommes di- re tes. Or il est lair d'après la dé�nition 2 que F = F (0) + F ǫ J et don : OX(− · rE) ◦ F op ≃ OY (−rF (0)E)⊗ (OX(− · rE) ◦ F opǫ Le lemme 2 montre qu'on doit al uler l'homologie du omplexe ∗ (F ⊗OY (−rF (0)E)⊗ Tot(⊗i∈J(OY (−ǫiEi)→ OY ))) Le fon teur p ∗ (F ⊗ OY (−rF (0)E)) ⊗ ·) étant exa t (en e�et p est a�ne don p∗ est exa t [[17℄ 1.3.2℄, et µr est diagonalisable), on trouve don ∗ (F ⊗OY (−rF (0)E)⊗Hl(Tot(⊗i∈J (OY (−ǫiEi)→ OY )))) Comme D = (Di)i∈I est une famille régulière de diviseurs sur X , il en est de même, d'après le lemme 7, pour E = (Ei)i∈I . La dé�nition 2 d'une fa ette impose que ∀i ∈ J , ǫi ≥ 1, et le lemme 5 montre que la famille ǫE|J = (ǫiEi)i∈J est également une famille régulière de diviseurs. On déduit du lemme 4 (ii) que Hl(Tot(⊗i∈J (OY (−ǫiEi) → OY ))) est nul pour l > 0, et est isomorphe à O∩i∈JǫiEi pour l = 0. On peut on lure à l'aide du lemme suivant : Lemme 8. Pour tout µ -fais eau F lo alement libre de rang �ni sur Y , le fais eau p ∗ (F ⊗ O∩i∈JǫiEi) est lo alement libre omme O∩i∈JDi-module. Démonstration. Vu la ommutativité du diagramme ∩i∈J ǫiEi jJ // Y ∩i∈JDi on a p ∗ (F ⊗ O∩i∈J ǫiEi) ≃ p ∗ jJ∗jJ ∗F ≃ iJ∗q ∗F , et il s'agit don de véri�er que q ∗F est lo alement libre. Comme µ est diagonalisable, 'est un fa teur dire t de q∗jJ ∗F , et vu que F lo alement libre de rang �ni sur Y , il su�t de prouver que q est �ni et plat. Il est lairement �ni omme omposé de l'immersion fermée ∩i∈J ǫiEi → p∗(∩i∈JDi) (on rappelle que pour tout i on a ǫi ≤ ri) et de p∗(∩i∈JDi) → ∩i∈JDi (�ni ar p l'est). Pour la platitude on peut lairement supposer X a�ne, soit X = specR. Mais alors ∩i∈J ǫiEi = spec(⊗i∈J Rsi ⊗R ⊗i/∈J R[ti]tri ). Or pour i ∈ J (resp. i /∈ J) tǫii (resp. trii − si) est unitaire, don Rsi (resp. R[ti] ) est plat sur (resp. sur R), don ⊗i∈J Rsi ⊗R⊗i/∈J R[ti]tri est plat sur ⊗i∈J Rsi , qui est l'anneau dé�nis- sant ∩i∈JDi. 2.4 Fibrés paraboliques et hamp des ra ines Soit S un s héma, etX → S un hamp de Deligne-Mumford, qu'on supposera toujours lo alement noethérien. 2.4.1 Champ des ra ines Soit r un entier supérieur ou égal à 1, inversible dans S. Dé�nition 8 ([5℄,[14℄). (i) Soit un ouple (L, s) onstitué d'un fais eau in- versible sur X et d'une se tion de e fais eau. Soit U = [A1|Gm] le hamp lassi�ant les fais eaux inversibles muni d'une se tion. On appelle hamp des ra ines r-ièmes de (L, s) le hamp (L, s)/X = X ×U U où le produit �bré est pris par rapport aux morphismes (L, s) : X → U , et l'élévation à la puissan e r : ·⊗r : U → U . (ii) Soit D un diviseur de Cartier e�e tif sur X, sD la se tion anonique de OX(D). On note r D/X le hamp r (OX(D), sD)/X. Soit I un ensemble �ni, r = (ri)i∈I un ensemble d'entiers ri ≥ 1, inversibles dans S. Dé�nition 9. 1. Soit (L, s) = (Li, si)i∈I un ensemble de fais eaux inver- sibles sur X muni ha un d'une se tion. On note r (L, s)/X le hamp ×i∈I ri (Li, si)/X. 2. Soit D = (Di)i∈I un ensemble de diviseurs de Cartier e�e tifs sur X. On D/X le hamp ×i∈I ri Di/X. Proposition 4. Soit (L, s) = (Li, si)i∈I un ensemble de fais eaux inversibles sur X munis de se tions. Supposons qu'on dispose sur X de fais eaux inversibles Ni et d'isomorphismes ψi : N⊗rii ≃ Li. Il existe alors un isomorphisme naturel de hamps sur X : (L, s)/X ≃ [Spec( Sym(⊕i∈IN (N⊗rii ≃ Li)i∈I Démonstration. Le as où #I = 1 est traité dans [14℄, version 1, Proposition 3.2, (voir aussi [10℄, théorème 4), et le as général en résulte immédiatement. Corollaire 1. Soit s = (si)i∈I un ensemble de se tions régulières de OX , Di = (si) les diviseurs de Cartier orrespondants. Alors il existe un isomorphisme naturel de hamps sur X : D/X ≃ [Spec(OX [(ti)i∈I ]/(trii − si)i∈I)|µr] Démonstration. Dé oule dire tement de la proposition 4. 2.4.2 La orrespondan e : énon é Soit (L, s) = (Li, si)i∈I un ensemble de fais eaux inversibles sur X munis de se tions. Sur (L, s)/X, on dispose d'une ra ine r-ième anonique (N , t) = (Ni, ti)i∈I de (L, s). On note π : r (L, s)/X → X le morphisme anonique. Dans le as parti ulier où les ouples (Li, si) sont asso iés à des diviseurs e�e tifs Di sur X (i.e. (Li, si) = (OX(Di), sDi) pour tout i ∈ I), les relations π∗si = t i , et le fait que π est plat, montrent que les ti : O r√D/X → Ni sont des monomorphismes (i.e. les ti sont des se tions régulières), si bien que les ouples (Ni, ti) sont asso iés à des diviseurs de Cartier e�e tifs Ei sur r En imitant la onstru tion du �2.3.4 on obtient un fon teur : ·̂ : MOD( D/X) // PAR 1 (X,D) F // F̂· = π∗(F ⊗O r√D/X(− · rE)) Théorème 2. On suppose que D est une famille régulière de diviseurs sur le hamp de Deligne-Mumford X. Alors le fon teur ·̂ induit une équivalen e de atégories tensorielles entre Vect( r D/X) et Par 1 (X,D). 2.4.3 Bonne dé�nition Il s'agit de voir que si F ∈ objVect( r D/X), alors le fais eau parabolique F̂· est lo alement libre. Comme il s'agit d'une question lo ale pour la topologie étale sur X , on peut, quitte à prendre un atlas étale, supposer que X est un s héma. Quitte à lo aliser en ore de façon à trivialiser ha un des diviseurs de la famille D, le orollaire 1 montre qu'on peut supposer que r D/X → X est du type [Y |µ ]→ X , où Y → X est un revêtement de Kummer rami�é le long de D. Mais alors la proposition 3 permet de on lure. 2.4.4 Équivalen e ré iproque Soit E· ∈ objPar 1 (X,D). On pose Ê· = π∗E· ⊗N⊗r· désigne la o�n ( oend), voir [27℄. C'est à priori un élément de MOD( r D/X), mais on va voir que 'est en fait un fais eau lo alement libre. Lemme 9. On �xe i ∈ I. Soient les fon teurs PAR 1 (X,D) PAR( 1 )j 6=i Di/X, (π iDj)j 6=i) Li : E· // LiE· = ∫ li∈ 1ri Z π∗i (Eli)· ⊗N ⊗rili RiF· : (li → πi∗(F· ⊗N⊗−lirii )) F· : Rioo où πi : Di/X → X désigne la proje tion anonique, Ni la ra ine ri-ème anonique de OX(Di) sur ri Di/X, et (Eli)· la restri tion de E· via le fon teur∏ j 6=i Z induit par li. Ces fon teurs sont adjoints, Li étant adjoint à gau he et Ri adjoint à droite. Démonstration. Cela résulte de la dé�nition des o�ns. Lemme 10. Le fon teur Li envoie Par 1 (X,D) sur Par( 1 )j 6=i Di/X, (π iDj)j 6=i). Démonstration. On pro ède par ré urren e sur #I. On ommen e par le as où #I = 1. Le as où X est un s héma est traité dans [10℄. Le as général s'y ramène grâ e au lemme suivant : Lemme 11. Soit X un hamp de Deligne-Mumford, E : K → VectX un dia- gramme, p : X0 → X un atlas étale. Si lim → p∗Ek existe dans VectX0, alors lim → Ek existe dans VectX. Démonstration. C'est à peu près dire t à partir de la des ription suivante des objets de VectX : soit X1 = X0 ×X X0, et s, b : X1 ⇒ X0 le groupoïde orrespondant. La atégorie VectX est équivalente à la atégorie des ouples (E0, α), où E0 ∈ objVectX0, et α : s∗E0 ≃ b∗E0 est une donnée de des ente, i.e. véri�e la ondition de des ente : m∗α = pr∗1α ◦ pr∗2α, où pr1, pr2,m : X1 ×X0 X1 → X1 désignent respe tivement, les proje tions et la multipli ation dans le groupoïde. On revient au as général (#I quel onque) : pour montrer que ∫ li∈ 1riZ π∗i (Eli)·⊗ N⊗rilii est un fais eau parabolique lo alement libre, le plus simple est d'appli- quer la proposition 1. La partie (i) du ritère résulte du as#I = 1. Pour véri�er la partie (ii) de e ritère, on �xe j 6= i, et l′j < lj ≤ lj + 1 ∈ 1rj Z. Il s'agit de voir que coker( ∫ li∈ 1riZ π∗i (Elil′j )· ⊗N ⊗rili ∫ li∈ 1riZ π∗i (Elilj )· ⊗N ⊗rili est un objet de Par( 1 )k 6=i,j (π∗iDj, (π i (Dk ∩Dj))k 6=i,j). Mais omme le fon - teur Li est adjoint à gau he (lemme 9), don exa t à droite, et π iDj = Dj ∩Di/Dj, ela résulte de l'hypothèse de ré urren e. 2.4.5 Preuve de l'équivalen e Lemme 12. Le fon teur Ri envoie Par( 1 )j 6=i Di/X, (π iDj)j 6=i) sur Par 1 (X,D), et est une équivalen e ré iproque de la restri tion de Li à Par 1 (X,D). Démonstration. On ommen e par remarquer que si la première assertion est véri�ée, la se onde à un sens, et est vraie : en e�et on peut se ontenter de véri�er que Li et Ri sont des isomorphismes après évaluation des variables, et on se ramène don au as où #I = 1. C'est de nouveau un problème lo al pour la topologie étale, et on peut don se ramener au as où X est un s héma. Pour e dernier as, on renvoie à [10℄. On montre l'ensemble des deux assertions par ré urren e sur #I. Pour le as où #I = 1 : la première assertion résulte de la partie 2.4.3, la se onde s'ensuit. Pour I quel onque on applique l'hypothèse de ré urren e qui permet de dire que Vect r D/X ≃ Par( 1 )j 6=i Di/X, (π iDj)j 6=i), et à nouveau la partie 2.4.3 permet de on lure à la validité de la première assertion, et don de la se onde. 2.4.6 Preuve du ara tère tensoriel Vu l'expression du produit tensoriel donnée �2.1.2, le fait que l'équivalen e soit ompatible au produit tensoriel résulte de la formule de Fubini pour les o�ns (voir [27℄, et [10℄ pour le détail dans le as où #I = 1). 2.4.7 Stru ture lo ale des �brés paraboliques Corollaire 2. Ave les notations du théorème 2, pour tout �bré parabolique E· ∈ obj(Par 1 (X,D)), et tout point x ∈ X, il existe un voisinage étale U → X de x tel que E·|U soit une somme dire te �nie de �brés paraboliques inversibles. Démonstration. Vu le théorème 2, il su�t de montrer la propriété orrespon- dante pour les �brés hampêtres, mais alors la preuve de [10℄, Proposition 3.2, s'adapte immédiatement. Remarque 3. 1. Si X est de plus un s héma, on peut imposer à U → X d'être un ouvert pour la topologie de Zariski. 2. La preuve montre qu'on peut hoisir les �brés paraboliques de la forme OX ·[l], pour l ∈ obj ZI . En termes hampêtres, si X = specR, où R est un anneau lo al, alors Pic( r D|X) ≃ , voir �2.4.8. 3. On suppose de plus que D une famille de diviseurs à roisements nor- maux simples. Alors la proposition 2 et le orollaire 2 montrent que si les omposantes du fais eau parabolique E· ∈ obj PAR 1 (X,D) sont lo alement libres, il est lo alement abélien au sens de [23℄, De�nition 2.2. 2.4.8 Groupe de Pi ard des hamps des ra ines Le orollaire suivant est énon é, dans le as parti ulier des ourbes tordues, dans [13℄. Le théorème 2 n'est pas indispensable pour le démontrer (on peut aussi utiliser [14℄, Corollary 3.1.2), mais en fournit une preuve ommode. Corollaire 3. Ave les notations du théorème 2, on a une suite exa te naturelle 0→ PicX → Pic( r D|X)→ H0(Di, Démonstration. On peut supposer les Di onnexes (en e�et si D et D sont deux diviseurs de Cartier e�e tifs à supports disjoints, et r ≥ 1 est un entier, alors D +D′|X ≃ r D|X ×X r D′|X). On note Ni la ra ine ri-ème anonique de OX(Di) sur r D|X , et π : D|X → X le morphisme naturel. On va montrer plus pré isément : il existe un unique morphisme surje tif Pic( r D|X)→ envoyant [Ni] sur le gé- nérateur anonique de la i-ème omposante, et dont le noyau est π∗ : PicX → Pic( r D|X). Il su�t de le montrer pour #I = 1. En e�et en supposant ette propriété véri�ée dans e as, le morphisme naturel Pic( r (rj)j 6=i (Dj)j 6=i|X envoie [Ni] sur le générateur anonique de la i-ème omposante, et est don surje tif. Comme une ré urren e immédiate donne l'égalité des ardinaux, 'est un isomorphisme. Reste à voir le as où #I = 1. Cela résulte immédiatement de l'assertion : pour tout fais eau inversible K sur r D|X, il existe un unique entier l dans {0, · · · , r − 1} tel qu'il existe M fais eau inversible sur X tel que K ⊗ N⊗l ≃ π∗M. En traduisant en termes de �brés paraboliques grâ e au théorème 25 on voit qu'il faut montrer : pour tout fais eau parabolique inversibleK· à poids dans Z, il existe un unique entier l dans {0, · · · , r − 1} tel qu'il existe M fais eau inversible sur X tel que K·[ lr ] ≃M· (le fais eau parabolique à stru ture spé iale asso ié àM, voir �2.1.2). La donnée de K· équivaut à elle d'une �ltration K0 ⊃ K 1 ⊃ · · · ⊃ K1− 1 ⊃ K1 ≃ K0 ⊗OX OX(−D) telle que pour l ≤ l′, le fais eau K l /K l′ est lo alement libre sur OD. Ce i implique l'égalité des rangs : rg(K0|D) = K l+1 mais omme K0 est inversible et D est onnexe, il existe un unique entier l dans {0, · · · , r − 1} tel que K l 6= K l+1 , et pour et entier K·[ lr ] ≃ K0·. 2.4.9 Image dire te de �brés paraboliques On se donne S un s héma de base, X → S (respe tivement Y → S) un S- hamp de Deligne-Mumford lo alement noethérien, D = (Di)i∈I (respe ti- vement E = (Ej)j∈J ) une famille régulière de diviseurs sur X (respe tivement sur Y ), et r = (ri)i∈I (respe tivement s = (sj)j∈J ) une famille d'entiers (su- périeurs à 1, inversibles dans S). On �xe de plus p : Y → X un S-morphisme représentable �ni et plat, α : J → I une appli ation, véri�ant 1. ∀j ∈ J sj |rα(j) 2. ∀i ∈ I p∗Di = j∈α−1(i) En�n, on note q : s E/Y → r D/X le morphisme naturel. Dé�nition 10. Sous les onditions i-dessus, on dé�nit l'image dire te d'un �bré parabolique par la formule : p∗ : Par 1 (Y,E) // Par 1 (X,D) E· // (p∗E·)· = ( lr → p∗(E l◦α en fait la version à indi e unique montrée dans [10℄. Proposition 5. Les fon teurs d'images dire tes hampêtre et parabolique sont ompatibles : on a un isomorphisme fon toriel en E· ∈ objPar 1 (Y,E) : (̂p∗E·)· ≃ q∗(Ê·). Démonstration. On note π : r D/X → X et ̟ : s E/Y → Y les morphismes anoniques. On a un 2-isomorphisme naturel π ◦ q ≃ p ◦̟. Pour tout i dans I (respe tivement j dans J), soitMi (respe tivementNj) la ra ine ri-ème (respe tivement sj-ème) anonique de OX(Di) (respe tivement de OY (Ej)) sur r D/X (respe tivement sur s E/Y ), elle est muni de sa se tion anonique. Le morphisme q est dé�ni par la ondition : pour tout i dans I, q∗(Mi) = ⊗j∈α−1(i)Nj , et la ondition orrespondante évidente sur les se tions. On �xe E· ∈ obj Par 1 (Y,E) et ( l ) = ( li )i∈I ∈ obj(1rZ), et on al ule : π∗(q∗(Ê·)⊗i∈IM⊗−lii ) ≃ π∗ ((∫ 1 ̟∗E· ⊗N⊗s· ⊗i∈I ⊗j∈α−1(i)N⊗−lij ̟∗E· ⊗N⊗s·−l◦α ̟∗E·[ l ◦ α] ]⊗N⊗s· ≃ p∗(E l◦α d'où la on lusion. 3 Groupe fondamental modéré omme groupe fon- damental hampêtre 3.1 Groupe fondamental hampêtre Noohi ([31℄) et Zoonekynd ([43℄) ont étendu la théorie lassique du groupe fondamental pro�ni de [2℄ du as d'un s héma à elui d'un hamp de Deligne- Mumford. On rappelle brièvement leur dé�nition. Dé�nition 11 ([2℄,[31℄,[43℄). Soit X un hamp de Deligne-Mumford. On note RevX la 2-sous- atégorie pleine de la 2- atégorie ChampsX des hamps sur X dont les objets sont les morphismes Y → X représentables étales �nis. On note CatRevX la atégorie asso iée (i.e. la atégorie dont les morphismes sont les lasses de 2-isomorphisme de 1-morphismes dans RevX). Théorème 3 ([31℄ Theorem 4.2, [43℄ �3). Si X est un hamp de Deligne- Mumford onnexe, et x : specΩ→ X un point géométrique, la paire (CatRevX, x∗) est une atégorie galoisienne au sens de [2℄. Dé�nition 12. Ave les notations du théorème, on notera π1(X, x) le groupe fondamental de la atégorie galoisienne (CatRevX, x∗). 3.2 Groupe fondamental modéré On rappelle le résultat suivant : Théorème 4 ([21℄ Theorem 2.4.2). Soit X un s héma lo alement noethérien, normal, onnexe, D un diviseur à roisements normaux, et x : specΩ → X un point géométrique, x /∈ D. La atégorie RevD(X) des revêtements de X modérément rami�és le long de X est une atégorie galoisienne, dont on note πD1 (X, x) le groupe fondamental. On va se restreindre à étudier le as d'une famille de diviseurs à roisements normaux simples (dé�nition 6). Le but de e paragraphe est de démontrer : Proposition 6. Ave les hypothèses du théorème 4, si on suppose de plus X dé�ni sur un orps k, et D est la réunion d'une famille D = (Di)i∈I de diviseurs irrédu tibles à roisements normaux simples, alors il existe un isomorphisme naturel : πD1 (X, x) ≃ lim←− D/X, x) où la limite est prise sur les multi-indi es r = (ri)i∈I d'entiers non divisibles par la ara téristique p de k. Dans le reste de e paragraphe 3, on onservera les hypothèses de la pro- position 6. On va montrer que l'on a une équivalen e naturelle de atégories galoisiennes RevD(X) ≃ lim−→ CatRev( r La ompatibilité de et isomorphisme aux fon teurs �bres induits par x impliquera bien la proposition 6 3.3 Le fon teur C Lemme 13. Soit π : Y → X dans objRevD(X) galoisien de groupe G (au sens de [21℄ 2.4.5), de multi-indi e de rami� ation r. Alors le morphisme naturel de hamps [Y |G]→ r D/X est un isomorphisme. Démonstration. Cela résulte, essentiellement, de l'hypothèse que D est à roi- sements normaux simples, et du lemme d'Abhyankar. En détail : on pré ise d'abord la dé�nition du morphisme naturel [Y |G] → D/X. Soit E = (Ei)i∈I la famille de diviseurs sur Y dé�ni par ∀i ∈ I Ei = (π∗Di)red, si bien que ∀i ∈ I π∗Di = riEi. la notion de 2-limite �ltrée de atégories utilisée i i et par la suite est pré isée dans l'appendi e A Comme D/X admet X pour espa e des modules, x dé�nit aussi un point géométrique de e hamp Soit S un s héma, et (f, p) un objet de [Y |G](S), i.e. une paire onstituée d'un G-torseur p : T → S et d'un morphisme G-équivariant f : T → Y . On a les quotients s hématiques S = T/G et X = Y/G, si bien qu'il existe un unique morphisme g : S → X tel que g◦p = π◦f . Pour tout i ∈ I, le fais eau OT (f∗Ei) dé�nit une ra ine ri-ème de Di sur T , omme e fais eau est G-équivariant, on peut le des endre anoniquement le long du G-torseur p : T → S, et les ra ines ri-ièmes p ∗ (OT (f∗Ei)) de Di sur S dé�nissent un objet de r D/X(S). Pour montrer que e morphisme [Y |G] → r D/X est un isomorphisme, il su�t de le véri�er sur les �bres géométriques, et on peut don supposer que X = specR, où R est un anneau lo al noethérien stri tement hensélien. Comme 'est évident en dehors du support de D, on peut supposer de plus R régulier. On hoisit pour tout i ∈ I une équation lo ale si de Di. On pose R′ = R[(Ti)i∈I ] −si)i∈I , où les (Ti)i∈I sont des indéterminées, etX ′ = specR′. Le morphisme X ′ → X est modérément rami�é ([21℄ Example 2.2.4). Comme les si sont tous dans l'idéal maximalm de R, R′ est un anneau lo al. Vu que la famille (Di)i∈I est à roisements normaux simples, R′ est même régulier ([21℄, Proposition 1.8.5). On a en ore Y = specS, où S est une R-algèbre �nie, don un produit �ni d'anneaux lo aux stri tement henséliens (en e�et l'hypothèse de modération impose que les extensions résiduelles sont séparables [[21℄ De�nition 2.1.2℄, don i i triviales). On réduit fa ilement le problème au as où Y est irrédu tible. Vu que les ri sont non nuls résiduellement, on peut hoisir des équations ti des Ei telles que trii = si dans S. Celles- i dé�nissent un X-morphisme Y → X ′. Soit Y ′ le produit �bré de Y et X sur X ′ dans RevD(X), 'est à dire la nor- malisation du produit �bré s hématique Y ×XX ′. On a don Y = specS′, où S′ est la l�ture intégrale de R′ dans l'extension des anneaux de fra tions totaux R(Y )/R(X). Comme elle- i est �nie étale d'après l'hypothèse de modération, le morphisme Y ′ → X ′ est �ni. Comme Y ′ → X est modéré, toute ompo- sante irrédu tible de Y ′ domine X , et don toute omposante irrédu tible de Y ′ domine X ′. On peut don appliquer le théorème de pureté de Zariski-Nagata ([2℄, X Théorème 3.1) pour voir que le lieu où Y ′ → X ′ est rami�é est de pure odimension 1. Mais le lemme d'Abhyankar ([2℄, X Lemme 3.6) montre que e morphisme n'est pas non plus rami�é en odimension 1. Il est don étale, et omme X ′ est stri tement lo al, 'est un revêtement trivial de X ′. Comme Y ′ → Y est séparé ( ar a�ne), la se tion dé�nie par le morphisme Y → X ′ i-dessus est une immersion fermée. L'égalité des dimensions, et le fait que Y et Y ′ sont réduits ( ar normaux), impliquent que ette se tion identi�e Y à une omposante irrédu tible de Y ′. Don le morphisme Y → X ′ onstruit au départ est en fait un isomorphisme au dessus de X . Le groupe G = AutX Y s'identi�e anoniquement à AutX X ′ = µ , et on on lut grâ e au orollaire 1 (ou plut�t, grâ e à sa preuve, qui donne une version expli ite de l'isomorphisme). Lemme 14. Soit Y → X dans un objet de RevD(X), Z → X un objet galoisien de RevD(X), de groupe G, dominant Y → X, r les indi es de rami� ation de Z → X, H le groupe de Galois de Z → Y . Le morphisme de hamps [Z|H ] → [Z|G] est étale, et omposé ave l'isomorphisme anonique [Z|G] ≃ r dé�ni dans la proposition 13 il dé�nit un objet de lim−→r CatRev( D/X), qui est, à isomorphisme près, indépendant du hoix de Z. Dé�nition 13. On notera C(Y → X) l'objet de lim−→r CatRev( D/X) dé�ni dans le lemme 14. Il est lair qu'on a en fait dé�ni un fon teur C : RevD(X)→ lim−→ CatRev( r 3.4 Le fon teur M Lemme 15. Soit r = (ri)i∈I une famille d'entiers non divisibles par la ara - téristique p de k et T → r D/X un revêtement étale. Soit N(T ) la fermeture intégrale de X dans R(T )/R( r D/X) = R(X). (i) Il existe un unique morphisme de hamps T → N(T ) faisant ommuter le diagramme N(T ) D/X // X et e morphisme est surje tif. (ii) Le morphisme anonique N(T ) → X est un revêtement modérément ra- mi�é de X le long de D, et le fon teur obtenu Rev( r D/X) // RevD(X) ommute au produit �bré. (iii) Si de plus T → r D/X est galoisien de groupe G, N(T ) → X l'est également, de multi-indi e de rami� ation r′ divisant r, et le morphisme N(T ) → [N(T )|G] ≃ r′ D/X dé�ni dans le lemme 13 s'ins rit dans un diagramme artésien : N(T ) D/X // r Démonstration. (i) Pour l'existen e et l'uni ité du morphisme : soit T0 → T un atlas étale, et s, b : T1 ⇒ T0 le groupoïde orrespondant. Il résulte de [21℄ Proposition 1.8.5 et [2℄ Exposé I, Corollaire 9.10, que T1 et T0 sont normaux. L'a�rmation résulte alors de la propriété universelle de la fermeture intégrale ([16℄ 6.3.9). Pour montrer la surje tivité, on peut supposer X = specR, où R est un anneau. On pose R′ = R[(Ti)i∈I ] −si)i∈I , où les (Ti)i∈I sont des indéterminées, et X ′ = specR′ (quitte à renommer I, on peut supposer qu'au un des si n'est inversible). D'après le orollaire 1 il existe un isomorphisme naturel de hamps sur X : r D/X ≃ [X ′|µ ]. Posons T ′ = X ′ ×[X′|µ ] T , 'est un atlas étale de T . De plus T ′ → X est �ni ( ar T ′ → T et T → X le sont) et N(T ) → X est séparé ( ar a�ne), et don T ′ → N(T ) est �ni, et en parti ulier entier. Comme e morphisme est de plus dominant ( ar T ′ → T est surje tif, et T → N(T ) est birationnel), il est surje tif, d'après le théorème de Cohen-Seidenberg. (ii) N(T )→ X est modérément rami�é le long de D : il faut véri�er les inq onditions de la Dé�nition 2.2.2 de [21℄. 1) N(T )→ X est �ni : ça résulte de la �nitude de la fermeture intégrale d'un anneau noethérien normal dans une extension séparable �nie ([12℄, V, Proposition 18, Corollaire 1). 2) N(T ) → X est étale au dessus de U = X − D : posons V = T × r√ U , 'est un revêtement étale de U , et omme U est normal, V s'identi�e d'après [20℄, Corollaire 18.10.12.à la fermeture intégrale de U dans R(V ) = R(T ), qui n'est autre que N(T )×X U . 3) Toute omposante irrédu tible de N(T ) domine X : les omposantes irrédu tibles de N(T ) sont les mêmes que elle de son ouvert dense V , or V → U étant étale, toute omposante irrédu tible de V domine 4) N(T ) est normal : lair d'après [16℄ Proposition 6.3.7, ou simplement d'après la propriété universelle dé�nissant N(T ). 5) Pour tout point générique x de D, N(T ) est modérément rami�é au dessus de OX,x : 'est également lair d'après (i), qui permet d'appliquer [21℄ Lemma 2.2.5, vu qu'un revêtement de Kummer est modérément rami�é ([21℄ Example 2.2.4). On en tire également la propriété sur les indi es de rami� ation du (iii). Le fon teur obtenu ommute au produit �bré : donnés T → r T ′ → r D/X deux revêtements étales, on a un morphisme anonique N(T × r√ T ′)→ N(T )×X N(T ′) (produit �bré s hématique), omme e morphisme est entier et birationnel, et N(T × r√ T ′) est normal, il identi�e N(T × r√ T ′) à la normalisation de N(T )×X N(T ′), qui par dé�nition est le produit �bré de N(T ) et N(T ′) sur X dans RevD(X). (iii) La première assertion résulte de la se onde a�rmation de (ii). Pour la se onde assertion, il su�t de remarquer que le morphisme naturel T → D/X× r′√ N(T ) est un morphisme birationnel de revêtements étales D/X, 'est don un isomorphisme d'après [20℄, Corollaire 18.10.12. Le lemme 15 permet de poser : Dé�nition 14. On dé�nit un fon teur M : lim−→r CatRev( D/X)→ RevD(X) sur les objets en posant, pour une famille r = (ri)i∈I d'entiers non divisibles par la ara téristique p de k : M(T → r D/X) = N(T )→ X. 3.5 Con lusion Preuve de la proposition 6. Il su�t de montrer que le fon teur C est une équi- valen e. Le fait qu'un revêtement modéré est par dé�nition normal permet de dé- �nir une transformation naturelle 1 → MC, et la propriété universelle de la normalisation montre que 'est un isomorphisme. Le lemme 15 (ii) permet à son tour de dé�nir une transformation naturelle (T → r D/X) → CM(T → r D/X) dans lim−→r CatRev( D/X), et le point (iii) montre que 'est un isomorphisme pour T → r D/X galoisien, et on en déduit que 'est vrai pour tout objet. Don C et M sont des équivalen es de atégories ré iproques. 4 Fais eaux lo alement onstants et �brés �nis sur un hamp de Deligne-Mumford Dans ette partie, on �xe à nouveau un hamp de Deligne-Mumford lo ale- ment noethérien X . 4.1 Topologies La dé�nition suivante est une adaptation de [43℄, Lemme 1.2. Dé�nition 15. On dé�nit le site étale Xet (resp. le site étale �ni Xetf ) du hamp X omme le site dont la atégorie sous-ja ente a pour objets les mor- phismes représentables étales f : T → X d'un hamp de Deligne-Mumford vers X, a pour �è hes les lasses d'isomorphismes de ouples (φ, α) px iiii f ′~~}} où φ est un morphisme représentable, α un 2-isomorphisme, et dont les re- ouvrements sont les familles épimorphiques (resp. les familles épimorphiques (Ti → T )i∈I onstituée de morphismes �nis). On parlera aussi de topologie étale pour Xet, et de topologie étale �nie globale pour Xetf . Remarque 4 ([43℄ Lemme 1.2). On obtient une topologie équivalente à Xet si dans la dé�nition des objets, on impose à T d'être un s héma. je remer ie l'auteur pour avoir fourni le � hier sour e de son texte, me permettant ainsi de reproduire les diagrammes 4.2 Systèmes lo aux ensemblistes et groupe fondamental Zoonekynd ([43℄) a remarqué que l'on pouvait interpréter la dé�nition 12 omme un as parti ulier du groupe fondamental d'un topos donnée par Leroy ([25℄). 4.2.1 Systèmes lo aux ensemblistes Dé�nition 16 ([25℄,[43℄). Donné un topos T , on dé�nit la sous- atégorie LC(T ) (resp. LCF(T )) omme elle des objets lo alement onstants (resp. lo alement onstants �nis) et SLC(T ) (resp. SLCF(T )) omme elle des unions disjointes d'objets de LC(T ) (resp. de LCF(T )). SLC(T ) (resp. SLCF(T )) est un topos galoisien (resp. un topos galoisien �ni). Si on suppose T onnexe, et que l'on �xe un point géométrique x de T , on asso ie anoniquement à ette atégorie un progroupe stri t π1(T , x) (resp. un groupe pro�ni π̂1(T , x)), véri�ant SLC(T ) ≃ π1(T , x) − Ens (resp. SLCF(T ) ≃ π̂1(T , x) − Ens), les ensembles étant munis d'a tion ontinues des pro-groupes onsidérés. De plus (LCF(T ), x∗) est une atégorie galoisienne dont le groupe fondamental s'identi�e à π̂1(T , x)− Ens. 4.2.2 Systèmes lo aux ensemblistes et revêtements On dispose d'un fon teur naturel CatRevX → LCF(X̃et) donné sur les objets par (Y → X)→ HomX(·, Y ), où HomX(·, Y ) est donné omme fon teur sur les objets par (T → X)→ HomT (T, T ×Y X). Pour voir que ela dé�nit bien un préfais eau d'ensembles, on peut, en utilisant la remarque 4, supposer que T est un s héma, mais ça résulte alors du fait que Y → X est représentable, don que T ×Y X est également un s héma. Le fait que HomX(·, Y ) est e�e tivement un fais eau sur Xet dé oule immédiatement du fait que 'est vrai lorsque X est un s héma ([4℄, VII.2, [38℄). On dispose également d'un fon teur naturel dans la dire tion opposée LCF(X̃et)→ CatRevX . En e�et, soit p : X0 → X un atlas étale, X1 = X0 ×X X0, et s, b : X1 ⇒ X0 le groupoïde orrespondant. Soit E ∈ objLCF(X̃et), on peut l'interpréter ([38℄, Example 4.11) omme un fais eau lo alement onstant �ni équivariant sur X0, 'est à dire un ouple (F, ψ), où F ∈ objLCF(X̃0et), et φ : s∗F→ b∗F est un isomorphisme véri�ant la ondition de o y le habituelle. L'équivalen e de atégories usuelle LCF(X̃iet) ≃ RevXi pour i ∈ {0, 1}, per- met d'interpréter ette donnée omme un revêtement étale Y0 → X0, muni d'un isomorphisme φ : X1ցs ×X0 Y0 ≃ X1ցb ×X0 Y0. En posant Y1 = X1ցs ×X0 Y0, on obtient un nouveau groupoïde (pr2, pr2 ◦ φ : Y1 ⇒ Y0) et l'on obtient ainsi un hamp Y = [Y1 ⇒ Y0] et même en fait un objet de CatRevX . Théorème 5 ([43℄, Théorème 3.1). Les fon teurs i-dessus dé�nissent des équi- valen e de atégories ré iproques CatRevX ≃ LCF(X̃et). Corollaire 4. Si X est un hamp de Deligne-Mumford onnexe, et x : specΩ→ X un point géométrique, on a un isomorphisme naturel π̂1(X̃et, x) ≃ π1(X, x) Remarque 5. π1(X̃et, x) porte le nom de groupe fondamental élargi de X (voir [1℄ X 7.6 pour le as d'un s héma). 4.2.3 Interprétation à l'aide de la topologie étale �nie globale Proposition 7. Soit X hamp de Deligne-Mumford onnexe. Le fon teur SLC(X̃etf )→ SLC(X̃et) induit un isomorphisme de SLC(X̃etf ) sur SLCF(X̃et). En parti u- lier, si x est un point géométrique, on a un isomorphisme naturel π̂1(X̃et, x) ≃ π1(X̃etf , x). Démonstration. Le morphisme de sites f : Xetf → Xet induit un morphisme de topos (f∗, f∗) : X̃et → X̃etf dont l'adjoint à gau he f∗ induit un fon teur �dèlement plein SLC(X̃etf )→ SLC(X̃et), et dont on va montrer que l'image es- sentielle est SLCF(X̃et). Via l'équivalen e SLC(T ) ≃ π1(T , x)−Ens, e fon teur s'interprète omme la restri tion le long du morphisme π1(X̃et, x)→ π1(X̃etf , x). Soit d'abord E ∈ objLC(X̃etf ). Il existe une famille ouvrante (Ti → X)i∈I onstituée de morphismes représentables étales �nis telle que pour tout i ∈ I, E|Ti soit onstant. L'image de Ti → X est ouverte ar le morphisme est étale, et fermé ar il est �ni, 'est don ∅ ou X . On peut don se ramener à une famille ouvrante à un élément T → X , revêtement qu'on peut supposer de plus galoisien. Soit G son groupe de Galois. Alors E orrespond (via l'équivalen e π1(X̃etf , x) − Ens ≃ SLC(X̃etf )) à un G-ensemble E, qui se dé ompose en j∈J Ej , ses orbites sous G. Chaque ensemble Ej est �ni ar G l'est, don orrespond (via l'équivalen e π1(X̃et, x) − Ens ≃ SLC(X̃et)) à un Ej ∈ objLCF(X̃et). Don E = j∈J Ej s'envoie sur un objet de SLCF(X̃et). Ré iproquement si E ∈ obj LCF(X̃et), alors le théorème 5 montre que E dé�nit un revêtement Y → X , dont une l�ture galoisienne trivialise E, don E vient bien d'un objet de LC(X̃etf ). Corollaire 5. Si X est un hamp de Deligne-Mumford onnexe, et x : specΩ→ X un point géométrique, on a un isomorphisme naturel π1(X̃etf , x) ≃ π1(X, x) 4.3 La atégorie tannakienne des systèmes lo aux de k- ve toriels Dé�nition 17 ([34℄, hapitre VI, 1.1.2). Donné un topos T onnexe lo alement onnexe, et un orps k, on dé�nit la atégorie LC(T , k) des systèmes lo aux de k-ve toriels de rang �ni. Si on hoisit de plus un point géométrique x, 'est une atégorie tannakienne. Son groupe de Tannaka est alors l'enveloppe k-algébrique du progroupe stri t π1(T , x), au sens suivant. Proposition 8 ([34℄, hapitre VI, 1.1.2.1). Si π1(T , x) = (Gi)i∈I et Hi est l'enveloppe k-algébrique de Gi, alors le groupe de Tannaka de (LC(T , k), x∗) est anoniquement isomorphe à lim←−i∈I Hi. On déduit du orollaire 5 et de la proposition 8 : Corollaire 6. Soit X un hamp de Deligne-Mumford onnexe, et x un point géométrique. Le groupe de Tannaka de (LC(X̃etf , k), x ∗) est anoniquement iso- morphe au groupe fondamental pro�ni π1(X, x). Remarque 6. 1. Il vaudrait mieux i i parler du k-groupe pro onstant asso- ié à π1(X, x). 2. Le groupe de Tannaka de (LC(X̃et, k), x ∗) est, d'après e qui pré ède, iso- morphe l'enveloppe k-algébrique du groupe fondamental élargi de X. Lemme 16. Soit X hamp de Deligne-Mumford onnexe. SiV ∈ obj LC(X̃etf , k), alors il existe un revêtement Y → X de X trivialisant V. Démonstration. C'est immédiat à partir du orollaire 6. 4.4 Fon teur à la Riemann-Hilbert 4.4.1 Dé�nition Donné un hamp de Deligne-Mumford X , on peut dé�nir la atégorie VectX des �brés ve toriels sur X omme la atégorie [X,Vect] des morphismes de hamps de X vers le hamp Vect des �brés ve toriels, parfois appelés représen- tations du hamp X , 'est le point de vue que l'on a adopté jusqu'à présent. La théorie de la des ente des �brés ve toriels (et plus généralement des fais eaux quasi- ohérents [2℄) fournit un point de vue alternatif, en e�et les fais eaux de OX -modules F sur Xet, tels qu'il existe un atlas étale X ′ → X , tel que F|X′ est libre, forment une atégorie équivalente ([24℄, hapitre 13). On utilisera librement ette équivalen e par la suite. La dé�nition suivante est inspirée de [34℄ VI 1.2.4. Dé�nition 18. Soit X un hamp de Deligne-Mumford lo alement noethérien sur un orps k. On dé�nit le fon teur à la Riemann-Hilbert RH : LC(X̃etf , k)→ VectX omme le fon teur omposé du fon teur anonique LC(X̃etf , k)→ LC(X̃et, k) et du fon teur LC(X̃et, k)→ VectX donné sur les objets par V→ OX ⊗k V. 4.4.2 Propriétés du fon teur RH Proposition 9. Soit X un hamp de Deligne-Mumford lo alement noethérien sur un orps k. 1. Le fon teur RH est �dèle. 2. Si X est de plus omplet, réduit, et k est algébriquement los, il est �dè- lement plein. Démonstration. Pour V ∈ obj LC(X̃etf , k), on note φX,V : V → OX ⊗k V le morphisme de fais eaux sur Xetf dé�ni à partir du morphisme anonique k→ OX . Quitte à rempla er V par Hom(V,W), il su�t de voir : 1. H0(X,φX,V) est inje tif. 2. Si X est omplet et réduit sur k algébriquement los, H0(X,φX,V) est bije tif. Le premier point est évident ar k→ OX est inje tif, et les fon teurs · ⊗k V et H0(X, ·) sont exa ts à gau he. Pour le se ond point, X étant lo alement noethérien ( e qui assure que les omposantes onnexes sont ouvertes), on peut supposerX onnexe. Le lemme 16 donne l'existen e d'un revêtement π : Y → X trivialisant V. On peut supposer π galoisien de groupe G. On a alors un diagramme ommutatif : 0 // H0(X,V) π−1 // H0(X,φX,V) H0(Y, π−1V) H0(Y,φ Y,π−1V 0(Y ×X Y, p−1V) H0(Y×XY,φY,p−1V) 0 // H0(X,OX ⊗k V) π∗ // H0(Y,OY ⊗k π−1V) pr∗1// 0(Y ×X Y,OY×XY ⊗k p−1V) où p désigne le morphisme anonique p : Y ×X Y → X . Y étant en ore propre ( ar �ni surX) et réduit ([2℄ I Proposition 9.2) on s'est don ramené au as où V est trivial. On peut à nouveau supposer X onnexe, et don V = s−1X V , où sX : X → spec k est le morphisme stru turel, et V un k-ve toriel de rang �ni. On est alors immédiatement ramené à V = k, et il s'agit de voir que le morphisme naturel k → H0(X,OX) est un isomorphisme, mais ça résulte du fait que X est propre, réduit, onnexe, et k algébriquement los. 4.5 Fibrés �nis Dé�nition 19. On appellera s héma tordu un hamp de Deligne-Mumford X admettant pour espa e des modules un s héma M , tel qu'il existe un ouvert dense U de M , tel que X →M soit un isomorphisme en restri tion à U . On adapte les dé�nitions de [32℄, [33℄ au as d'un s héma tordu X modéré (au sens de [7℄, dé�nition 2.3.2) réduit sur un orps k, dont l'espa e des modules M est propre et onnexe sur k. Dé�nition 20 ([32℄, [33℄). Un fais eau lo alement libre E sur X est dit �ni s'il existe deux polyn�mes distin ts P,Q à oe� ients entiers positifs tels que P (E) ≃ Q(E). Pour identi�er l'image essentielle du fon teur RH , on va suivre la stratégie de Nori, qui onsiste à plonger la atégorie des �brés �nis dans la atégorie abélienne des �brés semi-stables sur X . Dé�nition 21. Une orbi ourbe dans X est un morphisme birationnel sur son image D/C → X, où C est une ourbe proje tive, onnexe, et lisse sur k, D = (Di)i∈I un ensemble de diviseurs de Cartier e�e tifs réduits sur C. Dé�nition 22 ([32℄, [33℄). Un fais eau lo alement libre E sur X est dit semi- stable s'il est semi-stable de degré 0 en restri tion à toute orbi ourbe dans X. On notera SS0X la sous- atégorie pleine de VectX des fais eaux lo alement libres semi-stables sur X. Proposition 10. La atégorie SS0X est une atégorie abélienne. Démonstration. La preuve est identique à elle de [32℄, Lemma 3.6, (b) : étant donné un morphisme f : E → E ′ dans SS0X , le point lé est de voir que ker f et coker f sont lo alement libres. Il est aisé de voir qu'ils sont sans torsion 9, et don lo alement libres si X est une orbi ourbe. Dans le as général, X étant réduit ela revient à voir que la fon tion qui à un point géométrique x : spec k → X asso ie le rang de x∗f : x∗E → x∗E ′ est onstant sur X . Or, le as parti ulier envisagé i-dessus montre que ette fon tion est onstante sur toute orbi ourbe dans X . On peut don on lure à l'aide du lemme suivant : Lemme 17. La relation d'équivalen e sur les points x : spec k → X engendrée par x ∼ x′ s'il existe une orbi ourbe dans X dont l'image ontient x et x′ admet une unique orbite. Démonstration. Dans le as où X est un s héma, on se ramène au as où X est proje tif sur k grâ e au lemme de Chow ([16℄, 5.6), où 'est un fait lassique. Dans le as général, on note U omme dans la dé�nition 19 un ouvert de l'espa e de modules M de X tel que la �è he X → M de X vers son espa e de modules M soit un isomorphisme en restri tion à U . Soit y, y′ les images respe tives de x, x′ dans M , on peut supposer que y ∈ U . D'après le as par- ti ulier i-dessus, il existe une ourbe C dans M (au sens de [32℄, ou de la dé�nition 21) ontenant y et y′. Le hamp de Deligne-Mumford C ×M X ad- met C pour espa e des modules, et il en est de même de (C ×M X)red ([7℄ Lemma 2.3.3 ou [6℄ Corollary 3.3). D'après [14℄, Theorem 4.1, on a un isomor- phisme (C ×M X)red ≃ r D/C sur X pour un hoix onvenable d'une famille D de diviseurs e�e tifs et d'une famille d'entiers naturels r. Comme on dis- pose d'un morphisme birationnel surje tif Dred/C → r D/C, omme de plus les arguments généraux de [37℄, �3 s'appliquent i i, à l'aide de [10℄, �5 pour les adapter au as des orbi ourbes ; omme 'est par ailleurs bien onnu dans le adre -équivalent- des �brés paraboliques sur les ourbes (voir [36℄), nous ne rentrons pas dans les détails (C ×M X)red est birationnel sur son image dans X et que elle- i ontient x et x′, on a terminé. Proposition 11. Tout �bré �ni sur X est semi-stable. Démonstration. Comme la restri tion d'un �bré �ni l'est en ore, il su�t de le véri�er sur les orbi ourbes. Mais on peut alors adapter la preuve de [32℄ au as des orbi ourbes : voir [10℄, Proposition 6. Dé�nition 23 ([32℄, [33℄). Un fais eau lo alement libre E sur X est dit essen- tiellement �ni si 'est un quotient de deux sous-�brés semi-stables d'un �bré �ni. On notera EFX la sous- atégorie pleine de SS0X des fais eaux lo alement libres essentiellement �nis sur X. Théorème 6. Soit X un s héma tordu modéré et réduit sur un orps k, dont l'espa e des modules M est propre et onnexe sur k, et x ∈ X(k) un point rationnel. La paire (EFX, x∗) est une atégorie tannakienne. Démonstration. Compte tenu de la proposition 10, la preuve est la même que elle donnée dans [32℄, �3. Corollaire 7. Si on suppose, en plus des hypothèses du théorème 6, que k est algébriquement los de ara téristique 0, alors tout �bré essentiellement �ni est �ni, et le fon teur RH induit une équivalen e de atégories tensorielles entre LC(X̃etf , k) et FX. En parti ulier (FX, x ∗) est une atégorie tannakienne dont le groupe est anoniquement isomorphe à π1(X, x). Démonstration. SoitV ∈ objLC(X̃etf , k). Le fait que RH(V) soit �ni résulte du lemme 16 : si π : Y → X est un revêtement galoisien de groupe G trivialisantV, il existe une représentation V de G sur le orps k telle que π−1V = VY . On suit alors l'argument de [32℄ Proposition 3.8 : ette représentation se plonge dans un k[G]-module libre et est don essentiellement �nie, omme la ara téristique de k est 0, elle en en fait �nie. Or le morphisme naturel RH(V) → πG∗ (OY ⊗k V ) est un isomorphisme, et don RH(V) est lui-même �ni. La proposition 9 montre que le fon teur RH est �dèlement plein. Soit à présent E un �bré essentiellement �ni. Soit < E > la sous- atégorie tannakienne engendrée, et G son groupe de Tannaka (i.e. le groupe d'holonomie de E). Comme E est essentiellement �ni, G est un s héma en groupe �ni sur k ([32℄ Theorem 1.2), omme e orps est de ara téristique 0, G est réduit d'après un théorème de Cartier ([39℄, Chapter 11), don étale, et k étant de plus algébriquement los, G est don onstant. Le fon teur tensoriel GRep → VectX orrespond d'après [32℄ Proposition à un G-revêtement π : Y → X , et E s'identi�e via l'équivalen e < E >≃ qui, du fait de sa fon torialité, vaut aussi pour les hamps de Deligne-Mumford, voir sur le sujet [26℄ GRep à une représentation V de G. Si V est le système lo al orrespondant, on a vu i-dessus que RH(V) est isomorphe à πG∗ (OY ⊗k V ), lui-même isomorphe à E . D'où les deux premières assertions. La dernière résulte alors du orollaire 6. 5 Théorème de Weil-Nori 5.1 Fibrés paraboliques modérés Soit X un s héma lo alement noethérien sur un orps k, D une famille de diviseurs irrédu tibles à roisements normaux simples sur X . 5.1.1 Fibrés paraboliques �nis Dé�nition 24. 1. On dé�nit la atégorie Par(X,D) des �brés paraboliques modérés sur (X,D) par : Par(X,D) = lim−→ Par 1 (X,D) où les multi-indi es varient parmi les familles r = (ri)i∈I d'entiers non divisibles par la ara téristique p de k. 2. Par(X,D) est munie d'un produit tensoriel véri�ant, pour E·, E ′· ∈ obj Par 1 (X,D), la formule de onvolution suivante : (E· ⊗ E ′· )m = El ⊗ E ′m−l désigne la o�n ( oend), voir �2.1.2. 3. Un �bré parabolique modéré E· sur (X,D) est dit �ni s'il existe deux poly- n�mes distin ts P,Q à oe� ients entiers positifs tels que P (E·) ≃ Q(E·). On notera FPar(X,D) la atégorie des �brés paraboliques modérés sur (X,D). Remarque 7. Ces notions sont ompatibles, via l'équivalen e Vect( r D/X) ≃ Par 1 (X,D) du théorème 2, ave les notions hampêtres du �4.5, voir [10℄. 5.1.2 Fibrés paraboliques essentiellement �nis Dé�nition 25. 1. Un �bré parabolique modéré E· sur (X,D) à poids mul- tiples de est dit semi-stable si le fais eau lo alement libre sur D/X as- so ié par la orrespondan e du théorème 2 est semi-stable au sens de la dé- �nition 22. On notera SS0 Par(X,D) la sous- atégorie pleine de Par(X,D) dont les objets sont semi-stables. 2. Un �bré parabolique modéré semi-stable E· est dit essentiellement �ni si 'est un quotient de deux sous-�brés paraboliques modérés semi-stables d'un �bré parabolique modéré �ni. On notera EFPar(X,D) la sous- atégorie pleine de SS0 Par(X,D) dont les objets sont essentiellement �nis. Remarque 8. 1. La dé�nition de semi-stabilité est indépendante du hoix de r. 2. Il serait intéressant de donner une dé�nition de la semi-stabilité ne faisant intervenir que la topologie de Zariski. 5.2 Lien ave le groupe fondamental 5.2.1 Énon é Théorème 7. Soit X un s héma propre, normal, onnexe sur un orps k, D une famille de diviseurs irrédu tibles à roisements normaux simples sur X, D = ∪i∈IDi, x ∈ X(k) un point rationnel, x /∈ D. (i) La paire (EFPar(X,D), x∗) est une atégorie tannakienne. (ii) Si k est algébriquement los de ara téristique 0, tout �bré parabolique mo- déré essentiellement �ni est �ni, et le groupe de Tannaka de (FPar(X,D), x∗) est anoniquement isomorphe au groupe fondamental π1(X −D, x). Démonstration. On ommen e par remarquer que D/X est normal ([21℄, Pro- position 1.8.5). (i) Ce i résulte alors des théorèmes 2 et 6. (ii) La première assertion dé oule du théorème 2 et du orollaire 7. Pour la se onde, notons π e s héma en groupe. Alors π ≃ lim←−r πr, où πr est le groupe de Tannaka de la atégorie (FPar 1 (X,D), x∗), ave des notations évidentes. D'après la proposition 6, il su�t de voir qu'on a des isomor- phismes naturels πr ≃ π1( r D/X, x), ompatibles ave les systèmes pro- je tifs (vu qu'on est en ara téristique zéro, on a un isomorphisme naturel π1(X−D, x) ≃ πD1 (X, x) donné, au niveau des revêtements, par le fon teur de normalisation). On on lut don en appliquant à nouveau le théorème 2 et le orollaire 7. 5.2.2 S héma en groupe fondamental modéré On est naturellement onduit à poser : Dé�nition 26. Ave les notations du théorème 7, on appellera s héma en groupe fondamental modéré de (X,D) le groupe fondamental πD(X, x) de la atégorie tannakienne (EFPar(X,D), x∗). Remarque 9. 1. Ce s héma en groupe πD(X, x) est une limite inverse de s hémas en groupes �nis, se spé ialise sur le s héma en groupe fondamen- tal de Nori ([32℄) lorsque D = ∅, et sur le groupe fondamental modéré de Grothendie k-Murre ([21℄) lorsque k est algébriquement los de ara té- ristique 0, d'où son nom. 2. Lorsque k est quel onque, les arguments de orollaire 7 montrent qu'on a un morphisme πD(X, x) → πD1 (X, x) qui est un épimorphisme lorsque k est algébriquement los. 3. Toutefois, il onviendrait de pré iser la nature des �torseurs modérément rami�és� que πD(X, x) lassi�e. 6 Appli ation au al ul de �brés paraboliques �- nis de groupe d'holonomie résoluble 6.1 Introdu tion et notations On reprend les hypothèses de la partie 5 : X est un s héma propre, normal, onnexe sur un orps k, qu'on suppose de plus algébriquement los de ara - téristique 0, D une famille de diviseurs irrédu tibles à roisements normaux simples sur X , D = ∪i∈IDi, x ∈ X(k) un point rationnel, x /∈ D. Le but de ette partie est d'utiliser le théorème 7 pour onstruire expli itement ertains objets de FPar(X,D). Puisqu'on est en fait intéressé par le groupe fondamental modéré deX−D, on évite bien sûr de onstruire un tel �bré parabolique �ni à partir d'un revêtement de Y → X modérément rami�é le long de D le trivialisant. L'idée de la méthode présentée est de n'utiliser que des sous-revêtements d'un tel Y → X , et repose essentiellement sur la proposition 12. Cette observation est inspirée par une trans ription dire te de la méthode des petits groupes de Wigner et Ma key de la théorie des représentations à la théorie des revêtements, rendue possible grâ e au théorème 7. Par la suite, on note X = r D/X le hamp des ra ines. 6.2 Compléments sur les �brés �nis On ommen e par quelques remarques générales on ernant les �brés �nis. Comme la stru ture parabolique n'entre pas vraiment en jeu, e qui va être dit est aussi valable dans la situation lassique où X est un s héma, propre, réduit et onnexe sur un orps k algébriquement los de ara téristique 0. Dans ette situation, il su�t de rempla er l'utilisation du théorème 7 par le théorème de Nori originel [32℄. 6.2.1 Image dire te d'un �bré �ni La base de la méthode pour onstruire des �brés �nis est la remarque élé- mentaire suivante : Proposition 12. Soit p : Y → X un revêtement étale. Si F ∈ objFY, alors p∗F ∈ objFX . Démonstration. On peut hoisir Y onnexe, et aussi un point y ∈ Y(k) au dessus de x. Soit G un groupe �ni, BkG = [spec k|G] le hamp lassi�ant, πet1 (X , x)→ G un morphisme, Z → X le revêtement galoisien asso ié, orrespondant aussi à un morphisme m : X → BkG. Alors le fon teur RH induit une équivalen e entre la atégorie GRep des représentations de G et la atégorie FZX des �brés �nis sur X trivialisés par Z → X . Si V est une représentation de G, et V est le système lo al sur Xet asso ié, alors ette orrespondan e asso ie à V le �bré OX ⊗k V, qui est anoniquement isomorphe à m∗V . On hoisit à présent un morphisme πet1 (Y, y) → A, tel que le revêtement galoisien asso ié Z → Y trivialise F , et tel que le revêtement omposé Z → X soit galoisien, de groupe G. D'après e qui pré ède, il existe une représentation W du groupe A telle que F ≃ OY ⊗k W. La proposition résulte alors du Lemme 18. p∗(OY ⊗k W) ≃ OX ⊗k V où V = IndGAW . Démonstration. Il s'agit d'une formule de hangement de base dans le dia- gramme artésien : X // BkG Dans la pratique, il est utile de savoir al uler le produit tensoriel de deux �brés �nis obtenus par la méthode de la proposition 12. Pour ela, il su�t d'adapter à e ontexte les formules lassiques, dues à Ma key, donnant le pro- duit tensoriel de deux représentations induites omme somme dire te de repré- sentations induites (voir par exemple [15℄ �44). On obtient ainsi : Lemme 19. Soit Z → X un revêtement galoisien onnexe, G le groupe de Galois, H1, H2 deux sous-groupes. Pour i ∈ {1, 2}, on note pi : Yi → X le revêtement intermédiaire orrespondant au sous-groupe Hi, et Fi ∈ objFZYi. De plus pour g ∈ G, on note pg : Yg → X le revêtement orrespondant au sous-groupe H1 ∩ gH2g−1 et qg,i : Yg → Yi le morphisme naturel. Alors p1∗F1 ⊗OX p2∗F2 ≃ ⊕g∈H1\G/H2pg∗(q g,1F1 ⊗OXg q g,2F2) Démonstration. On a un isomorphisme naturel de BkG-groupoïdes BkH1×BkG BkH2 ≃ H1\G/H2 Bk(H1 ∩ gH2g−1). En le tirant par le morphisme X → BkG dé�nissantZ → X , on obtient un X -isomorphisme Y1×XY2 ≃ H1\G/H2 Yg . De plus si p : Y1×X Y2 → X est le morphisme anonique, la formule de hangement de base donne p1∗F1 ⊗ p2∗F2 ≃ p∗(pr∗1 F1 ⊗ pr∗2 F2), d'où la formule annon ée. 6.2.2 La méthode des petits groupes de Wigner et Ma key Comme appli ation de la partie pré édente, on dé rit les �brés �nis asso iés à une extension triviale d'un revêtement galoisien par un groupe abélien. On se donne don un revêtement galoisien onnexe Z → X de groupe G = A⋉H , où H est quel onque, et A est abélien, d'exposant n premier à l'ordre de H . On note Y → X le revêtement intermédiaire orrespondant à A, et on �xe un point z ∈ Z(k) au dessus de x, d'image y dans Y(k). D'après la dualité de Tannaka, on a une équivalen e naturelle FZ Y ≃ ARep. En parti ulier le groupe PicZ Y des lasses d'isomorphisme de �brés inversibles sur Y trivialisés par Z est anoniquement isomorphe au groupe  des ara tères de A, et détermine omplètement FZ Y. Le but est de dé rire omplètement la atégorie tannakienne FZ X en fon - tion de FY X ′ (pour les extensions X ′ intermédiaires entre Y et X ) et de PicZ Y. On ommen e par dé rire la stru ture additive. On remarque qu'on a une a tion naturelle de H sur  ≃ PicZ Y. Alors l'in lusion PicZ Y ⊂ H1et(Y,µn) est H-équivariante. Soit L un �bré inversible sur Y trivialisé par Z . On note HL le stabilisateur de sa lasse dans PicZ Y, et πL : Y → Y/HL le morphisme quotient. Comme elui- i est étale, on dispose de la suite spe trale de Ho hs hild-Serre, qui s'é rit i i : Hp(HL, H q(Y,µn)) =⇒ Hp+q(Y/HL,µn) L'hypothèse que n est premier à l'ordre de H et la suite exa te des termes de bas degré asso iée à la suite spe trale montrent que H1et(Y/HL,µn) ≃ H1et(Y,µn)HL , et don il existe, à isomorphisme près, un unique �bré inversible de n-torsion L̃ sur Y/HL tel que L ≃ π∗LL̃. On note de plus pL : Y/HL → X le morphisme anonique. Proposition 13. 1. Soit L un �bré inversible sur Y trivialisé par Z et E ∈ obj FY(Y/HL). Le �bré pL∗(L̃ ⊗ E) sur X est �ni. 2. Lorsque L varie dans un système de représentants de (PicZ Y)/H et E varie dans une base de générateurs additifs de FY(Y/HL), es fais eaux forment une base de générateurs additifs de FZ X . Démonstration. 1. C'est une appli ation dire te de la proposition 12. 2. C'est, en fait, via la orrespondan e de Tannaka entre �brés �nis et repré- sentations du groupe fondamental ( orollaire 7), un problème de théorie des groupes, pour lequel on renvoie à [35℄, 8.2, Proposition 25. La stru ture tensorielle de FZ X est alors omplètement déterminée par le lemme 19. 6.2.3 Fibrés �nis de groupe d'holonomie résoluble En poursuivant la même idée, on voit que la méthode onduit au al ul des �brés �nis dont le groupe d'holonomie (i.e. le groupe de Tannaka de la atégorie tannakienne engendrée) est résoluble. En e�et, soit p : Y → X un revêtement onnexe, étale, galoisien de groupe d'automorphismes H , et y ∈ Y(k) un point au dessus de x. On note Pic(Y)[n] la sous- atégorie pleine de FY dont les objets sont les �brés inversibles de n- torsion, où n ≥ 1 est un entier. Donné un groupe (abstrait, ou pro�ni) π, on note Dn(π) le noyau du mor- phisme de groupe π → π , 'est un sous-groupe ara téristique. Lemme 20. Le groupe de Tannaka de la atégorie tannakienne engendrée par l'image du fon teur p∗ : Pic(Y)[n] → FX est anoniquement isomorphe à π1(X ,x) Dn(π1(Y,y)) Démonstration. Le morphisme anonique π1(X , x) → π1(X ,x)Dn(π1(Y,y)) orrespond à un nouveau revêtement étale, galoisien, onnexe Y ′ → X , muni d'un point géométrique y′ ∈ Y ′(k) au dessus de x, et dominant Y → X . Par dualité de Tannaka, la atégorie Rep π1(X ,x) Dn(π1(Y,y)) est anoniquement isomorphe à la atégo- rie FY′X des �brés �nis sur X trivialisés par Y ′. Or, soit E un tel �bré, p∗E est un objet de FY′Y, dont le groupe de Tannaka est isomorphe à π1(Y,y) , don est abélien et de n-torsion. On peut don é rire p∗E ≃ ⊕Ni=1Li, où les Li sont dans Pic(Y)[n]. Mais omme k est supposé de ara téristique 0, E ≃ pH∗ p∗E est un fa teur dire t de p∗p ∗E ≃ ⊕Ni=1p∗Li. On garde les notations de la preuve, en parti ulier Y ′ est le plus grand revê- tement abélien n-élémentaire de Y. Soit de plus A le dual de Cartier du groupe Pic0(Y)[n]. La théorie de Kummer usuelle a�rme que π1(Y,y) ≃ A. L'avan- tage de la méthode utilisée i i, qui peut être vue omme une version relative de la théorie de Kummer, est qu'elle donne une interprétation tannakienne du groupe G = π1(X ,x) Dn(π1(Y,y)) : si l'on sait al uler la atégorie tannakienne engendrée par l'image du fon teur p∗ : Pic(Y)[n] → FX , on sait déterminer G omme extension de H par A. On peut en parti ulier itérer le pro édé, en partant de (Y, y) = (X , x), et en hoisissant une suite d'entiers n1, · · · , nm, e qui onduit au al ul des quotients π1(X ,x) Dnm ···Dn1(π1(X ,x)) . La limite naturelle de la méthode est bien le plus grand quotient pro-résoluble πres1 (X , x) de π1(X , x), vu qu'on obtient ainsi un sous-ensemble o�nal de l'ensemble de ses quotients �nis. 6.3 Fibrés paraboliques �nis de groupe d'holonomie réso- luble 6.3.1 Fibrés paraboliques �nis obtenus omme image dire te le long d'un morphisme modérément rami�é On onserve les notations de la partie 6.1. Soit p : Y → X dans objRevD(X), ave Y onnexe. On note (Ej)j∈J la fa- mille des omposantes irrédu tibles (munies de la stru ture réduite) de p−1(D). Soit Z → X une l�ture galoisienne, (ri)i∈I (respe tivement (sj)j∈J ) la famille des indi es de rami� ation de la famille (Di)i∈I (respe tivement (Ej)j∈J ) dans Lemme 21. Le morphisme naturel q : s E|Y → r D|X est �ni étale. Démonstration. Ce i résulte du lemme d'Abhyankar. En e�et si G (respe tive- ment A) est le groupe de Galois de Z → X (respe tivement Z → Y ) le lemme 13 montre que q s'identi�e au morphisme entre hamps quotients [Z|A] → [Z|G], qui est �ni étale, ar obtenu par hangement de base à partir du morphisme des hamps lassi�ants BkA→ BkG par le morphisme [Z|G]→ BkG orrespondant au G-torseur Z → [Z|G]. On peut don utiliser la proposition 12 pour onstruire des �brés parabo- liques �nis. De plus, es �brés �nis sont expli itement al ulables grâ e à la proposition 2.4.9. Plus pré isément, si on veut al uler des �brés paraboliques �nis de groupe d'holonomie résoluble, on peut appliquer le lemme 20 aux hamps des ra ines. Ainsi, il est naturel d'essayer de déterminer la n-torsion Pic( r D/X)[n] du groupe de Pi ard des hamp des ra ines, pour n ≥ 1 entier, e qui est l'objet de la partie suivante. 6.3.2 Fibrés inversibles de torsion sur les hamps des ra ines Le fon teur de Pi ard des hamps algébriques a été ré emment étudié par S.Bro hard (voir [13℄). Il a en parti ulier montré qu'on pouvait en étudier la omposante neutre omme dans le as lassique des s hémas. On rappelle briè- vement les dé�nitions dont on aura besoin. Dé�nition 27. 1. Deux fais eaux inversibles L et L′ sur le hamp X sont dits algébriquement équivalents s'ils sont équivalents pour la relation d'équi- valen e engendrée par la relation : L ∼ L′ s'il existe un k-s héma onnexe de type �ni T , des points géométriques t, t′ : specΩ → T , un fais eau inversible M sur X ×k T , et des isomorphismes (L ×k T )|Xt ≃ M|Xt , (L′ ×k T )|Xt′ ≃M|Xt′ . 2. On note Pic0 X le sous-groupe des éléments [L] de PicX tels que L est algébriquement équivalent à OX . 3. On appelle groupe de Néron-Severi le groupe NS(X ) = PicX/Pic0 X . 4. Si T (A) désigne la torsion du groupe abélien A, on note Picτ X = ker(PicX → NS(X )/T (NS(X )) Lemme 22. 1. PicX ∩ Pic0 r D|X = Pic0X 2. PicX ∩ Picτ r D|X = Picτ X dans la mesure où l'on sait déterminer l'image dire te d'un �bré ve toriel usuel par p, mais e problème peut-être pris en harge par le théorème de Grothendie k-Riemann-Ro h. Démonstration. 1. On note omme d'habitude π : r D|X → X le mor- phisme vers l'espa e des modules, et X = r D|X . Soient L, L′ deux fais eaux inversibles sur X tels que π∗L ∼ π∗L′, T un k-s héma onnexe de type �ni, t, t′ : specΩ → T des points géométriques, M un fais eau inversible sur X ×k T , et des isomorphismes (π∗L ×k T )|Xt ≃ M|Xt , (π∗L′ ×k T )|Xt′ ≃M|Xt′ . Alors (π×k T )∗M est un fais eau inversible sur X×k T : en e�et il est lo alement libre omme image dire te d'un fais eau lo alement libre par un morphisme �ni et plat, et omme π est généri- quement un isomorphisme, son rang est 1. En appliquant les formules de hangement de base pour un morphisme a�ne ou le long d'un morphisme plat on obtient des isomorphismes naturels (L×kT )|Xt ≃ ((π×kT )∗M)|Xt , (L′ ×k T )|Xt′ ≃ ((π ×k T )∗M)|Xt′ , e qui montre que L ∼ L 2. C'est une onséquen e laire du premier point et des dé�nitions. On a don des monomorphismes anoniques Pic0 r Pic0X Picτ r Picτ X Pic r On rappelle ( orollaire 3) qu'on a un isomorphisme anonique Pic( r On note ( les sous-groupes asso iés aux images des monomorphismes i-dessus. Comme X est propre sur le orps algébriquement los k, le groupe NS(X) est de type �ni ([3℄, XIII, Théorème 5.1), e qui prouve qu'il en est de même pour NS( r D|X) et donne un sens à l'énon é suivant. Proposition 14. Soit n premier à #T (NS(X)). Il y a une suite exa te natu- relle : 0→ Pic(X)[n]→ Pic( r D|X)[n]→ [n]→ 0 Démonstration. D'après les dé�nitions, on a lairement Picτ (X)[n] = Pic(X)[n] et Picτ ( r D|X)[n] = Pic( r D|X)[n]. Pour on lure, on doit justi�er que Ext1 (Z/n,Picτ (X)) = 0. Or (voir par exemple [40℄ 3.3.2) Ext1 (Z/n,Picτ (X)) ≃ Picτ(X)/nPicτ(X), et on peut on lure en appliquant le fon teur ⊗Z · à la suite exa te : 0→ Pic0(X)→ Picτ (X)→ T (NS(X))→ 0 En e�et Pic0(X) est un groupe divisible : ommeX/k est omplet le fon teur de Pi ard PicX/k est représentable par un s héma ([30℄), omme X est de plus normal (Pic0X/k)red est une variété abélienne sur k ([18℄, 236, Corollaire 3.2), et on peut appliquer [29℄ II, �6, Appli ation 2. Remarque 10. On a une interprétation assez dire te en termes de revête- ments : si Y → X est le revêtement étale galoisien de groupe abélien n-élémentaire maximal pour es propriétés, et de même Z → X en remplaçant étale par modé- rément rami�é de multi-indi e divisant r, alors par dualité de Tannaka on voit que la suite exa te duale (pour la dualité de Cartier) de la suite exa te pré édente est isomorphe à la suite exa te des groupes de Galois de la tour Z → Y → X. 6.3.3 Un exemple expli ite L'exemple le plus simple possible de onstru tion de �bré parabolique �ni indé omposable de rang plus grand que 1 est le suivant. On onsidère le morphisme p : Y = P1 → X = P1 donné par l'équation y2 = x , il est modérément rami�é le long du diviseur (0) + (1), ave indi es de rami� ation 2. p induit un morphisme �ni étale q : Y → (2,2) (0, 1)|X, et par hangement de base on en déduit un morphisme �ni étale r : (3,3) (1,−1)|Y → (2,2,3) (0, 1,∞)|X. A présent la proposition 14 montre que Pic0( (3,3) (1,−1)|Y ) ≃ et on voit fa ilement que = ker où Σ désigne la somme, don Pic0( (3,3) (1,−1)|Y ) est y lique d'ordre 3. Pour y ∈ {1,−1}, soit Ny une ra ine ubique de OY ((y)) sur (3,3) (1,−1)|Y , et soit L = N∨1 ⊗N−1. Alors le �bré r∗(L) est un �bré �ni indé omposable de rang 2 (2,2,3) (0, 1,∞)|X. Le �bré parabolique orrespondant a été onsidéré par L.Weng, voir [42℄, Appendix, �6, dans le langage du à Seshadri. Il nous semble que la des ription fournie i i en termes de hamps des ra ines permet de pré iser la des ription des drapeaux donnée par l'auteur, fa ilite le al ul de la stru ture tensorielle de la atégorie tannakienne engendrée (naturellement, dans e as pré is, le groupe fondamental est isomorphe au groupe symétrique S3), et en�n, donne une méthode de onstru tion des �brés paraboliques �nis de groupe d'holonomie résoluble. 6.3.4 Problème ouvert Dans [11℄ est donnée une preuve algébrique du théorème de stru ture du groupe fondamental pro-résoluble πres1 (X−D, x) pour X une ourbe proje tive et lisse sur un orps k algébriquement los de ara téristique 0 , et D un diviseur (non vide) sur elle- i. Peut on utiliser le lemme 20 pour donner une preuve alternative ? A 2-limite indu tive �ltrée de atégories A.1 2-limite On emprunte les dé�nitions de [22℄. On note Cat la 2- atégorie dont les objets sont les atégories, les 1-�è hes les fon teurs, les 2-�è hes les transformations naturelles. Soit C une 2- atégorie, I une atégorie usuelle. Pour tout objet D de C, on note cD le fon teur onstant I → C envoyant tout objet de I sur D, et toute �è he de I sur l'identité. On note de plus (I,C) la 2- atégorie des pseudo-fon teurs de I dans C. Si F ,F ′ : I → C sont deux pseudo-fon teurs, on note (I,C)(F ,F ′) la até- gorie des transformations naturelles de pseudo-fon teurs entre F et F ′. Dé�nition 28. Soit F : I → C un pseudo-fon teur. On appelle 2-limite indu - tive de F le 2-fon teur : D // (I,C)(F , cD) Si e fon teur est 2-représentable, on appelle aussi 2-limite indu tive et on note lim−→I F(i) l'objet de C le représentant. Plus pré isément, un représentant est un ouple (C, λ), où C est un objet de C, et λ : F → cC est une transformation naturelle entre pseudo-fon teurs, qui est 2-universelle au sens suivant : pour tout objet D de C, et toute transformation naturelle µ : F → cD, il existe un ouple (f, θ) formé d'un 1-morphisme f : C → D de C et d'un 2-isomorphisme θ de (I,C) : F λ // px iiii qui est unique à 2-isomorphisme unique près : si (f ′, θ′) est un autre tel ouple, il existe un unique 2-isomorphisme ρ de C : tel que θ ◦ (cρ ◦ λ) = θ′. A.2 Cas des atégories On suppose désormais que la atégorie I est �ltrante et que C = Cat. Dans e as, on dispose d'une des ription naturelle et probablement bien onnue de la 2-limite d'un pseudo-fon teur F : I → Cat. Pour f : i→ j, on note f∗ : F(i)→ F(j), plut�t que C(f). Soit C la atégorie 1. dont les objets sont les ouples (i, C), où i est un objet de I, et C est un objet de F(i), 2. dont les morphismes (i, C) → (j,D) sont les lasses d'équivalen e12 de triplets (f, g, α) : k, f∗C α // g∗D 66nnnnnn pour la relation QQ i f ′ k, f∗C α // g∗D ∼ k′, f ′∗C α′ // g′∗D 77nnnnnn 66nnnnnn s'il existe 66mmmmmm tel que h ◦ f = h′ ◦ f ′, h ◦ g = h′ ◦ g′ et h∗α = h′∗α′. Proposition 15. La atégorie C, munie de la transformation naturelle ano- nique F → cC, est une 2-limite pour F . Démonstration. C'est une véri� ation longue mais dire te de la dé�nition 28 dans e as pré is. Référen es [1℄ S hémas en groupes. II : Groupes de type multipli atif, et stru ture des s hémas en groupes généraux. Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendie k. Le - ture Notes in Mathemati s, Vol. 152. Springer-Verlag, Berlin, 1962/1964. [2℄ Revêtements étales et groupe fondamental. Springer-Verlag, Berlin, 1971. 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[arXiv:math.AG/0111073℄. http://arxiv.org/abs/math.AG/0412512 http://arxiv.org/abs/math.AG/0111073 Introduction Une description alternative Organisation de l'article Origines et liens avec des travaux existants Remerciements Fibrés paraboliques le long d'une famille régulière de diviseurs Faisceaux paraboliques Définition Opérations élémentaires sur les faisceaux paraboliques Fibrés paraboliques Facette Complexe associé à un faisceau parabolique et une facette Définition des fibrés paraboliques Fibrés paraboliques et revêtements Famille régulière de diviseurs Fibrés paraboliques relativement à une famille de diviseurs à croisements normaux simples Revêtements de Kummer Fibrés paraboliques associés à un revêtement de Kummer Fibrés paraboliques et champ des racines Champ des racines La correspondance : énoncé Bonne définition Équivalence réciproque Preuve de l'équivalence Preuve du caractère tensoriel Structure locale des fibrés paraboliques Groupe de Picard des champs des racines Image directe de fibrés paraboliques Groupe fondamental modéré comme groupe fondamental champêtre Groupe fondamental champêtre Groupe fondamental modéré Le foncteur C Le foncteur M Conclusion Faisceaux localement constants et fibrés finis sur un champ de Deligne-Mumford Topologies Systèmes locaux ensemblistes et groupe fondamental Systèmes locaux ensemblistes Systèmes locaux ensemblistes et revêtements Interprétation à l'aide de la topologie étale finie globale La catégorie tannakienne des systèmes locaux de k-vectoriels Foncteur à la Riemann-Hilbert Définition Propriétés du foncteur `39`42`"613A``45`47`"603ARH Fibrés finis Théorème de Weil-Nori Fibrés paraboliques modérés Fibrés paraboliques finis Fibrés paraboliques essentiellement finis Lien avec le groupe fondamental Énoncé Schéma en groupe fondamental modéré Application au calcul de fibrés paraboliques finis de groupe d'holonomie résoluble Introduction et notations Compléments sur les fibrés finis Image directe d'un fibré fini La méthode des petits groupes de Wigner et Mackey Fibrés finis de groupe d'holonomie résoluble Fibrés paraboliques finis de groupe d'holonomie résoluble Fibrés paraboliques finis obtenus comme image directe le long d'un morphisme modérément ramifié Fibrés inversibles de torsion sur les champs des racines Un exemple explicite Problème ouvert 2-limite inductive filtrée de catégories 2-limite Cas des catégories References

0704.1237

Infrared High-Resolution Spectroscopy of Post-AGB Circumstellar Disks. I. HR 4049 - The Winnowing Flow Observed?

Infrared High-Resolution Spectroscopy of Post-AGB Circumstellar Disks. I. HR 4049 – The Winnowing Flow Observed? Kenneth H. Hinkle National Optical Astronomy Observatory1, P.O. Box 26732, Tucson, AZ 85726-6732 [email protected] Sean D. Brittain2 National Optical Astronomy Observatory, P.O. Box 26732, Tucson, AZ 85726-6732 Clemson University, Department of Physics and Astronomy, Clemson, SC 29634; [email protected] David L. Lambert The W.J. McDonald Observatory, University of Texas, Austin, TX 78712 USA; [email protected] ABSTRACT High-resolution infrared spectroscopy in the 2.3-4.6 µm region is reported for the peculiar A supergiant, single-lined spectroscopic binary HR 4049. Lines from the CO fundamental and first overtone, OH fundamental, and several H2O vibration-rotation transitions have been observed in the near-infrared spectrum. The spectrum of HR 4049 appears principally in emission through the 3 and 4.6 µm region and in absorption in the 2 µm region. The 4.6 µm spectrum shows a rich ‘forest’ of emission lines. All the spectral lines observed in the 2.3-4.6 µm spectrum are shown to be circumbinary in origin. The presence of OH and H2O 1Operated by Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation 2Michelson Fellow http://arxiv.org/abs/0704.1237v1 – 2 – lines confirm the oxygen-rich nature of the circumbinary gas which is in contrast to the previously detected carbon-rich material. The emission and absorption line profiles show that the circumbinary gas is located in a thin, rotating layer near the dust disk. The properties of the dust and gas circumbinary disk and the spectroscopic orbit yield masses for the individual stars, MA I ∼0.58 M⊙ and MM V ∼0.34 M⊙. Gas in the disk also has an outward flow with a velocity of km s−1. The severe depletion of refractory elements but near-solar abundances of volatile elements observed in HR 4049 results from abundance winnowing. The separation of the volatiles from the grains in the disk and the subsequent accretion by the star are discussed. Contrary to prior reports, the HR 4049 carbon and oxygen isotopic abundances are typical AGB values: 12C/13C=6+9 and 16O/17O>200. Subject headings: accretion disks — stars:abundances — stars:AGB and post- AGB — stars:chemically peculiar — stars:evolution — stars:winds,outflows 1. Introduction HR 4049 is the prototype for a class of peculiar, post-AGB, single-lined, long-period, spectroscopic binaries (van Winckel et al. 1995). The primary star of these binaries is an early-type supergiant with very peculiar abundances. Most objects in this class exhibit strong infrared excesses of circumstellar origin with carbon-rich circumstellar dust typically present. The combination of carbon-rich material, high luminosity, and location out of the galactic plane forms the basis for the post-AGB designation. The peculiar designation stems from a photospheric abundance pattern characterized by a severe deficiency of refrac- tory (high dust condensation temperature) elements and a near-solar abundance for volatile (low dust condensation temperature) elements. The abundance anomalies indicate that the present photosphere contains material from which refractory elements have been very largely removed, i.e., a winnowing of dust from gas has occurred. The basic characteristics of the prototype object HR 4049 are well known. The photo- sphere has an effective temperature of about 7500 K and shows extreme abundance differ- ences between refractory and volatile elements: for example, HR 4049 has [Fe/H]∼-4.8 but [S/H]∼-0.2 (Waelkens et al. 1991b; Takada-Hidai 1990). The orbital period of HR 4049 is 430 days (Waelkens et al. 1991a), leading to a minimum separation between the two stars of 190 R⊙. Bakker et al. (1998) pointed out that the orbit requires a phase of common envelope evolution when the primary was at the tip of the AGB and had a radius of ∼ 250 R⊙. This phase altered the masses and abundances of the components. – 3 – The HR 4049 infrared excess is pronounced redward of ∼1 µm and is very well fit by a single blackbody at a temperature of about 1150 K and attributed to radiation from the tall inner walls of an optically thick circumbinary disk (Dominik et al. 2003). A circumbinary Keplerian rotating disk appears a common feature of the HR 4049 class of post-AGB binaries (De Ruyter et al. 2006). The inner walls of the HR 4049 circumbinary disk are ∼ 10 AU from the binary or 50 times the radius of the supergiant. Our line of sight to the binary nearly grazes the edge of the disk; the angle of inclination of the line of sight to the normal to the disk is about 60◦. A cartoon of the system is shown in Figure 1. Superimposed on the infrared dust continuum are emission features. Waters et al. (1989) detected features due to polycyclic aromatic hydrocarbons (PAHs). This result was confirmed by the ISO/SWS spectrum of Beintema et al. (1996). Geballe et al. (1989) con- firmed a C-rich circumstellar environment by detecting 3.43 and 3.53 µm emission features later identified with hydrogen-terminated crystalline facets of diamond (Guillois et al. 1999). Remarkably, Dominik et al. (2003) report that the gas molecular species seen in the infrared (ISO) spectrum are those expected of an O-rich mixture, suggesting that the HR 4049 cir- cumbinary environment is a blend of C-rich dust and O-rich gas. Optical spectroscopy offers some information on the disk’s gas. Bakker et al. (1998) re- port changes in the Hα line profile with orbital phase. Bakker et al. (1998) and Bakker et al. (1996) detected a broad (∼15 km s−1) stationary emission component in the Na I D2 and [O I] 6300 Å lines which they attributed to the circumbinary disk. However, the infrared offers much more readily interpreted signatures of gas in and around the binary. Lambert et al. (1988) detected the 12C16O first overtone spectrum in absorption with an excitation tem- perature of about 300 K. Cami & Yamamura (2001) on analyzing an ISO/SWS spectrum of CO2 emission bands found strong contributions from isotopomers containing 17O and 18O which they interpreted as 16O/17O = 8.3 ± 2.3 and 16O/18O = 6.9±0.9. The origins of the HR 4049 class of chemically peculiar supergiants and the structure of their circumbinary/circumstellar material remain ill-understood. New observational attacks appear to be essential. In this paper we report on a detailed look at several regions of the 2-5 µm infrared spectrum of HR 4049. 2. Observations & Data Reduction The spectrum HR 4049 was observed at high resolution at a number of near-infrared wavelengths in the 2.3 to 4.6 µm region using the 8m Gemini South telescope and the NOAO high-resolution near-infrared Phoenix spectrometer (Hinkle et al. 1998, 2000, 2003). – 4 – Phoenix is a cryogenically cooled echelle spectrograph that uses order separating filters to isolate sections of individual echelle orders. The detector is a 1024×1024 InSb Aladdin II array. Phoenix is not cross dispersed and the size of the detector in the dispersion direction limits the wavelength coverage in a single exposure to about 0.5%, i.e. 1550 km s−1, which is 0.012 µm at 2.3 µm (22 cm−1 at 4300 cm−1) and 0.024 µm at 4.6 µm (11 cm−1 at 2100 cm−1). One edge of the detector is blemished so the wavelength coverage is typically trimmed a few percent to avoid this area. Wavelength coverage is limited overall to 0.9-5.5 µm by the InSb detector material. All the spectra discussed here were observed with the widest (0.35 arcsecond) slit resulting in a spectral resolution of R=λ/∆λ= 50,000. The central wavelengths of the regions observed are listed in Table 1. The thermal brilliance of the sky makes observations longward of ∼4 µm much more difficult than in the non-thermal 1-2.4µm region. However, for a bright star like HR 4049 this only slightly increases the already short integration time. Thermal infrared observations were done using standard infrared observing techniques (Joyce 1992). Each observation consists of multiple integrations at several different positions along the slit, typically separated by 4′′ on the sky. At thermal infrared wavelengths the telluric lines are in emission. In order not to saturate the telluric emission lines the limiting exposure time is about 30 seconds at 4.6 µm. For longer integration times, multiple exposures can be coadded in the array controller to make up a single exposure. However, HR 4049 is so bright that total exposure times of only 10 to 20 seconds were required. The delivered image FWHM at the spectrograph varied from 0.25′′– 0.80′′during the nights that spectra were taken. With the positions along the slit separated by several arcseconds the resulting spectral images were well separated on the detector. An average flat observation minus an average dark observation was divided into each frame observed and frames with the star at different places along the slit were then differenced and the spectrum extracted using standard IRAF1 routines. A hot star, with no intrinsic spectral lines in the regions observed, was also observed at each wavelength setting. The hot star was observed at airmass near that of HR 4049 and the HR 4049 spectrum was later divided by the hot star spectrum to ratio the telluric spectrum from the HR 4049 star spectrum. Wavelength calibrations were computed by using a set of telluric wavelengths obtained from the hot star spectra. The wavelength calibration yielded residuals of typically 0.25 km s−1. Observations of HR 4049 were taken in the 2 and 3 µm region as well as the 4.6 µm 1The IRAF software is distributed by the National Optical Astronomy Observatories under contract with the National Science Foundation. – 5 – region. For a bright star the stellar signal is much stronger than background radiation in these spectral regions and as a result the observations are much less challenging than 4.6 µm observations. The Phoenix observing technique for this spectral region has previously been described in Smith et al. (2002). 3. Analysis of the Spectra Observations of the 2.3, 3.0, and 4.6 µm regions reveal lines from just three molecular species: CO, OH, and H2O. As discussed in §1, HR 4049 has a carbon-rich circumstellar envelope but we did not identify any molecules associated with conditions where C>O. In the 4.6 µm region we searched for C3 and CN which should be prominent if C>O. Rather, we report on the new detection of a rich 4.6 µm forest of CO fundamental and H2O lines. The 4.6 µm region atomic hydrogen lines Pfund β and Humphreys ǫ, if present, are blended with H2O lines. The 4.6 µm HR 4049 emission line spectrum is very rich making the identi- fication of occasional atomic features problematic. The circumstellar continuum at 4.6 µm is approximately 25 times more intense than the continuum of the supergiant (Dominik et al. 2003) so features of stellar origin will be highly veiled. At 3 µm we make a first detection of the OH fundamental vibration-rotation lines in HR 4049. Before exploring the emission line spectrum, we revisit in §3.1 the 2.3 µm CO first overtone spectrum detected previously by Lambert et al. (1988). All velocities in this paper are heliocentric. In order to compare heliocentric velocities of HR 4049 with microwave observations add -11.6 km s−1 to convert to the local standard of rest. 3.1. CO First Overtone Our observations confirm and extend the discovery of first-overtone (∆v=2) vibration- rotation CO lines in absorption (Lambert et al. 1988). At the 2.3 µm wavelength of the CO first overtone, the continuum from the dust is about four times that of the supergiant. Thus, the CO absorption lines should be formed along the lines of sight to the circumbinary disk. By inspection (Figure 2), it is apparent that the rotational and vibrational temperatures are low; high rotational lines of the 2-0 band are absent and the 3-1 (and higher) bands are weak or absent relative to the low rotational lines of the 2-0 band. All of the prominent lines are attributable to the most common isotopic variant, 12C16O, but weak 2-0 13CO lines are – 6 – detectable2. The first set of observations from February 2002 show the CO lines at a radial velocity of -33±0.5 km s−1. The lines reach maximum strength at J”∼7, suggesting a rotational excitation temperature of ∼300 K. At J”=7, the R branch lines are about 17% deep and are resolved with a FWHM of 16 km s−1 compared to the instrumental resolution of 6 km s−1. Weak 13CO lines are detected with depths for the R18 to R23 lines of about 4%. Comparing lines of similar excitation suggests that 12C/13C∼10. The observed regions cover the strongest predicted C17O lines (2-0 lines near J”∼ 7) but these lines can not be convincingly identified in the spectra (Figure 2) demanding C16O/C17O > 100. A yet more stringent limit can be applied (§4) by modeling. After the original observations, additional data were collected to extend the excitation range of the lines. For instance, observations made in December 2002 included higher J lines than observed previously. Observations in December 2005 covered the low J P branch required for curve-of-growth analysis. Some wavelength intervals were reobserved over the 2002 – 2005 interval to check for variability. The radial velocity was in all cases unchanged from the -33 km s−1 measured in the original data set. This velocity is nearly equal to the -32.09 ±0.13 km s−1 systemic velocity of the spectroscopic binary (Bakker et al. 1998). The CO first overtone line profiles are symmetrical with no hint of an emission compo- nent. The line strengths showed no temporal variability. In fact, the line intensities are quite similar to those reported by Lambert et al. (1988). Similarly, the velocity is identical to the earlier reported value. The profile of the lowest excitation line, 2-0 R0, differs from others in that it possibly has a weak blue-shifted component. However, the 2-0 R0 line lies in a region with a complex telluric spectrum which is difficult to ratio out of the HR 4049 spectrum. In the February 2003 spectrum, the 2-0 R0 line appears to have components at -23.6 and -32.9 km s−1. On other dates, the blue-shifted component is less clearly resolved suggesting that it is either of variable velocity, affected by overlying emission of variable intensity and/or velocity, or a relic of the reduction process. If this blue-shifted component exists, it must originate in very cold gas (T 5 K) because the component is not detectable in the 2-0 R1 line. 2We follow the convention of omitting the superscript mass number for the most common isotope. Hence 12C16O appears as CO, etc. – 7 – 3.2. CO Fundamental In sharp contrast to the spectrum at 2.3 µm where a sparse collection of weak absorption lines are found, the spectrum near 4.6 µm is rich in emission lines (Figure 3). Emission lines were identified from four isotopic variants: CO, 13CO, C17O, and C18O with roughly equal intensities for all variants. The rarer isotopic variants 13C17O, 13C18O, and 14CO were searched for but are not present. Absorption below the local continuum is also seen in the profiles of the lowest excitation 1-0 CO lines in this interval. The observed spectral interval provides lines mostly from the 1-0 and 2-1 CO bands but a few R branch lines of the 3-2 CO band are clearly present. The maximum observable rotational level, J”∼30, is similar to that seen in the CO first overtone. Table 2 lists the detected fundamental lines of the four CO isotopic forms. Blending with other lines of different CO isotopes, vibration-rotation transitions, or H2O lines is common and results in an apparent variety of profiles. All unblended emission lines are double-peaked – see Figures 4 – 8. For all but the lowest excitation lines the blue and red peaks are of similar intensity, but characteristically with the blue peak slightly weaker than the red, and occur at velocities of -38.9 ±0.4 and -28.4 ± 0.5 km s−1, respectively. The central valley of the emission profile has a velocity of -33.7 ± 0.3 km s−1. Velocities are not dependent on the isotopic species. The observed, i.e. uncorrected for instrumental profile, full-width at zero intensity (FWZI) of the emission profile for the weaker lines is ∼ 27 km s−1, with stronger lines having FWZI up to ∼ 35 km s−1. Due to the high line density, the FWZI is a difficult parameter to measure and our values carry an uncertainty of several km s−1. The observed full-width at half maximum (FWHM) similarly depends on the line strength but much less dramatically than the FWZI. Typical FWHM values are ∼ 19 km At the observed resolution of λ/∆λ=50000, the instrument profile has a significant impact on the observed line profiles and FWZI. Assuming a Gaussian instrumental profile equal to the 6 km s−1 FWHM spectral resolution, deconvolution of this instrumental profile from the observed line profile gives a true FWZI of 18 km s−1. The line profiles are strongly smoothed by the instrumental profile. The observations show blue and red sides of the CO lines rising ∼20% above the body of typical CO emission lines with the peak at each edge having a FWHM of ∼ 4 km s−1. For more strongly saturated lines, e.g. low excitation 13CO lines the FWHM of the blue and red emission spikes are ∼8 km s−1. The intrinsic profile clearly has much stronger emission peaks. The combined absorption-emission profile for the 1-0 CO lines is shown best by the R2 and R5 lines (Figure 9). After allowance for similar blends, the profiles of the CO R1, 2, 3, and 4 lines can be judged very similar to that of the R5 line. The profile of the R0 line is – 8 – possibly of the same type but blending is more severe. The R5 profile is almost a P Cygni profile: blue absorption accompanied by red emission. However, the absorption component is not strongly blue shifted but has a velocity very similar to that of the absorption lines in the other fundamental lines and to that of the 2-0 lines. Absorption below the continuum is seen only in the 1-0 P and R branch CO lines (Figure 3 – 8). The observed interval includes the P1, P2, P3 and R0, R1, R2, R3, R4, and R5 lines with definite absorption below the local continuum seen in all these lines except P1 and R0. The strength of the absorption at R5 suggests absorption below the local continuum should be detectable to higher J lines of the R branch. Due to the isotopic shifts, the low J 1-0 lines of the isotopic variants are not in the observed interval. The lowest member of the R branch in our spectra is J” = 9 for 13CO, J” = 10 for C18O, and J” = 3 for C17O. All the 1-0 lines regardless of isotopic species have a stronger central valley than the vibrationally excited transitions. The valley almost reaches the local continuum for the 1-0 13CO lines (note 13CO 1-0 R10 and R11 in Figure 7). The central valley in the line profiles becomes asymmetric for the stronger lines. Self- absorption of the blue emission is obvious when comparing the strength of the blue and red emission in profiles of the 1-0 CO lines (Figure 9). The 1-0 CO line central absorptions are on average 76% broader on the blue side than the red side. The weaker 13CO 1-0 central absorptions are 20% broader on the blue side. The central absorption is systemically blue shifted relative to the γ velocity of the binary (Bakker et al. 1998) for the very lowest excitation lines. The shift increases with decreasing J”, with a shift of 1 km s−1 for R5 and 3.5 km s−1 for R2 (Figure 10). The intensities of the emission lines of CO, 13CO, C17O, and C18O are remarkably similar and quite different from the abundance ratios estimated from the 2-0 lines. Peak intensities of the following representative unblended lines illustrate this point: CO: 1-0 R2 28%, 2-1 R11 27%, 3-2 R11 11% 13CO: 1-0 R12 24%, 2-1 R17 14% C17O: 1-0 R11 9%, 2-1 R17 14% C18O: 1-0 R14 14% In contrast to the ratio CO/13CO ∼ 10 from the 2-0 lines, the CO/13CO intensity ratio from 1-0 and 2-1 lines of similar J is about 1.5. Even more striking is the appearance of fundamental lines of C17O and C18O with intensities about one-half that of similar lines of CO. Yet, CO/C17O > 100 from the first-overtone lines. The simplest interpretation of these contrasting ratios is that emitting regions are optically thick in all the observed fundamental lines. As is well known, the first-overtone transitions are much weaker than the fundamental – 9 – lines. Optically thin emission in fundamental and first-overtone lines from a common upper state in the second vibrational level will differ by a factor of about 100 in flux. In the case of absorption from a common state in the ground vibrational level, the absorption coefficient of the 1-0 line is similarly about factor of 100 stronger than the 2-0 line. The strengthening of the central absorption for the lowest energy vibrational transition, the asymmetric absorption, and absorption below the continuum require the presence of an absorbing gas cooler than the emitting gas. This absorbing gas has a velocity shifted to the blue of the system barycentric velocity (-32.1 km s−1) by 1 to 3.5 km s−1. The emitting gas covers a ∼18 km s−1 range of velocity but is also shifted by ∼-1.5 km s−1 relative to the barycentric velocity. 3.3. OH Fundamental The lowest excitation OH vibration-rotation 1-0 lines are in a region of considerable telluric obscuration. J”=4.5 is the lowest OH level accessible under typical water vapor conditions (a few mm of precipitable H2O) at Gemini South. However, a suitable order sorting filter was not available for the J”=4.5 wavelength. An observation was made of the P branch line region for J”=5.5. The 2Π OH ground state results in Λ-doubled rotational levels, so each rotational line is divided into four components. As a result, in spite of the large rotational line spacing for OH, several OH lines can appear in a Phoenix spectrum taken with a single grating setting. The 3.0 µm 1-0 P2f5.5 and P2e5.5 lines were detected in the spectrum of HR 4049 (Fig. 11). This spectral region has considerable telluric absorption. The removal of this absorption results in variable noise in the ratioed spectrum. The OH lines are, as are the CO lines, seen in emission. The profiles are similar to those of CO with double peaked profiles of observed FWZI ∼26 km s−1 centered at -35 km s−1. The two OH lines observed are just 5% above the continuum. These were the only lines that were detected in the 3.0 µm spectral region observed. 3.4. H2O Vibration-Rotation Lines The asymmetric top molecule H2O is known for the complexity, apparent lack of rota- tional structure, and richness of its spectrum. As a result H2O lines are much more challeng- ing to identify than vibration-rotation lines of simple diatomic molecules (Hinkle & Barnes 1979). Emission lines are clearly present in the 4.6 µm HR 4049 spectrum from three vibration-rotation bands: ν2, ν1 − ν2, and ν3 − ν2. With the above caveats on the H2O – 10 – spectrum and based on the tentative identification of four lines, the vibrationally excited band 2ν2 − ν2 possibly also contributes to the 4.6 µm spectrum. The strongest observed H2O transitions are from the ν3 − ν2 band. A number of lines identified with this band have intensities ∼20% above continuum. Typical H2O lines are weaker than typical CO lines, with many of the H2O lines identified having intensity <10% above continuum. Table 3 presents a list of the H2O lines tentatively identified in the HR 4049 spectrum. In Figures 4 - 9 these lines are labeled on the spectrum of HR 4049. Many lines (e.g. Figure 9) are unblended and clearly present. However, a fairly large number are blended with CO or other H2O lines. Due to the overlapping H2O energy levels, the line strengths of H2O lines can vary significantly between adjacent vibration-rotational transitions and, hence, the contribution of a H2O line to a blend is uncertain. The band strength, S o , is a factor of five lower for the ν1 − ν2 band than the ν3 − ν2 band. However, the band strength for the ν2 band is more than 10 6 higher than that of either of these bands (McClatchey et al. 1973). The origin of the ν2 band is ∼1.5 µm red of the region observed. While it would be of interest to observe the lowest excitation ν2 lines, for ground based observers the telluric ν2 lines are very strong and prevent observations in 6 µm region. The 2ν2 − ν2 band has similar band strength to the ν1 − ν2 and ν3 − ν2 combination bands but, unlike these bands which have origins in the regions observed, 2ν2 − ν2 has an origin near that of ν2. This adds to our suspicion of the 2ν2 − ν2 identifications. Like CO and OH lines the 4.6 µm H2O lines have a double peaked profile with emission peaks at -38.6 and -29.2 km s−1 and absorption at -33.6 km s−1. The observed FWZI of the weaker H2O lines is ∼25 km s −1, perhaps slightly more narrow than the CO lines. In addition to the observed 4.6 µm transitions, H2O also has low excitation transitions in the 3.0 and 2.3 µm regions. In particular the ν3 band crosses the 3 µm region and has a band strength similar to that of the ν2. The ν1 band is also present in the 3 µm region and, while weaker than ν2 or ν3, is a much stronger transition than the combination bands seen at 4.6 µm. Our 3.0 µm observation has an uneven continuum perhaps as a result of weak emission features. We undertook a detailed search for H2O lines but failed to identify any 3.0 µm H2O lines. Future searches of this region for H2O lines using higher signal-to-noise data and wider wavelength coverage are justified. On the other hand, our spectra in the 2.3 µm region are of very high quality with broad wavelength coverage and this region is clearly devoid of any contribution from H2O. – 11 – 3.5. Line Profile Overview In summary of the above subsections, CO vibration-rotation fundamental lines of four differently isotopically substituted species and H2O vibration-rotation lines populate the 4.6 µm region. These lines all have double peaked emission profiles of total (including both peaks) FWHM ∼ 19 km s−1. There is little difference of intensity between lines of different isotopes. The lowest excitation lines, which are only seen in 12C16O in the wavelength range observed, have a central absorption as much as 20% below the local continuum. This central absorption overwhelms the bluest of the double peaks in the emission profile but the extreme bluest edge of the emission remains. Examples of observed CO fundamental and H2O line profiles are given in Fig. 9. OH lines from the fundamental vibration-rotation transition were seen in the 3 µm region. The OH lines are in emission with double peaked profiles similar to those seen in the CO fundamental and H2O lines. The CO vibration-rotation first-overtone transition appears in absorption in the 2.3 µm region. The absorption lines are nearly as broad as the CO emission lines, FWHM ∼ 16 km s−1, but have simple Gaussian profiles and exhibit a range of line depths suggesting the lines are optically thin. 12CO dominates but weaker 13CO is detectable. The oxygen isotopes are not present. All spectral lines in the 2-5 µm region have a small (> 1 km s−1) shift blue of the systemic velocity. 4. Modeling the Molecular Probes The fundamental spectrum presents a difficult analysis task. The CO is seen in both emission and, for the very lowest excitation CO lines, absorption. The small change in intensity for emission lines over the full range of isotopes and over a large range of excitation energy (rotational levels J”=0 to 30 and vibrational transitions 1-0 to 3-2) clearly indicates that the emission lines are very saturated. A detailed investigation of these strongly saturated lines would require detailed radiative transfer and disk modeling beyond the scope of this paper. However, analysis of the CO first overtone lines, which are seen in absorption, is a much more tractable problem. The observations cover nearly all 12C16O 2-0 R branch lines from J”=0 through the highest detectable R branch line at J”=35. The largest interval of the 2-0 R branch not observed is R24 through R28. The 2-0 P branch was observed from P1 through P8. The 3-1 R branch from J” ≥ 4 also lies in the observed region. Equivalent widths of the first overtone CO line profiles were measured from the fully processed, normalized spectra. The lines were measured both by summing the absorption area and second by Gaussian fits to the line profiles. Uncertainties were estimated from the mean deviations from the Gaussian fits and by the formal uncertainty in the fitted continuum level. – 12 – Equivalent width data was used to produce an excitation plot (Figure 12). The log- linear increase of line strength with excitation level demonstrates that the high-J v=2-0 lines (J>15) are optically thin (i.e. τ < 0.7). A least squares fit to the excitation plot of these data requires a temperature of ∼550 K. However, the fit to these lines underestimates the column density of the low-J lines. To infer the temperature and optical depth of the low-J lines, we extrapolated the column density of the hot gas (inferred from the high-J lines) and subtracted that from the measured column density. In order to correct for the effects of saturation, column densities and level populations were determined from a curve of growth (COG) analysis (c.f. Spitzer 1978; Brittain et al. 2005), which relates the measured equivalent widths to column densities by taking into account the effects of opacity on a Gaussian line profile. The derived column density for a measured equivalent width only depends upon one parameter, the Doppler broadening of the line, b = σRMS/1.665, where σRMS is the RMS linewidth. To find the value of b, we apply two complementary methods: comparison of the P and R branch lines and the linearization of the excitation plot. A key assumption is that the small scale line broadening results entirely from thermal broadening. The resolved line profiles which are seen in the spectra indicate an additional large scale broadening mechanism which will be discussed in §5.1. CO exhibits absorption lines in both P (J”=J’+1) and R (J”=J’-1) branches, which have different oscillator strengths yet probe the same energy levels, e.g., the P1 absorption line originates from the same J=1 level as the R1 absorption line. Any differences in the column density derived from lines that share a common level must be due to optical depth, which is related to b. The line width, b, can be used to determine the optical depth and adjusted so that the derived level populations from the two branches agree as closely as possible. The (v=0, low-J) transitions are thermalized at densities as low as nH ∼ 10 3−4 cm−3, and at even lower densities due to radiative trapping in the rotational lines with high opacity. Therefore, the low-J lines are the ones most likely to exhibit a thermal population distri- bution. The line width that best linearizes a plot of the level populations to a common temperature in an excitation diagram is used. Subject to the above constraints, the best fitting b value in the COG analysis for HR 4049 is 0.5±0.1 km s−1. The consistency of all data to this common velocity dispersion is depicted in the excitation plot of Figure 12. With a measurement of b, equivalent widths can be directly related to column density. The column density from fitting the 2-0 ‘high-J’ lines is 4.6±0.3×1017 cm−2 at a temperature of 530±20 K. The column density of the ‘low-J’ lines is 1.6±0.2×1018 cm−2 at a temperature of 40±10 K. Uncertainty in the hot N(12CO) from the overtone lines, estimated from the measurement of unsaturated lines, is small and – 13 – dominated by measurement errors in the equivalent widths of the lines. The uncertainty in the cold gas is dominated by the uncertainty in b. Assuming that the 0.5 km s−1 b value for the cold gas applies to all the spectral lines, the opacity of the most optically thick line is ∼1.5. Increasing the b value for the hot gas lowers the optical thickness of the higher excitation lines. The detection of weak 3-1 lines allows a check on vibrational LTE in the gas. The column of CO in the v=1 state (from the v=3-1 lines) is (5.7±1.3)×1015 and the temperature is 540±80K. This is consistent with the temperature for the hot component of the v=0 12CO and 13CO branches (530±20 and 570±40 respectively). The combined 2-0 and 3-1 data give a rotational temperature of 620±20K. The vibrational temperature is 700±50K. This is consistent with a slight overpopulation of the v=1 state although the relative rotational populations are consistent. The vibrational temperature is more uncertain than the other temperatures and evidence for non-LTE populations is weak. Using the best fitting b value, the column density can be determined for other isotopic lines in the spectrum. The corresponding column density of 13CO is 2.3±0.3×1017 cm−2 at a temperature of 570±40 K. Comparing lines of similar excitation, 12C/13C ratio is 6+9 . First overtone C17O lines could not be detected. The strongest lines, assuming a 550 K excitation temperature, that are clear of both major telluric features and blending CO lines are R5 and R8 (Fig. 2). A firm upper limit on the equivalent width of these lines is 1.7 mÅ which translates to a column density of 6×1014 cm−2. At a temperature of 550 K, this corresponds to a total column density of C17O of less than 1×1016 cm−2. Allowing for the temperature uncertainty a 3σ limit for 16O/17O is >200. 5. Discussion The basic characteristics of the HR 4049 system are well understood. At the heart of HR 4049 is a single-lined spectroscopic binary (Bakker et al. 1998). The visible early A/late B supergiant is a low mass, perhaps white-dwarf mass, post-AGB star. The unseen companion is an M dwarf or white dwarf of lower mass than the supergiant. The infrared prominent feature of the HR 4049 system is the circumbinary shell. Antoniucci et al. (2005) review the various geometries proposed for the circumbinary material. Considerable evidence now points to a thick disk geometry. Detailed arguments are presented by Dominik et al. (2003). In the following discussion we adopt the Dominik et al. (2003) disk model (Figure 1) with the following key points. The disk is optically thick with a height-to-radius ratio ∼1/3. The dust on the interior disk surface facing the star is approximately isothermal at 1150 K. – 14 – The temperature of the dust wall implies a distance between the star and dust of ∼10 AU. The variability of HR 4049 and the hydrostatic scale height suggests that the inclination of the disk is ∼60◦ (i.e. the plane of the disk is tipped 30◦ from the line of sight). The optical depth of the dust, the height of the disk, and the inclination result in only the far side of the disk being observable (Figure 1). 5.1. Circumbinary Flow Previous observations of the CO first overtone are reported by Lambert et al. (1988). Based on an excitation temperature of 300 ± 100 K and a non-stellar velocity Lambert et al. (1988) conclude the CO is circumstellar. The much higher resolution and S/N data analyzed above refine the excitation temperature to 520 ± 20 K for the higher excitation lines and 40±10 K for the low excitation lines. The velocities reported here and those reported by Lambert et al. (1988) show no change over nearly 20 years, as expected for lines of circumbi- nary origin. Although of the current data is of higher precision, both data sets are consistent with an outflow velocity of ∼1 km s−1. The column densities reveal that about four times more cool gas, ∼2×1018 cm−2, is present than hot gas, ∼5×1017 cm−2. The observations demonstrate that the gas is in rotational LTE and near or in vibrational LTE. For vibrational equilibrium the critical density is nH ∼ 10 10 cm−3 (Najita et al. 1996). Taking this density and a CO column density of 2×1018 cm−2, the thickness of the CO absorption line forming region is ∼4×1011 cm. So the gas is restricted to a zone radially ∼6 R⊙ from the disk inner dust wall. The CO appears to depart slightly from vibrational LTE, so the density is likely slightly lower than the critical density. In any case, the thickness of the gas layer is certainly thin compared to the 2150 R⊙ spacing between the binary and dust wall. If the gas layer is located radially just on the star side of the dust wall, adopting the Dominik et al. (2003) geometry permits the total gas mass to be calculated. Taking the radius to be 10 AU and the height of the disk to be 1/3 the radius, the surface area of the cylindrical wall follows. The column density then gives the total number of CO molecules. Since the gas is oxygen rich, the number of CO molecules is limited by the carbon abundance. Taking [C/H] for HR 4049 from Waelkens et al. (1996) and the solar carbon abundance of Grevesse et al. (1991), the mass of the gas disk is 6 ×1026 gm, i.e. ∼ 0.1 M⊕. A total disk mass of a 33 M⊕ was suggested by Dominik et al. (2003), so the mass estimates are in accord with a thin gas zone at the edge of a more massive dust disk. The CO first overtone lines are symmetric and ∼16 km s−1 across. In contrast, a line – 15 – width 20 times smaller, ∼0.5 km s−1, is required to model the curve of growth. A 0.5 km s−1 width is consistent with thermal broadening. We suggest that the broadening to 16 km s−1 is due to a systemic flow of gas in the circumstellar shell. As noted above, our view of the dust disk continuum is limited to the side opposite from the star (Figure 1), so many kinds of axisymmetric flows could result in the observed line broadening. We consider the line shapes to constrain further our understanding of the flow. All the unblended 4.6 µm emission line profiles, including those for weaker lines, are double peaked. Since all the lines are double peaked, we discount self-absorption as the principal cause for this line shape. Double peaked lines suggest an origin in a rotating ring or disk. The observation of CO fundamental band absorption demands a P-Cygni type geometry, i.e., an emitting area extended relative to the continuum forming area. While emission is very dominant in the 4.6 µm HR 4049 spectrum, there is a hint of underlying and offset absorption in all the lines with the emission line profiles having a lower peak on the blue side than on the red side. The spectrum is dominated by saturated lines 10-20% above the continuum. However, some lines are stronger and we assume these stronger lines result as optical thick transitions cover larger areas. At 3 µm radiation from a 550 K blackbody is about half that at 4.6 µm and, indeed, the 3 µm OH ∆v=1 lines are ∼5% above continuum. Ultimately saturation combined with the physical extent of the line forming region demands a limited range of emission line strengths. The strongest lines in the 4.6 µm CO spectrum of HR 4049 are ∼40% above the dust continuum. The simple model of a rotating disk can be applied to the existing disk model (Figure 1) and tested by producing model profiles of the lines. The CO first overtone lines suggest that most of the CO occurs in a relatively thin layer. An absorption line was modeled by assuming a continuum source of 10 AU radius. The layer of absorbing gas was divided into zones of 0.1 AU along the circumference. The line RMS was assumed to be 0.5 km s−1 and the gas was assumed to be rotating in a Keplerian orbit. The resulting profile is a double peaked absorption line. This profile was then convolved with a Gaussian instrumental profile with 6 km s−1 FWHM. The resulting synthetic profile, which is an excellent match to observations, is shown in Figure 13. Is this model also consistent with the 4.6 µm emission line shapes? To investigate this question five assumptions were made: (1) The emissivity of the gas is constant over the entire region modeled. (2) The gas is in Keplerian orbits. (3) Line broadening is limited to the thermal b value, 0.5 km s−1 discussed above and the broadening from the Keplerian motion. (4) Absorption is insignificant. (5) The gas originates at 10 AU and extends to larger radii. Since the dust disk is opaque this extension is along the top of the disk (Figure 1). An extension to larger radii was included since there is no requirement for a background – 16 – continuum source for the emission line spectrum. To fit the profiles with this model we found a maximum radius of 14 AU. As for the overtone model, the disk was divided into zones, the profile from each zone shifted and weighted by the viewing aspect, and then summed into velocity bins of 0.1 km s−1. The resultant double-peaked profile of the emission line is seen in Figure 14. Convolution with a Gaussian instrumental profile of FWHM = 6 km s−1 produced a good match to a typical emission line. While consistency between the modeled and observed profiles is satisfying, the funda- mental transitions clearly require much more refined modeling to address a number of details. For instance, there is a large difference between the 550 K CO and 1150 K dust temperatures. If, as is commonly assumed (see e.g. Glassgold et al. 2004), the gas and dust temperatures are in equilibrium within the disk then the 1150 K dust temperature applies only to a sur- face layer. Radiative cooling from CO fundamental emission (Ayres & Wiedemann 1989) is largely disabled by the large optical depth of the CO lines (Glassgold et al. 2004). It is plausible that the gas undergoes heating on exiting the disk. In an isothermal model optically thick CO self-absorption occurs for the fundamental transitions; the opacity in the low J 1-0 is ∼400. The fundamental lines are seen in emission because the the 550 K temperature of the CO makes optically thick CO lines brighter at 4.6 µm than the 1150 K continuum. At the resolution of the observations, 6 km s−1, narrow self-absorption lines of 0.5 km s−1 width are largely smeared out. Additionally, the gas is certainly not isothermal. If, for instance, the gas is heated as it leaves the disk, the temperature profile could increase toward the observer. Depending on the details of the spatial filling, optical depth, and temperature profile absorption is not a requirement. Two temperatures were measured in the CO first overtone, ∼40 and ∼550 K but no velocity differences were measured between the 40 K and 550 K regions, suggesting that these temperature regions are physically close together. Both the 40 K and 550 K CO are seen in absorption against the 1150 K continuum. H2O, on the other hand, is seen only in emission. Emission lines are not spatially limited to the 1150 K continuum forming region. If the absence of H2O absorption results from H2O existing only in the disk edge region and not in the disk mid-plane, the gas is differentiated vertically as well as horizontally relative to the plane of the disk. The measured decrease in the column density in between the v=0 and v=1 levels implies that there is ample population to produce the observed optically thick 2-1 lines. However, the observation of optically thick 3-2 emission suggests that the v=3 level is populated above that expected from LTE. Overtone transitions higher than 3-1 are outside of the region observered. It would be of interest to search for the strongest lines in the 4-2 band. – 17 – The emission profile is shifted relative to the center-of-mass velocity by ∼1.5 km s−1, suggesting an outflow. If the depression of the blue wing in the emission profiles is due to absorption in front of the dust disk, the absorption line profile is formed in a region with less outflow than the extended emission line forming region. Outflow was also noted for the first overtone CO lines. The outflow increases for the very lowest excitation lines, suggesting that the gas cools in the inner-disk region and is accelerated as it flows out. For the lowest excitation CO fundamental lines the cold outflow is seen in absorption with the outflow velocity increasing (Figure 10) as excitation energy decreases. Dominik et al. (2003) suggested that along the edges of the disk an outward flow results from radiation pressure erosion. Alternatively, a disk pressure gradient can result in an outward flow (Takeuchi & Lin 2002) without a need for small grains. Indeed, the outflow could be driven initially by either gas or dust since momentum is transferred between the gas and dust through collisions (Netzer & Elitzur 1993). 5.2. Comparison with Optical Spectra A detailed analysis of time series C I, Na I D (D1 and D2), and H α spectra of HR 4049 is presented by Bakker et al. (1998). The Na D lines contain a number of absorption components as well as weak emission. Bakker et al. (1998) identify two Na D absorption components with the circumstellar environment of the binary system. These are labeled as ‘A1’ and ‘A2’ (see Table 2 and Figure 4 of Bakker et al. 1998). A1 has a velocity of ∼ -5.0 km s−1 (mean of D1 and D2) relative to the systemic velocity. A2 has a velocity of -0.8 km s−1 again relative to the systemic velocity. A2 is stronger than A1 by about 50% and has a slightly greater FWHM. The continuum in the infrared is dominated by the dust continuum. However, in the optical the continuum is entirely from the stellar photosphere. Thus the optical absorption is formed along a pencil beam originating near the center of the circumbinary disk. The A1 velocity has similar velocity to the outflow seen in the lowest excitation 1-0 CO lines. This outflow is a cold wind perhaps leaving the system. The A2 outflow is close to the outflow velocity seen in the CO first overtone as well as the slightly higher excitation 1-0 lines (Figure 10). This flow is an outward flow of warmer gas perhaps associated with circulation in the disk. Given that the star is the continuum source of the Na absorption and the rear inner walls of the disk are the continuum source for the CO absorption, perfect agreement is not expected in either line of sight velocity or in FWHM. The overtone CO has a much larger FWHM than the Na D, as expected given the larger range of velocities sampled by the CO – 18 – along the lines of slight to the CO continuum forming area (Fig. 13). Na I also has an emission component (‘A3’ in Bakker et al. 1998). This is perhaps due to fluorescent emission from the gas interior to the disk. The line profile is disrupted by the Na D absorption components but the FWMH of the emission, ∼21 km s−1, is comparable to that of the CO emission. The Hα line profile is complex. Bakker et al. (1998) identified two components, ‘Cmax’ and ‘Rmin’ which are stationary and presumably are associated with the circumbinary envi- ronment. Both are seen in absorption. Cmax has a large outward velocity, -21.3 km s −1. The velocity of Rmin is much less, -7.5 km s −1. The energetics of the Hα line are very different from those of the cold gas lines discussed in this paper. Hα also has an absorption feature ‘Bmin’ which possibly varies in anti-phase with the primary. Detailed understanding of the excited gas sampled by Hα requires modeling beyond the current discussion. 5.3. Properties of the Binary Members The conceptual picture of a thin gas layer co-rotating just in front of the dust wall suggests the observed velocities result from Keplerian rotation. A FWZI of 18 km s−1 implies a rotational velocity of 9 km s−1. Assuming Keplerian rotation and a 10 AU disk radius, the total binary mass required is 0.9 M⊙. This is in agreement with the total binary mass suggested by Bakker et al. (1998). Bakker’s mass was based on the mass function from the spectroscopic orbit and the assumption that the A supergiant had a typical white dwarf mass. The mass function from Bakker et al. (1998), the total binary mass, and the orbital inclination allows a solution for the individual masses in the binary. We make the assumption, discussed in §5.6, that the binary orbit is co-planer with the disk. The definition of the mass function then yields the masses for the individual stars. The A supergiant has a very low mass of 0.58 M⊙ confirming the post-AGB state of this star. This mass, nearly equal to that of a typical white dwarf (Bergeron et al. 1992), implies that the mass-loss process for this star has terminated. The companion mass is 0.34 M⊙. This mass does not resolve the status of the companion. While a mass of 0.34 M⊙ is low for a white dwarf and strongly suggests an M-dwarf, it is possible that the companion mass has been altered by evolution (§5.6). 5.4. Winnowing Lambert et al. (1988) report quantitative abundances for HR 4049 revealing an ex- – 19 – tremely metal-poor star with [Fe/H] < -3 but near-solar C, N, and O: [C/H] = -0.2, [N/H]=0.0, [O/H]=-0.5. Lambert et al. (1988) argue that the ultra-low iron abundances found in a post- AGB star cannot be primordial since there are no known progenitor AGB stars with similar abundances. Venn & Lambert (1990) and Bond (1991) find similar abundance patterns to those in HR 4049 in the young main-sequence λ Boo stars and gas in the interstellar medium (ISM). In all three cases the abundance pattern is deficient in refractory (high condensation temperature) elements but nearly solar in volatile (low condensation temperature) elements. This abundance pattern is explained in the ISM by the locking up of refractory elements in grains. Five extremely iron-deficient post-AGB stars are known in the HR 4049 class (van Winckel et al. 1995). Lambert et al. (1988) and Mathis & Lamers (1992) have noted that all are A stars with no surface convection. A likely scenario is that the observed abundances result from peculiar abundances in little more than the observed photospheric layer. The very low re- fractory abundance in the HR 4049 stars results in a much lower opacity in the photospheric material than from a normal composition making this region additionally stable against convection (Mathis & Lamers 1992). Assuming a 47 R⊙ radius for HR 4049 (Bakker et al. 1998) and referring to a 7500 K Teff , log g = 1.0 model atmosphere (Kurucz 1979), the photosphere of HR 4049 above optical depth unity contains a few percent of an Earth mass of volatile material. Mathis & Lamers (1992) postulated that the HR 4049 abundance pattern results from the separation of mass-loss gas and dust by differential forces on the gas and dust in a circumstellar shell. Waters et al. (1992) further suggested that a circumbinary disk played a critical role. Winnowing occurs as gas is accreted to the stellar surface while dust remains in the circumstellar shell or is ejected. The λ Boo stars, which are also A stars without surface convection, have similar surface abundances due to winnowing of gas from dust in a pre- main sequence disk (Venn & Lambert 1990). Mathis & Lamers (1992) found the removal of refractory elements from a solar abundance gas to be a very inefficient winnowing process. They suggested that an efficient winnowing process is one that creates a gas with a low refractory abundance. Models of disks, created mainly to explore pre-main sequence evolution, provide a rich view of the basic disk physics. While the detailed physics of the winnowing are complex, these models, combined with the current observations, reveal the basics of the winnowing process. In optically thin circumstellar regions, the radiation pressure to gravity ratio for A stars drags grains with larger than 4 microns inward while expelling grains smaller than 4 microns (Takeuchi & Artymowicz 2001). Takeuchi & Lin (2002) extend this to disk models, showing that large particles accumulate in the inner part of the disk. These models also – 20 – apply to optically thick disks where the majority of the dust in the disk is not exposed to stellar radiation (Takeuchi & Lin 2003). In this case, interaction with the stellar radiation field at the inner disk edge drives flows in the disk with the dust-to-gas ratio increasing at the inner disk edge. Dominik et al. (2003) conclude that the grain size distribution in the HR 4049 disk is currently indeterminate. However, they note that in the case where the inner disk consists of small grains, these grains will be driven outward by radiation pressure exposing a dust free region of gas. This gas layer will be driven inwards by either the gas pressure gradient or sub- Keplerian rotation. In pre-main sequence disks it has been shown that the gas interior to the dust suffers turbulent viscosity and accretes onto the central star. The viscous time scale in pre-main sequence circumstellar disks is typically estimated at ∼ 106 years (Takeuchi & Lin 2003; Hartmann et al. 1998). The observations reported here of a sheet of gas at the inner disk edge support the model where separation of volatiles from grains occurs near the inner dust disk surface. The temperature of the gas released from the grains is far to low for evaporation of refractory elements to take place. The total gas mass interior to the HR 4049 dust disk is ∼0.1 M⊕. The gas required in the stellar photosphere to alter the observed stellar abundances is one tenth this. A naive interpretation is that the surface material required to match the observed abundances could be accreted in < 105 years. For post-AGB evolution this may be too long. An alternative is that the winnowing process currently observed is the termination of a very rapid clearing of the inner disk region that result in sudden accretion of the gas now present in the stellar photosphere. The observed CO, H2O, OH is near the disk. The flow to the stellar surface is presumed much more tenuous and is not observed. Takeuchi & Lin (2002) found that higher than a few disk scale heights from the disk midplane the gas rotates faster than the particles due to an inward pressure gradient. This drag causes particles to move outward in the radial direction. Takeuchi & Lin (2003) spec- ulate that in an optically thick disk, particles in the irradiated surface layer move outward, while beneath the surface layer, particles move inward. The outward flow seen in CO plus the driving entrained particles could rejoin the cool outer portions of the disk. In this case, the inward interior disk flow would move this material to the inner disk surface. The winnowing process could then be a distillation process resulting in a disk with increasingly refractory grains. – 21 – 5.5. Isotopic Abundances The oxygen isotopic ratios reported by Cami & Yamamura (2001) set HR 4049 apart as having by a factor >10 the smallest ratios of 16O/17O and 16O/18O known at that time. Our analysis of the optically thin CO first overtone transition does not support these results. There are no detectable C17O first overtone lines giving a 3σ limit of 16O/17O > 200. On the other hand, the fundamental spectrum of CO in HR 4049 consists of optically thick emission lines. Four isotopic variants (12C16O, 13C16O, 12C17O, and 12C18O) can be seen in the fundamental spectrum with lines of similar intensity. The CO2 bands measured by Cami & Yamamura (2001) were observed at low resolution by ISO and are in the 13 - 17 µm region of the infrared. These bands appear in emission. We contend that the oxygen isotope ratios appear small in these CO2 bands because, as for the CO fundamental, the emission lines are highly saturated. Cami & Yamamura (2001) warn that their isotopic ratios are in the optically thin limit. The carbon and oxygen isotopic ratios appear typical for an AGB star (Lambert 1988). While the oxygen isotopic ratio in the circumstellar environment of HR 4049 is not ab- normally low, there are stars that do have extreme oxygen isotope values. Some hydrogen deficient carbon stars have 16O/18O considerably less than 1 (Clayton et al. 2005, 2007). Clayton et al. (2007) suggest that the extreme overabundance of 18O observed in these ob- jects is the result of He-burning in white dwarf mergers. Meteoritic samples have been found with small 16O/18O ratios. These could result from processes in the pre-solar nebula or pollution from a stellar source. While the rarity of stellar sources with small oxygen isotope ratios suggests a stellar source is unlikely, the origin of exotic oxygen isotopic ratios detected in early solar system samples remains uncertain (Aleon et al. 2005). 5.6. Binary Evolution ISO data described by Dominik et al. (2003) contain features from oxygen-rich molecules implying that the disk is oxygen-rich. The lack of mid-infrared silicate features associated with oxygen-rich grains is attributed by Dominik et al. (2003) to high optical depth in the dust disk. Our observations reveal a 2.3 – 4.6 µm spectrum resulting from a mix of gas phase molecules, CO, OH, and H2O, typical for an oxygen-rich environment. If, as seems probable, the gas consists of volatiles evaporated from grains then the disk environment is oxygen rich. In contrast, as reviewed in §1, carbon-rich circumstellar grains have been observed. An explanation is that these grains are exterior to the disk. Carbon-rich chemistry is the result – 22 – of evolution in the AGB phase where CNO material processed in the stellar interior is mixed to the surface. For AGB stars of mass 2, the surface chemistry of the AGB star is converted to carbon-rich by the third dredge up. Rapid AGB mass loss then produces a carbon-rich circumstellar shell. In the case of HR 4049 the fossil carbon-rich shell of AGB mass loss is still observable although HR 4049 is now a post-AGB object. Why is the disk oxygen-rich? Bakker et al. (1996) noted that the current binary separa- tion is less than the radius required for the AGB phase of the current post-AGB star. Hence, prior to the post-AGB stage the HR 4049 system underwent common envelope evolution. Prior to the common envelope phase the system passed through a pre-AGB contact binary phase with the more massive star transferring mass onto the less massive member. An AGB star does not contract due to mass loss, so the AGB star continued to expand enveloping the dwarf companion. Common envelope systems rapidly eject mass from both members (Taam & Sandquist 2000). Most carbon stars have C/O near unity (Lambert et al. 1986). Mixing or mass transfer during the common-envelope stage converted the carbon-rich enve- lope of the AGB star back to an oxygen-rich envelope. Mass lost during the common-envelope phase formed the current circumbinary disk. In such a scenario a co-rotating circumbinary disk is formed surrounding the binary (Rasio & Livio 1996). This process is apparently not unusual. De Ruyter et al. (2006) find that circumbinary disks are a common feature of post-AGB stars. The current A-supergiant M-dwarf HR 4049 binary is rapidly evolving to a white-dwarf M-dwarf binary system. White-dwarf M-dwarf binary systems with a co-binary disk resulting from common-envelope evolution also appear to be common (Howell et al. 2006). 6. Conclusions The 2 to 5 µm spectrum of HR 4049 is formed in a circumbinary disk and wind. The optically thin 2.3 µm CO lines appear in absorption against the dust continuum, allowing the determination of the mass of gas. The gas forms a thin layer, of radial thickness ∼ 6 R⊙, lining the dust disk. This gas is composed of the volatiles separated in the disk from grains. The 4.6 µm emission spectrum requires a region of line formation extended beyond the continuum forming region. The circumbinary gas is rotating with a Keplerian velocity of ∼ 9 km s−1. Combined with the circumbinary disk radius and inclination derived from photometry, Keplerian rotation allows the determination of the masses of the individual binary stars. The very low mass of the A supergiant, 0.58 M⊙, confirms the post-AGB nature of this object. Gas is also flowing out of the system, perhaps as a result of a disk pressure gradient, at ∼1 km s−1. – 23 – Our observations show that the HR 4049 circumbinary disk has typical AGB abundances for the carbon and oxygen isotopes; 12C/13C = 6+9 and 16O/17O >200. Exotic mechanisms, as proposed e.g. by Lugaro et al. (2005), for production of the 16O/17O are not required. The widely quoted value of 16O/17O ∼ 8 reported by Cami & Yamamura (2001) results from the naive interpretation that the infrared emission lines are optically thin. Cami & Yamamura (2001) warn that their values were in the optically thin limit. The extreme saturation of lines in the 4.6 µm spectrum of HR 4049 results in nearly equal apparent strengths for isotopic variants of molecular species with abundances differing by factors of 103. The peculiar surface abundances of HR 4049 are likely the result of winnowing driven by the evaporation of volatiles in the disk and the viscous accretion of this gas onto the star. Detailed modeling of the process will be required to determine if there is adequate time for the current outgassing of the disk to fully explain the surfaces abundances of the A supergiant or if a sudden, post-common envelope clearing of the inner disk is required. The existence of the λ Boo stars shows that the winnowing process applies to pre-main sequence as well as post-AGB systems. The wider significance of the winnowing process may well be in systems where the convective nature of the stellar photosphere cancels any impact on stellar abundances. However, circumstellar grains in these systems are undergoing processes separating volatile and refractory elements. This winnowing could have general application to the chemical evolution of grains in pre-main sequence disks. Carbon-rich circumstellar material implies that the post-AGB star was a carbon-rich star on the AGB. The current oxygen-rich circumstellar disk likely evolved from common envelope mixing. HR 4049 is one of five known post-AGB stars with similar photospheric abundances. Of the other four at least one, the red-rectangle nebula/binary HD 44179, has a similar oxygen-rich circumbinary disk in a carbon-rich circumstellar shell (Waters et al. 1998). In future papers of this series we will explore the infrared spectra of the other members of the HR 4049 class of objects. This paper is based in part on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a coop- erative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Particle Physics and Astronomy Research Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Re- search Council (Australia), CNPq (Brazil), and CONICRT (Argentina). The observations were obtained with the Phoenix infrared spectrograph, which was developed and is oper- ated by the National Optical Astronomy Observatory. The spectra were obtained as part of programs GS-2002A-DD-1, GS-2002B-DD-1, GS-2003A-DD-1, GS-2004A-DD-1, and GS- 2005B-DD-1. We thank Drs. Claudia Winge and Bernadette Rodgers and the Gemini South – 24 – staff for their assistance at the telescope. We thank Dr. Richard Joyce for useful discussions. We thank the anonymous referee for a very detailed critical reading of the draft. S.D.B. ac- knowledges that work was performed under contract with the Jet Propulsion Laboratory (JPL) funded by NASA through the Michelson Fellowship Program. JPL is managed for NASA by the California Institute of Technology. REFERENCES Aleon, J., Robert, F., Duprat, J., & Derenne, S. 2005, Nature, 437, 385 Antoniucci, S., Paresce, F., & Wittkowski, M. 2005, A&A, 429, L1 Ayres, T. R. & Wiedemann, G. R. 1989, ApJ, 338, 1033 Bakker, E. J., van der Wolf, F. L. A., Lamers, H. J. G. L. M., Gulliver, A. F., Ferlet, R., & Vidal-Madjar, A. 1996, A&A, 306, 924 Bakker, E. J., Lambert, D. L., Van Winckel, H., McCarthy, J. K., Waelkens, C., & Gonzalez, G. 1998, A&A, 336, 263 Beintema, D. A., van den Ancker, M. E., Molster, F. J., Waters, L. B. F. 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R., & Waelkens, C. 1992, A&A, 262, L37 Waters, L. B. F. M., et al. 1998, Nature, 391, 868 Waelkens, C., Lamers, H. J. G. L. M., Walters, L. B. F. M., Rufener, F., Trams, N. R., LeBertre, T., Ferlet, R., & Vidal-Madjar, A. 1991a, A&A, 242, 433 Waelkens, C., Van Winckel, H., Bogaert, E., & Trams, N. R. 1991b, A&A, 251, 495 Waelkens, C., Van Winckel, H., Waters, L.B.F.M., & Bakker, E.J. 1996, A&A, 314, L17 This preprint was prepared with the AAS LATEX macros v5.2. – 27 – Table 1. Log of Observations Date Wavelength Frequency S/N (µm) (cm−1) 2002 Feb 13 2.3120 4324 290 2002 Feb 13 2.3233 4303 220 2002 Feb 13 2.3309 4289 280 2002 Feb 13 2.3421 4268 260 2002 Feb 13 2.3617 4233 400 2002 Dec 11 2.3405 4272 80 2002 Dec 12 2.3126 4323 90 2002 Dec 12 4.6434 2153 200 2002 Dec 12 4.6629 2144 190 2002 Dec 14 2.2980 4350 350 2002 Dec 14 2.9977 3335 130 2002 Dec 14 4.6219 2163 150 2002 Dec 14 4.6825 2135 170 2003 Feb 16 2.3416 4269 210 2004 Apr 3 4.8874 2045 220 2005 Dec 10 2.3634 4230 300 2005 Dec 10 2.3523 4250 350 – 28 – Table 2. CO ∆v=1 Line List Species Line ic 1 Line ic Line ic Line ic 12C16O 1-0 P3 1.36: 1-0 P2 1.25 1-0 P1 1.27 1-0 R0 1.28 1-0 R1 1.26 1-0 R2 1.25 1-0 R3 1.17 1-0 R4 1.36 1-0 R5 1.38 2-1 P18 1.36 2-1 R3 · · · 2-1 R4 1.27 2-1 R6 · · · 2-1 R7 1.37 2-1 R8 1.31 2-1 R8 1.31 2-1 R9 1.26 2-1 R10 · · · 2-1 R11 1.27 2-1 R12 1.27 2-1 R13 1.22 3-2 P11 1.13 3-2 R10 · · · 3-2 R11 1.08 3-2 R12 · · · 3-2 R13 · · · 3-2 R14 1.08 3-2 R15 · · · 3-2 R16 · · · 3-2 R18 · · · 3-2 R21 1.09 5-4 P5 1.03 13C16O 1-0 R9 · · · 1-0 R10 1.23 1-0 R11 1.20 1-0 R12 1.23 1-0 R13 1.22 1-0 R14 · · · 1-0 R15 1.27 1-0 R16 1.20 1-0 R17 1.20 1-0 R18 1.21 1-0 R19 1.17 1-0 R20 · · · 2-1 P7 1.28 2-1 P6 1.25 2-1 R17 1.12 2-1 R18 · · · 2-1 R19 1.16 2-1 R20 · · · 2-1 R21 · · · 2-1 R22 1.13 2-1 R23 · · · 2-1 R24 · · · 2-1 R25 1.13 2-1 R26 · · · 2-1 R27 1.09 2-1 R28 · · · 2-1 R29 · · · 2-1 R30 1.08 12C17O 1-0 P17 1.14 1-0 R3 · · · 1-0 R4 1.11 1-0 R5 1.15 1-0 R6 · · · 1-0 R8 · · · 1-0 R9 1.12 1-0 R10 · · · 1-0 R11 1.08 1-0 R12 · · · 1-0 R13 · · · 2-1 R17 · · · 2-1 R19 · · · 12C18O 1-0 P13 1.26 1-0 P12 1.24 1-0 R10 1.12 1-0 R11 1.14 1-0 R12 · · · 1-0 R13 · · · 1-0 R14 1.12 1-0 R15 1.12 1-0 R16 1.18 1-0 R17 1.13 1-0 R18 1.11 1-0 R19 · · · 1-0 R21 1.13 1-0 R22 1.12 2-1 R22 1.03 – 29 – Note. — Central intensities at the maximum emission strength. Continuum = 1.0. – 30 – Table 3. H2O Line List Vibrational Rotational ic Rotational ic Rotational ic Transition Transition Transition Transition (100)-(010) [2,2,0]-[1,1,1] 1.07 [2,2,1]-[1,1,0] 1.19: [3,1,3]-[2,0,2] · · · [3,2,1]-[4,1,4] 1.13 [3,2,2]-[2,1,1] · · · [4,0,4]-[3,1,3] · · · [4,1,4]-[3,0,3] 1.16 [4,2,3]-[4,1,4] 1.11 [5,0,5]-[4,1,4] 1.13 [5,1,5]-[4,0,4] 1.10 [5,2,4]-[5,1,5] · · · [5,4,2]-[5,3,3] 1.03 [5,5,0]-[5,4,1] 1.06 [5,5,1]-[5,4,2] · · · [6,1,5]-[5,2,4] 1.08 [6,1,5]-[6,0,6] · · · [6,2,5]-[6,1,6] 1.07: [6,3,4]-[6,2,5] · · · [7,2,5]-[6,3,4] 1.05 [7,3,5]-[7,2,6] 1.07: [8,2,6]-[8,1,7] 1.05 [8,3,6]-[8,2,7] · · · [9,2,7]-[9,1,8] 1.08 [9,3,6]-[8,4,5] · · · [9,3,7]-[9,2,8] · · · [10,4,7]-[10,3,8] 1.03 (010)-(000) [6,5,1]-[5,2,4] · · · [8,8,1&0]-[7,7,0&1] · · · [9,5,5]-[8,2,6] · · · [9,8,1&2]-[8,7,2&1] 1.04 [10,3,7]-[10,0,10] 1.02 [10,3,8]-[9,0,9] 1.04 [10,4,7]-[9,1,8] 1.06 [11,3,8]-[10,2,9] 1.08 [11,7,4]-[10,6,5] 1.04 [11,7,5]-[10,6,4] 1.03 [12,4,8]-[11,3,9] · · · [12,6,6]-[11,5,7] 1.06 [13,5,8]-[12,4,9] 1.05 [13,5,9]-[12,4,8] 1.07 [13,6,7]-[12,5,8] · · · [14,6,9]-[13,5,8] 1.06 [15,6,10]-[14,5,9] · · · [16,6,11]-[15,5,10] · · · (001)-(010) [0,0,0]-[1,0,1] · · · [1,1,0]-[1,1,1] · · · [1,1,1]-[1,1,0] 1.15 [2,1,2]-[2,1,1] · · · [2,2,0]-[2,2,1] 1.18 [2,2,0]-[3,0,3] · · · [2,2,1]-[2,2,0] · · · [3,0,3]-[2,2,0] 1.20 [3,2,1]-[3,2,2] · · · [3,2,2]-[3,2,1] 1.23: [3,2,2]-[4,2,3] 1.21 [4,0,4]-[3,2,1] · · · [4,2,2]-[4,2,3] 1.06 [5,0,5]-[4,2,2] 1.02 [5,1,4]-[4,3,1] · · · [6,1,5]-[5,3,2] 1.01: [6,3,3]-[6,3,4] 1.02 [7,1,6]-[6,3,3] · · · [7,3,4]-[7,3,5] 1.09 [8,1,7]-[7,3,4] · · · [8,2,6]-[7,4,3] · · · [8,3,5]-[8,3,6] 1.03 (020)-(010) [5,5,0]-[4,2,3] · · · [6,5,2]-[5,2,3] 1.01 [8,8,0&1]-[7,7,1&0] 1.05 [10,2,8]-[9,1,9] · · · [10,3,8]-[9,0,9] · · · [10,4,7]-[9,1,8] 1.02 [10,7,4]-[9,6,3] 1.03 [10,7,3]-[9,6,4] · · · [11,3,8]-[10,2,9] · · · – 31 – – 32 – Primary Secondary highly optically thick surfaces 1150 K surface stable dust disk gas outflow driven by radiation pressure on dust 1150 K Dust Surface Cold Dust Cold Disk Vignetting Li mitCold Disk Vignetting Li Fig. 1.— Cartoon of the ‘wall’ model for HR 4049 (top) taken from Dominik et al. (2003). Below the Dominik model the spatially resolved observer’s view of the system is shown in cross section. The disk is illuminated only from the inside. The cold disk blocks a large section of the inner 1150 K surface from view. The observer sees only that section of the 1150 K disk inside the oval labeled “Cold Disk Vignetting Limit.” – 33 – Fig. 2.— A selection of CO 2-0 R branch lines. The abscissa consists of ∼8 Å (1.5 cm−1 or ∼100 km s−1) increments of spectrum centered on each of the labeled lines. Top row shows 12C16O lines, middle row 13CO lines and bottom row C17O lines. The columns aline the rotation quantum number J for the isotopic lines to approximately equal excitation (within J”±1). The spectral region containing 13CO was not well covered, hence only a few lines are shown, however, 13CO lines are clearly present in the spectrum. Only limits to the C17O lines are detected. All CO first overtone lines have purely absorption profiles. – 34 – Fig. 3.— Overview of the 4.61 – 4.69 µm spectrum of HR 4049 showing the forest of CO and H2O emission lines. Gapped spectral regions indicate failure to restore the spectrum of HR 4049 due to optically thick telluric lines. – 35 – Fig. 4.— The 4.62µm region spectrum of HR 4049 with line identifications. – 36 – Fig. 5.— As per Figure 4 for the 4.64µm region. – 37 – Fig. 6.— As per Figure 4 for the 4.66µm region. – 38 – Fig. 7.— As per Figure 4 for the 4.68µm region. – 39 – Fig. 8.— As per Figure 4 for the 4.89µm region. – 40 – Fig. 9.— An enlarged view of the spectrum shown in Figures 4 and 5 showing the regions surrounding the 12C16O R 2 line (left) and the R 5 line (right). The R 2 line is unblended on the red wing while the R 5 line is unblended on the blue wing. Both lines have P-Cygni type profiles. Higher excitation CO lines as well as H2O lines shown in this Figure exhibit typical double peaked emission profiles. – 41 – Fig. 10.— Radial velocities of the absorption component of the CO 1-0 low excitation lines (J”=0 through 5) as a function of excitation energy of the lower level. A number of these lines are blended with other circumstellar lines. The bar to the right labeled ‘ high-J” ’ is at the value of the mid-emission absorption for higher excitation lines. There is a clear trend for the lowest excitation lines to have a larger outflow. The dashed line is the binary system γ-velocity (Bakker et al. 1998). – 42 – Fig. 11.— The 3µm region spectrum of HR 4049 showing the OH 1-0 P2f5.5 and 1-0 P2e5.5 lines. – 43 – E“/k (K) T=570±40K T=530±20KT=40±10K b=0.5km/s N13CO=(2.29±0.34)x10 17 cm-2 N12COcold=(1.65±0.28)x10 18 cm-2 /2J+1)-4 N12COv=0=(4.58±.26)x10 17 cm-2 N12COv=1=(5.7±1.3)x10 15 cm-2 T=540±80K Fig. 12.— Boltzman plot for HR 4049 first overtone CO lines. Data are shown for the two isotopic species of CO that were detected in the first overtone spectra, 12C16O and 13C16O. The 12C16O excitation temperature is 40±10 K the low excitation lines and 530±20 K for the high excitation 2-0 lines. The four 3-1 12C16O lines populate the 5000-6000 K region of the abscissa with an excitation temperature of 540±80K, suggesting a slight departure from vibrational LTE. However, a fit through all higher excitation lines remains within the uncertainties and gives an excitation temperature of 620±20 K. The 13CO lines have an excitation temperature of 570±40 K. – 44 – -40 -20 0 20 40 km s-1 Fig. 13.— Synthetic line profiles for the CO first overtone. The dot-dash line results from modeling a thin gas layer on the interior surface of the dust disk (see text). The dash line is the same model spectral line convolved to the instrumental resolution. The solid line is the observed profile of the CO R6 line. km s-1 Fig. 14.— A synthetic line profile for the CO fundamental lines compared to an observed profile. The dot-dash line is the synthetic line profile from an emitting zone near the ‘cold disk vignetting limit’ in Figure 1 (see text). The dash line shows this profile convolved to the instrumental resolution. The upper solid line is an observed CO line profile and the lower solid line is the difference between the model and observed line profile. Introduction Observations & Data Reduction Analysis of the Spectra CO First Overtone CO Fundamental OH Fundamental H2O Vibration-Rotation Lines Line Profile Overview Modeling the Molecular Probes Discussion Circumbinary Flow Comparison with Optical Spectra Properties of the Binary Members Winnowing Isotopic Abundances Binary Evolution Conclusions

0704.1238

Tannakian Categories attached to abelian Varieties

Tannakian Categories attached to abelian Varieties Rainer Weissauer August 22, 2021 Let k be an algebraically closed field k, where k is either the algebraic closure of a finite field or a field of characteristic zero. Let l be a prime different from the characteristic of k. Notations. For a variety X over k let Dbc(X,Ql) denote the triangulated cate- gory of complexes of etale Ql-sheaves on X in the sense of [5]. For a complex K ∈ Dbc(X,Ql) let D(K) denote its Verdier dual, and H ν(K) denote its etale cohomology Ql-sheaves with respect to the standard t-structure. The abelian subcategory Perv(X) of middle perverse sheaves is the full subcategory of all K ∈ Dbc(X,Ql), for which K and its Verdier dual D(K) are contained in the full subcategory pD≤0(X) of semi-perverse sheaves, where L ∈ Dbc(X,Ql) is semi- perverse if and only if dim(Sν) ≤ ν holds for all integers ν ∈ Z, where Sν denotes the support of the cohomology sheaf H−ν(L) of L. If k is the algebraic closure of a finite field κ, then a complex K of etale Ql- Weil sheaves is mixed of weight ≤ w, if all its cohomology sheaves Hν(K) are mixed etale Ql-sheaves with upper weights w(H ν(K)) − ν ≤ w for all integers ν. It is called pure of weight w, if K and its Verdier dual D(K) are mixed of weight ≤ w. Concerning base fields of characteristic zero, we assume mixed sheaves to be sheaves of geometric origin in the sense of the last chapter of [1], so we still dispose over the notion of the weight filtration and purity and Gabber’s decompo- sition theorem in this case. In this sense let Pervm(X) denote the abelian category of mixed perverse sheaves on X. The full subcategory P (X) of Pervm(X) of pure perverse sheaves is a semisimple abelian category. http://arxiv.org/abs/0704.1238v2 Abelian varieties. Let X be an abelian variety X of dimension g over an alge- braically closed field k. The addition law of the abelian variety a : X × X → X defines the convolution product K ∗ L ∈ Dbc(X,Ql) of two complexes K and L in Dbc(X,Ql) by the direct image K ∗ L = Ra∗(K ⊠ L) . For the skyscraper sheaf δ0 concentrated at the zero element 0 notice K ∗ δ0 = K. Translation-invariant sheaf complexes. More generally K ∗ δx = T ∗−x(K), where x is a closed k-valued point in X, δx the skyscraper sheaf with support in {x} and where Tx(y) = y+x denotes the translation Tx : X → X by x. In fact T ∗y (K ∗L) ∼= T ∗y (K) ∗ L ∼= K ∗ T ∗y (L) holds for all y ∈ X(k). For K ∈ D c(X,Ql) let Aut(K) be the abstract group of all closed k-valued points x of X, for which T ∗x (K) ∼= K holds. A complex K is called translation-invariant, provided Aut(K) = X(k). If f : X → Y is a surjective homomorphism between abelian varieties, then the di- rect image Rf∗(K) of a translation-invariant complex is translation-invariant. As a consequence of the formulas above, the convolution of an arbitrary K ∈ Dbc(X,Ql) with a translation-invariant complex on X is a translation-invariant complex. A translation-invariant perverse sheaf K on X is of the form K = E[g], for an or- dinary etale translation-invariant Ql-sheaf E. For a translation-invariant complex K ∈ Dbc(X,Ql) the irreducible constituents of the perverse cohomology sheaves pHν(K) are translation-invariant. Multipliers. The subcategory T (X) of Perv(X) of all perverse sheaves, whose ir- reducible perverse constituents are translation-invariant, is a Serre subcategory of the abelian category Perv(X). Let denote Perv(X) its abelian quotient category and P (X) the image of P (X), which is a full subcategory of semisimple objects. The full subcategory of Dbc(X,Ql) of all K, for which pHν(K) ∈ T (X), is a thick subcategory of the triangulated category Dbc(X,Ql). Let c(X,Ql) be the corresponding triangulated quotient category, which contains Perv(X). Then the convolution product ∗ : D c(X,Ql)×D c(X,Ql) → D c(X,Ql) still is well defined, by reasons indicated above. Definition. A perverse sheaf K on X is called a multiplier, if the convolution induced by K ∗K : Dbc(X,Ql) → D c(X,Ql) preserves the abelian subcategory Perv(X). Obvious from this definition are the following properties of multipliers: If K and L are multipliers, so are the product K ∗ L and the direct sum K ⊕ L. Direct summands of multipliers are multipliers. If K is a multiplier, then the Verdier dual D(K) is a multiplier and also the dual K∨ = (−idX) ∗(D(K)) . Examples: 1) Skyscraper sheaves are multipliers 2) If i : C →֒ X is a projective curve, which generates the abelian variety X, and E is an etale Ql-sheaf on C with finite monodromy, then the intersection cohomology sheaf attached to (C,E) is a multiplier. 3) If : Y →֒ X is a smooth ample divisor, then the intersection cohomology sheaf of Y is a multiplier. The proofs. 1) is obvious. For 2) we gave in [7] a proof by reduction mod p using the Cebotarev density theorem and counting of points. Concerning 3) the morphism j : U = X \Y →֒ X is affine for ample divisors Y . Hence λU = Rj!Ql[g] and λY = i∗Ql,Y [g − 1] are perverse sheaves, which coincide in Perv(X). The morphism π = a◦(j×idX ) is affine. Indeed W = π−1(V ) is affine for affine subsets V of X, W being isomorphic under the isomorphism (u, v) 7→ (u, u + v) of X2 to the affine product U × V . By the affine vanishing theorem of Artin: For perverse sheaves L ∈ Perv(X) we get λU ⊠ L ∈ Perv(X2) and pHν(Rπ!(λU ⊠ L)) = 0 for all ν < 0. The distinguished triangle Ra∗(λY ⊠ L), Rπ!(λU ⊠ L), Ra∗(δX ⊠ L) for δX = Ql,X [g] and the corresponding long exact perverse cohomology sequence gives isomorphisms pHν−1(δX ∗ L) ∼= pHν(λY ∗ L) for the integers ν < 0. Since Ra∗(δX ⊠ L) = δX ∗ L is a direct sum of translates of constant perverse sheaves δX , we conclude pHν(λY ∗ L) for ν < 0 to be zero in Perv(X). For smooth Y the intersection cohomology sheaf is λY = i∗Ql,Y [g − 1], and it is self dual. Hence by Verdier duality i∗Ql,Y [g − 1] ∗ L has image in Perv(X). Thus i∗Ql,Y [g − 1] is a multiplier. � Let M(X) ⊆ P (X) denote the full category of semisimple multipliers. Let M(X) denote its image in the quotient category P (X) of P (X). Then, by the definition of multipliers, the convolution product preserves M(X) ∗ : M(X)×M(X) → M(X) . Theorem. With respect to this convolution product the category M(X) is a semisimple super-Tannakian Ql-linear tensor category, hence as a tensor cate- gory M(X) is equivalent to the category of representations Rep(G, ε) of a projec- tive limit G = G(X) of supergroups. Outline of proof. The convolution product obviously satisfies the usual commuta- tivity and associativity constraints compatible with unit objects. See [7] 2.1. By [7], corollary 3 furthermore one has functorial isomorphisms (K,L) ∼= Γ{0}(X,H 0(K ∗ L∨)∗) , where H0 denotes the degree zero cohomology sheaf and Γ{0}(X,−) sections with support in the neutral element. Let L = K be simple and nonzero. Then the left side becomes End M(X)(K) ∼= Ql. On the other hand K ∗ L∨ is a direct sum of a perverse sheaf P and translates of translation-invariant perverse sheaves. Hence H0(K ∗ L∨)∨) is the direct sum of a skyscraper sheaf S and translation-invariant etale sheaves. Therefore Γ{0}(X,H 0(K ∗ L∨)∨) = Γ{0}(X,S). By a comparison of both sides therefore S = δ0. Notice δ0 is the unit element 1 of the convolution product. Using the formula above we not only get (K,L) ∼= HomM(X)(K ∗ L ∨, 1) , but also find a nontrivial morphism evK : K ∗K ∨ → 1 . By semisimplicity δ0 is a direct summand of the complex K ∗ K∨. In particular the Künneth formula implies, that the etale cohomology groups do not all vanish identically H•(X,K) 6= 0 . Therefore the arguments of [7] 2.6 show, that the simple perverse sheaf K is du- alizable. Hence M(X) is a rigid Ql-linear tensor category. Let T be a finitely ⊗-generated tensor subcategory with generator say A. To show T is super- Tannakian, by [4] it is enough to show for all n lenghtT (A ∗n) ≤ Nn , where N is a suitable constant. For any B ∈ M(X) let B, by abuse of no- tation, also denote the perverse semisimple representative in Perv(X) without translation invariant summand. Put h(B, t) = ν dimQl(H ν(X,B))tν . Then lenghtT (B) ≤ h(B, 1), since every summand of B is a multiplier and there- fore has nonvanishing cohomology. For B = A∗n the Künneth formula gives h(B, 1) = h(A, 1)n. Therefore the estimate above holds for N = h(A, 1). This completes the outline for the proof of the theorem. � Principally polarized abelian varieties. Suppose Y is a divisor in X defining a principal polarization. Suppose the intersection cohomology sheaf δY of Y is a multiplier. Then a suitable translate of Y is symmetric, and again a multiplier. So we may assume Y = −Y is symmetric. Let M(X,Y ) denote the super-Tannakian subcategory of M(X) generated by δY . The corresponding super-group G(X,Y ) attached to M (X,Y ) acts on the super-space W = ω(δY ) defined by the underlying super-fiber functor ω of M(X). By assumption δY is self dual in the sense, that there exists an isomorphism ϕ : δ∨ ∼= δY . Obviously ϕ∨ = ±ϕ. This defines a nondegenerate pairing on W , and the action of G(X,Y ) on W respects this pairing. Curves. If X is the Jacobian of smooth projective curve C of genus g over k, X car- ries a natural principal polarization Y = Wg−1. If we replace this divisor by a sym- metric translate, then Y is a multiplier. The corresponding group G(X,Y ) is the semisimple algebraic group G = Sp(2g−2,Ql)/µg−1[2] or G = Sl(2g−2,Ql)/µg−1 depending on whether the curve C is hyperelliptic or not. The representation W of G(X,Y ) defined by δY as above is the unique irreducible Ql-representation of G(X,Y ) of highest weight, which occurs in the (g − 1)-th exterior power of the (2g − 2)-dimensional standard representation of G. See [7], section 7.6. Conjecture. One could expect, that a principal polarized abelian variety (X,Y ) of dimension g is isomorphic to a Jacobian variety (Jac(C),Wg−1) of a smooth projective curve C (up to translates of the divisor Y in X as explained above) if and only if Y is a multiplier with corresponding super-Tannakian group G(X,Y ) equal to one of the two groups Sp(2g − 2,Ql)/µg−1[2] or Sl(2g − 2,Ql)/µg−1 . References [1] Beilinson A., Bernstein J., Deligne P., Faisceaux pervers, Asterisque 100 (1982) [2] Deligne P., Milne J.S., Tannakian categories, in Lecture Notes in Math 900, p.101 –228 [3] Deligne P., Categories tannakiennes, The Grothendieck Festschrift, vol II, Progr. Math, vol. 87, Birkhäuser (1990), 111 – 195 [4] Deligne P., Categories tensorielles, Moscow Math. Journal 2 (2002) no.2, 227 – 248 [5] Kiehl R., Weissauer R., Weil conjectures, perverse sheaves and l-adic Fourier transform, Ergebnisse der Mathematik und ihrer Grenzgebiete 42, Springer (2001) [6] Weissauer R., Torelli’s theorem from the topological point of view, arXiv math.AG/0610460 [7] Weissauer R., Brill-Noether Sheaves, arXiv math.AG/0610923 http://arxiv.org/abs/math/0610460 http://arxiv.org/abs/math/0610923

0704.1239

On the Entropy Function and the Attractor Mechanism for Spherically Symmetric Extremal Black Holes

On the Entropy Function and the Attractor Mechanism for Spherically Symmetric Extremal Black Holes Rong-Gen Cai∗ Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China Li-Ming Cao† Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China Graduate School of the Chinese Academy of Sciences, Beijing 100039, China In this paper we elaborate on the relation between the entropy formula of Wald and the “entropy function” method proposed by A. Sen. For spherically symmetric extremal black holes, it is shown that the expression of extremal black hole entropy given by A. Sen can be derived from the general entropy definition of Wald, without help of the treatment of rescaling the AdS2 part of near horizon geometry of extremal black holes. In our procedure, we only require that the surface gravity approaches to zero, and it is easy to understand the Legendre transformation of f , the integration of Lagrangian density on the horizon, with respect to the electric charges. Since the Noether charge form can be defined in an “off-shell” form, we define a corresponding entropy function, with which one can discuss the attractor mechanism for extremal black holes with scalar fields. e-mail address: [email protected] e-mail address: [email protected] http://arxiv.org/abs/0704.1239v4 I. INTRODUCTION The attractor mechanism for extremal black holes has been studied extensively in the past few years in supergravity theory and superstring theory. It was initiated in the context supersymmetric BPS black holes [1, 2, 3, 4, 5, 6] and generalized to more general cases, such as supersymmetric black holes with higher order corrections [7, 8, 9, 10] and non-supersymmetric attractors [11, 12, 13, 14, 15]. Recently, A. Sen has proposed a so-called “entropy function” method for calculating the entropy of n-dimensional extremal black holes, where the extremal black holes are defined to be the space- times which have the near horizon geometry AdS2 × Sn−2 and corresponding isometry [16, 17, 18, 19]. It states that the entropy of such kind of extremal black holes can be obtained by extremizing the “entropy function” with respect to some moduli on the horizon, where the entropy function is defined as 2π times the Legendre transformation ( with respect to the electric charges ) of the integration of the Lagrangian over the spherical coordinates on the horizon in the near horizon field configurations. This method does not depend upon supersymmetry and has been applied or generalized to many solutions in supergravity theory, such as extremal black objects in higher dimensions, rotating extremal black holes, various non-supersymmetric extremal black objects and even near-extremal black holes [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. In general, for spherically symmetric extremal black holes in a theory with Lagrangian L = L(gab, Rabcd,Φs, AIa), the near horizon geometry of these black holes has the form AdS2 × Sn−2 [17, 18]. Due to SO(1, 2) × SO(n − 1) isometry of this geometry, the field configuration have the form as follows: The metric can be written down as ds2 = gabdx adxb = v1 −ρ2dτ2 + 1 + v2dΩ n−2 , (1.1) where v1, v2 are constants which stand for the sizes of AdS2 and S n−2. Some other dynamical fields such as the scalar fields and U(1) gauge fields are also taken to be constant: Φs = us and F ρτ = eI . The magnetic-type fields are also fixed with magnetic-charges pi. Then, for this configuration, defining f(v1, v2, us, eI ; pi) = dx2 ∧ · · · ∧ dxn−1 −gL , (1.2) where the integration is taken on the horizon, and {x2 · · · xn−1} are angle coordinates of Sn−2, those constant moduli can be fixed via the equations of motion = qI , (1.3) where qI are electrical-like charges for U(1) gauge fields A a. To relate the entropy of the black holes to these definitions, one defines fλ as (1.2) with the Riemann tensor part in L multiplied by a factor λ, and then one finds a relation between fλ and the Wald formula for spherically symmetric black holes [44]: SBH = −2π∂fλ/∂λ|λ=1. Consider the structure of the Lagrangian, one can find − fλ = 0 . (1.4) When the equations of motion are satisfied, the entropy of black holes turns out to be SBH = 2π(eIqI − f). Therefore, one can introduce the “entropy function” for the extremal black holes E(v1, v2, us, eI ; pi) = 2π (eIqI − f(v1, v2, us, eI ; pi)) , (1.5) which is obtained by carrying an integral of the Lagrangian density over Sn−2 and then taking the Legendre transformation with respect to the electric fields eI . For fixed electric changes qI and magnetic charges pi, these fields us and v1 and v2 are determined by extremizing the entropy function with respect to the variables us and v1 and v2. And then the entropy of the extremal black holes is given by the extremum of the entropy function by substituting the values of v1, v2 and us back into the entropy function. In addition, let us notice that if the moduli fields us are only dependent of the charges qI and pi, the attractor mechanism is then manifested, and the entropy is a topological quantity. This is a very simple and powerful method for calculating the entropy of such kind of extremal black holes. In particular, one can easily find the corrections to the entropy due to the higher derivative terms in the effective action. However, we notice that this method is established in a fixed coordinate system (1.1). If one uses another set of coordinates for the AdS2 part, instead of the coordinates {ρ, τ}, it seems that one can not define an entropy function as (1.5) because the function f is not invariant under the coordinate transformation. In addition, the reason that to get the entropy of black holes, one should do the Legendre transformation with respect to the electric charges, but not include magnetic charges seems unclear in this procedure. Some authors have pointed out that the entropy function E resulting from this Legendre transformation of the function f with respect to electric charges transforms as a function under the electric-magnetic dual, while the function f does not [37]. But it is not easy to understand the Legendre transformation with respect to the angular-momentum J in the rotating attractor cases [32]. There might be a more general formalism for the entropy function, and the Legendre transformation can be naturally understood in this frame. In this paper, we will elaborate these issues in the “entropy function” method and show that a general formalism of the “entropy function” method can be extracted from the black hole entropy definition due to Wald et al. [44, 45, 46]. In this procedure, we only require that the surface gravity of the black hole approaches to zero. Our entropy expression will reduce to the expression of A. Sen if we choose the same coordinates as in [17, 18]. The extremal black holes are different objects from the non-extremal ones due to different topological structures in Euclidean sector [49, 50, 51]. The extremal black hole has vanishing surface gravity and has no bifurcation surface, so the Noether charge method of Wald can not be directly used [44]. Thus, in this paper we regard the extremal black holes as the extremal limit of non-extremal black holes as in [17, 18, 42]. That is, we will first consider non-extremal black holes and then take the extremal limit. In this sense, the definitions of Wald are applicable. The paper is organized as follows. In section II, we make a brief review on the entropy definition of Wald and give the required formulas. In section III, we give the near horizon analysis for the extremal black holes and derive the general form of the entropy. In section IV, we define the entropy function and discuss the attractor mechanism for the black holes with various moduli fields. The conclusion and discussion are given in section V. II. THE DEFINITION OF WALD In differential covariant theories of gravity, Wald showed that the entropy of a black hole is a kind of Noether charge [44, 45]. In this paper, we will use the Wald’s method to define the entropy functions for spherically symmetric black holes. Assume the differential covariant Lagrangian of n-dimensional space-times (M,gab) is L = L(gab, Rabcd,Φs, AIa) ǫ, (2.1) where we have put the Lagrangian in the form of differential form and ǫ is the volume element. Rabcd is Riemann tensor (since we are mainly concerning with extremal black holes, therefore we need not consider the covariant derivative of the Riemann tensor). {Φs, s = 0, 1, · · · } are scalar fields, {AIa, I = 1, · · · } are U(1) gauge potentials, and the corresponding gauge fields are = ∂aA − ∂bAIa. We will not consider the Chern-Simons term as [18]. The variation of the Lagrange density L can be written as δL = Eψδψ + dΘ, (2.2) where Θ = Θ(ψ, δψ) is an (n− 1)-form, which is called symplectic potential form, and it is a local linear function of field variation (we have denoted the dynamical fields as ψ = {gab,Φs, AIa}). Eψ corresponds to the equations of motion for the metric and other fields. Let ξ be any smooth vector field on the space-time manifold, then one can define a Noether current form as J[ξ] = Θ(ψ,Lξψ)− ξ · L , (2.3) where “ · ” means the inner product of a vector field with a differential form, while Lξ denotes the Lie derivative for the dynamical fields. A standard calculation gives dJ[ξ] = −ELξψ . (2.4) It implies that J[ξ] is closed when the equations of motion are satisfied. This indicates that there is a locally constructed (n − 2)-form Q[ξ] such that, whenever ψ satisfy the equations of motion, we have J[ξ] = dQ[ξ] . (2.5) In fact, the Noether charge form Q[ξ] can be defined in the so-called “off shell” form so that the Noether current (n− 1)-form can be written as [46] J[ξ] = dQ[ξ] + ξaCa , (2.6) where Ca is locally constructed out of the dynamical fields in a covariant manner. When the equations of motion hold, Ca vanishes. For general stationary black holes, Wald has shown that the entropy of the black holes is a Noether charge [44], and may be expressed as SBH = 2π Q[ξ] , (2.7) here ξ be the Killing field which vanishes on the bifurcation surface of the black hole. It should be noted that the Killing vector field has been normalized here so that the surface gravity equals to “1”. Furthermore, it was shown in [45] that the entropy can also be put into a form SBH = −2π EabcdR ǫabǫcd, (2.8) where ǫab is the binormal to the bifurcation surface H, while EabcdR is the functional derivative of the Lagrangian with respect to the Riemann tensor with metric held fixed. This formula is purely geometric and does not include the surface gravity term. In this paper, since we will treat a limit procedure with surface gravity approaching to zero, we will not normalize the Killing vector such that the surface gravity equal to one. So we use the formula (2.8) to define the entropy of black holes as in [17, 18, 42]. For an asymptotically flat, static spherically symmetric black hole, one can simply choose ξ = ∂t = For the Lagrangian as (2.1), we have δL = Eabδgab +E sδΦs + dΘ , (2.9) where I = −2ǫ∇b , (2.10) s = ǫ ∂∇aΦs , (2.11) ab = ǫ gabL+ ∂L ∂Rcdea b + 2∇c∇d ∂Rcabd (2.12) are the equations of motion for the U(1) gauge fields, the scalar fields and the metric gab, respec- tively. The symplectic potential form has the form Θa1···an−1 = ∂∇aΦs δΦs + 2 ∂Rabcd ∇dδgbc − 2∇d ∂Rdbca ǫaa1···an−1 . (2.13) Let ξ be an arbitrary vector field on the space-time, The Lie derivative of ξ on the fields are LξΦs = ξa∇aΦs, Lξgab = ∇aξb +∇bξa , LξAIa = ∇a(ξbAIb) + ξbF Iba . (2.14) Substituting these Lie derivatives into the symplectic potential form, we find Θa1···an−1 = ∂∇aΦs ξb∇bΦs + 2 ∇b(ξcAIc) + 2 ξcF Icb ∂Rabcd ∇d(∇bξc +∇cξb)− 2∇d ∂Rdbca (∇bξc +∇cξb) ǫaa1···an−1 ∂∇aΦs ξb∇bΦs + 2∇b ξcAIc − 2∇b ξcAIc + 2 ξcF Icb ∂Rabcd ∇d(∇bξc +∇cξb)− 2∇d ∂Rdbca (∇bξc +∇cξb) ǫaa1···an−1 . (2.15) Then, we have Θa1···an−1 = ξcAIc ∂Rabcd ∇[cξd] ǫaa1···an−1 + · · · ∂∇aΦs ξb∇bΦs + 2 ξcF Icb + · · · · · · ǫaa1···an−1 − 2∇b ξcAIcǫaa1···an−1 . (2.16) The first line in the above equation will give the Noether charge form, while the second line together with the terms in ξ · L in Eq. (2.3) will give the constraint which corresponds to the equations of motion for the metric. For example, the first term in the second line combined with scalar fields terms in ξ · L will give the energy-momentum tensor for scalar fields. Similarly the second term in the second line will enter the energy-momentum tensor for the U(1) gauge fields in the equations of motion for the metric. The last line in the above equation will give the constraint which corresponds to the equations of motion for the U(1) gauge fields. Thus, we find J[ξ] = dQ[ξ] + ξaCa , (2.17) where Q = QF +Qg + · · · (2.18) a1···an−2 ξcAIcǫaba1···an−2 , (2.19) a1···an−2 = − ∂L ∂Rabcd ∇[cξd]ǫaba1···an−2 . (2.20) The “ · · · ” terms are not important for our following discussion, so we brutally drop them at first. We will give a discussion at the end of the next section for these additional terms. Especially, the constraint for the U(1) gauge fields is simply ca1···an−1 = −2∇b AIcǫaa1···an−1 . (2.21) The term QF in the Q was not discussed explicitly in the earlier works of Wald et al. [44, 45, 46]. This is because that the killing vector vanishes on the bifurcation surface and the dynamical fields are assumed to be smooth on the bifurcation surface. However, in general, the U(1) gauge fields are singular on the bifurcation surface, so one have to do a gauge transformation, A→ A′ = A−A|H, such that the ξaA′a are vanished on this surface, and then Q F . This gauge transformation will modify the data of gauge potential at infinity and an additional potential-charge term ΦδQ into the dynamics of the charged black holes from infinity, where Φ = ξcAc|H is the electrostatic potential on the horizon of the charged black hole and Q is the electric charge [47]. Another treatment is: We only require the smoothness of the gauge potential projecting on the bifurcation surface, i.e., ξaAa instead of the gauge potential itself, so QF will generally not vanish on the bifurcation surface, and then Φ = ξcAc|H is introduced into the law of black hole without help of gauge transformation [48]. Similarly, in the next sections of this paper we only require that the projection of the gauge potential on the bifurcation surface is smooth. Since our final result will not depend on the gauge potential, the gauge transformation mentioned above will not effect our discussion. One can do such gauge transformation if necessary. In this paper, however, we will merely use the explicit form of the Noether charge (n − 2)-form and we will not discuss the first law. Certainly, it is interesting to give a general discussion on the thermodynamics of these black holes. The relevant discussion can be found in a recent paper [43]. III. ENTROPY OF EXTREMAL BLACK HOLES In this section, we will use the formulas above to give the general entropy function for static spherically symmetric extremal black holes. Assume that the metric for these black holes is of the ds2 = −N(r)dt2 + dr2 + γ(r)dΩ2n−2 , (3.1) where N, γ are functions of radial coordinate r, and dΩ2n−2 is the line element for the (n − 2)- dimensional sphere. The horizon r = rH corresponds to N(rH) = 0. If the equations of motion are satisfied, the constraint Ca = 0, and we have J[ξ] = dQ[ξ] . Consider a near horizon region ranged from rH to rH +∆r, we have rH+∆r Q[ξ]− Q[ξ] = Θ− ξ · L . (3.2) If ξ is a Killing vector, then Θ = 0, and rH+∆r Q[ξ]− Q[ξ] = − ξ · L . (3.3) Thus we arrive at rH+∆r g[ξ]− rH+∆r F [ξ] + F [ξ]− ξ · L . (3.4) Taking ξ = ∂t, (since we consider the asymptotically flat space-time, N(r) has the property limr→∞N(r) = 1, such that ∂t has a unit norm at infinity.), we have ∇[aξb] = 12N ǫab, and rH+∆r g[∂t]− g[∂t] N ′(rH +∆r)B(rH +∆r)−N ′(rH)B(rH) N ′′(rH)B(rH) +N ′(rH)B ′(rH) +O(∆r2) , (3.5) where we have defined a function B(r) B(r) ≡ − (n− 2)! ∂Rabcd ǫcdǫaba1···an−2dx a1 ∧ · · · ∧ dxan−2 . (3.6) Note that the QF terms in the right hand side of Eq. (3.4) can be written as rH+∆r F [∂t] + F [∂t] = AIt (rH +∆r)qI −AIt (rH)qI = qIA t (rH)∆r +O(△r2) = qIF rt(rH)∆r +O(△r2) = qI ẽI∆r +O(△r2) , (3.7) where AIt = (∂t) aAIa, ẽI ≡ F Irt(rH), and the U(1) electrical-like charges are defined to be qI = − (n− 2)! ǫaba1···an−2dx a1 ∧ · · · ∧ dxan−2 . (3.8) They do not change with the radii r. This is ensured by the Gaussian law. Note that there is an integration on the sphere part in (3.8), therefore the only F Irt in F is relevant, so that we can simply write F I (rH) as −ẽIǫab. Considering −2ẽI2 = ẽIǫabẽIǫab we have = − ∂L ∂(ẽIǫab) ab . (3.9) Substituting this result into the definition of the electric charges, we find qI = − 2(n− 2)! ǫaba1···an−2dx a1 ∧ · · · ∧ dxan−2 = ∂f̃(rH) . (3.10) Here f̃(rH) will be defined below in Eq. (3.12). The last term in the right hand side of Eq. (3.4) can be written as ∂t · L = ∫ rH+∆r dx2 ∧ · · · ∧ dxn−1 −gL = ∫ rH+∆r drf̃(r) , (3.11) where f̃(r) = dx2 ∧ · · · ∧ dxn−1 −gL . (3.12) Thus we arrive at ∂t · L = ∆rf̃(rH) +O(△r2) , (3.13) up to the leading order of △r. Substituting Eqs. (3.5), (3.7) and (3.13) into Eq. (3.4), we get N ′′(rH)B(rH) +N ′(rH)B ′(rH) +O(∆r2) = ∆rqI ẽI −∆rf̃(rH) . (3.14) Considering the limit ∆r → 0, we find N ′′(rH)B(rH) +N ′(rH)B ′(rH) = qI ẽI − f̃(rH) . (3.15) So far, we have not specialized to extremal black holes; therefore, the above results hold for general non-extremal black holes. For the extremal black holes limit with N ′(rH) → 0, while N ′′(rH) 6= 0, from (3.15) we have B(rH) = N ′′(rH) qI ẽI − f̃(rH) . (3.16) Since we view the extremal black holes as the extremal limit of non-extremal black holes, the entropy formula of Wald is applicable for the extremal black holes. Note that B(rH) is nothing but the integration in Eq. (2.8) without the 2π factor. Thus, the entropy of the extremal black holes can be expressed as SBH = 2πB(rH) = N ′′(rH) qI ẽI − f̃(rH) . (3.17) This is one of main results in this paper. It is easy to see that this entropy form is very similar to the one in the “entropy function” method of A. Sen. But some remarks are in order: (i). We have not stressed that the extremal black holes have the near horizon geometry AdS2 ×Sn−2 as in [17, 18] although the vanishing surface gravity and the metric assumption (3.1) may coincide with the definition through the near horizon geometry. However, let us notice that some extremal black holes have near horizon geometries of the form AdS3 products some compact manifold X. In our procedure, the near horizon geometry is not necessary to be AdS2 × Sn−2 and the only requirement is to have vanishing surface gravity. Therefore our procedure can be used to discuss that kind of extremal black holes whose near horizon geometry is of the form AdS3 ×X by simply modifying the metric assumption in Eq.(3.1). (ii). Our result is explicitly invariant under coordinate transformation, and this can be easily seen from the above process. We have not used the treatment method Eq.(1.4) employed by A. (iii). The Legendre transformation with respect to the electric charges appears naturally in this procedure, while the Legendre transformation with respect to the magnetic charges does not appear. (iv). If we choose a set of coordinates as the one in [17, 18], our expression for the entropy is exactly same as the one given by A. Sen. This can be seen as follows. In the extremal limit N ′(rH) = 0, we can rewrite the metric near the horizon as ds2 = − N ′′(rH)(r − rH)2dt2 + N ′′(rH)(r − rH)2 dr2 + γ(rH)dΩ n−2 . (3.18) Redefine the coordinates as ρ = r − rH , τ = N ′′(rH)t . (3.19) Then, the near horizon metric can be further rewritten as ds2 = N ′′(rH) −ρ2dτ2 + + γ(rH)dΩ n−2 . (3.20) The components of gauge fields F Irt and f̃ are dependent of coordinates, in this new set of coordi- nates they are ẽI = N ′′(rH)eI , (3.21) f̃(rH) = N ′′(rH)f . (3.22) where eI = F ρτ (rH), f = dx2 ∧ · · · ∧ dxn−1 −g′L . (3.23) Since the entropy is invariant under the coordinate transformation, we find in these coordinates like {τ, ρ, · · · }, SBH = 2π (qIeI − f) . (3.24) This is nothing but the entropy formula given by A. Sen for extremal black holes. Since the factor 2/N ′′(rH) in (3.17) disappears in this new set of coordinates, the entropy formula becomes more simple and good look. This is an advantage of this set of coordinates. But we would like to stress that the entropy expression with the factor “2/N ′′(rH)” makes it invariant under coordinate transformation. (v). Finally the function f̃(rH) is evaluated for the solution of the equations of motion, i.e. all the fields: {gab,Φs, F Iab} are on shell. For example, if the near horizon geometry has the form ds2 = v1(−ρ2dτ2 + dρ2) + v2dΩ n−2 , (3.25) and the equations of motion are satisfied, then we can express the entropy in the form (3.24). There v1 and v2 should equal to 2/N ′′(rH) and γ(rH). N , γ, and other fields, should satisfy the equations of motion. One may worry about that the conserved charge form Q in Eq.(2.18) is not complete: For example, we will have an additional term ǫaba1···an−2ξ a∇bD(φ) if the action has a dilaton coupling termD(φ)R. In general, the conserved charge form can be written asQ = QF+Qg+ξaWa+Y+dZ, where Wa, Y and Z are smooth functions of fields and their derivatives, and Y = Y(ψ,Lξψ) is linear for the field variation [45, 46]. Obviously, Y and dZ will not give contributions to the near horizon integration (3.2) if ξ is a killing vector. It seems that ξaWa will give an additional contribution to this integration. For the extremal case, this contribution will vanish due to the smoothness of Wa and the vanishing surface gravity. For example, the term corresponding to the dilaton coupling mentioned above will vanish in the near horizon integration. So the final form of the entropy (3.17) will not change. For the non-extremal case, this term essentially appear in the near horizon integration if we add the ξaWa into Q. However, if necessary, we can always change the Lagrangian L to be L+dµ and put the conserved charge form Q into the form of (2.18) without the “ · · · ” terms, where µ is a (n − 1)-form. This change of Lagrangian will not affect the equations of motion and the entropy of the black holes [45, 46]. Then, the formulas (3.4) and therefore (3.15) are still formally correct for the non-extremal case after considering that ambiguity of the Lagrangian and therefore f̃(rH). But this ambiguity has no contribution to Eq. (3.17) which describes the entropy of the black hole in the extremal case. IV. ENTROPY FUNCTION AND ATTRACTOR MECHANISM In this section we show further that one can define an entropy function with the help of the entropy definition of Wald. The Noether current can always be written as J[ξ] = dQ[ξ] + ξaCa where Ca corresponds to constraint. The constraint for the U(1) gauge fields is (2.21). If the equations of motion for the U(1) gauge fields hold, this constraint vanishes. In this section, we will assume the equations of motion for the U(1) gauge fields are always satisfied, but not for the metric and scalar fields. In other word, we will not consider the constraint for the gauge fields. Assuming that the metric of the extremal black holes has the form ds2 = −N(r)dt2 + dr2 + γ(r)dΩ2n−2 , on the horizon r = rH of an extremal black hole, one has N(rH) = 0, N ′(rH) = 0, but N ′′(rH) 6= 0. Thus the near horizon geometry will be fixed if N ′′(rH) and γ(rH) are specified. This means the “off-shell” of the near horizon geometry corresponds to the arbitrariness of the parameter N ′′(rH) and γ(rH). In the near horizon region ranged from rH to rH +∆r, we have rH+∆r Q[ξ]− Q[ξ] + J[ξ] = Θ− ξ · L . (4.1) If ξ is a Killing vector for the field configuration space for our discussion (the solution space is a subset of this space), then Θ = 0, and we have rH+∆r Q[ξ]− Q[ξ] + ξaCa = − ξ · L . (4.2) With this, we obtain rH+∆r g[ξ]− g[ξ] + rH+∆r F [ξ] + F [ξ]− ξ · L . (4.3) Define our “entropy function” as E = lim N ′′(rH)∆r rH+∆r g[∂t]− g[∂t] + . (4.4) If the equations of motion are satisfied, obviously, this E will reduce to the entropy of extremal black holes given in the previous section. Therefore this definition is meaningful. Further, from Eq. (4.3), we have E = lim N ′′(rH)∆r rH+∆r F [∂t] + F [∂t]− ∂t · L . (4.5) Recalling that the equations of motion for the U(1) gauge fields have been assumed to hold always, and following the calculations in the previous section, we have N ′′(rH) ẽIqI − f̃(rH) . (4.6) This expression looks the same as the one given in the previous section. However, a crucial difference from the one in the previous section is that here the fields need not be the solutions of the equations of motion. To give the entropy of the extremal black holes, we have to solve the equations of motion or extremize the entropy function with respect to the undetermined values of fields on the horizon. It is easy to find that entropy function has the form E = E(N ′′, γ, us, ẽI ; pi) = ẽIqI − f̃H(N ′′, γ, us, ẽI ; pi) , (4.7) where, for simplicity, we have denoted the N ′′(rH) and γ(rH) by N ′′ and γ, respectively. The terms u′s will not appear because those kinetic terms of scalar fields in the action always have a vanishing factor N(rH) = 0 on the horizon. Similarly, γ ′(rH), γ ′′(rH) will not appear because that the components of the Riemann tensor which include these terms have to contract with the vanished factors N(rH) or N ′(rH). Certainly, this point can be directly understood from the near horizon geometry in Eq. (3.20). So, extremizing the entropy function becomes ∂N ′′ = 0 . (4.8) The electric charges are determined by = 0 or qI = ∂f̃(rH) . (4.9) The entropy of the black hole can be obtained by solving these algebraic equations, and substituting the solutions for N ′′, γ, us back into the entropy function. If the values of moduli fields on the horizon are determined by charges of black holes, then the attractor mechanism is manifest. Then the entropy has the form SBH = SBH(qI ; pi) = E|extremum piont , (4.10) a topological quantity which is fully determined by charges [17, 18]. These definitions will become more simple if one chooses the coordinates {τ, ρ, · · · } so that one can define N ′′(rH) , v2 = γ(rH) , (4.11) then, the entropy function can be written as E = E(v1, v2, us, eI ; pi) = 2π (eIqI − f(v1, v2, us, eI ; pi)) , (4.12) where eI are gauge fields on the horizon in this set of coordinates, and qI = are electric charges which are not changed with the coordinate transformation. So, in this set of coordinates, our entropy function form reduces to the entropy function defined by A. Sen [17, 18]. V. CONCLUSION AND DISCUSSION In this paper, we have shown that the “entropy function” method proposed by A. Sen can be extracted from the general black hole entropy definition of Wald [44]. For a spherically symmetry extremal black hole as described by metric (3.1), we find that the entropy of the black hole can be put into a form SBH = N ′′(rH) ẽIqI − f̃(rH) which is similar to the one given in Ref. [17, 18]. To get this entropy form, we have regarded the extremal black hole as the extremal limit of an non-extremal black hole, i.e., we have required (and only required) that the surface gravity approaches to zero. In a special set of coordinates, i.e., {τ, ρ · · · }, this entropy is exactly of the same form as the one given by A. Sen. We have obtained a corresponding entropy function (4.7). After extremizing this entropy function with respect to N ′′, γ and other scalar fields, one gets the entropy of the extremal black holes. Similarly, in the coordinates {τ, ρ · · · }, our entropy function reduces to the form of A. Sen. Note that in our procedure, we have neither used the treatment of rescaling AdS2 part of the near horizon geometry of extremal black holes, nor especially employed the form of the metric in the coordinates {τ, ρ, · · · } as Eq.(1.1). In this procedure, it can be clearly seen why the electric charge terms eIqI appear, but not the magnetic charges terms in the entropy function. Recently it was shown that for some near-extremal black holes with BTZ black holes being a part of the near horizon geometry, that the “entropy function” method works as well [40]. A similar discussion for non-extremal D3,M2 and M5 branes has also been given in [41]. Therefore it is interesting to see whether the procedure developed in this paper works or not for near-extremal black holes. In this case, N ′(rH) is an infinitesimal one instead of vanishing. Eq. (3.15) then gives SBH = 2πB(rH) = S0 N ′(rH) N ′′(rH) , (5.1) where N ′′(rH) (ẽIqI − f̃(rH)) , (5.2) and r∗ = B(rH)/B ′(rH) approximately equals to “ · radius of the black hole” if the higher derivative corrections in the effective action are small. Thus, after considering that ambiguity in f̃(rH) becomes very small and for large r∗ (sometimes, this corresponds to large charges), the entropy function method gives us an approximate entropy for near-extremal black holes, but the attractor mechanism will be destroyed [15]. In addition, it is also interesting to discuss the extremal rotating black holes with the procedure developed in this paper. Certainly, in this case, the Killing vector which generates the horizon should be of the form χ = ∂t+ΩH∂φ instead of ξ = ∂t. 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W. Hawking, G. T. Horowitz and S. F. Ross, “Entropy, Area, and black hole pairs,” Phys. Rev. D 51, 4302 (1995) [arXiv:gr-qc/9409013]. [51] G. W. Gibbons and R. E. Kallosh, “Topology, entropy and Witten index of dilaton black holes,” Phys. Rev. D 51, 2839 (1995) [arXiv:hep-th/9407118]. http://arxiv.org/abs/hep-th/0703260 http://arxiv.org/abs/0704.0955 http://arxiv.org/abs/gr-qc/9307038 http://arxiv.org/abs/gr-qc/9403028 http://arxiv.org/abs/gr-qc/9503052 http://arxiv.org/abs/gr-qc/0106071 http://arxiv.org/abs/gr-qc/0304094 http://arxiv.org/abs/hep-th/9410103 http://arxiv.org/abs/gr-qc/9409013 http://arxiv.org/abs/hep-th/9407118 Introduction The Definition of Wald Entropy of extremal black holes Entropy function and attractor mechanism Conclusion and discussion Acknowledgements References

0704.1240

Dynamical layer decoupling in a stripe-ordered, high T_c superconductor

Dynamical layer decoupling in a stripe-ordered, high T superconductor E. Berg,1 E. Fradkin,2 E.-A. Kim,1 S. A. Kivelson,1 V. Oganesyan,3 J. M. Tranquada,4 and S. C. Zhang1 Department of Physics, Stanford University, Stanford, California 94305-4060 Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080 Department of Physics, Yale University, New Haven, Connecticut 06520-8120 Brookhaven National Laboratory, Upton, New York 11973-5000 (Dated: October 11, 2018) In the stripe-ordered state of a strongly-correlated two-dimensional electronic system, under a set of special circumstances, the superconducting condensate, like the magnetic order, can occur at a non-zero wave-vector corresponding to a spatial period double that of the charge order. In this case, the Josephson coupling between near neighbor planes, especially in a crystal with the special structure of La2−xBaxCuO4, vanishes identically. We propose that this is the underlying cause of the dynamical decoupling of the layers recently observed in transport measurements at x = 1/8. High-temperature superconductivity (HTSC) was first discovered [1] in La2−xBaxCuO4. A sharp anomaly [2] in Tc(x) occurs at x = 1/8 which is now known to be indica- tive [3, 4] of the existence of stripe order and of its strong interplay with HTSC. Recently, a remarkable dynamical layer decoupling has been observed [5] associated with the superconducting (SC) fluctuations below the spin- stripe ordering transition temperature, Tspin = 42K. While Tc(x), as determined by the onset of a bulk Meissner effect, reaches values up to Tc(x = 0.1) = 33 K for x somewhat smaller and larger than x = 1/8, Tc(x) drops to the range 2–4 K for x = 1/8. However, in other respects, superconductivity appears to be optimized for x = 1/8. The d-wave gap determined by ARPES has recently been shown [6] to be largest for x = 1/8. More- over, strong SC fluctuations produce an order of magni- tude drop [5] in the in-plane resistivity, ρab, at T ≈ Tspin, which is considerably higher than the highest bulk SC. The fluctuation conductivity reveals heretofore un- precedented characteristics (as described schematically in Fig. 1): 1) ρab drops rapidly with decreasing tem- perature from Tspin down to TKT ≈ 16K, at which point it becomes unmeasurably small. In the range Tspin > T > TKT , the temperature dependence of ρab is qualitatively of the Kosterlitz-Thouless form, as if the SC fluctuations were strictly confined to a single copper- oxide plane. 2) By contrast, the resistivity perpendicular to the copper-oxide planes, ρc, increases with decreasing temperatures from T ⋆ >∼ 300 K, down to T ⋆⋆ ≈ 35 K. For T < T ⋆⋆, ρc decreases with decreasing temperature, but it only becomes vanishingly small below T3D ≈ 10 K. Within experimental error, for TKT > T > T3D, the re- sistivity ratio, ρc/ρab, is infinite! 3) The full set of usual characteristics of the SC state, the Meissner effect and perfect conductivity, ρab = ρc = 0, is only observed be- low Tc = 4K. Thus, for T3D > T > Tc, a peculiar form of fragile 3D superconductivity exists. The above listed results are new, so an extrinsic explanation of some aspects of the data is possible. Here we assume that the measured properties do re- flect the bulk behavior of La2−xBaxCuO4. We show that there is a straightforward way in which stripe or- 35 K= Meissner State ~ 0 ~ 0 ab ~ 0 ρc ~ 10 m Ω cm ~ 10 x 10 m Ω cm }max { ~ 8 x 10 m Ω cm ~ 8 x 10 5 x 10 2 x 10 3 FIG. 1: Summary of the thermal phase transitions and trans- port regimes in x = 1/8 doped La2−xBaxCuO4. der can lead to an enormous dynamical suppression of interplane Josephson coupling, particularly in the charge ordered low-temperature tetragonal (LTT) phase of La15/8Ba1/8CuO4, i.e. T ≤ Tco = 54 K. The LTT structure has two planes per unit cell. In alternating planes, the charge stripes run along the x or y axes, as shown in Fig. 3. Moreover, the parallel stripes in second neighbor planes are thought to be shifted over by half a period (so as to minimize the Coulomb interac- tions [7]) resulting in a further doubling of the number of planes per unit cell, as seen in X-ray scattering studies. Below Tspin, the spins lying between each charge stripe have antiferromagnetic (AFM) order along the stripe di- rection, which suffers a π phase shift across each charge http://arxiv.org/abs/0704.1240v2 stripe, resulting in a doubling of the unit cell within the plane, see Fig. 2c. Hence, the Bragg scattering from the charge order in a given plane occurs at (2π/a)〈±1/4, 0〉 while the spin-ordering occurs at (2π/a)〈1/2± 1/8, 1/2〉. SC order should occur most strongly within the charge stripes. Since it is strongly associated with zero center-of- mass momentum pairing, one usually expects, and typi- cally finds in models, that the SC order on neighboring stripes has the same phase. However, as we will discuss, under special circumstances, the SC order, like the AFM order, may suffer a π phase shift between neighboring stripes if the effective Josephson coupling between stripes is negative. Within a plane, so long as the stripe order is defect free, the fact that the SC order occurs with k = (2π/a)〈±1/8, 0〉 has only limited observable conse- quences. However anti-phase SC order within a plane results in an exact cancellation of the effective Josephson coupling between first, second and third neighbor planes. This observation can explain an enormous reduction of the interplane SC correlations in a stripe-ordered phase. Before proceeding, we remark that there is a preexist- ing observation, concerning the spin order, which sup- ports the idea that interplane decoupling is a bulk fea- ture of a stripe-ordered phase. Specifically, although the in-plane spin correlation length measured in neutron- scattering studies in particularly well prepared crystals of La2−xBaxCuO4 is ξspin ≥ 40a [8], there are essentially no detectable magnetic correlations between neighboring planes. In typical circumstances, 3D ordering would be expected to onset when (ξspin/a) 2J1 ∼ T , where J1 is the strength of the interplane exchange coupling. How- ever, the same geometric frustration of the interplane couplings that we have discussed in the context of the SC order pertains to the magnetic case, as well. Thus, we propose that the same dynamical decoupling of the planes is the origin of both the extreme 2D character of the AFM and SC ordering. We begin with a caricature of a stripe ordered state, consisting of alternating Hubbard or t–J ladders which are weakly coupled to each other (Fig. 2). Such a car- icature, which has been adopted in previous studies of superconductivity in stripe ordered systems [9, 10, 11], certainly overstates the extent to which stripe order pro- duces quasi-1D electronic structure. However, we can learn something about the possible electronic phases and their microscopic origins, in the sense of adiabatic conti- nuity, by analyzing the problem in this extreme limit. As shown in the figure, distinct patterns of period 4 stripes can be classified by their pattern of point group symme- try breaking as being “bond centered” or “site centered.” Numerical studies of t–J ladders [12] suggest that the dif- ference in energy between bond and site centered stripes is small, so the balance could easily be tipped one way or another by material specific details, such as the specifics of the electron-lattice coupling. The simplest caricature of bond centered stripes is an array of weakly coupled two-leg ladders with alternately larger and smaller doping, as illustrated in Fig. 2a. This a) Bond centered b) Site centered c) Magnetic striped superconductor Figure 2 (a) Bond centered a) Bond centered b) Site centered c) Magnetic striped superconductor Figure 2 (b) Site centered a) Bond centered b) Site centered c) Magnetic striped superconductor Figure 2 (c) Magnetic striped FIG. 2: a) Pattern of a period 4 bond centered and b) site centered stripe, with nearly undoped (solid lines) and more heavily doped (hatched lines) regions. c) Sketch of the pair- field (lines) and spin (arrows) order in a period 4 site centered stripe in which both the SC and AFM order have period 8 due to an assumed π phase shift across the intervening regions. Solid (checked) lines denote a positive (negative) pair-field. problem was studied in Ref. 10. Because a strongly inter- acting electron fluid on a two-leg ladder readily develops a spin-gap,[13] i.e. forms a LE liquid, this structure can exhibit strong SC tendencies to high temperatures. Weak electron hopping between neighboring ladders produces Josephson coupling which can lead to a “d-wave like” SC state.[14] However, the spin-gap precludes any form of magnetic ordering, even when the ladders are weakly cou- pled, and there is nothing about the SC order that would prevent phase locking between neighboring planes in a 3D material. For both these reasons, this is not an attractive model for the stripe ordered state in La15/8Ba1/8CuO4. (There is, however, evidence from STM studies on the surface of BSCCO [15] of self-organized structures sug- gestive of two-leg ladders.) By contrast, a site-centered stripe is naturally related to an alternating array of weakly coupled three and one leg ladders, as shown in Fig. 2(b). Because the zero-point kinetic energy of the doped holes is generally large com- pared to the exchange energy, it is the three-leg ladder that we take to be the more heavily doped. The three leg ladder is known [9, 16] to develop a spin-gapped LE liquid above a rather small [16] critical doping, xc (which de- ������������������������� ������������������������� ������������������������� ������������������������� ������������������������� ������������������������� ������������������������� ������������������������� ������������������������� ������������������������� ������������������������� ������������������������� ������������������������� ������������������������� ������������������������� ������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� �������������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ����������������������� ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ������������������������ ��������������������������������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� ��������������������� FIG. 3: Stacking of stripe planes. pends on the interactions). An undoped or lightly doped one-leg ladder, by contrast, is better thought of as an incipient spin density wave (SDW), and has no spin-gap. Where the one-leg ladder is lightly doped it forms a Lut- tinger liquid with a divergent SDW susceptibility at 2kF . The phases of a system of alternating, weakly coupled LE and Luttinger liquids were analyzed in [11]. How- ever, the magnetic order in La15/8Ba1/8CuO4 produces a Bragg peak at wave-vector (2π 〉 in a coordinate system in which y is along the stripe direction. There- fore, it is necessary to consider the case in which, in the absence of inter-ladder coupling, the one-leg ladder is ini- tially undoped, and the three leg ladder has x = 1 > xc. Our model of the electronic structure of a single charge- stripe-ordered Cu-O plane is thus an alternating array of LE liquids, with a spin-gap but no charge gap, and spin- chains, with a charge gap but no spin gap. None of the obvious couplings between nearest-neighbor subsystems is relevant in the renormalization group sense, because of the distinct character of their ordering tendencies. How- ever, certain induced second neighbor couplings, between identical systems, are strongly relevant, and, at T = 0, lead to a broken symmetry ground-state. The induced exchange coupling between nearest- neighbor spin-chains leads to a 2D magnetically ordered state. The issue of the sign of this coupling has been addressed previously [17, 18, 19] and found to be non- universal, as it depends on the doping level in the inter- vening three-leg ladder. For x = 0, the preferred AFM order is in-phase on neighboring spin-chains, consistent with a magnetic ordering vector of (2π/a)〈1/2, 1/2〉. For large enough x (probably, x > xc), the ordering on neighboring spin-chains is π phase shifted, resulting in a doubling of the unit-cell size in the direction perpen- dicular to the stripes, and a magnetic ordering vector (2π/a)〈1/2± 1/8, 1/2〉. This ordering tendency has also been found in studies of wide t–J ladders [12]. A question that has not been addressed systematically until now is the sign of the effective Josephson coupling between neighboring LE liquids. In the case of 2-leg lad- ders, it was found [10, 12] that the effective Josephson coupling is positive, favoring a SC state with a spatially uniform phase. It is possible, in highly correlated sys- tems, especially when tunneling through a magnetic im- purity [20], to encounter situations in which the effective Josephson coupling is negative, therefore producing a π- junction. Zhang [21] has observed that, regardless the microscopic origin of the anti-phase character of the mag- netic ordering in the striped state, if there is an approx- imate SO(5) symmetry relating the antiferromagnetism to the superconductivity, one should expect an anti-phase ordering of the superconductivity in a striped state. The example of tunneling through decoupled magnetic impu- rities [20] is a proof in principle that such behavior can occur. However, interplane decoupling associated with the onset of superconductivity is not seen in experiments in other cuprates, and states with periodic π phase shifts of the SC order parameter have not yet surfaced in nu- merical studies of microscopic models [12]; this suggests anti-phase striped SC order is rare. The new proposal in the present paper is that, for the reasons outlined above, the SC striped phase of La15/8Ba1/8CuO4 has anti-phase SC and anti-phase AFM order, whose consequences we now outline. We can express the most important possible interplane Josephson-like coupling terms compactly as Hinter = ∆⋆j∆j+m + h.c. where ∆j is the j-th plane SC order parameter. The term proportional to the usual (lowest order) Josephson coupling, J1,1, and indeed, J1,2 and J1,3 all vanish by sym- metry. The most strongly relevant residual interaction is the Josephson coupling between fourth-neighbor planes, J1,4. Double-pair tunnelling between nearest-neighbor planes, J2,1, is more weakly relevant, but it probably has a larger bare value since it involves half as many pow- ers of the single-particle interplane matrix elements than J1,4. J1,4 and J2,1 have scaling dimensions 1/4 and 1 at the (KT) critical point of decoupled plains, so both are relevant. Thus, they become important when the in- plane SC correlation length ξ ∼ ξ1,4 ∼ [Jo/J1,4] 1/4 and ξ2,1 ∼ [Jo/J2,1], where Jo is the in-plane SC stiffness. We can make a crude estimate of the magnitude of the residual interplane couplings by noting that the same interplane matrix elements (although not neces- sarily the same energy denominators) determine the in- terplane exchange couplings between spins and the in- terplane Josephson couplings. Defining Jm to be the exchange couplings between spins m planes apart, this estimate suggests that Jn,m/J0 ∼ [Jm/J0] n. In undoped La2CuO4, it has been determined [22] that J1/J0 ≈ 10 which is already remarkably small. Although in-plane translation invariance forbids direct Josephson coupling between adjacent planes, there is an allowed biquadratic inter-plane coupling involvingM and ∆, the SDW and the SC order parameters, δHinter = J1,s ∆∗j∆j+1Mj ·Mj+1 + h.c. Even though M 6= 0 for T < Tspin, this term vanishes because, not only the direction of the stripes, but also the axis of quantization of the spins (due to spin-orbit coupling) rotates [23] by 90◦ from plane to plane, i.e. Mj · Mj+1 = 0. However, a magnetic field, H ∼ 6T , induces a 1st order spin-flop transition to a fully collinear spin state [23] in which Mj ·Mj+1 6= 0. Thus, for perfect stripe order, the anti-phase SC or- der would depress, by many orders of magnitude, of the interplane Josephson couplings, which explains the exis- tence of a broad range of T in which 2D physics is ap- parent. Accordingly, there still would be a transition to a 3D superconductor at a temperature strictly greater than TKT , when ξ(T ) ∼ ξ1,4 or ξ2,1, whichever is smaller. The only evidence for the growth of ξ comes indirectly from the measurement of ρab; by the time ρab is “un- measurably small,” it has dropped by about 2 orders of magnitude from its value just below Tspin, which implies (since ρab ∼ ξ −2) that ξ has grown by about 1 order of magnitude. Thus, if some other physics cuts off the growth of in-plane SC correlations at long scales, we may be justified in neglecting the effects of Hinter. Defects in the pattern of charge stripe order have con- sequences for both magnetic and SC orders. A dislo- cation introduces frustration into the in-plane ordering, resulting in the formation of a half-SC vortex bound to it. For the single-plane problem, this means that the long- distance physics is that of an XY spin-glass. Since there is no finite T glass transition in 2D, the growth of ξ will be arrested at a large scale determined by the density of dislocations. The same is true of the in-plane AFM correlations. Both ξ and ξspin should be bounded above by the charge stripe correlation length, ξch. From X- ray scattering studies it is estimated that ξch ≈ 70a [24]. This justifies the neglect ofHinter. Conversely, any defect in the charge-stripe order spoils the symmetry responsi- ble for the exact cancellation of the Josephson coupling between neighboring planes. Finite T ordering of an XY spin-glass is possible in 3D. We tentatively identify the temperature at which ρc → 0 as a 3D glass transition. A SC glass would result in the existence of equilibrium cur- rents (spontaneous time-reversal breaking) and in glassy long-time relaxations of the magnetization or ρc. For x 6= 1/8, there is a tendency to develop discom- mensurations in the stripe order, which, in turn, produce regions of enhanced (or depressed) SC order with relative sign depending on the number of intervening stripe peri- ods. So long as the stripes are dilute, the energy depends weakly on their precise spacing. Thus, to gain inter- layer condensation energy, the system can self-organize so that there are always an even number of intervening stripes, thus producing an interplane Josephson coupling J1,1 ∼ |x − 1/8| 2. This, in turn, will lead to a dramatic increase of the 3D SC Tc. An enhancement of interplane coherence in any range of T triggered by the magnetic field induced spin-flop transition would be a dramatic confirmation of the physics discussed here. Note added: It was pointed out to us that the state dis- cussed here was considered by A. Himeda et al.[25] They found that this is a good variational state for a t− t′ −J model at x ∼ 1/8 for a narrow range of parameters. We thank P. Abbamonte, S. Chakravarty, R. Jamei, A. Kapitulnik, and D. J. Scalapino for discussions. This work was supported in part by the National Sci- ence Foundation, under grants DMR 0442537 (EF), DMR 0531196 (SAK), DMR 0342832 (SCZ), and by the Office of Science, U.S. Department of Energy un- der Contracts DE-FG02-91ER45439 (EF), DE-FG02- 06ER46287 (SAK) DE-AC02-98CH10886 (JT) and DE- AC03-76SF00515 (SCZ), by the Stanford Institute for Theoretical Physics (EAK), and by a Yale Postdoctoral Prize Fellowship (VO). [1] J. G. Bednorz and K. A. Mueller, Z. Phys. B: Condens. Matter 64, 189 (1986). [2] A. R. Modenbaugh et al., Phys. Rev. B 38, 4596 (1988). [3] J. M. Tranquada et al., Nature, 375, 561 (1995). [4] P. Abbamonte et al., Nature Phys. 1, 155 (2005). [5] Q. Li, M. Hücker, G. D. Gu, A. M. Tsvelik, and J. M. Tranquada, cond-mat/0703357. [6] T. Valla et al, Science 314, 1914 (2006). [7] M. v. Zimmermann et al, Europhys. Lett. 41, 629 (1998). [8] M. Fujita et al., Phys. Rev. B 70, 104517 (2004). In 1/8 doped La1.6−xNd0.4SrxCuO4, J. M. Tranquada et al., Phys. Rev. B 59, 14712 (1999) found ξspin ≈ 50a (with no substantial interplane correlations). In stage IV O-doped La2CuO4 (which does not exhibit the LTT structure), Y. S. Lee et al., Phys. Rev. B 60, 3643 (1999), found ξspin > 100a with c-axis correlations of 2–3 planes. [9] V. J. Emery, S. A. Kivelson and O. Zachar, Phys. Rev. B 56, 6120 (1997). [10] E. Arrigoni, E. Fradkin, and S. A. Kivelson, Phys. Rev. B 69, 214519 (2004). [11] M. Granath et al., Phys. Rev. Lett. 87, 167011 (2001). [12] S. R. White and D. J. Scalapino, Phys. Rev. Lett. 80, 1272 (1998). [13] S. R. White, I. K. Affleck and D. J. Scalapino, Phys. Rev. B 65, 165122 (2002). [14] A “d-wave like” gap changes sign as well as magnitude under rotation by π/2. [15] Y. Kohsaka et al., Science 315, 1380 (2007). [16] S. R. White and D. J. Scalapino, Phys. Rev. B 57, 3031 (1998). [17] O. Zachar, Phys. Rev. B 65, 174411 (2002). [18] L. Pryadko et al., Phys. Rev. B 60, 7541 (1999). [19] W. V. Liu and E. Fradkin Phys. Rev. Lett. 86, 1865 (2001). [20] B. I. Spivak and S. A. Kivelson,Phys. Rev. B 43, 3740 (1991). [21] S-C. Zhang, J. Phys. Chem. Solids 59, 1774 (1998). [22] B. Keimer et al., Phys. Rev. B 46, 14034 (1992). [23] M. Hücker, G. Gu and J. M. Tranquada, cond-mat/0503417. [24] J. Kim, A. Kagedan, G. D. Gu, C. S. Nelson, T. Gog, D. Casa, and Y.-J. Kim, cond-mat/0703265. [25] A. Himeda, T. Kato and M. Ogata, Phys. Rev. Lett. 88, 117001 (2002). http://arxiv.org/abs/cond-mat/0703357 http://arxiv.org/abs/cond-mat/0503417 http://arxiv.org/abs/cond-mat/0703265

0704.1241

Cooling and heating by adiabatic magnetization in the Ni$_{50}$Mn$_{34}$In$_{16}$ magnetic shape memory alloy

Cooling and heating by adiabatic magnetization in the Ni50Mn34In16 magnetic shape memory alloy Xavier Moya, Llúıs Mañosa∗ and Antoni Planes Departament d’Estructura i Constituents de la Matèria, Facultat de F́ısica, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Catalonia, Spain Seda Aksoy, Mehmet Acet, Eberhard F. Wassermann and Thorsten Krenke† Fachbereich Physik, Experimentalphysik, Universität Duisburg-Essen, D-47048 Duisburg, Germany (Dated: November 22, 2021) We report on measurements of the adiabatic temperature change in the inverse magnetocaloric Ni50Mn34In16 alloy. It is shown that this alloy heats up with the application of a magnetic field around the Curie point due to the conventional magnetocaloric effect. In contrast, the inverse magnetocaloric effect associated with the martensitic transition results in the unusual decrease of temperature by adiabatic magnetization. We also provide magnetization and specific heat data which enable to compare the measured temperature changes to the values indirectly computed from thermodynamic relationships. Good agreement is obtained for the conventional effect at the second- order paramagnetic-ferromagnetic phase transition. However, at the first order structural transition the measured values at high fields are lower than the computed ones. Irreversible thermodynamics arguments are given to show that such a discrepancy is due to the irreversibility of the first-order martensitic transition. PACS numbers: 75.30.Sg,64.70.Kb,81.30Kf I. INTRODUCTION When the magnetization of any magnetic material is changed isothermally under the application of a mag- netic field, heat is exchanged with the surroundings. If the change is performed adiabatically, the temperature changes. This is the magnetocaloric effect (MCE), which provides the basis of the adiabatic demagnetization cool- ing technique [1]. This technique was developed to reach mK temperatures soon after the pioneering work by De- bye [2] and Giauque [3], who independently suggested such a possibility. The discovery in the nineties of the gi- ant magnetocaloric effect associated with first-order mag- netostructural transitions in a number of intermetallic alloy families [4] opened up the possibility of using this technique in room temperature refrigeration applications and, thus, yielded renewed interest in the subject [5]. It has been known for a long time that the isother- mal reduction of a magnetic field gives rise to a de- crease in entropy in some antiferromagnetic and ferri- magnetic systems, [6, 7]. This inverse magnetocaloric phenomenon was supposed to produce small effects and has been largely ignored. Recently, however, it has been shown that in some ferromagnetic [8] and metamagnetic [9] systems, inverse MCE can have an amplitude com- parable to the conventional effect detected in giant mag- netocaloric intermetallic materials. The inverse effect is related to the existence of regions in phase space where ∗Electronic address: [email protected] †Present address: ThyssenKrupp Electrical Steel GmbH, D-45881 Gelsenkirchen, Germany ζ = (∂M/∂T )H is positive. In a paramagnetic system, ζ is always negative, and thus, the origin of a positive ζ must be ascribed to coupling between magnetic moments. The inverse MCE can occur in the vicinity of magne- tostructural and metamagnetic phase transitions due to changes in the magnetic coupling driven by the interplay between magnetic and structural degrees of freedom [10]. In the present paper, we study the MCE in a Ni50Mn34In16 alloy. This is a magnetic shape-memory alloy which undergoes a martensitic transition from a cubic (L21) to a monoclinic (10M) structure below its Curie temperature [11]. Interestingly, the sample shows both inverse and conventional MCE in rather close tem- perature intervals. While the conventional effect arises from the continuous transition from paramagnetic to fer- romagnetic states, the inverse effect is associated with the martensitic transition at which the magnetic moment of the system decreases. This decrease originates from the tendency of the excess of Mn atoms (with respect to 2-1- 1 stoichiometry) to introduce antiferromagnetic coupling. The antiferromagnetic coupling is caused by the change in the Mn-Mn distance as the martensitic phase of lower symmetry gains stability [12]. While most of the reported data on giant MCE ma- terials refer to the isothermal entropy change, the most relevant parameter for actual applications of this effect is the adiabatic temperature change [13]. This value is usually computed from entropy data by means of equilib- rium thermodynamic relationships. However, irreversible effects are expected to take place at first-order phase transitions which can yield discrepancies between the computed temperature change and the directly measured one. Actually, direct measurements of the temperature change in giant MCE compounds are scarce, and the re- http://arxiv.org/abs/0704.1241v1 mailto:[email protected] ported values in many cases do not seem to be consistent with those indirectly computed [14, 15, 16]. Here, we report on adiabatic temperature measurements, which provide direct evidence of cooling by adiabatic magne- tization in an inverse magnetocaloric material. It is also shown that heating is achieved at the paramagnetic- ferromagnetic phase transition. We focus on moderate magnetic fields which are readily available for applica- tions of giant MCE materials [13]. Furthermore, data obtained from magnetization and heat capacity experi- ments have enabled us to compare the measured tem- perature change with that computed from entropy data. Irreversible thermodynamics arguments are provided to account for the discrepancies observed at the first-order structural phase transition. II. EXPERIMENTAL DETAILS A polycrystalline Ni50Mn34In16 ingot was prepared by arc melting the pure metals under argon atmosphere in a water-cooled Cu crucible and subsequently re-melted in order to ensure homogeneity. The ingot was sealed under argon in a quartz recipient and annealed at 1073 K for 2 hours. Finally, it was quenched in ice-water. The composition of the alloy was determined by energy dispersive X-ray photoluminescence analysis (EDX). For calorimetric and magnetization measurements, a small sample (61.5 mg) was cut using a low-speed diamond saw. The remaining button (13 mm in diameter, 6 mm thickness and 4.6 g) was used for the adiabatic tempera- ture change measurements. Magnetization was measured by means of a SQUID magnetometer, and differential scanning calorimetric (DSC) measurements were conducted using a high- sensitivity calorimeter. Specific heat measurements were performed using a modulated differential scanning calorimeter (MDSC), and data were taken with the con- stant temperature method [17] starting from the lowest temperature (190 K). Adiabatic temperature changes were measured at at- mospheric pressure using a specially designed set-up. A thin (0.75 mm diameter) Ni-Cr/Ni-Al thermocouple was used to measure the temperature. The output of this thermocouple was continuously monitored by means of a multimeter that also electronically compensates for the reference junction. Measurements without any specimen confirmed that the recorded values were not affected by magnetic fields up to 1.3 T. The thermocouple was em- bedded within the sample and good thermal contact be- tween the sample and the thermocouple was ensured by Ariston conductive paste. The sample is situated inside a copper container (sample holder), which is placed on the top face of a Peltier element. The bottom surface sits on a copper cylinder, which acts as a heat sink. The bottom end of the cylinder is in contact with a nitrogen bath. By controlling the current input into the Peltier element, it is possible to achieve fine tuning of the temperature 0 5 10 15 20 25 150 200 250 300 350 x (at. %) Cooling T (K) Heating FIG. 1: (Color online) Phase diagram of Ni50Mn50−xInx, ob- tained using the data in ref. [11]. MS indicates the marten- sitic transition line and TC indicates the Curie point line.The inset shows DSC curves for heating and cooling runs for the x = 16 sample. in the 200-320 K interval. Temperature oscillations were less than 0.05 K. Thermal insulation (adiabaticity) be- tween the sample and sample holder was ensured by a polystyrene layer. The sample holder was placed in be- tween the poles of an electromagnet (28 mm gap), which enabled fields up to 1.3 T to be applied. A major advan- tage of using an electromagnet is the short rising time in the application of the field (the field rises from 0 to 1 T in about 0.5 s). Such a field rise time is several orders of magnitude shorter than the thermal relaxation time of the sample-holder system (∼ 100 s), thus ensuring the adiabaticity of the process. In order to check the reliability of the device, we mea- sured the MCE of commercial pure (99.9 wt %) Gd. The measured temperature changes obtained around the Curie point for a magnetic field of 1T agree with those reported in the literature [18]. III. RESULTS AND DISCUSSION For the present study we selected a composition with the para-ferromagnetic and martensitic transition tem- peratures close to each other. This is illustrated in fig- ure 1, which shows the Curie and martensitic transi- tion start temperatures as a function of In content for Ni50Mn50−xInx alloys. Continuous lines are polynomial fits to the data given in ref. [11]. The arrows indi- cate the composition of the studied sample. The in- set presents DSC curves (heating and cooling) for the present Ni50Mn34In16 alloy [19]. The peaks at higher temperature correspond to the Curie point and those at lower temperatures correspond to the martensitic tran- sition (which occurs with 15 K thermal hysteresis). In- tegration of the peaks associated with the martensitic transition renders latent heats of -1750 ± 100 J/kg for 210 220 230 240 200 240 280 320 T (K) 0.5 T 0.7 T 1 T 1.3 T T (K) FIG. 2: (Color online) (a) Measured adiabatic temperature change and (b) computed isothermal entropy change, as a function of temperature at selected values of the magnetic field. The inset shows an enlarged view for the 0.5 and 1.3 T fields which illustrates the shift in the inverse MCE with magnetic field. the cooling run (forward transition) and 1850 ± 100 J/kg for the heating run (reverse transition). The adiabatic temperature changes measured over the 200-320 K temperature range for selected values of the magnetic field are shown in figure 2(a). Data points were obtained according to the following procedure, which en- sures the suppression of any history dependent effect: first, the sample is heated up to 320 K (above the Curie point) and then cooled down to the fully martensitic state at 170 K. Subsequently, it is heated up to the desired tem- perature and the magnetic field is switched on for 20 s. After switching off the field, the sample is heated again above the Curie point and the protocol is repeated for the next data point. The measured adiabatic tempera- ture changes shown in fig. 2(a) prove unambiguously that the sample cools down upon adiabatic application of the field in the temperature range 200-245 K, while it heats up in the temperature range 245-320K. The positive tem- perature change has its maximum value (∆T ≃1.5 K for 1.3 T) at the Curie point. The maximum temperature decrease (∆T ≃ – 0.6 K for 1.3 T) occurs at a tem- perature that shifts with magnetic field [see inset in fig- ure 2(a)], in agreement with the decrease in the marten- sitic transition temperature reported for Ni-Mn-In alloys [11]. The values found for ∆T at their corresponding peak temperatures are comparable to those reported for other giant MCE materials. However, a novel feature for 200 240 280 320 T (K) 0.5 T 0.7 T 1.3 T ) (a) FIG. 3: (Color online) (a) Temperature dependence of the magnetization for selected values of the magnetic field. (b) Specific heat as a function of temperature. Arrows indicate the region of the reverse martensitic transition. Ni50Mn34In16 is that these relatively large temperature changes can be either positive or negative. In order to correlate the measured temperature changes with those indirectly computed from entropy data, we measured the magnetization of the sample as a function of temperature and magnetic field. Results at selected fields are shown in fig 3(a). In the temperature range 245-320 K, ζ is negative, while a positive ζ is ob- tained in the range 200-245 K. From these data, we com- puted the magnetic field-induced entropy change by using the Maxwell relation ∆S = µ0 ζdH . Results are shown in Fig. 2(b). Excellent qualitative agreement is observed between the two quantities (∆T and ∆S) characterizing giant MCE. Conventional MCE is observed within the 245-320 K temperature range, i.e. a negative entropy change with the associated positive temperature change, while in the 200-245 K interval, the sample exhibits in- verse MCE: an increase in entropy with the associated negative temperature change. It is customary to compute the adiabatic temperature change from isothermal entropy data by means of the following relationship, ∆Trev = − ∆S, (1) which is expected to be valid in equilibrium. C is the spe- cific heat at constant magnetic field and is assumed to be independent of the magnetic field. In order to check the 200 240 280 320 (a) µ H=0.5 T H=0.7 T H=1 T H=1.3 T T (K) FIG. 4: (Color online) Adiabatic temperature change, as a function of temperature, for different magnetic fields. Black symbols stand for measured data and red symbols correspond to data indirectly computed using equilibrium thermodynam- ics relationships. validity of this approach, we measured the specific heat of our Ni50Mn34In16 sample. Results are shown in figure 3(b). The large lambda-type peak at 302 K corresponds to the second-order para-ferromagnetic phase transition. In the temperature range 216-257 K a small bump is observed, which coincides with the reverse martensitic transition. No latent heat contributions are expected for the isothermal-modulated method we have used. In Fig. 4, we compare the measured adiabatic temper- ature changes (black symbols) with those computed from the entropy [Fig. 2(b)] and specific heat data [Fig. 3(b)] (red symbols) for different values of the applied field. Good agreement between measured and computed values over the complete temperature range is obtained at low magnetic fields. As the magnetic field is increased, there is still good agreement between the data corresponding to conventional MCE [20], but the absolute value of the measured temperature change becomes smaller than the computed one in the inverse MCE region. Such a dif- ference is due to the irreversibility associated with the first-order phase transition. In order to consider the effect of dissipation, we start from the Clausius inequality ≤ 0, which can be ex- pressed as δq = dS − δSi, where dS is a reversible dif- ferential change of entropy and δSi is the entropy pro- duction (δSi ≥ 0). When the magnetic field is adiabat- ically changed, δq = 0, and under the assumption of a quasistatic, continuous process with hysteresis [21], the adiabatic temperature change is expressed as [−∆S + Si] = ∆Trev + , (2) where TSi is the dissipated energy (Ediss). For an in- verse magnetocaloric effect, there is an increase of en- tropy by the application of the field, i.e. ∆S > 0. On the other hand, Si is always positive. Hence, for an out- of-equilibrium process, the two terms within brackets in equation 2 will partially cancel each other when the field is swept from zero to a given value, and therefore, the measured temperature change will always be less than the value computed using equilibrium thermodynamics (see equation 1). Such a difference is expected to be small at low fields (close to equilibrium conditions), but it be- comes larger at higher fields. Note that for conventional MCE, when the field changes from 0 to H, ∆T ≥ ∆Trev, which is consistent with the data around the Curie point. At each temperature, the dissipated energy is given by Ediss = T∆S + C∆T . A value of 158 J/kg is found at 225 K for a field of 1.3 T. This value amounts to about 10 % of the latent heat of the martensitic transition in this alloy. In giant magnetocaloric materials for which the MCE is associated with a first-order transition, the giant effect relies on the possibility of inducing the phase transition by application of a magnetic field. The martensitic tran- sition is driven by phonon instabilities in the transverse TA2 phonon branch ([110] propagation and [11̄0] polar- ization) [22, 23]. Recent ab-initio calculations for cubic Ni2MnIn have shown that increasing the magnetization due to an external field favors the cubic structure and leads to a gradual vanishing of the phonon instability [24] due to the coupling between vibrational and mag- netic degrees of freedom. This effect results in a marked decrease of the martensitic transition temperature with increasing field that enables to induce the transition by the application of a field at a temperature close to the zero field transition temperature. Hence, the microscopic origin of the inverse MCE in Ni50Mn34In16 must be as- cribed to such magnetoelastic coupling responsible for the change in the relative stability of the martensitic and cubic phases. IV. CONCLUSION By directly measuring the adiabatic temperature change in the Ni50Mn34In16 alloy, we provide experi- mental evidences of both cooling and heating in a giant inverse magnetocaloric compound. It has been shown that the irreversibility associated with the first-order structural transition gives rise to measured temperature changes which are lower than those indirectly computed using equilibrium thermodynamics. The existence of a temperature region where the magnetocaloric effect re- verses sign under weakly applied magnetic fields opens up the possibility of new applications of this fascinating property. Acknowledgments This work received financial support from the CICyT (Spain), Project No. MAT2004–01291, DURSI (Catalo- nia), Project No. 2005SGR00969, and from the Deutsche Forschungsgemeinschaft (GK277). XM acknowledges support from DGICyT (Spain). We thank Peter Hinkel for technical support. [1] A.M. Tishin, Y.I. Spinchkin, The Magnetocaloric Ef- fect and its Applications, Institute of Physics Publishing, Bristol, 2003. [2] P. Debye, Ann. Phys. 81, 1154 (1926). [3] W.F. Giauque, J. Amer. Chem. Soc. 49, 1864 (1927). [4] V.K. Pecharsky and K.A. Gschneidner, Phys. Rev. Lett. 78, 4494 (1997); for a thorough and comprehensive re- view on new magnetocaloric materials see, K.A. Gschnei- dner, V.K. Pecharsky and A.O. Tsokol, Rep. Prog. Phys. 68, 1479 (2005). [5] E. Brück, J. Phys. D: Appl. Phys. 38, R381 (2005). [6] R.J. Joenk, Phys. Rev. 128, 1634 (1962). [7] A.E. Clark and E. Callen, Phys. Rev. Lett. 23, 307 (1969). [8] T. Krenke, M. Acet, E. F. Wassermann, X. Moya, L. Mañosa and A. Planes, Nature Materials 4, 450 (2005). [9] K.G. Sandeman, R. Daou, S. Özcan, J.H. Durrell, N.D. Mathur, and D.J. Fray, Phys. Rev. B 74, 224436 (2006). [10] A. Planes, L. Mañosa, and A. Saxena, Interplay of Mag- netism and Structure in Functional Materials edited by A. Planes, L. Mañosa, and A. Saxena, Materials Science Series Vol. 79 (Springer-Verlag, Berlin, 2005). [11] T. Krenke, M. Acet, E. F. Wassermann, X. Moya, L. Mañosa, A. Planes, Phys. Rev. B. 73, 174413 (2006). [12] P. J. Brown, A. P. Gandy, K. Ishida, R. Kainuma, T. Kanomata, K. U. Neumann, K. Oikawa, B. Ouladdiaf, and K. R. A. Ziebeck, J. Phys.:Condens. Matter 18, 2249 (2006). [13] V.K. Pecharsky, K.A. Gschneidner, Int. J. Refigeration 29 1239 (2006). [14] A. Giguère, M. Foldeaki, B.Ravi Gopal, R. Chahine, T.K. Bose, A. Frydman, J.A. Barclay, Phys. Rev. Lett. 83, 2262 (1999). [15] K.A. Gschneidner, V.K. Pecharsky, E. Brück, H.G.M Duijn, E.M. Levin, Phys. Rev. Lett. 85, 4190 (2000). [16] M. Pasquale,C.P. Sasso, L.H. Lewis, L. Giudici, T. Lo- grasso, D. Schlagel, Phys. Rev. B 72, 94435 (2005). [17] A. Boller, Y. Jin, B. Wunderlich, J. Thermal. Anal. 42, 307 (1994). [18] S.Y. Dan’kov, A.M. Tishin, V.K. Pecharsky, K.A. Gschneidner, Phys. Rev. B 57, 3478 (1998). [19] The thermograms shown in fig. 1 differ with those re- ported in [11]. Unavoidable small composition inhomo- geneities, impurities, etc. are know to affect the actual transition path. Therefore, different thermograms can be obtained in different specimens with the same nominal composition. [20] The small discrepancies in the data around the Curie point arise from the fact that the specific heat has been considered to be independent of magnetic field. For fer- romagnetic materials the effect of magnetic field is to smooth out the peak in the specific heat. [21] J. Ort́ın, A. Planes, L. Delaey in The Science of Hystere- sis Vol 3, p. 467. ed. by I. Mayergoyz and G. Bertotti, Elsevier 2005. [22] A. Planes and L. 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0704.1242

Giant Fluctuations of Coulomb Drag in a Bilayer System

7 Giant Fluctuations of Coulomb Drag in a Bilayer System A. S. Price,1 A. K. Savchenko,1∗ B. N. Narozhny,2 G. Allison,1 D. A. Ritchie3 1School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK 2The Abdus Salam ICTP, Strada Costiera 11, Trieste I-34100, Italy 3Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, UK ∗To whom correspondence should be addressed: [email protected]. The Coulomb drag in a system of two parallel layers is the result of electron- electron interactions between the layers. We have observed reproducible fluc- tuations of the drag, both as a function of magnetic field and electron concen- tration, which are a manifestation of quantum interference of electrons in the layers. At low temperatures the fluctuations exceed the average drag, giving rise to random changes of the sign of the drag. The fluctuations are found to be much larger than previously expected, and we propose a model which explains their enhancement by considering fluctuations of local electron properties. In conventional measurements of the resistance of a two-dimensional (2D) layer an electrical current is driven through the layer and the voltage drop along the layer is measured. In contrast, Coulomb drag studies are performed on two closely spaced but electrically isolated layers, where a current I1 is driven through one of the layers (active layer) and the voltage drop V2 is measured along the other (passive) layer (Fig. 1). The origin of this voltage is electron-electron (e-e) interaction between the layers, which creates a ‘frictional’ force that drags electrons in http://arxiv.org/abs/0704.1242v1 the second layer. The ratio of this voltage to the driving current RD = −V2/I1 (the drag resistance) is a measure of e-e interaction between the layers. The measurement of Coulomb drag in systems of parallel layers was first proposed in Ref. (1,2) and later realised in a number of experiments (3, 4, 5, 6, 7) (for a review see Ref. (8)). As Coulomb drag originates from e-e interactions, it has become a sensitive tool for their study in many problems of contemporary condensed matter physics. For example, Coulomb drag has been used in the search for Bose- condensation of interlayer excitons (9), the metal-insulator transition in two-dimensional (2D) layers (10), and Wigner crystal formation in quantum wires (11). Electron-electron scattering, and the resulting momentum transfer between the layers, usu- ally creates a so-called ‘positive’ Coulomb drag, where electrons moving in the active layer drag electrons in the passive layer in the same direction. There are also some cases where unusual, ‘negative’ Coulomb drag is observed: e.g. between 2D layers in the presence of a strong, quan- tising magnetic field (6, 7); and between two dilute, one-dimensional wires where electrons are arranged into a Wigner crystal (11). All previous studies of the Coulomb drag, however, refer to the macroscopic (average) drag resistance. Recently there have been theoretical predictions of the possibility to observe random fluctuations of the Coulomb drag (12, 13), where the sign of the frictional force will change randomly from positive to negative when either the carrier concentration, n, or applied (very small) magnetic field, B, are varied. Drag fluctuations originate from the wave nature of electrons and the presence of disorder (impurities) in the layers. Electrons travel around each layer and interfere with each other, after collisions with impurities, over the characteristic area ∼ L2ϕ, where Lϕ is the coherence length (Fig. 1). This interference is very important for conductive properties of electron waves. For example, the interference pattern is changed when the phase of electron waves is varied by a small magnetic field, producing universal conductance fluctuations (UCF) seen in small samples with size L ∼ Lϕ. There is, however, a significant difference between UCF and the fluctuations of the drag resistance. The former are only a small correction to the average value of the conductance: in our experiment the single-layer resistance fluctuates by ∼ 200 mOhm around an average resistance of approximately 500 Ohm. In contrast, the drag fluctuations, although small in absolute magnitude (∼ 20 mOhm) are able to change randomly, but reproducibly the sign of the Coulomb drag between positive and negative. Surprisingly, we have found that these fluctuations of the Coulomb drag, observed at temperatures below 1 K, are four orders of magnitude larger than predicted in Ref. (12). Our explanation of the giant drag fluctuations takes into account that, unlike the UCF, the drag fluctuations are not only an interference but also fundamentally an interaction effect. In conventional drag structures the electron mean free path l is much larger than the separation d between the layers, and therefore large momentum transfers h̄q between electrons in the layers become essential. According to the quantum mechanical uncertainty principle, ∆r∆q ∼ 1, electrons interact over small distances ∆r ≪ l when exchanging large values of momentum (Fig. 1). As a result the local properties of the layers, such as the local density of electron states (LDoS), become important in the interlayer e-e interaction. These local properties at the scale ∆r ≪ l exhibit strong fluctuations (14) that directly manifest themselves in the fluctuations of the Coulomb drag. The samples used in this work are AlGaAs-GaAs double-layer structures, in which the car- rier concentration of each layer can be independently controlled by gate voltage. The two GaAs quantum wells of the structure, 200 Å in thickness, are separated by an Al0.33Ga0.67As layer of thickness 300 Å. Each layer has a Hall-bar geometry, 60µm in width and with a distance between the voltage probes of 60µm (15). Figure 2 shows the appearance of the fluctuations in the drag resistivity, ρD, at low temper- atures. At higher temperatures, the drag resistance changes monotonically with both T and n: the insets to Fig. 2 show that ρD increases with increasing temperature as T 2 and decreases with increasing passive-layer carrier concentration as nb , where b ≈ −1.5. These results are consistent with existing experimental work on the average Coulomb drag (4, 16). Figure 3A shows a zoomed-in view of the reproducible fluctuations as a function of n2. These fluctuations result in an alternating sign of the drag, which is demonstrated in the inset to Fig. 3 where the temperature dependence of the drag is shown at two different values of n2. The drag is seen first to decrease as the temperature is decreased, but then become either in- creasingly positive or increasingly negative, dependent upon n2. The reproducible fluctuations of the drag resistivity have also been observed as a function of magnetic field (Fig. 3B). For a fixed temperature, the magnitude of the drag fluctuations as a function of n2 is roughly the same as that as a function of B. The theory of Ref. (12) calculates the variance of drag fluctuations in the so-called diffusive regime, l < d. In this case the drag is determined by global properties of the layers, aver- aged over a region ∆r ≫ l. The expected variance of drag fluctuations (at low T when the fluctuations exceed the average) in the diffusive regime is 〈∆σ2D〉 ≈ A ET (L)τϕ ln κd g4h̄(κd)3 , (1) where σD ≈ ρD/(ρ1ρ2), and ρ1 and ρ2 are the active and passive layer resistivities, respec- tively; ET (L) is the Thouless energy, ET (L) = h̄D/L 2, D is the diffusion coefficient; τϕ is the decoherence time; κ is the inverse screening length; A = 4.9 × 10−3 and g = h/(e2ρ) is the dimensionless conductivity of the layers. Using the parameters of our system, this expres- sion gives a variance of ∼ 6 × 10−11 µS2, which is approximately eight orders of magnitude smaller than the variance of the observed drag fluctuations. The fluctuations in ρD have been measured in two different samples, and their variance is seen to be similar in magnitude and T -dependence, confirming the discrepancy with the theoretical prediction (12). The expected fluctuations of the drag conductivity share the same origin as the UCF in the conventional conductivity: coherent electron transport over Lϕ in the layers prior to e-e interaction between the layers (Fig. 1). For this reason we have compared the drag fluctuations with the fluctuations seen in the single-layer resistivity of the same structure (Fig. 3B, inset), which have shown the usual behaviour (17). We estimate the expected variance of the single- layer conductance fluctuations using the relation 〈∆σ2xx〉 = (e (LT /L) 2, where LT = h̄D/kBT is the thermal length (17). This expression produces a value of 0.8µS 2, which is in good agreement with the measured value of 0.6µS2. The typical ‘period’ of the drag fluctuations (the correlation field, ∆Bc) is similar to that of the UCF (15), indicating that both depend upon the same Lϕ and have the same quantum origin. To address the question of the discrepancy between the magnitude of drag fluctuations in theory (12) and our observations, we stress that the theoretical prediction for the variance, Eq. 1, was obtained under the assumption of diffusive motion of interacting electrons, with small interlayer momentum transfers, q ≪ 1/l. As the layers are separated by a distance d, the e-e interactions are screened at distances ∆r > d. Therefore, in all regimes the maximum momentum transfers are limited by q < 1/d. In the diffusive regime, l < d, this relation also means that q < 1/l, that is, interlayer e-e interactions occur at distances ∆r > l and involve scattering by many impurities in the individual layers. In the opposite situation, l ≫ d, the transferred momenta will include both small and large q-values: q < 1/l and 1/l < q < 1/d. We have seen that small q cannot explain the large fluctuations of the drag (12), and so argue that it is large momentum transfers with q > 1/l which give rise to the observed effect. In this case the two electrons interact at a distance ∆r that is smaller than the average impurity separation and, therefore, it is the local electron properties of the layers which determine e-e interaction. In Ref. (14) it is shown that the fluctuations of the local properties are larger compared to those of the global properties that are responsible for the drag in the diffusive case. A theoretical expression for the drag conductivity is obtained by means of a Kubo formula analysis (18, 19, 20, 21) (detailed description in supporting text). For a qualitative estimate, three factors have to be taken into account: (i) the inter-layer matrix elements of the Coulomb interaction Dij ; (ii) the phase space (the number of electron states available for scattering); and (iii) the electron-hole (e-h) asymmetry in both layers. Point (iii) takes into account that in a quantum system the current is carried by both electron-like (above the Fermi surface) and hole- like (below the Fermi surface) excitations. If they were completely symmetric with respect to each other, then the current-carrying state of the active layer would have zero total momentum and thus no drag effect would be possible. The physical quantity that measures the degree of e-h asymmetry is the non-linear susceptibilityΓ of the 2D layer. Theoretically, the drag conductivity is represented in terms of the non-linear susceptibilities of each layer and dynamically screened interlayer Coulomb interaction Dij(ω) as σD ∝ dωD12(ω)Γ2(ω)D21(ω)Γ1(ω) (indices 1 and 2 correspond to the two layers) (18, 12). The e-h asymmetry appears in Γ as a derivative of the density of states ν and the diffusion coefficient D: Γ ∝ ∂ (νD) /∂µ, and it is this quantity that is responsible for the fact that drag fluctuations can exceed the average. As Dν ∼ g and the typical energy of electrons is the Fermi energy, EF , we have ∂(νD)/∂µ ∼ g/EF for the average drag. The typical energy scale for the interfering electrons, however, is ET (Lϕ) (17), which is much smaller than EF and therefore a mesoscopic system has larger e-h asymmetry. Under the condition of large momentum transfer between the layers, fluctuations in Γ are similar to the fluctuations of the LDoS, which can be estimated as δν2 ∼ (ν2/g) ln (max(Lϕ, LT )/l) (14). Also, the interaction in the ballistic regime can be assumed to be constant, Dij ≈ −1/νκd, as q is limited by q ≤ 1/d. Finally, to average fluctuations of the drag over the sample with size L we should divide it into coherent patches of size Lϕ that fluctuate independently and thus decrease the total variance: 〈∆σ2D〉 = 〈∆σ D(Lϕ)〉 (Lϕ/L) 2. If kBT > ET (Lϕ), fluctuations are further averaged on the scale of ∼ kBT , and therefore the variance is suppressed by an additional factor of ET (Lϕ)/kBT . Combining the above arguments we find 〈∆σ2D〉 = N g2h̄2(κd)4 (kBT ) E2T (Lϕ) l4L2ϕ , (2) where N is a numerical coefficient. Compared to the diffusive situation (Eq. 1) the fluctuations described by our model are greatly enhanced. The difference between Eqs. 2 and 1 comes from the fact that in the ballistic regime electrons are not scattered by impurities between events of e-e scattering. Large momen- tum transfers correspond to short distances, and thus in the ballistic regime drag measurements explore the local (as opposed to averaged over the whole sample) non-linear susceptibility. This leads to the appearance of three extra factors in Eq. 2: (i) the factor l4/d4 (which is also present in the average drag in the ballistic regime – see Ref. (18)); (ii) the phase space factor T/ET (which appears due to the fact that interaction parameters Dij are now energy-independent); and (iii) the extra factor g2 due to fluctuations of the local non-linear susceptibility. Local fluc- tuations are enhanced since the random quantity Γ is now averaged over a small part of the ensemble, allowing one to detect rare impurity configurations. Our model not only explains the large magnitude of the fluctuations, but also predicts a non-trivial temperature dependence of their magnitude. The latter comes from the change in the temperature dependence of Lϕ (22): at low temperatures, kBTτ/h̄ ≪ 1, the usual result is Lϕ ∝ T −1/2, while for kBTτ/h̄ > 1 the temperature dependence changes to Lϕ ∝ T −1 (23). Consequently, the temperature dependence of the variance of the drag fluctuations is expected to change from T−1 at low T , to T−4 at high T . This temperature dependence is very different from the T -dependence of drag fluctuations in the diffusive regime, 〈∆σ2D〉 ∝ T −1. To test the prediction of Eq. 2, the T -dependence of 〈∆σ2D〉 has been analysed (Fig. 4). The variance is calculated in the limits of both the diffusive τϕ (solid line, τ ϕ ∝ T ) and ballistic τϕ (dashed line, τ−1ϕ ∝ T 2), using N ≃ 10−4. In fitting the drag variance we have found τϕ to agree with theory to within a factor of two (15), which is typical of the agreement found in other experiments on determining τϕ (24). (The single-layer values of τϕ found from our analysis of the UCF agree with theory to within a factor of 1.5.) Thus, the temperature dependence of the observed drag fluctuations strongly supports the validity of our explanation. We have observed reproducible fluctuations of the Coulomb drag and demonstrated that they are an informative tool for studying wave properties of electrons in disordered materials, and the local properties in particular. Contrary to UCF which originate from quantum interference, fluctuations of drag result from an interplay of the interference and e-e interactions. More the- oretical and experimental work is required to study their manifestation in different situations. For instance, similarly to the previous extensive studies of the evolution of UCF with increasing magnetic field, such experiments can be performed on the fluctuations of drag. One of the im- portant developments in the field of Coulomb drag fluctuations can be their study in quantising magnetic fields, including the regimes of integer and fractional quantum Hall effects. References 1. M. B. Pogrebinskii, Sov. Phys. Semicond. 11, 372 (1977). 2. P. J. Price, Physica 117B, 750 (1983). 3. P. M. Solomon, P. J. Price, D. J. Frank, D. C. La Tulipe, Phys. Rev. Lett. 63, 2508 (1989). 4. T. J. Gramila, J. P. Eisenstein, A. H. MacDonald, L. N. Pfeiffer, K. W. West, Phys. Rev. Lett. 66, 1216 (1991). 5. U. Sivan, P. M. Solomon, H. Shtrikman, Phys. Rev. Lett. 68, 1196 (1992). 6. M. P. Lilly, J. P. Eisenstein, L. N. Pfeiffer, K. W. West, Phys. Rev. Lett. 80, 1714 (1998). 7. J. G. S. Lok et al., Phys. Rev. B 63, 041305 (2001). 8. A. G. Rojo, J. Phys.: Condens. Matter 11, R31 (1999). 9. D. Snoke, Science 298, 1368 (2002). 10. R. Pillarisetty et al., Physical Review B 71, 115307 (2005). 11. M. Yamamoto, M. Stopa, Y. Tokura, Y. Hirayama, S. Tarucha, Science 313, 204 (2006). 12. B. N. Narozhny, I. L. Aleiner, Phys. Rev. Lett. 84, 5383 (2000). 13. N. A. Mortensen, K. Flensberg, A.-P. Jauho, Phys. Rev. B 65, 085317 (2002). 14. I. V. Lerner, Phys. Lett. A 133, 253 (1988). 15. The details of the structures, measurements and the model are available as supporting ma- terial on Science Online. 16. M. Kellogg, J. P. Eisenstein, L. N. Pfeiffer, K. W. West, Solid State Communications 123, 515 (2002). 17. B. L. Altshuler, P. A. Lee, R. A. Webb, eds., Mesoscopic Phenomena in Solids (North- Holland, New York, 1991). 18. A. Kamenev, Y. Oreg, Phys. Rev. B 52, 7516 (1995). 19. L. Zheng, A. H. MacDonald, Phys. Rev. B 48, 8203 (1993). 20. A.-P. Jauho, H. Smith, Phys. Rev. B 47, 4420 (1993). 21. K. Flensberg, B. Y.-K. Hu, A.-P. Jauho, J. M. Kinaret, Phys. Rev. B 52, 14761 (1995). 22. B. N. Narozhny, G. Zala, I. L. Aleiner, Phys. Rev. B 65, 180202 (2002). 23. B. L. Altshuler, A. G. Aronov, Electron-Electron Interactions in Disordered Systems, A. L. Efros, M. Pollak, eds. (North-Holland, Amsterdam, 1985). 24. C. W. J. Beenakker, H. van Houten, Solid State Physics, H. Ehrenreich, D. Turnbull, eds. (Academic Press Limited, 1991). 25. Authors thank I.L. Aleiner, M. Entin, I.L. Lerner, A. Kamenev, and A. Stern for numerous helpful discussions. Supporting Online Material www.sciencemag.org Materials and Methods SOM text Figs. S1 to S3 Fig. 1. Schematic showing the origin of the drag signal V2 induced by the current I1. The fluctuations of the drag arise from the interference of electron waves in each layer, before the two electrons take part in the interlayer interaction. 0.5 1.0 1.5 2.0 (1011cm-2) T2 (K2) (1011 cm-2) Fig. 2. Drag resistivity as a function of passive-layer carrier concentration for different temperatures: T = 5, 4, 3, 2, 1, 0.4, and 0.24 K, from top to bottom. Inset (A): ρD as a function of T2. Inset (B): ρD as a function of n2, with n1 = 1.1× 1011 cm−2; dashed line is a n−1.5 0.5 1.0 0 10 20 20 30 -0.07 (1011cm-2) B (mT) T (K) Fig. 3. (A) Drag resistance measured at low temperatures as a function of passive- layer concentration; T = 1, 0.4, and 0.24 K, from top to bottom. Inset: ρD as a function of T for two values of n2 denoted by the dotted lines in Fig. 3A; solid line is the expected 2 dependence of the average drag. (B): ρD as a function of B; T = 0.4, 0.35, and 0.24 K, from top to bottom. (Graphs for higher T are vertically offset for clarity.) Single-layer concentration for each layer is 5.8× 1010cm−2. Inset: The UCF of the single-layer, with an average background resistance of 500 Ohm subtracted. Fig. 4. The variance of the drag conductivity fluctuations (squares) plotted against temperature. The solid and dashed lines are calculated using Eq. 2 with the diffusive and ballistic asymptotes of τϕ, respectively. Inset: τϕ extracted from the correlation magnetic field of the single-layer fluctuations, plotted against temperature.

0704.1243

Magnetic superelasticity and inverse magnetocaloric effect in Ni-Mn-In

Magnetic superelasticity and inverse magnetocaloric effect in Ni-Mn-In Thorsten Krenke, Eyüp Duman, Mehmet Acet, and Eberhard F. Wassermann Experimentalphysik, Universität Duisburg-Essen, D-47048 Duisburg, Germany Xavier Moya, Llúıs Mañosa and Antoni Planes Facultat de F́ısica, Departament dEstructura i Constituents de la Matèria, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Catalonia, Spain Emmanuelle Suard and Bachir Ouladdiaf Institut Laue-Langevin BP 156, 38042 Grenoble Cedex 9, France (Dated: October 26, 2018) Applying a magnetic field to a ferromagnetic Ni50Mn34In16 alloy in the martensitic state induces a structural phase transition to the austenitic state. This is accompanied by a strain which re- covers on removing the magnetic field giving the system a magnetically superelastic character. A further property of this alloy is that it also shows the inverse magnetocaloric effect. The magnetic superelasticity and the inverse magnetocaloric effect in Ni-Mn-In and their association with the first order structural transition is studied by magnetization, strain, and neutron diffraction studies under magnetic field. PACS numbers: 75.80.+q , 61.12.-q I. INTRODUCTION Shape memory alloys exhibit unique thermomechanical properties which originate from a martensitic transition occurring between the austenite state with high crystallo- graphic symmetry and a lower symmetry martensite state [1]. These materials are superelastic and can remember their original shape after severe deformation. Superelas- ticity is related to the stress induced reversible structural transition. Research on shape memory alloys received significant stimulus after the discovery of the magnetic shape mem- ory (MSM) effect in Ni2MnGa [2]. This effect arises from a magnetic field induced reorientation of twin- related martensitic variants and relies on high magne- tocrystalline anisotropy. The driving force is provided by the difference between the Zeeman energies of neigh- boring martensite variants [3, 4]. Giant strains up to 10% have been reported for off stoichiometric Ni-Mn-Ga single variant crystals with the 14M modulated marten- sitic structure [5]. Over the past decade, vast amount of knowledge accumulated on the properties of Ni-Mn-Ga Heusler alloys has enabled to foresee the possibility of employing these alloys in device applications [6]. In applied magnetic fields, the martensitic start tem- perature Ms of Ni-Mn-Ga shifts to higher temperatures along with the other characteristic temperatures marten- site finish Mf , austenite start As, and austenite finish Af [7]. With this feature, it is possible to induce a reversible structural phase transformation, whereby strain can be fully recovered on removing the field without the neces- sity of prestraining the specimen [8]. In such magnetic field induced superelasticity, the maximum field induced strain relies on the difference in the crystallographic di- mensions in the martensitic and austenitic states. When a field of sufficient strength is applied at a temperature corresponding initially to the austenitic state, the shift in all characteristic temperatures (therefore the shift in the hysteresis associated with the transformation) can be large enough so that the martensitic state is stabi- lized. However, experiments performed in fields up to 10 T have shown that in the case of Ni54Mn21Ga25 the rate of shift is only about ∼1 KT−1 [8]. Neutron diffraction experiments under magnetic field on an alloy with similar composition confirm these results [9]. Parallel to the development of the understanding of the MSM effect in Ni-Mn-Ga and exploiting giant strains for applications, search for other MSM material also took up considerable place in the research agenda [10]. In Ni-Co- Mn-In, it has recently been reported that when a mag- netic field is applied to a pre-strained single crystal spec- imen, the strain is recovered with a value that is nearly equal to the size of the pre-strain [11]. Although this is a considerable step in the search for magnetic supere- lasticity, a system in which considerable length change occurs reversibly by applying and removing a magnetic field without requiring pre-strain is still to be found. Recently, we have investigated a number of Ni-Mn based Heusler systems other than Ni-Mn-Ga with the aim of finding ferromagnetic alloys that undergo marten- sitic transformations and understanding their properties around the transformation point [12, 13]. In Ni-Mn-Sn [14], we have come across an inverse magnetocaloric effect at temperatures in the range of the first order martensitic transition with size comparable to that of the archetype Gd5(Si1−xGex)4 system, which exhibits the conventional giant magnetocaloric effect [15]. Here, we demonstrate the presence of both magnetic superelasticity and the inverse magnetocaloric effect in Ni-Mn-In in the range of the martensitic transition. Large field induced strains in polycrystalline Ni-Mn-In of magnitude similar to that in polycrystalline Ni-Mn- http://arxiv.org/abs/0704.1243v1 Ga are found. We show in Ni-Mn-In that instead of the large field induced strain being due to twin boundary motion in the martensitic phase, it relies essentially on the reverse field induced martensite-to-austenite transi- tion. Below, we present results of field dependent mag- netization, calorimetry, neutron diffraction, strain, and length change measurements in magnetic field on a Ni- Mn-In alloy and discuss the field induced strain and the inverse magnetocaloric effect in relation to the field in- duced martensite-to-austenite transition. II. EXPERIMENTAL Arc melted samples were annealed at 1073 K under argon atmosphere for two hours and quenched in ice wa- ter. Magnetization measurements were carried out us- ing a superconducting quantum interference device mag- netometer, and calorimetric measurements in magnetic field were performed using a high sensitivity differential scanning calorimeter [16]. Neutron diffraction in mag- netic fields up to 5 T was performed on the D2B powder diffractometer at ILL, Grenoble. The strain measure- ments were made using conventional strain-gage tech- nique in magnetic fields up to 5T. III. RESULTS A. Calorimetry and magnetization Ni50Mn50−xInx alloys undergo martensitic transitions for about x < 16 [13, 17]. Here we concentrate on the magneto-structural coupling in the ferromagnetic Ni50.3Mn33.8In15.9 alloy, which has a Curie point TC = 305 K and transforms martensitically on cooling at Ms = 210 K. The other characteristic temperatures defining the temperature limits of the transition are Mf = 175 K, As = 200 K, and Af = 230 K. These temperatures are determined from the calorimetry data in Fig. 1a. In order to search for the presence of a coupling of the structure with the magnetic degree of freedom within the temperature range of the martensitic transition, we have studied the field dependence of the magnetizationM(H). The data shown in Fig. 1b are obtained in increasing field and decreasing temperature. Here, the magnetizations in the temperature interval 160 ≤ T ≤ 210 K initially show a tendency to saturate, but, then, exhibit metamagnetic transitions in higher fields. The characteristic field Hc defining the transition point is determined as the crossing point of the linear portions of the curves. The transition shifts to higher fields with decreasing temperature, and on removing the field, the magnetization returns to its original value (see Fig. 7). As will be shown with neutron diffraction in external field, the metamagnetic transition is associated with the onset of a field induced reverse martensitic transition. 0 1 2 3 4 5 160 180 200 220 240 220 K 160 K H (T) 170 K 210 K Temperature (K) FIG. 1: Features in the martensitic transformation associated with temperature and magnetic field dependence. a) Calori- metric curves across the martensitic transition. The horizon- tal arrows indicate the direction of temperature change. b) Magnetization as a function of magnetic field measured on in- creasing field in the vicinity of the martensitic transition. The red lines drawn through the data points (shown only for the 200 K data) cross at a point corresponding to the characteris- tic field around which the metamagnetic transition begins to occur. 0 100 200 300 T (K) M 5TS =150 K M 2TS =185 K M 0TS =205 K FIG. 2: The relative length change in constant applied mag- netic fields of 0 T, 2 T and 5 T. Arrows indicate the positions of Ms. In Fig. 2, we show the relative length change ∆l/l as a function of temperature in different constant applied magnetic fields. As the field increases, Ms (indicated by arrows) decreases by an amount of about −10 KT−1. The other characteristic temperatures are positioned in the conventional manner around the temperature hys- teresis loop, and all shift by nearly the same amount in a given field. With increasing measuring field, the dif- ference in ∆l/l between the austenitic and martensitic states decreases. The cause of this decrease is associated with the crystallographic orientation of the easy axis of magnetization within the orthorhombic structure of the martensitic phase. This property is discussed separately in reference [18]. 0 1 2 3 4 5 160 180 200 220 H (T) T (K) 0 1 2 3 4 5 6 [13] H (T) FIG. 3: Characteristic field dependent properties around the martensitic transition. a) Shift in the martensitic transition temperature as a function of magnetic field, for Ni50.3Mn33.8In15.9, Ni2MnGa, and Ni53.5Mn19.5Ga27. b) Temperature dependence of the magnetization obtained at selected fields from Fig. 1b. The numbers refer to magnetic field values in Tesla. c) The difference between the magneti- zations in the cubic and martensite phases. ∆M saturates at about 0.5 T. The lines drawn through the data are guides. In Fig. 3a, we compare the magnitude of the shift of the transition temperatures represented by ∆T as a func- tion of the external magnetic field µ0H of the present Ni-Mn-In alloy with those reported for Ni2MnGa and Ni53.5Mn19.5Ga27; the latter exhibiting the strongest field dependent transition temperature [19, 20]. Since the ap- plied field shifts all characteristic temperatures by the same amount, ∆T is the change in any one of the charac- teristic transition temperatures in applied magnetic field with respect to zero field. The shifts in Ms and As of Ni-Mn-In determined from Fig. 2 are shown with up and down triangles respectively. We have also included data for µ0H ≤ 1 T obtained from calorimetric measurements under constant magnetic field [13]. Two significant features show up from Fig. 3a: (i) The rate of change of the transition temperature in Ni50.3Mn33.8In15.9 (−10 KT −1) is higher than in Ni53.5Mn19.5Ga27 (6 KT −1), and (ii) in Ni-Mn-In, ∆T is negative, i.e., the transition temperature decreases with increasing field. This is consistent with the fact that the magnetization in the high temperature cubic phase is larger than the magnetization in the martensitic phase as seen in Fig. 3b, where the temperature dependence of the magnetizations at constant fields obtained from Fig. 1b at selected fields are plotted. The difference in the magnetization of the martensitic and austenitic states around the transition temperature ∆M is plotted as a function of applied field in Fig. 3c. B. Neutron diffraction in magnetic field To understand the properties of the transition observed inM(T ) andM(H), we have undertaken powder neutron 0 1 2 3 4 5 200 K T=4 K 140 K 180 K µ0H (T) FIG. 4: The magnetic field dependence of the magnetization for the sample used in the neutron diffraction experiments. The crossing point of the red lines determine Hc. diffraction experiments as a function of temperature and magnetic field. The Ni49.7Mn34.3In16.0 sample used for these experiments has a composition that differs slightly from that used in the measurements presented above and, therefore, the transition temperatures are slightly shifted. Therefore, we plot in Fig. 4 the M(H) isotherms and will compare these data to the neutron diffraction data. Fig. 5a shows neutron diffraction patterns at 5 K and 317 K. The pattern at 317 K generates from an L21 struc- ture with a lattice constant a = 0.6011 nm. At 5 K, the pattern is that of a 10M modulated martensite struc- ture having a monoclinic unit cell with β = 86.97 and lattice constants a = 0.4398 nm, b = 0.5635 nm, and c = 2.1720 nm. Other than this slight monoclinicity, the structure is orthorhombic having a shuffle periodicity of 10 lattice planes in the [110] direction. In the pattern at 317 K, some additional weak reflections can be iden- tified around 36 and 47. These lie close to the positions of the (1 0 −7) and (2 1 3) reflections of the martensitic phase at 5 K, but at slightly smaller angles due to ther- mal expansion, and their presence is attributed to small amounts of mechanically induced martensite formed on grinding the ingot for powder specimen preparation. At 180 K, which is a temperature that falls well in the range of the transformation (Fig. 4), we have studied the evolution of the diffraction spectrum with applied mag- netic field. Fig. 5b shows the spectrum in 2θ ranges that encompass the vicinity of the positions of the (200) and (220) reflections of the L21 phase. As the magnetic field increases, the intensities of these reflections grow at the expense of the intensities of the (1 0 −5) and (1 2 5) reflections, which lie nearly at the same positions as 20 30 40 50 60 24 30 40 50 2 (deg) 5) (0 317 K H (T) 2 (deg) 180 K FIG. 5: Neutron diffractograms. a) Patterns at 317 K (L21) and 5 K (10M martensite). b) The field dependence of the diffraction pattern at 180 K showing the field induced trans- formation from the martensite to the austenite state. the (200) and the (220) reflections of the L21 phase re- spectively. This shows that a progressive magnetic field induced structural transition from the martensitic to the austenitic state is taking place with increasing magnetic field. In cases where the positions of the reflections per- tain only to the martensitic phase, e.g. (1 0 5), (1 2 −5), etc., the intensity first increases with increasing mag- netic field up to 2 T and, then, decreases. The initial increase is related to the increase in the magnetization in the martensitic state at 180 K up to around µ0Hc ≈ 2 T (Fig. 4). The subsequent decrease is associated with the gradual decrease in the amount of martensite and the stabilization of the L21 phase. However at 5 T, the reflections associated with the martensitic phase do not disappear, although their intensities are reduced. This indicates that the transition is not complete in this field, and larger magnetic fields are required to fully restore the L21 state at 180 K. The neutron diffraction data indicate clearly a magnetic field induced reverse transition and give evidence that the observed metamagnetic transition -5 -4 -3 -2 -1 0 1 2 3 4 5 T (K) µ0H (T) FIG. 6: Magnetic field dependence of strain at 195 K (T < Ms) and 295 K (L21). The strain recovers on removing the field indicating magnetically superelastic behavior. Arrows show the direction of field change. in M(H) is related to the reverse martensitic transition. C. Magnetic field induced strain The magnetic field induced structural phase transition in the present alloy can have important consequences on macroscopic strains occurring during the application of the field. Fig. 6 shows the results of magnetic field depen- dent strain measurements, where the strain is defined as ∆l/l = [l(H)−l0]/l0. Here, l0 is the length in the absence of field and l(H) the length in field. The sample is well within the austenitic temperature range at 295 K (filled circles) and is within the structural transition region at 195 K (open circles). The small field induced strain at 295 K increasing negatively with increasing field corre- sponds to the intrinsic magnetostriction of the austenite, while at 195 K, a strain of about 0.14% is reached in the initial curve. After the first field cycle is completed, the strain reduces to about 0.12 % and remains constant at this value, which is roughly the same as that attained in polycrystalline Ni-Mn-Ga. However, the effect here is due to the crystallographic transformation from marten- site to austenite with increasing field instead of a field induced twin boundary motion that occurs within the martensitic state as in Ni-Mn-Ga. Although hysteresis is observed in Fig. 6, the sample returns to its zero-field length on removing the field. The features in the field dependence of the strain is reflected in the field dependence of the magnetization at the same temperature as seen in Fig. 7. The M(H) curves show metamagnetic transitions around 2 T and 1 T for the increasing and decreasing field branches respec- tively. These points correspond to the fields where ∆l/l also changes rapidly. As in the case of ∆l/l, M(H) also shows essentially no remanence and recovers its zero-field value. 0 1 2 3 4 5 195 K µ0H (T) FIG. 7: The magnetic field dependence of the magnetization at 195 K. 140 160 180 200 T (K) 0H (T) FIG. 8: Temperature dependence of ∆S. D. Magnetocaloric effect Due to the first order magnetic field induced transition and considerable difference in the magnetization of the martensitic and L21 states at the transition temperature, substantial magnetocaloric effects can be expected. The field induced entropy change ∆S around the martensitic transition temperature can be estimated from magneti- zation measurements by employing the Maxwell equation ∆S(T,H) = µ0 dH, (1) from which the magnetocaloric effect can be evaluated by numerical integration using the data in Fig. 1b. The re- sulting ∆S in the temperature range 170 K ≤ T ≤ 225 K is plotted in Fig. 8. The sign of ∆S is positive for all tem- peratures indicating that an inverse magnetocaloric effect is present, i.e. the sample cools on applying a magnetic field adiabatically as in Ni-Mn-Sn [14]. The maximum value of ∆S = 12 JK−1kg−1 is reached in a magnetic field of 4 T at about 205 K. Since there is no substantial change in the magnetization above 4 T at this tempera- ture, increasing the magnetic field any further does not lead to any further increase in ∆S. As expected for mag- netostructural transitions [22], this value is larger than the transition entropy change [13] since it also includes the effect of magnetization changes with temperature be- yond the transition region. IV. DISCUSSION The origins of both distinct properties of the studied alloy, namely, the field induced martensite to austen- ite transition and the inverse magnetocaloric effect, are related to the lower value of the magnetization in the martensitic phase with respect to that in the austenitic phase. The difference in the magnetization can be as- cribed to the fact that in Mn based Heusler alloys, the magnetic moments are localized mainly on the Mn atoms and the exchange interaction strongly depends on the Mn-Mn distance. Hence, any change in the dis- tance caused by a change in the crystallographic con- figuration can modify the strength of the interactions leading to different magnetic exchange in each of the phases. Indeed, it has been shown that in the case of a Ni0.50Mn0.36Sn0.14 alloy that short range antiferromag- netism is present between Mn atoms located at the 4(b) positions of the austenite phase which is then strength- ened in the martensitic state [23]. The present Ni-Mn- In alloy transforms to the same martensitic structure as in Ni0.50Mn0.36Sn0.14. Therefore, the strong reduction of the magnetization belowMs in Ni-Mn-In can be expected to be due to a similar cause. The presence of short range antiferromagnetism in the ferromagnetic matrix leads to frustration in the temperature range of the martensitic transition. In such a frustrated system, the application of a magnetic field can lead to the degeneracy of the spin states giving rise to an increase in the configurational entropy that is required for the observed positive ∆S. However, the quantitative details of the frustrated state and the microscopic process leading to a positive ∆S re- main to be described. In Ni-Mn-Ga, giant strains are due to the reorientation of twin-related variants in the martensitic state and re- covery of this strain is, in general, not achieved by simply removing the field. By contrast, the magnetic supere- lastic effect and the associated strain reported here for Ni-Mn-In is related to the field induced structural tran- sition. This enables to reversibly induce and recover the strain by simply applying and removing the field. V. CONCLUSION Ni0.50Mn0.34In0.16 exhibits a magnetic field induced structural transition from the martensitic state to the austenitic state. The transition is directly evidenced by neutron diffraction measurements under magnetic field. Here, other than in Ni-Mn-Ga alloys, where the magneti- zation of the martensitic state is higher than that in the cubic phase, the austenite is stabilized by the application of a magnetic field. The shift of the transition temper- atures was found to be large and negative with values up to about -50 K in 5 Tesla. Due to the reversible magnetic field induced transition, magnetic superelastic- ity with 0.12% strains occur. Other than in magnetic shape memory alloys, where strain is mainly created by twin boundary motion, strain in Ni0.50Mn0.34In0.16 is caused by changes in lattice parameters during the transition. Additionally, an inverse magnetocaloric ef- fect with a maximum value in the entropy change of about 12 Jkg−1K−1 at 190 K and a minimum entropy change of 8 Jkg−1K−1 in a broad temperature range 170K ≤ T ≤ 190K is also found in this alloy. Acknowledgments We thank Peter Hinkel and Sabine Schremmer for tech- nical support. This work was supported by Deutsche Forschungsgemeinschaft (GK277) and CICyT (Spain), project MAT2004-1291. XM acknowledges support from DGICyT (Spain). [1] K. Otsuka, C. M. Wayman, Eds. Shape Memory Materi- als (Cambridge Univ. Press, Cambridge, 1998). [2] K. Ullakko, J. K. Huang, C. Kantner, R. C. O’Handley, V. V. Kokorin, Appl. Phys. Lett. 69, 1966 (1996). [3] R. C. O’Handley, J. Appl. Phys. 83, 3263 (1998). [4] R. D. James, M. Wuttig, Philos. Mag. A 77, 1273 (1998). [5] A. Sozinov, A. A. Likhachev, N. Lanska, K. Ullakko, Appl. Phys. Lett. 80, 1746 (2002). [6] K. Kakeshita, K. Ullakko, MRS Bulletin 27, 105 (2002). [7] A. D. Bozhko, A. N. Vasil’ev, V. V. Khovailo, I. E. Dik- shtein, V. V. Koledov, S. M. Seletskii, A. A. Tulaikova, A. A. Cherechukin, and V. D. Buchel’nikov, JETP 88, 954 (1999). [8] I. E. Dikshtein, D. I. Ermakov, V. V. Koledov, L. V. Koledov, T. Tagaki, A. A. Tulaikova, A. A. Cherechukin, and V. G. Shavrov, JETP Lett. 72, 373 (2000). [9] K. Inoue, K. Enami, Y. Yamaguchi, K. Ohoyama, Y. Morii, Y. Matsuoka, and K. Inoue, J. Phys. Soc. Japan 69, 3485 (2000). [10] M. Wuttig, L. Liu, K. Tsuchiya, R. D. James, J. Appl. Phys. 87, 4707 (2000). [11] R. Kainuma, Y. Imano, W. Ito, , Y. Imano, W. Ito, Y. Sutou, H. Morito, S. Okamoto, O. Kitakami, K. Oikawa, A. Fujita, T. Kanomata, and K. Ishida, Nature 439, 957- 960 (2006). [12] T. Krenke, M. Acet, E. F. Wassermann, X. Moya, L. Maosa, A. Planes, Phys. Rev. B. 72, 14412 (2005). [13] T. Krenke, M. Acet, E. F. Wassermann, X. Moya, L.Maosa, and A. Planes, Phys. Rev. B 73, 174413 (2006). [14] T. Krenke, M. Acet, E. F. Wassermann, X. Moya, L. Maosa, A. Planes, Nature Materials 4, 450 (2005). [15] V. K. Pecharsky, K. A. Gschneidner Jr., Phys. Rev. Lett. 78, 4494 (1997). [16] J. Marcos, F. Casanova, X. Batlle, A. Labarta, A. Planes, L. Maosa, Rev. Sci. Inst. 74, 4768 (2003). [17] Y. Sutou, Y. Imano, N. Koeda, T. Omori, R. Kainuma, K. Ishida, K. Oikawa, Appl. Phys. Lett. 85, 4358 (2004). [18] T. Krenke and M. Acet (unpublished). [19] J. Kim, F. Inaba, T. Fukuda, T. Kakeshita, Acta mater. 54, 493- 499 (2006). [20] S. Jeong, K. Inoue, S. Inoue, K. Koterazawa, M. Taya, K. Inoue, Mat. Sci. Eng. A 359, 253 (2003). [21] O. Tegus, E. Brck, L. Zhang, Dagula, K. H. J. Buschow, and F. R. de Boer, Physica B 319, 174-192 (2002) [22] F. Casanova, X. Batlle, A. Labarta, J. Marcos, L. Maosa, and A. Planes, Phys. Rev. B, 66 100401(R)(2002). [23] P. J. Brown, A. P. Gandy, K. Ishida, R. Kainuma, T. Kanomata, K. U. Neumann, K. Oikawa, B. Ouladdiaf, and K. R. A. Ziebeck, J. Phys.: Condens. Matter 18, 2249 (2006).

0704.1244

SN 2003du: 480 days in the Life of a Normal Type Ia Supernova

arXiv:0704.1244v1 [astro-ph] 10 Apr 2007 Astronomy & Astrophysics manuscript no. m6020 c© ESO 2018 September 20, 2018 SN 2003du: 480 days in the Life of a Normal Type Ia Supernova V. Stanishev1 ,⋆, A. Goobar1, S. Benetti2, R. Kotak3,4, G. Pignata5, H. Navasardyan2 , P. Mazzali6,7, R. Amanullah1, G. Garavini1, S. Nobili1, Y. Qiu8, N. Elias-Rosa2,9, P. Ruiz-Lapuente10 , J. Mendez10,11, P. Meikle12, F. Patat3, A. Pastorello6,4, G. Altavilla10, M. Gustafsson13 , A. Harutyunyan2 , T. Iijima2, P. Jakobsson14 , M.V. Kichizhieva15, P. Lundqvist16, S. Mattila12, J. Melinder16, E.P. Pavlenko17, N.N. Pavlyuk18, J. Sollerman16,14, D.Yu. Tsvetkov18, M. Turatto2, W. Hillebrandt7 1 Physics Department, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden 2 INAF, Osservatorio Astronomico di Padova, vicolo dell’Osservatorio 5, 35122 Padova, Italy 3 European Southern Observatory, Karl-Schwarzschild-Strasse 2, D-85748 Garching, Germany 4 Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN, UK 5 Departamento de Astronomı́a y Astrofı́sica, Pontificia Universidad Católica de Chile, Campus San Joaquı́n. Vicuña Mackenna 4860 Casilla 306, Santiago 22, Chile 6 INAF Osservatorio Astronomico di Trieste, Via Tiepolo 11, 34131 Trieste, Italy 7 Max-Planck-Institut für Astrophysik, PO Box 1317, 85741 Garching, Germany 8 National Astronomical Observatories, Chinese Academy of Sciences, 100012 Beijing, China 9 Universidad de La Laguna, Av Astrofı́sico Fransisco Sánchez s/n, E-38206. La Laguna, Tenerife, Spain 10 Department of Astronomy, University of Barcelona, Marti i Franques 1, E-08028 Barcelona, Spain 11 Isaac Newton Group of Telescopes, Apartado de correos 321, E-38700 Santa Cruz de La Palma, Canary Islands, Spain 12 Astrophysics Group, Imperial College London, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, UK 13 Department of Physics and Astronomy, University of Aarhus, 8000 Aarhus C, Denmark 14 Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen Ø, Denmark 15 Tavrida State University, Simferopol, Ukraine 16 Department of Astronomy, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden 17 Crimean Astrophysical Observatory, Ukraine 18 Sternberg Astronomical Institute, Moscow State University, Universitetskii pr. 13, Moscow 119992, Russia Received ;accepted ABSTRACT Aims. We present a study of the optical and near-infrared (NIR) properties of the Type Ia Supernova (SN Ia) 2003du. Methods. An extensive set of optical and NIR photometry and low-resolution long-slit spectra was obtained using a number of facilities. The observations started 13 days before B-band maximum light and continued for 480 days with exceptionally good time sampling. The optical photometry was calibrated through the S-correction technique. Results. The UBVRIJHK light curves and the color indices of SN 2003du closely resemble those of normal SNe Ia. SN 2003du reached a B-band maximum of 13.49±0.02 mag on JD2452766.38 ±0.5. We derive a B-band stretch parameter of 0.988±0.003, which corresponds to ∆m15 = 1.02 ± 0.05, indicative of a SN Ia of standard luminosity. The reddening in the host galaxy was estimated by three methods, and was consistently found to be negligible. Using an updated calibration of the V and JHK absolute magnitudes of SNe Ia, we find a distance modulus µ = 32.79 ± 0.15 mag to the host galaxy, UGC 9391. We measure a peak uvoir bolometric luminosity of 1.35(±0.20) × 1043 erg s−1 and Arnett’s rule implies that M56Ni ≃ 0.68 ± 0.14 M⊙ of 56Ni was synthesized during the explosion. Modeling of the uvoir bolometric light curve also indicates M56Ni in the range 0.6 − 0.8 M⊙. The spectral evolution of SN 2003du at both optical and NIR wavelengths also closely resembles normal SNe Ia. In particular, the Si ii ratio at maximum R(Si ii)= 0.22 ± 0.02 and the time evolution of the blueshift velocities of the absorption line minima are typical. The pre-maximum spectra of SN 2003du showed conspicuous high-velocity features in the Ca ii H&K doublet and infrared triplet, and possibly in Si ii λ6355, lines. We compare the time evolution of the profiles of these lines with other well-observed SNe Ia and we suggest that the peculiar pre-maximum evolution of Si ii λ6355 line in many SNe Ia is due to the presence of two blended absorption components. Key words. stars: supernovae: general – stars: supernovae: individual: SN 2003du – methods: observational – techniques: photometric – techniques: spectroscopic 1. Introduction Type Ia supernovae (SNe Ia) form a relatively homogeneous class of objects with only a small scatter in their observed ab- solute peak magnitudes (∼ 0.3 mag). Moreover, their spectra ⋆ E-mail: [email protected] and light curves are strikingly similar (e.g. Branch & Tammann, 1992). Theoretical investigations strongly suggest that SNe Ia are thermonuclear explosions of carbon/oxygen white dwarfs (WD) with masses close to the Chandrasekhar limit ∼ 1.4M⊙ (for a review see Hillebrandt & Niemeyer, 2000). In the fa- vored model, the WD mass grows via accretion from a com- http://arxiv.org/abs/0704.1244v1 2 Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova panion star until the mass reaches the Chandrasekhar limit and the WD ignites at (or near) its center. The light curves of SNe Ia are powered by the energy released from the decay of radioac- tive 56Ni produced during the explosion (typically a few tenths of M⊙) and its daughter nuclides, and the scatter of the abso- lute magnitudes is mostly due to the different amounts of syn- thesized 56Ni. However, it has been shown that the peak lumi- nosity of SNe Ia correlates with the luminosity decline rate af- ter maximum light; the slower the decline, the greater the peak luminosity (Pskovskii, 1977; Phillips, 1993; Hamuy et al., 1995, 1996; Riess et al., 1995). After correcting for the empirical ”light curve width – peak luminosity” relation and for the extinction in the host galaxy, the dispersion of the SN Ia absolute peak B magnitudes is ∼ 0.14 mag (Phillips et al., 1999). This prop- erty combined with their high intrinsic luminosity (MV ≃ −19.2 mag), make SNe Ia ideal for measuring relative cosmological distances. Observations of SNe Ia out to a redshift of z ∼ 1.0 led to the surprising discovery that the expansion of the Universe is accelerating, and that ∼ 70% of the Universe consists of an unknown constituent with effective negative pressure, dubbed ”dark energy” (Riess et al., 1998; Perlmutter et al., 1999; Knop et al., 2003; Riess et al., 2004; Astier et al., 2006; Riess et al., 2007; Wood-Vasey et al., 2007). Currently, the favored model for dark energy is a non-zero positive cosmological constant Λ (or vacuum energy), but more exotic models have also been pro- posed (for a review see Peebles & Ratra, 2003). There are sev- eral observational programs planned or in progress that aim to discover and observe hundreds of SNe Ia up to z ∼ 1.7, with the goal of measuring cosmological parameters with greatly im- proved accuracy. This will enable distinctions to be made be- tween the large number of proposed models for dark energy. Although these programs will be able to greatly reduce the sta- tistical uncertainties on the measured cosmological parameters, the output will still be limited by systematic errors due to our poor knowledge of some aspects of SNe Ia and their environ- ment. Two of the major concerns are the possible evolution of the brightness or colors of SNe Ia with redshift and the esti- mation of the reddening in the host galaxy. There are indica- tions that the amount of 56Ni synthesized during the explosion is sensitive to the metallicity, carbon-to-oxygen (C/O) ratio and the central density of the exploding WD (Hoeflich et al., 1998; Umeda et al., 1999; Timmes et al., 2003; Röpke & Hillebrandt, 2004; Röpke et al., 2006), although based on three-dimensional simulations Röpke & Hillebrandt (2004) and Röpke et al. (2006) found that the C/O ratio has little effect on the 56Ni production. These quantities may, however, evolve with redshift and might therefore introduce some evolution of the observed SNe Ia prop- erties. However, our poor knowledge of the details of the physics of the explosion, the progenitor systems and how the WD mass grows to the Chandrasekhar limit (e.g., Hillebrandt & Niemeyer, 2000)) prevents us from accurately estimating the magnitude of the effect, and the extent to which it could affect the derived cosmological parameters. The difficulties in accurately estimat- ing the reddening in the SN host galaxies arise mostly from the uncertainty in the intrinsic colors of SNe Ia (e.g., Nobili et al., 2003) and the calibration of the photometry (Suntzeff, 2000), combined with poor knowledge of the dust properties. In this paper we present observations of the nearby Type Ia SN 2003du. It was discovered by The Lick Observatory and Tenagra Observatory Supernova Searches (Schwartz & Holvorcem, 2003) in the nearby (recession velocity of 1914 km s−1) SBd galaxy UGC 9391 on 2003 April 22.4 UT. Kotak et al. (2003) classified SN 2003du as a normal SN Ia at about two weeks before maximum light and an intensive optical and NIR observational campaign was initiated by the European Supernova Collaboration (ESC). The optical and NIR observa- tions were carried out until 466 and 30 days after B-band max- imum light, respectively; throughout this paper we define the phase of the supernova as the time in days from the B-band maximum. The goal of the ESC is to make progress in our un- derstanding of the physics of the thermonuclear SN explosions by collecting and analyzing early-time observations of nearby SNe Ia. Since 2002 the ESC has obtained via coordinated obser- vations using a large number of telescopes optical and IR ob- servations for 15 nearby SNe Ia. First results of the observations have already been published (SN 2002bo – Benetti et al. 2004, Stehle et al. 2005; SN 2002dj – Pignata et al. 2005; SN 2002er – Pignata et al. 2004, Kotak et al. 2005; SN 2003cg – Elias-Rosa et al. 2006; SN 2004eo – Pastorello et al. 2007a; SN 2005cf – Pastorello et al. 2007b, Garavini et al. 2007; Benetti et al. 2005; Mazzali et al. 2005a). Optical observations of SN 2003du have also been presented by Gerardy et al. (2004), Anupama et al. (2005) and Leonard et al. (2005). 2. Observations and data reduction 2.1. Optical spectroscopy The optical spectroscopy log of SN 2003du is given in Table 1. The spectra were reduced1 following the algorithm of (Horne, 1986). The images were first bias and flat-field corrected. The 1D spectra were then optimally extracted from the 2D images, simultaneously identifying and removing the cosmic rays and bad pixels. The spectra were wavelength calibrated using arc- lamp spectra. The wavelength calibration was checked against the night-sky emission lines and, when necessary, small addi- tive corrections were applied. Spectrophotometric standard stars were used to flux calibrate the SN spectra. Telluric absorp- tion features were removed from the supernova spectra follow- ing Wade & Horne (1988). On a number of nights two differ- ent spectrometer settings were used to cover the whole optical wavelength range, and the two spectra were combined into a single spectrum. Most of the spectra have dispersion between ∼ 1 Å pixel−1 and ∼ 5 Å pixel−1, except for the few red spec- tra taken at Asiago 1.82m telescope, which have a dispersion of ∼ 15 Å pixel−1 and one WHT spectrum with ∼ 0.23 Å pixel−1. The spectra were obtained with the slit oriented along the parallactic angle in order to minimize differential losses due to atmospheric refraction (Filippenko, 1982). Nevertheless the rel- ative flux calibration was not always sufficiently accurate and the final flux calibration was achieved by slightly correcting the spectra to match the observed photometry. This step was done alongside the calibration of the photometry and is discussed in detail in the Appendix. 2.2. Optical photometry The optical photometric observations of SN 2003du were ob- tained with a number of instruments equipped with broadband UBVRI filters. The CCD images were bias and flat-field cor- rected. Cosmic ray hits were identified and cleaned with the 1 All data reduction and calibration was done in IRAF and with our own programs written in IDL. IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under co- operative agreement with the National Science Foundation. Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova 3 Table 1. Log of the optical spectroscopy Date (UT) JD Phase Wavelength Telescopea [day] range [Å] 2003 Apr 23 2452753.58 −12.8 4900-7500 INT 2003 Apr 25 2452755.43 −10.9 3450-10400 AS1.8 2003 Apr 25 2452755.58 −10.8 4250-7500 NOT 2003 Apr 28 2452758.54 −7.8 3200-7500 INT 2003 Apr 30 2452760.56 −5.8 3230-8060 TNG 2003 May 02 2452762.39 −4.0 3230-8060 TNG 2003 May 04 2452764.48 −1.9 3436-7776 AS1.8 2003 May 05 2452765.39 −1.0 3500-7776 AS1.8 2003 May 06 2452766.39 +0.0 3447-7776 AS1.8 2003 May 07 2452767.55 +1.2 3500-9590 AS1.8 2003 May 08 2452768.54 +2.2 5860-7060 AS1.2 2003 May 09 2452769.54 +3.2 3500-10010 AS1.8 2003 May 10 2452770.64 +4.3 3300-10000 CA2.2 2003 May 12 2452772.57 +6.2 4680-7017 AS1.2 2003 May 13 2452773.61 +7.2 3300-7200 NOT 2003 May 14 2452774.59 +8.2 3300-7200 NOT 2003 May 15 2452775.48 +9.1 3250-7200 NOT 2003 May 16 2452776.47 +10.0 3260-9800 NOT 2003 May 21 2452781.51 +15.1 3800-6130 AS1.2 2003 May 23 2452783.55 +17.2 3600-10100 AS1.8 2003 May 24 2452784.60 +18.2 4260-6595 AS1.2 2003 May 25 2452785.39 +19.0 3700-7776 AS1.8 2003 May 27 2452787.52 +21.1 3400-8830 CA2.2 2003 Jun 01 2452792.52 +26.1 3240-8060 TNG 2003 Jun 06 2452797.60 +31.2 3240-8060 TNG 2003 Jun 09 2452800.45 +34.1 3500-9500 WHT 2003 Jun 14 2452805.38 +39.0 3700-10000 WHT 2003 Jun 20 2452811.54 +45.2 3880-7770 AS1.8 2003 Jun 26 2452817.52 +51.1 3350-10000 WHT 2003 Jul 08 2452829.44 +63.1 3700-9850 NOT 2003 Jul 17 2452838.41 +72.0 3500-10000 WHT 2003 Jul 29 2452850.42 +84.0 3600-10000 WHT 2003 Aug 23 2452875.32 +108.9 3000-7820 AS1.8 2003 Sep 25 2452907.82 +141.4 3500-8800 CA2.2 2003 Nov 18 2452962.25 +195.9 4370-7050 WHTb 2003 Dec 01 2452975.70 +209.3 3000-7600 CA3.5 2003 Dec 13 2452987.72 +221.3 3500-8820 CA2.2 2004 Feb 02 2453038.71 +272.3 3800-8000 CA3.5 2004 May 17 2453143.30 +376.9 3500-8060 TNG a AS1.8 = Asiago 1.82m + AFOSC; AS1.2 = Asiago 1.22m + B&C; TNG = TNG 3.58m + DOLORES; NOT = NOT 2.6m + ALFOSC; CA2.2 = Calar Alto 2.2m + CAFOS; CA3.5 = Calar Alto 3.5m + MOSCA; WHT =WHT 4.2m + ISIS; INT = INT 2.5m + IDS b average of spectra obtained on 17 and 18 Nov. 2003; these spec- tra cover the ranges 4370–5220 Å and 6200–7050 Å with dispersion 0.23 Å pixel−1. Laplacian detection algorithm of van Dokkum (2001). The ob- servations consist of single exposures at early times and dithered multiple exposures at late epochs. In the latter case, the images in each filter were combined to form a single image. For the I- band, we also corrected for fringing in the individual exposures. The SN lies only 15′′ from the host galaxy nucleus, on a complex background (Fig. 1). The background contamination may significantly degrade the photometry, especially at late epochs when the SN has faded considerably. The approach com- monly used is to subtract the background using template galaxy images without the SN, taken either before or a few years af- ter the SN explosion. The galaxy template, preferably with bet- ter seeing and signal-to-noise ratio (S/N) than the SN images, is aligned with the SN image, convolved with a suitable kernel so that the point-spread functions (PSF) of the two images are the same, then scaled to match the flux level of the SN image and Fig. 1. A B-band finding chart of SN 2003du with the comparison stars labeled by numbers. The image was obtained 87 days after B-band max- imum. Table 2. Calibrated magnitudes of the local stars around SN 2003du. The number in parentheses are the uncertainties in mmag. Star U B V R I 1 13.864 (39) 13.848 (22) 13.309 (13) 12.960 (13) · · · 2 15.004 (39) 14.920 (22) 14.310 (14) 13.911 (13) 13.624 (12) 3 16.562 (40) 16.428 (22) 15.792 (13) 15.400 (13) 15.077 (13) 4 16.930 (40) 17.024 (23) 16.462 (14) 16.113 (14) 15.792 (12) 5 18.261 (45) 17.611 (24) 16.251 (14) 15.258 (15) 14.117 (13) 6 18.254 (45) 17.909 (24) 17.011 (15) 16.478 (14) 16.012 (12) 7 17.660 (42) 17.552 (23) 16.893 (14) 16.468 (14) 16.129 (12) 8 17.114 (40) 16.993 (23) 16.307 (15) 15.829 (14) 15.504 (12) 9 17.806 (42) 17.951 (24) 17.518 (15) 17.179 (16) 16.875 (13) 10 17.809 (42) 18.092 (24) 17.675 (16) 17.357 (16) 17.057 (13) 11 17.775 (42) 17.586 (23) 16.874 (15) 16.418 (15) 16.107 (14) 12 18.328 (46) 18.636 (27) 18.158 (18) 17.799 (18) 17.487 (14) subtracted. The SN flux can then be correctly measured on the background-subtracted image. Lacking pre-explosion observations of the host galaxy of SN 2003du, we constructed template images using observations which we obtained more than one year after SN maximum light. The SN magnitudes were measured by PSF-fitting. The small SN contribution was then subtracted and the images were visu- ally inspected for over- or under-subtraction (none was noticed). The best seeing images were then combined to form the tem- plates. The subtraction of the host galaxy from the ”SN + host” images was done with Alard’s (Alard & Lupton, 1998; Alard, 2000) optimal image subtraction software, slightly modified and kindly made available to us by B. Schmidt. When using galaxy templates built in this way, any improperly subtracted SN light will introduce systematic errors into the subsequent photometry. In the case of SN 2003du this should, however, be negligible be- cause at the epochs used to build the templates, the SN was much fainter than on the images to which the template subtraction was applied (at least 2 mag fainter at +220 days and 4–5 mag fainter over the first three months after maximum). Even if we conser- vatively assume that the final templates still contained 20% of 4 Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova the SN light, the error introduced would be at most 0.03 mag at +220 days and clearly negligible during the first 3-4 months after maximum. The SN magnitudes were measured differentially with re- spect to the field stars indicated with numbers in Fig. 1. The instrumental magnitudes were measured by aperture photometry for observations before September 2003 and by PSF fitting at the later epochs. The magnitudes of the field stars were calibrated for two photometric nights at the Nordic Optical Telescope – May 15 and 16, 2003. On each night, the field of the globular cluster M92 that includes the standard stars listed in Majewski et al. (1994) was observed at four airmasses between 1.06 and 1.8. The BVRI magnitudes of the standard stars were taken from (Stetson, 2000)2, while the U magnitudes were calculated from the U − B values given in Majewski et al. (1994). The standard star magnitudes were measured with PSF photometry and aper- ture corrections were applied to convert the PSF magnitudes to magnitudes in an aperture with a radius of five times the see- ing. Following Harris et al. (1981), all measured magnitudes were fitted simultaneously (with 3σ clipping) to derive linear transformation equations, with the additional requirement that the color-terms and the zero-points to be the same for the two nights. Second-order extinction terms were not included. The calibrations for the two nights agree very well within the es- timated photometric (statistical) errors. The weighted average magnitudes from the calibration in the two nights and the corre- sponding errors are given in Table 2. Note that the uncertainties of the calibrated magnitudes are donated by the uncertainty of the zero-point and not by the statistical uncertainty. A compari- son between the stars in common with Leonard et al. (2005) and Anupama et al. (2005) reveals that there are small systematic dif- ferences between the photometry; ours being generally brighter. The mean differences with the BVRI photometry of Leonard et al. (2005) are, respectively, 0.010 ± 0.020, 0.013 ± 0.020, 0.039 ± 0.010 and 0.013 ± 0.027 mag. Excluding star #1 which is brighter in Anupama et al. (2005) in all bands, the mean dif- ferences are 0.00±0.05, 0.06±0.01, 0.04±0.01, 0.06±0.02 and 0.015 ± 0.010 mag for the UBVRI bands, respectively. Some of these differences are non-negligible and we have no explanation of why they appear in the comparison stars calibrations. This is clearly worrisome and emphasizes one important source of systematic errors when different SN data sets are combined and used to derive cosmological parameters. Landolt (1992) standard fields were observed to derive the instrument color-terms (ct), allowing us to transform the pho- tometry of SN 2003du to the standard Johnson-Cousins system. The instrumental magnitudes of the standard stars were mea- sured by aperture photometry with large apertures. All measure- ments for a given instrument were fitted simultaneously (with 3σ clipping) with linear equations of the form: U − u = ctU(U − B) + zp , B − b = ctB(B − V) + zp V − v = ctV (B − V) + zp , R − r = ctR(V − R) + zp (1) I − i = ctI(V − I) + zp to determine the cts. The upper-case and lower-case letters de- note the standard and instrumental magnitudes, respectively. For each SN image, a zero-point was calculated for each calibrated star by applying Eqs. 1. The final image ZP and its uncertainty are, respectively, the average of the individual ZPs 2 Available at http://cadcwww.hia.nrc.ca/standards/ and as discussed by Stetson (2000) this photometry is essentially in the Landolt (1992) system. (with 3σ outliers removed if present) and the standard deviation. The measured scatter for the brightest stars was always larger than expected from Poisson statistics. This indicates that there are additional sources of uncertainties: imperfect flat-fielding, presence of non-uniform scattered light, CCD non-linearity, etc. Considering the magnitude scatter of the brightest stars we esti- mate that these effects contribute ≤ 0.01 mag to the error budget. Finally, the ZPs were added to the measured SN magnitudes to obtain the magnitudes in the natural systems of the instruments used, mnat. The SN magnitudes can be transformed to a standard photo- metric system using the color corrections obtained with Eqs. 1. It is, however, well known that these color corrections do not work well for SNe and significant systematic differences be- tween photometry obtained with different instruments are of- ten observed (Suntzeff, 2000; Stritzinger et al., 2002; Krisciunas et al., 2003). The reason is that the SN spectral energy distribu- tion (SED) is very different from that of normal stars. Another consequence of this is that if a given band is calibrated against different color indices, e.g. V(B − V) and V(V − R), one would get the same magnitude for normal stars but slightly different magnitudes for objects with non-stellar SEDs. This is because the color-terms are determined with normal stars, but SNe oc- cupy a different region in the color-color diagrams. The pho- tometric observations of SN 2003du were collected with many different instruments and we chose to standardize the photome- try using the S-correction method described by Stritzinger et al. (2002) coupled with our very well-sampled spectral sequence of SN 2003du. The S-correction method assumes that the SED of the SN and the response of the instruments used for the ob- servations are both accurately known. Then one can correct the photometry to any well-defined photometric system by means of synthetic photometry. If f phot (λ) is the photon flux of the ob- ject per unit wavelength, mnat the object magnitude as defined above, Rnat(λ) the response of the natural system and Rstd(λ) the response of the standard system, then the object standard mag- nitude mstd is: mstd = mnat −2.5 log (λ)Rstd(λ)dλ +2.5 log (λ)Rnat(λ)dλ +const (2) The constant in Eq. 2 is such that the correction is zero for A0 V stars with all color indices zero. This ensures that for normal stars the synthetic S-correction gives the same results as the lin- ear color-term corrections (Eq. 1). The constant can be deter- mined from synthetic photometry of stars for which both pho- tometry and spectrophotometry is available. The details of the application of the S-corrections are given in the Appendix. In Fig. 2 we only show the time evolution of the difference between the S-correction and the linear color-term correction. Note the particularly large difference for Calar Alto I, and NOT R and I-bands, as well as the rather large scatter for the V-band at all epochs and for the B-band after +20 days. The final photometry of SN 2003du is given in Table 3. Note that none of the U-band and part of the BVRI photometry could be S-corrected. Additional B and V photometry obtained at Moscow and Crimean Observatories is given in Table 4. Figure 3 shows a comparison between the S-corrected and color-term cor- rected B − V color index and I magnitudes. It is evident that in the color-term corrected photometry small systematic differ- ences between the various setups exist. It is also evident that the S-correction removes those differences to a large extent, the exception being the BAO data at early epochs. Significant im- Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova 5 −0.06 −0.04 −0.02 B−band Days from Bmax Asiago Calar Alto −0.06 −0.04 −0.02 0.04 V−band 0 50 100 −0.04 −0.02 R−band 0 50 100 I−band Fig. 2. Time evolution of the difference be- tween the S-correction and the linear color- term correction. provement is also achieved for the I-band, which required the largest S-corrections. 2.3. Near infrared photometry and spectroscopy Near infrared JHK photometry of SN 2003du was obtained on six nights at TNG and NOT. The two telescopes use identical J and H filters, but the TNG uses a K′ while the NOT has a Ks filter (Tokunaga et al., 2002). The observations were reduced in the standard way, using the XDIMSUM package in IRAF. The two nights at the NOT were photometric and standard stars from the list of Hunt et al. (1998) were observed in or- der to calibrate a local sequence of stars. However, only star #3 (Fig. 1) could be reliably calibrated because it is the only one that is faint enough to be in the linear range of the de- tector and is bright enough to give an adequate S/N. The av- erage NIR magnitudes of star #3 are J = 14.67, H = 14.38 and K = 14.37, all with uncertainties of ∼ 0.03 mag. The cali- brated magnitudes are in good agreement with the 2MASS val- ues, which are J = 14.633 ± 0.037, H = 14.362 ± 0.056 and K = 14.311 ±0.062. Star #3 was used to calibrate the TNG pho- tometry. No color terms were applied. The NIR photometry of SN 2003du is given in Table 5. Eleven low-resolution NIR spectra of SN 2003du were ob- tained at UKIRT and TNG (Table 6). At UKIRT, the spectral range was covered by using different instrument settings. At TNG an AMICI prism was used as disperser. In this mode the whole NIR spectral range is provided in one exposure at the ex- pense of having very low resolving power (≤ 100). Both sets of observations were performed in ABBA sequences, where A and B denote two different positions along the slit. After bias/dark and flat field corrections, for each pair of AB images, the B im- age was subtracted from the A image. The negative spectrum was shifted to the position of the positive one and subtracted from it. This resulted in an image with the sum of the spectra but minus the sky background. All such images were summed into a single image and the 1D spectra were then optimally ex- tracted. We note that the optimal extraction algorithm has to be applied on images where the pixel levels are given in the form of actual detected counts, and so it will not work quite correctly if applied to background-subtracted images. Special care was thus taken to calculate the optimal extraction weights correctly. The Table 4. Additional photometry SN 2003du. JD Phase B V Telescope 2452765.38 −1.0 13.45 (0.01) 13.61 (0.01) 1 2452768.33 +2.0 13.49 (0.02) 13.57 (0.02) 1 2452775.38 +9.0 13.95 (0.01) 13.83 (0.01) 1 2452782.37 +16.0 14.65 (0.05) 14.17 (0.02) 2 2452786.33 +20.0 14.99 (0.06) 14.43 (0.02) 2 2452792.31 +25.9 · · · 14.73 (0.06) 3 1 – 70-cm Moscow reflector + CCD Pictor 416; 2 – 30-cm Moscow refractor + CCD AP-7p; 3 – 38-cm Crimean reflector + CCD ST-7; UKIRT spectra were wavelength calibrated with arc-lamp spec- tra, while for the TNG spectra a tabulated dispersion solution relating pixel number to wavelength was used. The dispersion of the UKIRT spectra ranges from ∼ 5 Å pixel−1 to ∼ 25 Å pixel−1, while for the TNG spectra, the dispersion is in the ∼ 30 Å pixel−1 – ∼ 100 Å pixel−1 range. The A5 V star AS-24 (Hunt et al., 1998) and the F7 V star BS5581 (from the list of UKIRT standard stars) were observed at TNG and UKIRT respectively. The standard stars were ob- served close in time and airmass to the SN observations. The SN spectra were first divided by the spectra of the comparison stars to remove the strong telluric absorption features. The result was multiplied by a model spectrum of the appropriate spectral type, smoothed to the instrumental resolution, to remove any residual features due to the absorption lines of the standard, simultane- ously providing the relative flux calibration. The UKIRT spec- tra from the different instrument settings that did not overlap were combined using the SN 2003du photometry and average NIR color indices of normal SNe Ia. 3. Results 3.1. Light curves The UBVRIJHK light curves (LCs) of SN 2003du are shown in Fig. 4. The light curves morphology resemble that of a nor- mal SN Ia with a well-pronounced secondary maximum in the I-band and a shoulder in the R-band. The J-band also shows a strong rise towards a secondary maximum. Comparison with the photometry of Leonard et al. (2005) and Anupama et al. 6 Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova −15 −10 −5 0 5 10 Asiago Calar Alto S−corrected a) color−termcorrected + 0.1 40 50 60 70 80 90 S−corrected color−term corrected + 0.1 −10 0 10 20 30 40 Days from Bmax NOT Asiago Calar Alto NOT BAO S−corrected Fig. 3. Comparison between the S-corrected and color-term corrected a- b B − V color index and c I-band magnitude. The color-term corrected B − V data are shifted by 0.1 mag. A polynomial fit to the S-corrected B − V data is overplotted. To highlight the differences the fit is also plotted shifted by 0.1 mag. Table 5. NIR photometry of SN 2003du. The observations on 10.5 and 11.5 days are from NOT. The other four are from TNG. JD Phase J H K [day] 2452755.41 −11.5 14.96 (0.04) 15.02 (0.04) 15.02 (0.04) 2452768.68 +1.7 14.42 (0.04) 14.66 (0.04) 14.38 (0.04) 2452773.43 +6.5 14.92 (0.04) 14.77 (0.04) 14.53 (0.04) 2452777.46 +10.5 15.67 (0.04) 14.86 (0.04) 14.70 (0.04) 2452778.51 +11.5 15.84 (0.04) 14.86 (0.04) 14.75 (0.04) 2452782.58 +15.6 16.12 (0.04) 14.85 (0.04) 14.65 (0.04) (2005) reveals fairly good consistency. However, systematic dif- ferences between the data sets do exist and our photometry is generally brighter. This is probably due to the differences in the comparison star calibrations, as well as to the fact that our pho- tometry was S-corrected, unlike those of Leonard et al. (2005) and Anupama et al. (2005). To estimate the differences we fit- ted a smoothing spline function to our data and computed the mean difference and its standard deviation from Leonard et al. (2005) and Anupama et al. (2005) photometry. The difference slightly varies with the SN phase. Up to 30 days after maximum light the mean differences and standard deviations in BVRI are, respectively, 0.068 ± 0.030, 0.046 ± 0.029, 0.047 ± 0.022 and Table 6. Log of the NIR spectroscopy Date (UT) JD Phase Coverage Telescopea [day] [µm] 2003 Apr 25 2452754.89 −11.5 0.8-2.5 UK-1 2003 Apr 25 2452755.47 −10.9 0.75-2.45 TNG 2003 May 01 2452760.89 −5.5 0.8-2.5 UK-1 2003 May 04 2452763.88 −2.5 1.42-2.4 UK-2 2003 May 08 2452768.68 +2.3 0.9-2.3 TNG 2003 May 10 2452769.79 +3.4 1.39-2.50 UK-2 2003 May 11 2452770.90 +4.5 0.8-2.5 UK-1 2003 May 19 2452778.80 +12.4 1.48-2.30 UK-2 2003 May 22 2452782.58 +16.2 0.9-2.48 TNG 2003 May 27 2452786.77 +20.4 0.8-2.5 UK-1,2 2003 Jun 06 2452796.80 +30.4 0.8-2.5 UK-1,2 aTNG = TNG + NICS, UK-1/2 = UKIRT + CGS4/UIST 0.042±0.037 mag with Anupama et al. (2005) and 0.026±0.025, 0.026±0.015, 0.014±0.015 and 0.071±0.030 mag with Leonard et al. (2005). The difference with the Anupama et al. (2005) U- band photometry is 0.007 ± 0.085 mag. We fitted the B-band template of Nugent et al. (2002) to the data to determine the B-band light curve parameters. This pro- vided the time of B maximum light tBmax (JD)=2452766.38 (2003 May 6.88 UT), stretch factor sB = 0.988±0.003 and peak magni- tude Bmax = 13.49 ± 0.02 mag. The peak VRI magnitudes were estimated by fitting low-order polynomials to the data around maximum, giving Vmax = 13.57± 0.02, Rmax = 13.57± 0.02 and Imax = 13.83 ± 0.02 mag. The U-band maximum was estimated by fitting our own template derived from the SNe published by Jha et al. (2006a): Umax = 13.00 ± 0.05 mag. The optical pho- tometric coverage around 15 days after Bmax is rather sparse. However, the B-band template matches the observed photome- try very well, thus we are able to use this to determine the decline rate parameter. We find ∆m15 = 1.02±0.05. BVRI template light curves with ∆m15 = 1.02 were also generated using the data and the method described by Prieto et al. (2006). These light curves are also shown in Fig. 4, shifted to match SN 2003du peak mag- nitudes. The resemblance between SN 2003du light curves and the templates is excellent. The NIR templates from Krisciunas et al. (2004b) were fitted to the first three JHK photometric points (Fig. 4) to estimate the peak magnitudes: Jmax = 14.21, Hmax = 14.56 and Kmax = 14.29 mag. The rms around the fits are fairly small 0.03, 0.02 and 0.04 mag, respectively, but because the LCs are undersampled the un- certainties in the peak magnitudes should exceed these values. To derive the templates, Krisciunas et al. (2004b) fitted third- order polynomials to the photometry of a number of SNe. The rms around the fits are 0.062, 0.080 and 0.075 mag for J, H and K, respectively. These numbers were added in quadrature to the rms around the fits to the SN 2003du data to obtain the uncer- tainties of the JHK peak magnitudes, 0.07, 0.08 and 0.09 mag, respectively. The entire light curves are shown in the inset of Fig. 4. The late-time HS T data from Leonard et al. (2005) are also shown (open symbols); these are consistent with our ground based pho- tometry. After ∼ +180 days the magnitudes of SN 2003du de- cline linearly, following the expected form of an exponential ra- dioactive decay chain. The decline rates in magnitudes per 100 days in UBVRI-bands (as determined by linear least-squares fit- ting) are 1.62 ± 0.12, 1.47 ± 0.02, 1.46 ± 0.02, 1.70 ± 0.06 and 1.00 ±0.03, respectively. The decline rates in the B- and V-bands are virtually the same. The I-band decline on the other hand is much slower than in the other bands. Many other normal SNe Ia Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova 7 0 50 100 150 Days from Bmax 0 50 100 150 200 250 300 350 400 450 500 Days after Bmax Fig. 4. UBVRIJHK light curves of SN 2003du. The error bars are not plotted because they are typically smaller than the plot symbols. For the BVRI bands the filled and the open symbols are the S-corrected and non-S-corrected photometry, respectively. The open symbols for J band are synthetic photometry from the combined optical-NIR spectra. Overplotted are the B-band template of Nugent et al. (2002), the JHK templates from Krisciunas et al. (2004b) (solid lines), as well as our U-band template derived from Jha et al. (2006a) data (dashed line). The dotted lines are a light curve template with ∆m15 = 1.02 calculated as described in Prieto et al. (2006) using a program provided by the authors. Inset: The full light curves. The late-time HS T data from Leonard et al. (2005) are also shown with the open symbols. The linear fits to the late-time photometry are also shown. (e.g., Lair et al., 2006) and the peculiar SN 2000cx (Sollerman et al., 2004) also show similar behavior. 3.2. Reddening in the host galaxy Figure 5 shows that the time evolution of the color indices (CIs) of SN 2003du closely follows the reddening corrected CIs of normal SNe such as 1990N, 1998aq, and 1998bu, as well as the Nobili et al. (2003) templates. This implies that SN 2003du was probably not reddened within its host galaxy. Nevertheless, the reddening in the host galaxy was estimated with three different methods. The CIs of SN 2003du were first corrected for the small Milky Way reddening of E(B−V) = 0.01 (Schlegel et al., 1998) assuming RV = 3.1. i) Phillips et al. (1999) use the observed Bmax − Vmax and Vmax−Imax indices, and the evolution of B−V between 30 and 90 days after maximum to derive E(B− V). The first two quantities are weak functions of ∆m15. The time evolution of B−V (known as the Lira relation) seems to hold for the majority of SNe Ia (Phillips et al., 1999; Jha et al., 2006b). Following Phillips et al. (1999), for SN 2003du we obtain E(B − V)max = −0.01 ± 0.04, E(V − I)max = 0.07 ± 0.05 and E(B − V)tail = 0.05 ± 0.07. 8 Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova Lira−Phillips relation 2003du 1990N 1998bu 1998aq Nobili et al. (2003) −20 0 20 40 60 80 100 Days from Bmax −20 0 20 40 60 80 100 Days from Bmax Fig. 5. The evolution of the optical color indices of SN 2003du, compared with those of other well-observed SNe Ia. When necessary the colors were de-reddened with the appropriate E(B − V). The Nobili et al. (2003) B − V , V − R and V − I templates are also shown. The errors indicate the intrinsic accuracy of the three methods as given in Phillips et al. (1999), viz. 0.03, 0.04 and 0.05, added in quadrature to the uncertainties of the observed CIs. Note that the B − V evolution of SN 2003du has a different slope from that of the Lira relation, leading to rather a large scatter of ∼ 0.07 mag. We averaged the above estimates of E(B − V)max, 0.8×E(V − I)max 3 and E(B−V)tail weighted by their respective uncertainties to obtain the final reddening estimate: E(B−V) = 0.027 ±0.026. ii) Wang et al. (2003b) introduced a novel method, CMAGIC, to estimate the brightness and the reddening of SNe Ia. It is based on the observation that between 5-10 to 30- 35 days after maximum the B magnitude is a linear function of B − V with a fairly uniform slope. Applying this method, we obtain E(B − V) = 0.00 ± 0.05. iii) Krisciunas et al. (2000, 2001, 2004b) have shown that the intrinsic V − (JHK) CIs of SNe Ia are very uniform and can be used to estimate the reddening of the host galaxy. Figure 6 shows the V − (JHK) CIs of SN 2003du overplotted with the unreddened loci for mid- (∆m15 = 1.0 − 1.3) and slow-declining SNe (∆m15 = 0.8− 1.0) of Krisciunas et al. (2004b). Most of the V − (HK) data of SN 2003du fall between the two loci. This is consistent with the fact that its ∆m15 = 1.02 lies between these two groups of SNe Ia. Although the V − J CI is slightly redder than the locus, overall the V − (JHK) CIs of SN 2003du suggest little reddening. Combining the results of the three estimates we conclude that SN 2003du suffered negligible reddening within the host galaxy. The main parameters of SN 2003du that we derived from photometry are summarized in Table 7. 3 the factor 0.8 serves to convert E(V − I) to E(B − V) assuming the standard Milky Way extinction law with RV = 3.1. slow-decliners (∆m15=0.8-1.0) mid-decliners (∆m15=1.0-1.3) -10 0 10 20 30 Days from Bmax Fig. 6. V − (JHK) color indices of SN 2003du. The unreddened loci for mid- and slow-declining SNe of Krisciunas et al. (2004b) are over- plotted. The open symbols are estimates based on synthetic photometry from the combined optical-NIR spectra. 3.3. Spectroscopy Our collection of optical spectra of SN 2003du is shown in Figs. 7 and 8. The spectra marked with an asterisk have been smoothed using the á trous wavelet transform (Holschneider et al., 1989). The optical spectral evolution of SN 2003du is that of a normal SN Ia. In the earliest spectrum at −13 days the Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova 9 4000 6000 8000 10000 Rest frame wavelength [Å] −12.8 * −10.9 −10.8 −7.8 * +10.0 +15.1 +17.2 +18.2 +19.0 +21.1 +26.1 +31.2 +34.1 +39.0 +45.2 +51.1 +63.1 +72.0 +84.0 +108.9 +141.4 high−velocity Ca II HV Ca II Fig. 7. Evolution of the optical spectra of SN 2003du. The spectra marked with an asterisk were slightly smoothed (see text for details). The noticeable telluric features are marked with Earth symbols; the connected symbols mark the region of strong telluric absorption. Si ii λ6355 line is strong and broad (∼ 10000 km s−1 full-width at half-depth), and the S ii λ5454 and λ5640 lines are well devel- oped. In the −11 day spectrum the Ca ii H&K and the IR triplet lines are also very strong. In all the spectra until one week after maximum light, Si ii λ4129 and λ5972 lines are clearly visible. Mg ii λ4481, Si iii λλ4553,4568 and the blend of Fe ii, Si ii and S ii lines around 4500–5000 Å are also prominent. A few days after Bmax the spectrum starts to be dominated by Fe group ele- ments and gradually evolves into a nebular spectrum. The ratio between the depth of the Si ii λ5972 and λ6355 lines, R(Si ii) (Nugent et al., 1995), at maximum is R(Si ii) = 0.22 ± 0.02, typical for normal SN Ia. R(Si ii) does not change significantly in the pre-maximum spectra, remaining at ∼ 0.2. In Fig. 9 three of the pre-maximum spectra of SN 2003du are compared with spectra of other normal SNe Ia observed at sim- ilar epochs and appropriately de-reddened. For this and other comparison plots we use published optical spectra of SN 1994D (Patat et al., 1996; Filippenko, 1997; Meikle et al., 1996), SN 1990N (Leibundgut et al., 1991), SN 1996X (Salvo et al., 2001), SN 1999ee (Hamuy et al., 2002), SN 1998aq (Branch et al., 2003), SN 1998bu (Jha et al., 1999; Hernandez et al., 2000), SN 2002er (Kotak et al., 2005), SN 2001el (Wang et al., 10 Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova Table 7. Main photometric parameters of SN 2003du from this work. tBmax [JD] 2452766.38 ± 0.50 tBmax (UT date) May 6.88, 2003 B-band stretch, sB 0.988±0.003 B-band decline rate, ∆m15 1.02±0.05 Peak magnitudes U = 13.00 ± 0.05 B = 13.49 ± 0.02 V = 13.57 ± 0.02 R = 13.57 ± 0.02 I = 13.83 ± 0.02 J = 14.21 ± 0.07 H = 14.56 ± 0.08 K = 14.29 ± 0.09 Late-time decline γU = 1.62 ± 0.12 γB = 1.47 ± 0.02 rate γ [mag/100 days] γV = 1.46 ± 0.02 γR = 1.70 ± 0.06 γI = 1.00 ± 0.03 γBol = 1.40 ± 0.01 E(B − V)host 0.00±0.05 4000 5000 6000 7000 8000 Rest frame wavelength [Å] +195.9 * +195.9 +209.3 * +221.3 +272.3 * +376.9 SN 1998bu Fig. 8. Nebular spectra of SN 2003du. The spectra marked with an aster- isk were slightly smoothed. A nebular spectrum of SN 1998bu is shown for comparison. 2003a; Mattila et al., 2005) and SN 2005cg (Quimby et al., 2006). The spectra at about 10 days before maximum show sig- nificant differences. The spectra have not been taken at exactly the same phase and the rapid spectral evolution at such early phases may partly be responsible for the differences. However, most of the differences are most likely intrinsic. It worths noting that the weak feature at ∼6300 Å that is visible in the two earliest spectra of SN 2003du is present in other SNe Ia as well (Fig. 10) and has been attributed to C ii λ5860 (Mazzali, 2001; Branch et al., 2003; Garavini et al., 2004, 2005). At one week before maximum the spectra are more similar to each other. It is inter- esting to note that at these epochs the largest differences between the SNe are seen in the strengths and profiles of the Si ii λ6355, Ca ii H&K and Ca ii IR3 lines. Starting from one week before maximum the spectra of most SNe Ia are very homogeneous. The NIR spectra of SN 2003du are shown in Fig. 11. The earliest spectra at −11.5 and −11 days are rather featureless with only hints at weak broad P-Cygni profiles. The weak ∼ 1.05µm absorption could be due to Mg ii λ10926 or He i λ10830 (or a combination of the two) (Meikle et al., 1996; Mazzali & Lucy, 1998; Branch et al., 2004; Marion et al., 2003). The strength of this absorption in the earliest two spectra is quite different, despite the fact that they have been taken only half a day apart. In the −11.5 days spectrum, however, the absorption is likely 99ee −9d 90N −14d 03du −12.8d 03du −10.9d 01el −9d 98aq −9d 02er −11d 94D −12d Early spectra (day −11) Ca II H&K Ca II IR3 Si II 4000 6000 8000 10000 Rest frame wavelength [Å] 99ee −7d 90N −7d 03du −5.8d 98bu −6.5d 98aq −8d 02er −6.3d 94D −7.5d one week before maximum Fig. 9. Comparison of optical spectra of normal SNe Ia at two pre- maximum epochs. The arrows in the lower panel mark the lines whose velocities have been measured. enhanced by a noise spike due to the low instrument response at this wavelength. In the day −5.5 spectrum an absorption due to Mg ii λ9226 (Marion et al., 2003) is clearly seen. In the earlier IR spec- tra there are only hints of its presence and it may be just de- tectable in the optical spectrum at day −11. Our experiments with the SN spectral synthesis code SYNOW (see for details, e.g. Branch et al., 2003) show however, that Si iii and possibly Si ii may contribute to the red part of this line. No other fea- tures are detected in the 0.9-1.2µm spectral region. In particular, no C i or O i lines are observed, in accordance with the findings of Marion et al. (2006). The absorption at ∼ 1.21µm is due to Ca ii according to Wheeler et al. (1998), but the associated emis- sion peak at ∼ 1.24 µm was attributed to Fe iii by Rudy et al. (2002) in SN 2000cx. The 1.6µm absorption seen in the spectra until maximum light is due to Si ii with a possible contribution from Mg ii (Wheeler et al., 1998; Marion et al., 2003). The broad features beyond ∼ 1.7µm lack clear identification. Possible con- tributors are Si iii at ∼ 2 µm (Wheeler et al., 1998) and Co ii at ∼ 2 − 2.05µm and ∼ 2.3 µm (Marion et al., 2003). By day +12, two strong emission features at ∼ 1.55µm and ∼ 1.75µm dominate the 1.4-1.8µm spectral region. These two features are formed by blending of many Fe ii, Co ii and Ni ii emission lines (Wheeler et al., 1998). Lines of Fe ii, Co ii, Ni ii and Si ii dominate the spectral region beyond 2 µm. From day +15, a number of lines, with uncertain identifications also de- velop in the J band. One can also clearly see how a flux deficit at ∼ 1.35µm develops. This causes the very deep minimum ob- served in the J-band light curves of most SNe Ia around 20 days after maximum. Figure 12 presents a comparison of several IR spectra of SN 2003du with those of other normal SNe: SN 1994D (Meikle Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova 11 5600 6000 6400 Rest frame wavelength [Å] 03du −12.8 03du −10.9 99ac −15 98aq −9 96X −4 94D −10 90N −14 C II 6580? Fig. 10. Early spectra of SN 2003du and several other SNe Ia zoomed at the Si ii 6355 line. The dotted lines mark the weak absorption features that may be due to C ii λ6580. et al., 1996), SN 1999ee (Hamuy et al., 2002), SN 1998bu (Jha et al., 1999; Hernandez et al., 2000; Hamuy et al., 2002), and SN 2002bo (Benetti et al., 2004). Similarly to the optical, the IR spectra of normal SNe taken at similar epochs are very homoge- neous, even the spectra taken 6–12 days before maximum. The only significant difference is in the J-band, where the Mg ii lines of SN 2002bo are stronger compared to other SNe. 3.4. Blueshifts of absorption-line minima We have measured the blueshifts of the absorption-line min- ima of Si ii λ6355, S ii λ5640 and Si iii λλ4553,4568, which are thought to be relatively un-blended (Branch et al., 2003), by fit- ting a Gaussian to the line absorption troughs. In the rest of the paper we report the velocities that correspond to the mea- sured blueshifts of the absorption-line minima (unless otherwise stated) and will refer to these as velocities of the absorption lines. By convention, these velocities are negative and we say that the velocity of a line increases from, e.g. −20000 to −10000 km s−1. The velocities inferred from an explosion model will be reported as positive numbers. The velocities of the Si ii λ6355, S ii λ5640 and Si iii λλ4553,4568 lines are shown in Fig. 13 and it is evi- dent that the time evolution is very similar to that in other normal SNe Ia (see, e.g. Benetti et al., 2005). The Si ii λ6355 velocity initially increases rapidly, but 7–5 days before max- imum the increase rate slows down and the velocity remains almost constant thereafter. The velocities of the S ii λ5640 and Si iii λλ4553,4568 lines increase at nearly constant rate; however, there is a hint that the S ii λ5640 velocity remains constant after maximum, similarly to Si ii λ6355. Benetti et al. (2005) measured a post-maximum velocity increase rate of the Si ii λ6355 line to be v̇ = 31 ± 5 km s−1d−1 and classified SN 2003du as a Low Velocity Gradient SN Ia, along with other normal and all overluminous SNe Ia. It can be seen from Fig. 1 in (Benetti et al., 2005) that before maximum the Si ii λ6355 velocities of SN 2003du are systematically higher by 500-2000 km s−1 compared to all other SNe. During the SN photospheric epochs the main source of continuum opacity at the optical wavelengths is electron scat- tering and following Jeffery et al. (1992) we adopt that the 1.0 1.5 2.0 2.5 Rest frame wavelength [µm] −11.5 −10.9 −5.5 −2.5* +2.3 +3.4* +4.5 +12.4* +16.2 +20.4 +30.4 Mg II + He I? Mg II Si II/Mg II? Ca II Ca II Fe III? Si III? Co II? Co II? Ca II Fe/Si edge Fe/Ni/Co Fe II Fe/Ni/Co/Si Fe/Ni/Co/Si? Fig. 11. NIR spectral evolution of SN 2003du. The spectra marked with an asterisk have been smoothed (only the J band of the −2.5 days spec- trum is smoothed). (continuum) photosphere is at electron scattering optical depth 2/3. However, the velocity gradient in the expanding SN ejecta causes many weak lines to overlap which gives rise to strong pseudo-continuum (e.g., Pauldrach et al., 1996), and the so- called expansion opacity (Karp et al., 1977; Pinto & Eastman, 2000) is an analytical description of this effect. This expan- sion opacity may exceed electron scattering opacity by orders of magnitude. The velocity of the pseudo-photosphere thus cre- ated is wavelength-dependent. Besides, strong absorption lines may form in a large volume above the continuum photosphere. For these reasons, the line velocities we measure most likely do not trace the velocity of the continuum photosphere and should be interpreted with caution. Lentz et al. (2000) have computed a grid of photospheric phase atmospheres of SNe Ia with differ- ent metallicities in the C+O layer and computed non-LTE syn- thetic spectra. It would be more reasonable for us to compare the Si ii λ6355 line velocities in SN 2003du with the measurements from the Lentz et al. (2000) synthetic spectra. The time evolution is qualitatively similar and in Fig. 13 we also show the measure- ments for the 1/3 Solar metallicity models, which best follows the SN 2003du Si ii λ6355 line blueshift. 12 Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova 1.0 1.5 2.0 2.5 Rest frame wavelength [µm] 03du −11.5 99ee −9 02bo −9.2 94D −8.5 03du −5.5 03du −2.5 98bu −3 03du +2.3 99ee +2 03du +4.5 99ee +5 03du +16.2 98bu +15 Fig. 12. Comparison with NIR spectra of other normal SNe Ia. Marion et al. (2003) showed that the velocities of lines in NIR spectra could be used to constrain the location of the tran- sition region between the layers of explosive carbon and oxygen burning, and incomplete to complete silicon burning, and hence place constraints on the explosion models. We measured the ve- locities of the blue edges of the absorptions at ∼ 0.9 µm and 1.05µm in our optical and IR spectra between −11.5 and +4.5 days. Both lines show constant velocities of ∼ −11000 km s−1 and ∼ −13000 km s−1, respectively, assuming that the lines are formed by Mg ii λ9226 and Mg ii λ10926. The constant veloc- ity indicates that the continuum photosphere is well beneath the Mg-rich layers (Meikle et al., 1996). The velocity of the sharp edge at ∼ 1.55 µm in the spectra between +10 and +20 days can be used to estimate the transition between the layers of incomplete and complete silicon burning. We measure veloci- ties ≤ −9800 km s−1 which is similar to the results of Marion et al. (2003) and is also broadly consistent with their refer- ence explosion model in which Si is completely consumed be- low ∼ 8500 km s−1. This ties in with the measurements of the Si ii λ6355 line velocity, which is always ≤ −9300 km s−1. 4. Discussion 4.1. The distance to SN 2003du We have shown that SN 2003du was a spectroscopically and photometrically normal SN Ia, and furthermore that it was not reddened within its host galaxy. The distance to UGC 9391 has not been measured using direct techniques, and the only avail- −10 0 10 20 30 40 Days from Bmax − Si II λ6355 − Si III λ4560 − S II λ5640 Lentz model 1/3 Solar metallicity Fig. 13. The evolution of the velocity of the absorption lines of SN 2003du. able information is from its recession velocity. The observed velocity is 1914 km s−1, which after correcting for the Local Group in-fall onto Virgo becomes 2195 km s−1 (from the LEDA database) or a distance modulus of µ = 32.42 mag on the scale of H0 = 72 km s −1,Mpc−1. Recently, Riess et al. (2005) calibrated the luminosities of SN 1998aq and SN 1994ae by observing Cepheids in their host galaxies with the Hubble Space Telescope. Including two other SNe Ia with Cepheid calibrated distances, they estimated the absolute magnitude of a typical SN Ia to be MV = −19.20 ±0.10(statistical)±0.115(systematic) mag. Meikle (2000) and Krisciunas et al. (2004a,c) presented evidence that SNe Ia are standard candles in the NIR and that no correction for the light curve shape is needed for SNe with ∆m15 < 1.7 mag. Krisciunas et al. (2004c) derived the following absolute peak JHK mag- nitudes for H0 = 72 km s −1,Mpc−1: −18.61,−18.28 and −18.44 mag all with statistical uncertainty of ∼ 0.03 mag. The system- atic uncertainty of MV is mostly due to the 0.1 mag uncertainty in the distance to the Large Magellanic Cloud (LMC) and hence it also affects the NIR absolute magnitudes and the distance mod- ulus derived from the host galaxy recession velocity (through H0). The light curve decline rate parameter ∆m15 = 1.02 ± 0.05 and the normal spectral evolution suggest that SN 2003du is very similar to normal SNe Ia. If one assumes that SN 2003du had the above-mentioned absolute VJHK magnitudes, a distance mod- ulus of µ = 32.79 ± 0.04 (or a radial velocity of ∼ 2600 km s−1 with H0 = 72 km s −1,Mpc−1) is obtained 5. This estimate is the average of the four individual estimates weighted by their sta- tistical uncertainties, i.e. the errors of SN 2003du peak VJHK magnitudes added in quadrature to the statistical uncertainties of the absolute magnitudes. The difference between the two distance moduli is 0.37 mag (it will further increase if the Meikle 2000 absolute NIR mag- nitudes are used) and indicates that SN 2003du was fainter than the average of SNe with ∆m15 = 1.02. The 1σ dispersion of SNe Ia absolute magnitudes in both, optical and IR, is ∼ 0.15 mag (e.g., Phillips et al., 1999; Krisciunas et al., 2004c). The 4 Riess et al. (2005) estimated H0 = 73 km s −1,Mpc−1 and MV = −19.17, and we converted their MV to the scale of H0 = 72 km s−1,Mpc−1 5 The absolute JHK magnitudes of Meikle (2000) are by 0.4 mag brighter than those of Krisciunas et al. (2004c). The distance moduli derived with the values from the latter paper are consistent with the estimates of absolute V magnitude from Riess et al. (2005); we therefore adopt the Krisciunas et al. (2004c) values. Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova 13 uniformity and the small dispersion of the V − [JHK] colors of SNe Ia (Krisciunas et al., 2004b) indicates that the intrinsic scatter in the VJHK bands is correlated, and so cannot be re- duced by averaging observations in different bands. Therefore, the distance modulus we estimate, µ = 32.79 ± 0.04 mag, has an additional ∼ 0.15 mag uncertainty from the intrinsic disper- sion of SNe Ia luminosity. The fact that SN 2003du is 0.37 mag fainter than expected for SNe with ∆m15 ∼ 1.02 may thus be due to the intrinsic scatter (2.5σ from of the mean). It is also possible that UGC 9391 may not be in the undisturbed Hubble flow: if it has vr = 2600 km s −1 and a peculiar velocity compo- nent of ∼ 400 km s−1 toward the Earth, it may seem closer than it really is. UGC 9391 is nearly face-on and the contribution of the galaxy rotation should be small. 4.2. The bolometric light curve In order to compute the uvoir ”bolometric” light curve (i.e. the flux within the 0.2-2.4µm interval) of SN 2003du we proceeded as follows. First, our U-band template LC was fitted to the U photometry in order to estimate the U magnitudes when only BVRI were available. The magnitudes were corrected for the small Galactic reddening and transformed to flux densities us- ing the absolute calibration of the UBVRI system by Bessell et al. (1998). A cubic spline was fitted through the data points and the resulting fit was integrated numerically over the interval 3500-9000 Å. Most of the early-time SN Ia luminosity is emitted at optical wavelengths, however, a non-negligible correction for the flux emitted outside the optical wavelengths is also needed (see, e.g. Suntzeff, 1996). The flux emitted beyond 9000 Å was estimated by integrating the combined optical-NIR spectra of SN 2003du. The filled circles in Fig. 14a show the time evolution of the ra- tio of the flux emitted in the 9000-24000Å range to that emit- ted within 3500-9000 Å. Suntzeff (1996) finds that at +80 days less than 10% of the flux is emitted in the IR. We estimate from the photometry of SN 2001el (Krisciunas et al., 2003) that the contribution of the IR flux is ∼ 25% and ∼ 15% at +28 and +64 days, respectively. This is consistent with our estimates for SN 2003du and the findings of Suntzeff (1996), and indicates that the contribution of the IR flux decreases roughly linearly between days +30 and +80. As there are no UV spectra of SN 2003du observed, we used UV spectra of other SNe Ia to estimate the contribution of the UV flux. These comprised combined de-reddened UV-optical spectra of SN 1990N at −14 and −7 days (Leibundgut et al., 1991), SN 1989B at −5 (Wells et al. 1994 and UV spectra from the IUE archive), SN 1981B (Branch et al., 1983), SN 1992A at +5, +9 and +17 (Kirshner et al., 1993), and SN 2001el be- tween +30 and +66 (from HS T archive). For spectra that did not cover the full 2000–9000 Å range we extrapolated to 9000 Å using spectra of SN 2003du. The spectra of SN 2001el were lin- early extrapolated from ∼ 2900 Å down to 2000 Å assuming that the flux approached zero at 1000 Å. In Fig. 14a we show the ra- tios of the fluxes in the 2000–3500Å range to those in the 3500– 9000 Å range (open symbols). The total contribution of the UV and IR fluxes is plotted as a dashed line in Fig. 14a, and one can see the particularly large corrections needed before the B-band maximum and around the secondary I-band maximum. Beyond +80 days we assumed a constant IR contribution of 10% and that the UV contribution decreases linearly from 5% at +80 days to zero at +500 days. This correction was applied to the optical fluxes to derive the uvoir fluxes of SN 2003du. These were converted to luminos- ity assuming a distance modulus µ = 32.79 mag. The uvoir ”bolometric” light curve of SN 2003du is shown in Fig. 14b. For comparison, we also show the bolometric light curve of SN 2005cf (Pastorello et al., 2007b), which is very similar to that of SN 2003du. The maximum uvoir ”bolometric” luminos- ity of SN 2003du is 1.35(±0.20) × 1043 erg s−1 at ∼ 2 days before the B-band maximum. Using Arnett’s Rule as formu- lated by Stritzinger & Leibundgut (2005, their Eq. 7)) we es- timate the amount of 56Ni synthesized during the explosion, M56Ni = 0.68 ±0.14 M⊙. The error is a simple propagation of the uncertainty of the bolometric peak luminosity and the relation of Stritzinger & Leibundgut (2005). However, Khokhlov et al. (1993) have shown that the simplifying assumptions made in the derivation of Arnett’s rule may lead to errors as large as 50%. Combined with the uncertainty of the distance to SN 2003du, clearly this estimation of the 56Ni mass is subject to large sys- tematic uncertainty. Note, however, that Stritzinger et al. (2006) have analyzed a nebular spectrum and the optical photometry of SN 2003du, and derived M56Ni ≃ 0.6 M⊙, which is in good agreement with our estimate. If one accepts a distance modulus of µ = 32.42 mag (∼ 30.4 Mpc), then the estimated peak lumi- nosity and M56Ni should be reduced by ∼ 30%. 4.3. Bolometric light curve modeling To further estimate the amount of 56Ni synthesized we mod- eled the bolometric light curve of SN 2003du for both distance moduli µ = 32.42 and µ = 32.79 mag. We used the Monte Carlo light curve code described by Cappellaro et al. (1997) and Mazzali et al. (2001). Starting from an explosion model and a given 56Ni content the code computes the transport and deposi- tion of the γ-rays and the positrons generated by the decay chain 56Ni→56Co→56Fe in a grey atmosphere. The optical photons that are generated by the thermalization of the energy carried by the γ-rays and the positrons are then followed as they propagate through the SN ejecta. The optical opacity encountered by these photons is again assumed to be grey and to depend primarily on the relative abundance of iron-group elements. The opacity also decreases with time as (td/17) −3/2, td being the time since the explosion in days, to mimic the effect of the decreasing tem- perature. For more details on the adopted parametrization of the opacity see, e.g. Mazzali et al. (2001). This simple approxima- tion works well (e.g. Mazzali et al., 2001) but an alternative view that the opacity depends primarily on temperature has been sug- gested (Kasen & Woosley, 2007). In Mazzali et al. (2000) the Monte Carlo code was compared with the results from the radia- tion hydrodynamics code of Iwamoto et al. (2000), finding very good agreement. We followed the approach of Mazzali & Podsiadlowski (2006), who assumed that stable Fe-group isotopes (e.g. 54Fe, 58Ni) may be present not only in the innermost part of the ejecta (≤ 0.2 M⊙), but also in the 56Ni zone between ∼ 0.2 M⊙ and ∼ 0.8 M⊙. Mazzali & Podsiadlowski (2006) suggested that the scatter of SNe Ia luminosity at a given ∆m15 may be reproduced by changing the ratio of the amount of radioactive 56Ni and the stable isotopes in the 56Ni zone, while keeping the total mass of the Fe-group elements constant. This ratio may be sensitive, for example, to the metallicity of the progenitor white dwarf (Timmes et al., 2003). The SN Ia light curve width is mainly determined by the opacity of the ejecta, which in turn is mostly determined by the total amount (stable and radioactive) of Fe- group elements synthesized, provided the temperature is above ∼ 104 K (e.g. Khokhlov et al., 1993). The peak luminosity on the 14 Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova −20 0 20 40 60 80 model SN 2005cf SN 2003du FIR0.9−2.4µm FUV<0.35µm FUV+IR −10 0 10 20 0 100 200 300 400 Days from Bmax MC ke+=10 p.i. model no p.i. model SN 2003du Co decay Fig. 14. a: The ratio between the UV and IR fluxes to the flux within 3500-9000 Å; b: The uvoir bolometric light curves of SN 2003du, SN 2006cf and best model. The inset shows an expansion of the SN 2003du light curve and model around maximum; c: The entire uvoir bolometric light curve with the models overplotted. other hand is determined by the amount of 56Ni. Therefore, if the fraction of stable Fe-group isotopes is varied within reasonable limits (∼ 20%) the temperature may not be affected significantly, and thus the opacity may be effectively unchanged. This would lead to light curves with the same width, but different luminosi- ties. As shown in Fig. 14b, the uvoir ”bolometric” light curve of SN 2003du is remarkably similar to that of SN 2005cf (Pastorello et al., 2007b) if µ = 32.79 mag is adopted. Therefore, a model similar to that adopted for SN 2005cf can be used also to reproduce the light curve of SN 2003du. In this case the best fit, shown in Fig. 14b, is obtained for a model with 0.69M⊙ of 56Ni and 0.42M⊙ of stable Fe-group isotopes using the W7 ex- plosion model (Nomoto et al., 1984) as an input. This estimate of the amount of 56Ni is in excellent agreement with the esti- mate derived above using Arnett’s rule. However, mixing out of a sufficient amount of 56Ni is necessary to reproduce the early rise of the light curve. This is a feature that is not present in one-dimensional explosion models, but is often inferred from SN data. For example, for SN 2002bo, using the abundance dis- tribution and the amount of 56Ni mixed out as derived from an abundance tomography experiment (Stehle et al., 2005) gave a much better reproduction of the bolometric light curve. What is interpreted as mixing in one-dimensional models may be related to the presence of high velocity features (Mazzali et al., 2005b), which affect the early spectra of SN 2003du quite heavily. If the true distance modulus were µ = 32.42, the light curve could only be reproduced if the total mass of iron group elements was the same as above (i.e. 1.11M⊙) but the 56Ni content was ∼ 0.45 M⊙. While this may still be a possibility, with such a low 56Ni mass (less than half of the total Fe-group content) it can be expected that the heating by radioactive decay is not sufficient to keep the gas at a sufficiently high temperature (∼ 104K) that the opacity is unchanged. At lower temperatures, the opacity rapidly drops (Khokhlov et al., 1993), and thus the light curve would not be as broad as observed. We therefore suggest that a reasonable range of distances for SN 2003du is between µ = 32.7 and 33.0 mag, implying a 56Ni mass between 0.6 and 0.8M⊙ for a total Fe-group elements mass of ∼ 1.1M⊙. Roughly 200 days after maximum SN Ia ejecta become transparent to the γ-rays and the main source of energy is the positrons produced by the decay of 56Co. If the positrons are fully trapped and deposit all their kinetic energy, the true bolo- metric LC should have a decline rate of ∼ 1 mag per 100 days. Larger decline rates are typically found in SNe Ia, and assum- ing that the optical flux follows the true bolometric flux, this is usually interpreted as evidence for positron escape (see, e.g., Colgate et al., 1980; Cappellaro et al., 1997; Ruiz-Lapuente & Spruit, 1998; Milne et al., 1999). The uvoir ”bolometric” luminosity decline rate of SN 2003du after 200 days is 1.4 mag per 100 days. However, late-time NIR observations of few SNe Ia have recently been published (SN 1998bu – Spyromilio et al. 2004; SN 2000cx – Sollerman et al. 2004; SN 2004S – Krisciunas et al. 2007) and indicate that after 300–350 days the NIR luminosity does not decline but stays nearly constant. The contribution of the NIR flux therefore increases with time and if accounted for may lead to decline rates lower than the observed ones and closer to the full positron trapping value. Motohara et al. (2006) obtained late-time NIR spectra (1.1-1.8µm) and H- band photometry of SN 2003du. At +330 days SN 2003du had an H magnitude of 20.12±0.17 (Motohara et al. private commu- nication) and we calculate the integrated flux across the H-band to be ∼ 3% of the optical flux at that epoch. The late-time NIR spectra of SN 2003du indicate that the integrated J and H band fluxes are nearly equal, implying that the contribution of the NIR flux is at least 6%. If we adopt a 10% NIR contribution at +330 days and assume that the total NIR flux did not change after- wards, we obtain a decline rate of 1.2 mag per 100 days, which is still larger than the full positron trapping value. In Fig. 14c we compare the uvoir ”bolometric” LC of SN 2003du with the two models presented by Sollerman et al. (2004). The models are in the form of broadband U-to-H mag- nitudes. For a consistent comparison with SN 2003du we used only the UBVRI model fluxes to compute the model uvoir LC in exactly the same way as for SN 2003du. The models are generic, and have not been tuned to any particular SN. They have been computed with 0.6M⊙ 56Ni and assume full positron trapping, and differ only in the treatment of the photoionization repre- senting two extreme cases that the UV photons either escape or are fully redistributed to lower energies (for more details see Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova 15 Sollerman et al. 2004 and references therein). For a comparison with SN 2003du the models were only re-scaled to a distance modulus µ = 32.79 mag, and yet they fit the absolute flux level of the LC of SN 2003du quite well. It is evident from Fig. 14c that a model with an intermediate treatment of the photoion- ization could reproduce the SN 2003du light curve. Figure 14c also shows a model computed with the Monte Carlo code us- ing the best parameters we estimated above. Only the opacity to positrons kβ+ was adjusted to fit the late-time decline rate (Cappellaro et al., 1997). The best value is kβ+ = 10 cm −2 g−1, which is well within the range of values found by Cappellaro et al. (1997). Both late-time LC models we discussed are based on the 1D W7 explosion model and do not include a contribution from magnetic fields. However, detailed calculations (Ruiz-Lapuente & Spruit, 1998; Milne et al., 1999) show that the positron depo- sition rate is quite sensitive to the magnetic field configuration in the ejecta and the actual explosion model. Clearly, to fully ex- ploit the information in the bolometric LC a more detailed study is needed, but this is beyond the scope of this paper. 4.4. Evolution of Si ii λ6355, Ca ii H&K and IR triplet Figure 15 shows the pre-maximum evolution of the absorption lines Si ii λ6355, Ca iiH&K and Ca ii IR3 in SN 2003du (here we also use a few spectra of SN 2003du from Gerardy et al. 2004 and Anupama et al. 2005) and other SNe Ia. In the −13 days spectrum of SN 2003du the Si ii λ6355 line is broad and rather symmetric. In the −11 day spectrum the line is asymmetric and narrower, but around a week before maximum becomes sym- metric again and the profile does not change much until max- imum. The line evolution in SN 1994D is very similar, but is delayed with respect to SN 2003du: the −13 and −11 day spec- tra of SN 2003du are most similar to those of SN 1994D at −11 and −9 days. Similar evolution is also observed in SN 2001el, SN 1990N, SN 1999ee and SN 2005cg, but the pre-maximum coverage of these SNe is rather sparse. Nevertheless, this profile evolution may be explained if the Si ii λ6355 line is a blend of two components. At 10− 14 days before maximum, the strength of the two components should be nearly equal. The blue compo- nent then decreases very rapidly, disappearing by ∼ 7 − 5 days before maximum, while the red component increases in strength. In SN 2003du, the blue component was last seen in the −7.8 day spectrum as a weak feature on the blue wing of the line, and in SN 2001el it may be still present in the −2 day spectrum. The peculiar flat-bottom line shape in the early spectra of SN 2001el and SN 1990N is thus due to the blue component extending over a larger velocity interval compared to other SNe. The −9 day spectrum of SN 1999ee on the other hand, has a stronger blue component such that the line is asymmetric with an extended red wing. Note that Mazzali (2001) and Mazzali et al. (2005b) find that a two-component model is needed to explain the pe- culiar Si ii λ6355 line shape in SN 1990N and SN 1999ee, the high-velocity (HV) component being carbon/silicon and a thin pure Si shell, respectively. It is also clear that the early-time evo- lution of the blueshift of the line-profile minimum will be largely determined by the evolution of relative strength of the two com- ponents, and therefore will be very difficult to interpret. Mattila et al. (2005) suggest that the flat-bottomed shape of Si ii λ6355 in SN 2001el and its disappearance over a few days can be explained by the effects of scattering within a thin re- gion moving at the continuum photospheric velocity, thus re- quiring no absorbing HV material to produce the line shape. SN 2003du -12.8 -10.9 SN 1994D -12.0 -11.0 -10.2 SN 1990N SN 2001el SN 1999ee SN 2005cg -10.9 SN 2003du Si II 3850? -12.0 -11.0 -10.2 Si II 3850? SN 1994D SN 1998bu Si II 3850? SN 2001el -9 -20000 0 -10.9 SN 2003du -20000 0 Velocity [km s-1] SN 1994D SN 2001el -20000 0 SN 1998bu Fig. 15. Comparison of the evolution of Si ii λ6355, Ca ii H&K and Ca ii IR3 lines in SN 2003du with those of other normal SNe Ia. Quimby et al. (2006) argue that the triangular shape of the pro- file in SN 2005cg with an extended blue wing (see, Fig. 15) may be due to absorption by Si in the HV part of the ejecta. The line profile may be reproduced if the Si abundance slowly de- creases toward high velocities, which is typical for the delayed- detonation models (Khokhlov, 1991). However, these both sug- gestions may have difficulties to explain asymmetric line pro- files with a stronger blue component as observed in SN 1999ee. SN 1999ee is not unique. SN 2005cf, observed by the ESC with daily sampling starting from 12 days before maximum (Garavini et al., 2007), shows Si ii λ6355 line that consists of two dis- tinct components with profile evolution similar to SN 1999ee. It is therefore likely that the ”peculiar” profiles in SN 2001el, SN 1990N and SN 2005cg are just snapshots of this common 16 Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova evolutionary pattern. In addition, if more SNe Ia like SN 2005cf and SN 1999ee are found, the suggestion of Quimby et al. (2006) that the SNe Ia with a flat-bottomed Si ii λ6355 line may consti- tute a separate sub-class of SNe Ia, possibly produced from dif- ferent progenitors and/or explosion models can be ruled out. In the −11 day spectrum of SN 2003du the Ca ii H&K line is a broad, single absorption with high velocity of ∼ −21000 km s−1. In the −7.8 days spectrum another, less blueshifted Ca ii H&K component is also visible at velocity of ∼ −10000 km s−1. In the subsequent spectra, the HV compo- nent decreases in strength, while the low-velocity one grows stronger. Qualitatively the same evolution of also observed in SN 1994D. In the near maximum spectra, the HV component is much weaker, if present at all, in SN 1994D and SN 1998bu than in SN 2003du and SN 2001el. It can be seen in Fig 15 that the strength of the HV component in SN 1994D decreases faster than in SN 2003du and SN 2001el, thus qualitatively following the evolution of the Si ii λ6355 HV component. On the other hand, SN 1998bu either lacked HV components altogether, or they disappeared faster than in SN 1994D. The evolution of the Ca ii IR3 line is shown for few epochs only, but it is evident that a strong HV component with velocity of ∼ −21000 km s−1 is also present and that this component disappears at different time, earliest in SN 1994D, followed by SN 2003du, and latest in SN 2001el. It is also interesting to note that there is a segrega- tion of SNe Ia according to Ca iiH&K line profile: (i) SNe with a single-component line at all epochs, SN 2004S (Krisciunas et al., 2007), SN 1999ee and SN 2002bo being examples, and (ii) SNe like SN 2003du and SN 1994D with double-component line af- ter maximum. In SN 1994D the blue component of the post- maximum Ca ii H&K-split is already visible in the −9 spectrum as a weak feature superimposed on the broad HV component, while in SN 2003du it becomes apparent only around maximum, possibly because the HV component remains visible longer than in SN 1994D. Possible identification for this feature is Si ii λ3850 (Nugent et al., 1997; Lentz et al., 2000), which is also supported by the identification of strong Si ii λ3850 line in the early spec- trum of SN 2004dt (Wang et al., 2006). Due to severe line blending it is difficult to quantify the strength of the HV components at different epochs. However, the qualitative comparison strongly suggests that the strength of the HV components in the Si ii λ6355, Ca ii H&K and Ca ii IR3 line in given SN are correlated and evolve similarly. The HV features in the Ca ii lines are stronger and more separated from the lower- velocity components than in the Si ii λ6355 line. Comprehensive spectral modeling of the line profiles evolution is therefore needed to verify the two-component hypothesis for Si ii λ6355 and further investigate the HV features (e.g. Mazzali et al., 2005b). Such an analysis of the SN 2003du spectra will be pre- sented elsewhere. Currently, there is no consensus on the origin of the HV features. Interaction of the ejecta with circumstellar matter close to the SN (e.g. Gerardy et al., 2004) or the clumpy ejecta structure found in some explosion models (e.g., Mazzali et al., 2005b; Plewa et al., 2004; Kasen & Plewa, 2005) could cause the observed HV features. The continuum polarization in SNe Ia is typically low, but much higher polarization across the lines including the HV features is often observed, which favors the clumpy ejecta model rather than a global asymmetry (Wang et al., 2003a, 2006, 2007; Leonard et al., 2005). The HV features may thus carry information about the 3D structure of the ejecta and the environment close to the SN explosion site. Modeling of time sequences of flux and polarization spectra (e.g., Kasen et al., 2003; Thomas et al., 2004; Wang et al., 2007) may al- low us to recover this information and help to impose additional constraints on the SN Ia explosion and progenitor models. 5. Summary We present an extensive set of optical and NIR observations of the bright nearby Type Ia SN 2003du. The observations started 13 days before B-band maximum light, and continued for 480 days after with exceptionally good sampling. The optical pho- tometry was performed after the background contamination from the host galaxy had been removed by subtraction of tem- plate images. The photometry was obtained using a number of instruments with different filter responses. In order to properly account for deviations from the standard system responses, the optical photometry was calibrated by applying S-corrections. Our observations show that the spectral and photometric evolution of SN 2003du in both, optical and NIR wavelengths, closely follow that of the normal SNe Ia. The luminosity decline rate parameter ∆m15 is found to be 1.02 ±0.05, the ratio between the depth of the Si ii λ5972 and λ6355 linesR(Si ii) = 0.22 ±0.02 and the velocity of the Si ii λ6355 line is ∼ −10000 km s−1 around maximum light. The analysis of the uvoir light curve sug- gests that ∼ 0.6− 0.8 M⊙ of 56Ni was synthesized during the ex- plosion. All this indicates an average normal SN Ia. We also find that SN 2003du was unreddened in its host galaxy. This property is important for better understanding of the intrinsic colors of SNe Ia in order to obtain accurate estimates of the dust extinction to the high-redshift SNe Ia, which is one of the major systematic uncertainties in their cosmological use. SN 2003du also showed strong high-velocity features in Ca ii H&K and Ca ii IR3 lines, and possibly in Si ii λ6355. The excellent temporal coverage al- lowed us to compare the time evolution of the line profiles with other well-observed SNe Ia and we found evidence that the pe- culiar pre-maximum evolution of Si ii λ6355 line in many SNe Ia is due to the presence of two blended absorption components. The well-sampled and carefully calibrated data set we present is a significant addition to the well-observed SNe Ia and the data will be made publicly available for further analysis. For example comprehensive modeling of the extensive spectral data set, e.g. by the abundance tomography method (Stehle et al., 2005), may eventually help to achieve a better understanding of the physics of SNe Ia explosions and their progenitors. Acknowledgements. This work is partly supported by the European Community’s Human Potential Program “The Physics of Type Ia Supernovae”, under contract HPRN-CT-2002-00303. V.S. and A.G. would like to thank the Göran Gustafsson Foundation for financial support. The work of D.Yu.T. and N.N.P. was partly supported by the grant RFBR 05-02-17480. The work of S.M. was supported by a EURYI scheme award. This work is based on observations collected at the Italian Telescopio Nazionale Galileo (TNG), Isaac Newton (INT) and William Herschel (WHT) Telescopes, and Nordic Optical Telescope (NOT), all located at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias (La Palma, Spain), the 1.82m and 1.22m telescopes at Asiago (Italy), the 2.2m and 3.5m telescopes at Calar Alto (Spain), the United Kingdom Infrared Telescope (UKIRT) at Hawaii and the 60-cm telescope of the Beijing Astronomical Observatory (China). We thank the support astronomers of these telescopes for performing part of the observations. We also thank the director of the Calar Alto Observatory Roland Gredel for allocating additional time at the 2.2m telescope in May 2003. We thank all observers that gave up part of their time to observe SN 2003du. Thomas Augusteijn and Amanda Djupvik are acknowledged for observing dur- ing two technical nights at the NOT. Observations were also obtained at the NOT during a student training course in Observational Astronomy provided by Stockholm Observatory and the NorFA Summer School in Observational Astronomy. We thank Geir Oye for excellent support and close collaboration dur- ing this course. We also thank O.A.Burkhanov, S.Yu.Shugarov and I.M.Volkov for carrying out observations at Maidanak, Slovakia, Moscow and Crimea. We Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova 17 thank Cecilia Kozma for making available to us her late-time light curve mod- els. We thank Aaron Barth for providing us with the spectra of SN 1994D col- lected by the Alexei Filippenko group at UC Berkeley, and the people who did the observations: Aaron Barth, Alexei Filippenko, Tomas Matheson, Xiaoming Fan, Michael Gregg, Vesa Junkkarinen, Brian Espey, Matt Lehnert, Lee Armus, Graeme Smith, Greg Wirth, David Koo, Abe Oren and Vince Virgilio. We thank K. 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(2004) have calculated the responses of most of the instruments used by the ESC. However, we repeated the process for the 5 instruments most frequently used to observe SN 2003du: AFOSC at Asiago 1.82m telescope, DOLORES at TNG, ALFOSC at NOT, CAFOS at Calar Alto 2.2m telescope and BAO 60-cm telescope imager, using the new extensive spectrophotometry of Landolt stars in Stritzinger et al. (2005). The first four of the five instruments are combined spec- trographs/imagers with design that allows the grisms, the slits and the imaging filters to be inserted into the light beam simul- taneously. This made possible to measure the filter transmissions in situ as the filters are mounted in the instrument and used dur- ing the photometric observations. This measurement is straight- forward and consists of taking spectral flat-fields with and with- out the filter in the beam. The flat taken with the filter is divided by the one taken without, giving the filter transmission. Before doing this, the bias and any reflected light present was carefully removed. The latter can be important in the blue part of the spec- trum where the sensitivity of the system is low and the scattered light can be a significant fraction of the useful signal; this can af- fect the measured transmission. The wavelength calibration was done with arc-lamp spectra taken without the filter in the beam. When filters are introduced in the beam small shifts of the wave- length solution can be expected. After the measurements we checked this for few filters at AFOSC at Asiago 1.8m telescope and did indeed find shifts of a few pixels. Hence, the measured filter transmissions might be shifted by up to 20–30 Å, but the shape is accurately determined. Generally, we found good agree- ment with the filter transmissions available from the instrument web-pages. However, we found significant discrepancies for the Calar Alto 2.2m + CAFOS B and I filters, and minor differences for the TNG+DOLORS I-band. For the U-bands we used the transmissions available from the instrument web-pages. For the BAO 60-cm telescope the filter transmissions specified by the manufacturer were used. The total system responses were computed by multiplying the filter transmissions by (a) the CCD quantum efficiency (QE), (b) the reflectivity of at least two aluminum surfaces, (c) the continuum transmission of the Earth atmosphere at airmass one (the extinction laws were provided by the observatories), and (d) a telluric absorption spectrum, which we derived from the spectrophotometric standards observed at WHT close to airmass one. The lens and window transmissions were not included be- cause this information was unavailable. Synthetic magnitudes were calculated from Stritzinger et al. (2005) spectrophotometry of Landolt standard stars, msyn = −2.5 log f phot (λ)Rnat(λ)dλ The difference between the synthetic and the observed photome- try was fitted as a function of the observed color indices to com- pute synthetic color-terms (ctsyn), e.g. for B we have Bstd − Bsyn = ctsyn(Bstd − V std) + const. (A.1) For the VRI-bands, the ctsyn’s were close to the observed ones ctobs. In some cases small differences exceeding the uncertainty were accounted for by shifting the filter transmissions until ctsyn matched ctosb. Small shifts of up to ∼ 20 − 30 Å were required. These discrepancies could easily have arisen from the way in which the transmissions were determined, as discussed above. For the U and B-bands however, we found large differences which would have required an unacceptably large shift to cor- rect for them. The synthetic U and B bands were always too blue. This is, to some extent, to be expected because the neglected op- tical elements like lenses or windows, anti-reflection and other coatings will tend to reduce the system sensitivity shortward of ∼4000 Å. The uncertainty in the CCD QEs and the extinction laws may also contribute to this effect. To account for the net effect of these uncertainties we modified the U and B bands by multiplying them with a smooth monotonic function of wave- length so that ctsyn matched ctobs. We used the Sigmoid function F(λ; λ0,∆) = 1 + exp(−(λ − λ0)/∆) , (A.2) that changes smoothly from 0 to 1. The parameters λ0 and ∆ control the position and the width of the transition; for small ∆ the Sigmoid function approaches a Heaviside step function at λ0. We proceeded as follows: λ0 and ∆ were varied in the wavelength intervals 3200–4200Å and 100–500 Å, respectively, and the set of parameters that brought the synthetic U and B- band color-terms into accord with the observed ones was chosen. Note that independent modification of U and B results in degen- eracy in the (λ0,∆) parameter space, and it was only when the U and B-bands were considered together that an unique solution for λ0 and ∆ could be obtained. As standard Johnson-Cousins system responses we use the Bessell (1990) filters but following Stritzinger et al. (2005) we first modified them so that they could be used with photon fluxes and included the telluric absorptions. Small shifts were also applied to account for the small color- terms that are noticeable when compared with the Landolt pho- tometry. Bessell (1995) suggested correcting Landolt photome- try to bring it into the original Cousins system. The synthetic photometry with the original Bessell filters does match the cor- rected magnitudes. However, for sake of comparability with the existing SN photometry, we use the original Landolt photometry and modify the Bessell filters so that the synthetic color-terms are zero. The constant terms derived from the fits with Eq. A.1 are the filter zero-points for synthetic photometry. The constant in Eq. 2 is the difference between the zero-point for the Bessell and natural passbands. The reconstructed bands are shown in Fig. A.1 together with the modified Bessell filters, demonstrating the variety of pass- bands one may encounter at different telescopes. Note particu- larly the non-standard form of the Calar Alto I-band and NOT R-band. We note that the reconstructed responses should be re- garded only as approximations of the real responses. A given passband can be modified in many ways to match the observed and the synthetic color-term, and we would consider the pro- cedure we used as the most appropriate one given the available information. We also note that fitting the U-band synthetic color- term is ambiguous. Because of the Balmer discontinuity even small deviation from the Bessell passband changes U std − U syn such that it needs no longer be a simple linear function of Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova 19 4000 6000 8000 10000 Wavelength [Å] modified Bessell Asiago TNG NOT Calar Alto BAO U B V R I Fig. A.1. Reconstructed system responses for the five instruments studied compared to the modified Bessell passbands. U std−Bstd. This also affects the derivation of the observed color- terms and as a result the U-band photometry should be in general considered significantly less accurate than other bands. A.2. Computing the S-corrections We used our spectra obtained earlier than 110 days after max- imum, and spectra from Gerardy et al. (2004) and Anupama et al. (2005) to compute the S-corrections and to transform the BVRI photometry of SN 2003du into the Johnson-Cousins system. The TNG, Calar Alto and BAO I-bands extend out to 1.1 µm and to compute the S-corrections, we also used our NIR spectra of SN 2003du (Sec. 2.3). To compute the BAO I-band S- corrections between +30 and +63 days we also used NIR spectra of SN 1999ee (Hamuy et al., 2002) and SN 2000ca (Stanishev et al., in preparation) taken at ∼ +40 days. The U-band could not be S-corrected because no UV spectra of SN 2003du were avail- able. The relative spectrophotometry of SN 2003du was not al- ways sufficiently accurate for the purpose of computing S- corrections. It was thus necessary to slightly modify some of the spectra so that the synthetic photometry with the modified Bessell BVRI bands matched the observed one. To achieve that, the spectra were multiplied by a smooth correction function de- termined by fitting the ratio between the observed and the syn- thetic fluxes. When the ratio varied monotonically with wave- length, a second-order polynomial was used. When a more com- plex function was required, a spline fit was used. At the first it- eration the synthetic magnitudes were compared with the linear color-term corrected magnitudes of SN 2003du, and the spectra were only modified if the observed and the synthetic color in- dices differed by more than 0.05 mag for B − V and V − R, and 0.1 mag for V − I. These corrected spectra were used to compute S-corrected photometry of SN 2003du. The flux correction of the spectra was then repeated using the S-corrected rather than the color-term corrected photometry. Spectra were only corrected if the color discrepancies were greater than 0.03 mag for B − V and V − R, or 0.05 mag for V − I. New S-corrected photome- try was then computed and the process repeated to obtain the final S-corrected photometry and calibrated spectra. A number of spectra have a wavelength coverage that only allows B and V synthetic magnitudes to be computed. In these cases, only a sim- ple linear correction was applied to match the observed B and V magnitudes. We note that because the instrumental responses are fairly close to those of Bessell filters, the S-corrections are almost en- tirely determined by the SN spectral features and are practically insensitive to small changes of the SN colors. It was found that the initial correction of the spectra yielded spectrophotometry which was already accurate to a few per cent and that the subse- quent iterations had very little effect on the final calibrated pho- tometry. We therefore conclude that the few percent uncertain- ties in the spectrophotometry, which might have arisen from the way the spectra were corrected, should have little effect on the final photometry. List of Objects ‘SN 2003du’ on page 1 ‘SN 2003du’ on page 1 ‘UGC 9391’ on page 1 ‘SN 2003du’ on page 1 ‘SN 2003du’ on page 1 ‘SN 2003du’ on page 2 ‘UGC 9391’ on page 2 ‘SN 2003du’ on page 2 ‘SN 2002bo’ on page 2 ‘SN 2002dj’ on page 2 ‘SN 2002er’ on page 2 ‘SN 2003cg’ on page 2 ‘SN 2004eo’ on page 2 ‘SN 2005cf’ on page 2 ‘SN 2003du’ on page 2 ‘SN 2003du’ on page 2 ‘SN 2003du’ on page 2 ‘SN 2003du’ on page 3 ‘SN 2003du’ on page 3 ‘SN 2003du’ on page 3 ‘SN 2003du’ on page 3 ‘M92’ on page 4 ‘SN 2003du’ on page 4 ‘SN 2003du’ on page 4 ‘SN 2003du’ on page 4 ‘SN 2003du’ on page 4 ‘SN 2003du’ on page 5 ‘SN 2003du’ on page 5 ‘SN 2003du’ on page 5 ‘SN 2003du’ on page 5 20 Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova ‘SN 2003du’ on page 5 ‘SN 2003du’ on page 6 ‘SN 2003du’ on page 6 ‘SN 2003du’ on page 6 ‘SN 2003du’ on page 6 ‘SN 2003du’ on page 7 ‘SN 2000cx’ on page 7 ‘SN 2003du’ on page 7 ‘1990N’ on page 7 ‘1998aq’ on page 7 ‘1998bu’ on page 7 ‘SN 2003du’ on page 7 ‘SN 2003du’ on page 7 ‘SN 2003du’ on page 7 ‘SN 2003du’ on page 8 ‘SN 2003du’ on page 8 ‘SN 2003du’ on page 8 ‘SN 2003du’ on page 8 ‘SN 2003du’ on page 8 ‘SN 2003du’ on page 8 ‘SN 2003du’ on page 8 ‘SN 2003du’ on page 8 ‘SN 2003du’ on page 8 ‘SN 2003du’ on page 8 ‘SN 2003du’ on page 9 ‘SN 2003du’ on page 9 ‘SN 1994D’ on page 9 ‘SN 1990N’ on page 9 ‘SN 1996X’ on page 9 ‘SN 1999ee’ on page 9 ‘SN 1998aq’ on page 9 ‘SN 1998bu’ on page 9 ‘SN 2002er’ on page 9 ‘SN 2001el’ on page 9 ‘SN 2003du’ on page 10 ‘SN 2003du’ on page 10 ‘SN 1998bu’ on page 10 ‘SN 2005cg’ on page 10 ‘SN 2003du’ on page 10 ‘SN 2003du’ on page 10 ‘SN 2000cx’ on page 10 ‘SN 2003du’ on page 10 ‘SN 1994D’ on page 10 ‘SN 2003du’ on page 11 ‘SN 1999ee’ on page 11 ‘SN 1998bu’ on page 11 ‘SN 2002bo’ on page 11 ‘SN 2002bo’ on page 11 ‘SN 2003du’ on page 11 ‘SN 2003du’ on page 11 ‘SN 2003du’ on page 11 ‘SN 2003du’ on page 11 ‘SN 2003du’ on page 11 ‘SN 2003du’ on page 12 ‘SN 2003du’ on page 12 ‘UGC 9391’ on page 12 ‘SN 2003du’ on page 12 ‘SN 1998aq’ on page 12 ‘SN 1994ae’ on page 12 ‘SN 2003du’ on page 12 ‘SN 2003du’ on page 12 ‘SN 2003du’ on page 12 ‘SN 2003du’ on page 12 ‘SN 2003du’ on page 13 ‘UGC 9391’ on page 13 ‘UGC 9391’ on page 13 ‘SN 2003du’ on page 13 ‘SN 2003du’ on page 13 ‘SN 2001el’ on page 13 ‘SN 2003du’ on page 13 ‘SN 2003du’ on page 13 ‘SN 1990N’ on page 13 ‘SN 1989B’ on page 13 ‘SN 1981B’ on page 13 ‘SN 1992A’ on page 13 ‘SN 2001el’ on page 13 ‘SN 2003du’ on page 13 ‘SN 2001el’ on page 13 ‘SN 2003du’ on page 13 ‘SN 2003du’ on page 13 ‘SN 2005cf’ on page 13 ‘SN 2003du’ on page 13 ‘SN 2003du’ on page 13 ‘SN 2003du’ on page 13 ‘SN 2003du’ on page 13 ‘SN 2003du’ on page 13 ‘SN 2003du’ on page 14 ‘SN 2006cf’ on page 14 ‘SN 2003du’ on page 14 ‘SN 2003du’ on page 14 ‘SN 2005cf’ on page 14 ‘SN 2005cf’ on page 14 ‘SN 2003du’ on page 14 ‘SN 2002bo’ on page 14 ‘SN 2003du’ on page 14 ‘SN 2003du’ on page 14 ‘SN 2003du’ on page 14 ‘SN 1998bu’ on page 14 ‘SN 2000cx’ on page 14 ‘SN 2004S’ on page 14 ‘SN 2003du’ on page 14 ‘SN 2003du’ on page 14 ‘SN 2003du’ on page 14 ‘SN 2003du’ on page 14 ‘SN 2003du’ on page 14 ‘SN 2003du’ on page 14 ‘SN 2003du’ on page 15 ‘SN 2003du’ on page 15 ‘SN 2003du’ on page 15 ‘SN 2003du’ on page 15 ‘SN 2003du’ on page 15 ‘SN 2003du’ on page 15 ‘SN 1994D’ on page 15 ‘SN 2003du’ on page 15 ‘SN 2003du’ on page 15 ‘SN 1994D’ on page 15 ‘SN 2001el’ on page 15 ‘SN 1990N’ on page 15 ‘SN 1999ee’ on page 15 ‘SN 2005cg’ on page 15 ‘SN 2003du’ on page 15 ‘SN 2001el’ on page 15 ‘SN 2001el’ on page 15 ‘SN 1990N’ on page 15 ‘SN 1999ee’ on page 15 ‘SN 1990N’ on page 15 ‘SN 1999ee’ on page 15 ‘SN 2001el’ on page 15 Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova 21 ‘SN 2003du’ on page 15 ‘SN 2005cg’ on page 15 ‘SN 1999ee’ on page 15 ‘SN 1999ee’ on page 15 ‘SN 2005cf’ on page 15 ‘SN 1999ee’ on page 15 ‘SN 2001el’ on page 15 ‘SN 1990N’ on page 15 ‘SN 2005cg’ on page 15 ‘SN 2005cf’ on page 16 ‘SN 1999ee’ on page 16 ‘SN 2003du’ on page 16 ‘SN 1994D’ on page 16 ‘SN 1994D’ on page 16 ‘SN 1998bu’ on page 16 ‘SN 2003du’ on page 16 ‘SN 2001el’ on page 16 ‘SN 1994D’ on page 16 ‘SN 2003du’ on page 16 ‘SN 2001el’ on page 16 ‘SN 1998bu’ on page 16 ‘SN 1994D’ on page 16 ‘SN 1994D’ on page 16 ‘SN 2003du’ on page 16 ‘SN 2001el’ on page 16 ‘SN 2004S’ on page 16 ‘SN 1999ee’ on page 16 ‘SN 2002bo’ on page 16 ‘SN 2003du’ on page 16 ‘SN 1994D’ on page 16 ‘SN 1994D’ on page 16 ‘SN 2003du’ on page 16 ‘SN 1994D’ on page 16 ‘SN 2004dt’ on page 16 ‘SN 2003du’ on page 16 ‘SN 2003du’ on page 16 ‘SN 2003du’ on page 16 ‘SN 2003du’ on page 16 ‘SN 2003du’ on page 16 ‘SN 2003du’ on page 18 ‘SN 2003du’ on page 19 ‘SN 2003du’ on page 19 ‘SN 1999ee’ on page 19 ‘SN 2000ca’ on page 19 ‘SN 2003du’ on page 19 ‘SN 2003du’ on page 19 ‘SN 2003du’ on page 19 ‘SN 2003du’ on page 19 ‘SN 2003du’ on page 22 22 Stanishev et al.: SN 2003du: 480 days in the Life of a Normal Type Ia Supernova Table 3. Optical photometry SN 2003du. The measurements on the dates marked with ”∗” and all U magnitudes are not S-corrected. Date (UT) Phase [day] JD U B V R I Telescope 2003-04-25 −11.0 2452755.39 · · · 14.737 (0.018) 14.854 (0.014) 14.728 (0.010) 14.798 (0.017) AS1.8 2003-04-25 −10.8 2452755.61 14.382 (0.011) 14.629 (0.010) 14.774 (0.010) 14.672 (0.010) 14.756 (0.021) NOT 2003-04-29 −7.3 2452759.06 · · · 13.974 (0.014) 14.072 (0.022) 13.963 (0.015) 14.052 (0.017) BAO 2003-04-30 −6.2 2452760.17 · · · 13.820 (0.024) 13.920 (0.012) 13.852 (0.010) 13.936 (0.024) BAO 2003-04-30 −5.8 2452760.54 13.193 (0.034) 13.719 (0.018) 13.860 (0.010) 13.754 (0.010) 13.921 (0.021) TNG 2003-05-02 −4.0 2452762.38 13.077 (0.015) 13.589 (0.021) 13.712 (0.027) 13.639 (0.022) 13.834 (0.023) TNG 2003-05-04 −1.9 2452764.46 · · · 13.496 (0.010) 13.614 (0.018) 13.592 (0.010) 13.841 (0.010) AS1.8 2003-05-05 −1.0 2452765.41 · · · 13.489 (0.012) 13.595 (0.019) 13.569 (0.010) 13.857 (0.011) AS1.8 2003-05-06 +0.0 2452766.40 · · · 13.489 (0.019) 13.566 (0.025) 13.575 (0.011) 13.870 (0.010) AS1.8 2003-05-07 +1.1 2452767.51 · · · 13.506 (0.015) 13.575 (0.014) 13.590 (0.018) 13.927 (0.016) AS1.8 2003-05-09 +3.1 2452769.51 · · · 13.566 (0.010) 13.587 (0.021) 13.600 (0.010) 14.009 (0.010) AS1.8 2003-05-10 +4.2 2452770.61 13.234 (0.014) 13.605 (0.010) 13.620 (0.016) 13.600 (0.010) 13.982 (0.016) CA2.2 2003-05-11 +5.1 2452771.51 13.316 (0.040) 13.643 (0.011) 13.642 (0.017) · · · 14.017 (0.010) CA2.2 2003-05-13 +7.2 2452773.59 13.597 (0.021) 13.764 (0.010) 13.712 (0.010) 13.786 (0.011) 14.205 (0.018) NOT 2003-05-14 +8.2 2452774.56 13.651 (0.017) 13.845 (0.011) 13.758 (0.010) 13.838 (0.010) 14.271 (0.010) NOT 2003-05-15 +9.1 2452775.44 13.749 (0.017) 13.908 (0.010) 13.795 (0.011) 13.908 (0.011) 14.352 (0.015) NOT 2003-05-16 +10.1 2452776.45 13.838 (0.021) 14.004 (0.014) 13.842 (0.010) 13.979 (0.010) 14.443 (0.011) NOT 2003-05-17 +10.7 2452777.08 · · · 14.066 (0.018) 13.862 (0.010) 14.040 (0.011) 14.448 (0.010) BAO 2003-05-22 +15.8 2452782.20 · · · 14.596 (0.013) 14.186 (0.010) 14.342 (0.011) 14.588 (0.017) BAO 2003-05-23 +17.1 2452783.49 · · · 14.722 (0.023) 14.268 (0.028) 14.332 (0.012) 14.626 (0.031) AS1.8 2003-05-24 +18.0 2452784.42 · · · 14.833 (0.014) 14.302 (0.014) 14.350 (0.016) 14.611 (0.010) AS1.8 2003-05-25 +19.0 2452785.37 · · · 14.942 (0.012) 14.350 (0.013) 14.361 (0.012) 14.594 (0.010) AS1.8 2003-05-26 +20.0 2452786.38 · · · 15.043 (0.011) · · · · · · · · · CA2.2 2003-05-27 +21.0 2452787.45 15.144 (0.017) 15.146 (0.014) 14.463 (0.017) 14.392 (0.012) 14.464 (0.018) CA2.2 2003-05-29 +22.8 2452789.20 · · · · · · · · · 14.493 (0.031) 14.451 (0.014) BAO 2003-06-01 +26.1 2452792.51 15.809 (0.023) 15.648 (0.010) 14.724 (0.010) 14.453 (0.019) 14.410 (0.027) TNG 2003-06-06 +31.2 2452797.58 16.172 (0.033) 16.041 (0.011) 15.018 (0.014) 14.658 (0.010) 14.451 (0.019) TNG 2003-06-10 +34.7 2452801.06 · · · 16.363 (0.085) 15.223 (0.047) 14.881 (0.018) 14.540 (0.035) BAO 2003-06-14 +38.7 2452805.04 · · · · · · 15.477 (0.131) 15.140 (0.013) 14.829 (0.024) BAO 2003-06-15 +39.7 2452806.04 · · · 16.472 (0.036) 15.473 (0.011) 15.187 (0.018) 14.929 (0.018) BAO 2003-06-20 +45.1 2452811.51 · · · 16.606 (0.021) 15.676 (0.010) 15.415 (0.017) 15.265 (0.010) AS1.8 2003-06-26 +51.1 2452817.52 16.796 (0.023) 16.727 (0.011) 15.859 (0.015) 15.613 (0.011) 15.584 (0.022) TNG 2003-06-28 +52.7 2452819.04 · · · 16.745 (0.023) 15.892 (0.012) 15.679 (0.018) 15.633 (0.038) BAO 2003-06-30 +54.6 2452821.03 · · · 16.753 (0.039) 15.924 (0.018) 15.776 (0.032) 15.715 (0.039) BAO 2003-07-04 +58.7 2452825.06 · · · 16.835 (0.016) 16.046 (0.019) 15.874 (0.011) 15.920 (0.022) BAO 2003-07-05 +60.0 2452826.38 · · · · · · 16.088 (0.016) 15.926 (0.010) 16.026 (0.019) NOT 2003-07-08 +62.7 2452829.06 · · · · · · 16.159 (0.036) 16.010 (0.017) 16.096 (0.029) BAO 2003-07-08 +63.1 2452829.53 17.004 (0.026) 16.924 (0.011) 16.173 (0.010) 16.036 (0.011) 16.138 (0.017) NOT 2003-07-09 +63.7 2452830.06 · · · · · · · · · · · · 16.194 (0.045) BAO 2003-07-12∗ +66.8 2452833.19 · · · · · · 16.290 (0.020) 16.140 (0.020) 16.270 (0.030) MDK 2003-07-17 +72.0 2452838.33 · · · 17.082 (0.015) 16.410 (0.015) 16.275 (0.010) 16.477 (0.012) AS1.8 2003-08-01 +87.0 2452853.34 · · · 17.319 (0.010) 16.769 (0.011) 16.719 (0.011) 17.010 (0.011) AS1.8 2003-08-22 +108.0 2452874.40 · · · 17.618 (0.030) 17.266 (0.026) 17.309 (0.028) · · · AS1.8 2003-08-23 +109.0 2452875.37 · · · 17.688 (0.011) 17.281 (0.012) 17.379 (0.014) 17.756 (0.018) AS1.8 2003-09-16∗ +132.9 2452899.32 18.763 (0.079) 17.961 (0.022) 17.780 (0.021) 18.040 (0.023) 18.454 (0.075) CA2.2 2003-09-19∗ +136.0 2452902.38 18.844 (0.032) 18.059 (0.012) 17.851 (0.017) 18.077 (0.029) 18.329 (0.020) WHT 2003-09-26∗ +143.0 2452909.34 19.140 (0.072) 18.114 (0.021) 17.956 (0.043) 18.330 (0.062) 18.570 (0.067) CA2.2 2003-11-22∗ +199.2 2452965.62 · · · 18.660 (0.100) 18.670 (0.100) 19.090 (0.120) · · · CRM 2003-11-23∗ +200.3 2452966.63 · · · 18.990 (0.060) 18.930 (0.060) 19.210 (0.110) · · · CRM 2003-11-25∗ +202.2 2452968.61 · · · 18.900 (0.040) 18.950 (0.050) 19.330 (0.110) · · · CRM 2003-12-01∗ +208.3 2452974.63 · · · 19.020 (0.070) 19.090 (0.100) 19.370 (0.120) · · · CRM 2003-12-12∗ +220.4 2452986.75 20.688 (0.038) 19.304 (0.016) 19.313 (0.012) 19.899 (0.041) 19.673 (0.031) CA3.5 2003-12-19∗ +227.4 2452993.75 · · · 19.384 (0.010) 19.419 (0.011) 19.979 (0.044) 19.910 (0.033) CA3.5 2004-05-10∗ +370.2 2453136.62 · · · 21.476 (0.019) 21.453 (0.020) 22.202 (0.044) 21.192 (0.029) WHT 2004-05-11∗ +371.2 2453137.62 22.638 (0.079) 21.531 (0.011) 21.493 (0.018) 22.190 (0.028) 21.320 (0.030) WHT 2004-06-22∗ +413.4 2453179.54 · · · 22.100 (0.044) 22.010 (0.026) 23.010 (0.052) 21.720 (0.038) NOT 2004-08-11∗ +463.0 2453229.43 · · · 22.771 (0.027) 22.827 (0.026) · · · · · · NOT 2004-08-14∗ +466.0 2453232.39 · · · · · · · · · · · · 22.212 (0.048) TNG AS1.8 – Asiago 1.82m + AFOSC; NOT – Nordic Optical Telescope + ALFOSC; CA2.2 – Calar Alto 2.2m + CAFOS; TNG – Telescopio Nazionale Galileo + DOLORES; WHT – William Herschel Telescope + PFIP; CA3.5 – Calar Alto 3.5m + LAICA; BAO – Beijing Astronomical Observatory 60cm + CCD; MDK – Maidanak Observatory 1.5m + SITe CCD; CRM – 60-cm Crimean reflector + CCD.

0704.1245

Outflow and Infall in a Sample of Massive Star Forming Regions

Outflow and Infall in a Sample of Massive Star Forming Regions. P. D. Klaassen & C. D. Wilson Dept. of Physics and Astronomy, McMaster University, Hamilton, ON, Canada [email protected] ABSTRACT We present single pointing observations of SiO, HCO+ and H13CO+ from the James Clerk Maxwell Telescope towards 23 massive star forming regions previously known to contain molecular outflows and ultracompact HII regions. We detected SiO towards 14 sources and suggest that the non-detections in the other nine sources could be due to those outflows being older and without ongoing shocks to replenish the SiO. We serendipitously detected SO2 towards 17 sources in the same tuning as HCO+. We detected HCO+ towards all sources, and suggest that it is tracing infall in nine cases. For seven infall candidates, we estimate mass infall rates between 1×10−2 and 2×10−5 M⊙ yr−1. Seven sources show both SiO detections (young outflows) and HCO+ infall signatures. We also find that the abundance of H13CO+ tends to increase along with the abundance of SiO in sources for which we could determine abundances. We discuss these results with respect to current theories of massive star formation via accretion. From this survey, we suggest that perhaps both models of ionized accretion and halted accretion may be important in describing the evolution of a massive protostar (or protostars) beyond the formation of an HII region. Subject headings: Stars: Formation — ISM: Jets and Outflows — Accretion — HII regions — Submillimeter — Molecular Processes 1. Introduction The dynamics in massive star forming regions are, in general, much more complex than in regions which form only low mass stars. For instance, in the early stages within a low mass star forming region, the dynamics can be understood in terms of a few broad categories: large scale infall, which causes a disk to form, accretion through the disk, and outflow to release angular momentum (see for example, Di Francesco et al. 2001, André et al. 1993, Muzerolle et al. 2003). In intermediate and high mass star forming regions, http://arxiv.org/abs/0704.1245v1 – 2 – turbulence, stellar winds, multiple sites of star formation, and, for regions with massive star formation, the presence of HII regions, all contribute to the dynamics in these regions as well (i.e. Beuther et al. 2006, Shepherd & Churchwell 1996, McKee & Tan 2003, Krumholz et al. 2005). The more complicated source dynamics make the processes involved in the formation of the most massive stars much more difficult to understand than those involved in the formation of lower mass stars. Adding to the complexity, massive stars do not form as often as their lower mass counterparts and so we must look to larger distances before finding examples of high mass star formation. For instance, the average distance to the 63 sources in Shirley et al. (2003, hereafter S03) is 5.3 kpc. If we assume massive stars form through accretion, that this accretion occurs in the inner regions of disks (i.e. Pudritz & Norman 1986, or Shu et al. 1994), and that these disks have radii of a few thousand AU (Chini et al. 2004, Beltran et al. 2004, Cesaroni et al. 2005), we do not yet quite have the resolving power to detect accretion directly at the distances to massive star forming regions (1000 AU at 5.3 kpc is ∼ 0.2′′). Here, we define accretion as the infall motions from the disk onto the forming star, in contrast to the larger scale motions of envelope material falling onto the disk. However, while we cannot observe accretion directly, its presence can be inferred from the presence of accretion tracers such as larger scale infall and outflow. Infall can act to replenish disk material as mass accretes onto a protostar (Nakamura 2000), while molecular outflows serve as a release mechanism for the angular momentum which builds up during the accretion process (e.g Arce et al. 2006). These large scale motions are seen in star forming regions of all mass scales (see for instance Beuther & Shepherd 2005). In massive star forming regions, the accretion rates are orders of magnitude greater than in low mass star forming regions (i.e. Beuther et al. 2002), while the accreted masses are only approximately one order of magnitude greater. These accretion rates and masses result in accretion timescales that are much shorter than in low mass star forming regions, which allows the Kelvin-Helmholtz timescale to become important in the evolution of the protostar (e.g., there is no pre-main sequence stage for massive star formation). The outward radiation and thermal pressure from the forming star becomes strong enough that it can ionize the surrounding medium and a small, highly ionized HII region (either hypercompact (HCHII) or ultracompact (UCHII) region, Keto 2003) can form. It is still unclear whether the outward pressure needed to create the HII region is strong enough to halt accretion, or whether accretion can continue in some form (either through a molecular or ionized disk, or through an ionized accretion flow) after the formation of an HII region. Some models suggest that accretion must halt before the onset of a visible UCHII region (i.e. Garay & Lizano 1999, Yorke 2002), while other models suggest that an ionized accretion flow can continue through an HII region (i.e. Keto 2003, 2006). – 3 – There is now also observational evidence which suggests that, once the protostar be- comes hot enough to ionize its surroundings, both modes of massive star formation (halted and ionized accretion) are possible. G10.6-0.4 has been shown to have an ionized accretion flow by Sollins et al. (2005) and Keto & Wood (2006), while accretion in G5.89-0.39 seems to have halted at the onset of the UCHII region (Klaassen et al. 2006). Although sample statistics at this point are still quite small, these two examples pose interesting questions. We do not yet have enough data to determine whether the apparently conflicting models of halted and ionized accretion can both be correct. However, we can begin with a uniform survey of infall and outflow tracers in massive star forming regions in order to constrain massive star formation scenarios. In this paper, we present a survey of 23 massive star forming regions. Because we are interested in the relationship between accretion and outflow after the formation of an HII region, our source selection criteria include (1) the presence of an UCHII region, which indicates that there is a massive protostar forming, and (2) previous evidence of outflows, which suggests ongoing accretion in most formation scenarios. Sources were selected based on inclusion in the Wood & Churchwell (1989) and Kurtz et al. (1994) catalogs of UCHII regions as well as having molecular outflow signatures in the Plume et al. (1992) survey of massive star forming regions. Additional sources were taken from Hunter (1997) which were shown to have both UCHII regions and molecular outflows. We describe the observations collected for this survey in Section 2, we discuss the results of these observations in Section 3, and present our conclusions in Section 4. 2. Observations Observations of SiO (J=8-7), HCO+, and H13CO+ (J=4-3) were taken at the James Clerk Maxwell Telescope (JCMT)1 in 2005 (as parts of projects M05AC11 and M05BC04). SiO (347.330 GHz) and H13CO+ (346.999 GHz) were observed simultaneously in the same sideband by tuning the receiver to 347.165 GHz. Thirteen or twenty minute observations, depending on the source elevation, were taken towards each source with a velocity resolution of 1.08 km s−1, which resulted in rms noise levels of TMB <0.07 K. Separately, we observed HCO+ (356.370 GHz) with a velocity resolution of 0.53 km s−1 to an rms noise limit of TMB <0.13 K in twenty minute integrations. Both sets of observations were taken in position 1The James Clerk Maxwell Telescope is operated by The Joint Astronomy Centre on behalf of the Particle Physics and Astronomy Research Council of the United Kingdom, the Netherlands Organisation for Scientific Research, and the National Research Council of Canada. – 4 – switching mode with dual mixers and a sideband rejection filter in place. Table 1 shows the positions, rms noise limits of both tunings, the local standard of rest velocity, and distances to all sources in this survey. The half power beam width for these observations is 15′′, and the main beam efficiency is ηmb=0.62. Data were obtained using the DAS autocorrelator system and reduced using the SPECX software package. Linear baselines were removed from all spectra except for those towards G10.47. For this source, there were so many different chemical species in the observed spectrum that we were unable to fit a linear baseline over the entire ∼ 700 km s−1 bandwidth of the 347 GHz observations or the ∼ 450 km s−1 bandwidth of the 356 GHz observations. In this case, no baseline was removed. We also present 9 × 9 maps of one source (G45.07) in the same emission lines. These raster maps are sampled every 5” and have rms noise limits of 0.10 and 0.14 K (TMB) for the 347 and 356 GHz tunings, respectively. Note that these values are different than the ones reported in Table 1 for the single pointing observations. The DAS autocorrelator was configured with the same tunings as were described above for the single pointing observations, and the maps were centered at the same position. 3. Results This survey of single pointing observations towards 23 massive star forming regions is meant as an initial, uniform survey from which to base future observations. With these observations, we can only comment on the molecular gas component within our beams; we cannot discuss the larger scale molecular dynamics, or the ionized gas components of these regions. For our sources, the selection criteria of having an HII region confirms that these regions are forming massive stars. Figures 1 through 4 show the single pointing SiO and HCO+/H13CO+ spectra towards all sources; the spectra are ordered according to SiO integrated intensity. For each panel in these figures, line brightnesses have been corrected for the JCMT main beam efficiency and centered on the VLSR of the source (Table 1). Figures 1 and 2 show SiO and HCO +/H13CO+ spectra, respectively, towards the sources with no SiO detections. Figures 3 and 4 show the SiO and HCO+/H13CO+ spectra, respectively, towards sources with SiO detections. Peak line strengths and integrated intensities are given for all three lines in Table 2. SiO was only detected in 14 out of our 23 sources, where we define a detection as a minimum of 4 σ in integrated intensity. The rms noise limits in integrated intensity were calculated using ∆I = Trms∆v Nchan where Trms is the rms noise level in K, ∆v is the velocity – 5 – resolution of the observations, and Nchan is the number of channels over which the integrated intensity is calculated. HCO+ was detected in all sources and H13CO+ was detected in all but two sources. Along with HCO+, we serendipitously observed SO2 (J=104,6-103,7 at 356.755 GHz) in 17 of our sources. The HCO+ and SO2 lines are only separated by 17 km s −1 and thus the lines were blended in eight sources. For the sources with SO2 detections, we have also plotted (in gray) the lower intensity HCO+ observations in order to highlight the SO2 emission (Figures 2 and 4). Double peaked HCO+ line profiles were observed towards 10 sources, with nine of them having stronger blue peaks than red. This blue line asymmetry in an optically thick tracer such as HCO+ is often suggestive of infall (i.e Myers et al. 1996). We discuss the possibility of our observations tracing large scale infall further in Section 3.2. The distances to our sources, as taken from the literature, are shown in Table 1. The average distance is 5.7 ± 3.8 kpc, where the error quoted reflects the 1 σ dispersion in the distances. Since our observations were taken with a 15′′ beam, this resolution corresponds an average linear size of 0.4 pc for our observations. SiO is a well known outflow tracer, since in the general interstellar medium, Si is frozen out onto dust grains. When the gas in a region is shocked (i.e. the gas through which a protostellar outflow is passing) the dust grains can sublimate and Si is released into the gas phase. After the passage of a shock, the SiO abundance ([SiO]/[H2]) can jump to almost 10−6, whereas the dark cloud abundance of SiO is often closer to 10−12 (see for example Schilke et al. 1997, Caselli et al. 1997, or van Dishoeck & Blake, 1998). While SiO is easily identified as an outflow tracer, the emitting region for HCO+ is much less certain. Many authors suggest that HCO+ can be used to trace the envelope material surrounding a protostellar region (i.e. Hogerheijde et al. 1997, Rawlings et al. 2004), while others suggest that it traces disk material (i.e. Dutrey et al. 1997). One thing that is apparent, however, is that it becomes optically thick very quickly and readily self absorbs. We detected SO2 in 17 of our sources, suggesting that our beam contains at least some molecular gas at temperatures greater than 100 K (see for instance Doty et al. 2002, Charnley 1997). However, Fontani et al. (2002) have determined the average temperature in twelve massive star forming regions to be 44 K, using observations with beam sizes comparable to those presented here. For seven of our sources, which were observed in the Fontani et al (2002) sample, the average temperature is also 44 K. Thus, in the following analysis, we adopt an ambient temperature of 44 K for all sources. Table 3 shows the column densities for each region derived for both SiO and H13CO+. The column density was calculated assuming that each tracer is optically thin, in local – 6 – thermodynamic equilibrium, and at an ambient temperature of 44 K. For optically thin lines, the column density of the observed transition scales directly with the integrated intensity of the line (see for instance, Tielens 2005): 8kπν2 TMBdv (1) where Nu is the column density in the upper state of the transition, Auℓ is the Einstein A coefficient, ν is the frequency of the J=u-ℓ transition, and TMBdv is the integrated intensity of the line. The column density of this one state can then be related to the total column density of that molecule through the partition function. It is the total column density for the molecule (not the observed state) that is presented in Table 3. 3.1. Source Properties derived from SiO observations For each source we determined the column density, or upper limit to the column density, in SiO (Table 3) using the methods described above. These column densities can be compared to the column densities of other molecules (i.e. CS) for the same regions in order to determine the fractional abundance of SiO, if the abundance of the other molecule is known. We were able to obtain the CS or C34S column densities for fifteen of our sources from the literature. Column densities for fourteen sources were taken from Plume et al. (1997, hereafter P97), with the column density for one additional source taken from Wang et al. (1993). For those sources with C34S column densities instead of CS column densities, we assumed an abundance ratio of [CS]/[C34S] = 22 (Wilson & Rood, 1994) to determine a CS column density. The abundance of CS, relative to H2, was calculated by S03 for 13 of these sources, and we assume a CS abundance of 1.2 × 10−9 for the other two source for which the CS column density is known, since this was the average CS abundance as calculated by S03. We then compare the column density and abundance of CS to our observed SiO column density, or column density upper limit, to determine the abundance of SiO relative to H2 in our sources (Table 3). Despite our source selection criteria requiring previous evidence of outflows, we detected SiO towards only 14 of our 23 sources. This raises a number of questions, such as: is the observed SiO in fact tracing outflow if we do not detect it in all sources? Why do we not detect SiO in all sources? Is the signal being beam diluted at large distances? Has the Si evolved into other species? Below we first address whether the detected SiO can be used as an outflow tracer, and then discuss reasons for our non-detections of SiO in nine of our sources. – 7 – Si is liberated in shocks, and if these shocks are not due to the outflow, they must be due to the photo dissociation region (PDR) surrounding the UCHII region. Evidence for diffuse (not collimated) SiO can be seen in W75N (Shepherd, Kurtz & Testi, 2004) suggesting that the SiO may be due to the PDR and not an outflow. Models and observations of SiO in the PDRs around high mass star forming regions suggest moderate SiO enhancement, and that the SiO abundance is independent of the ambient radiation field (i.e. Schilke et al. 2001). Schilke et al. (2001) find SiO column densities of ∼ 1012 cm−2 in their observed PDRs. This is, admittedly, below our detection threshold; however, we detect average SiO column densities of ∼ 1014 cm−2. This suggests a possibly higher SiO abundance than found in PDRs. The enhanced SiO column density alone is not enough to discount the origin of the SiO in our sources as the PDR and so we can consider how our SiO abundance varies with the ambient radiation field. For the gas near an HII region, we can approximate the strength of the ambient radiation field using the Far Infrared (FIR) luminosity of the region. For twelve of our sources with SiO detections and abundance calculations, we obtained the FIR luminosity from either Wood & Churchwell (1989), Kurtz et al. (1994) or Evans et al. (1981). We then scaled their values for the different source distances used in this study (see Table 3). Comparing the SiO abundance to the source luminosity (see Figure 5), we find that the SiO abundance increases with source luminosity. There is a 10% chance that this relationship could arise from uncorrelated data. Thus, it is possible that our result is contrary to the findings of Schilke et al (2001), and we suggest that the SiO we observe does come primarily from outflow shocks. We can also compare our detection rate of SiO to that of the SiO survey towards maser sources of Harju et al (1998) who observed SiO (J=2-1) and SiO (J=3-2). For comparison to our results, we only consider the sources in Harju et al. which are listed as UCHII regions. Our detection rate is 61%, compared to their rate of 29%. While our detection threshold is slightly lower (our observations have rms noise levels generally below 0.05 K at 347 GHz, while their rms noise levels are generally below 0.08 K), we suggest that the different detection rates are due to differences in the source selection criteria. Although both samples contain UCHII regions, our sample contains sources with previous observations of outflows, while Harju et al. have selected sources based on previous observations of masers. For the 12 sources which overlap between the two studies, we detect SiO towards 9 sources, while they detect SiO towards 10. They detected SiO (J=2-1) in G31.41 while we did not detect it in SiO (J=8-7). Based on this comparison to SiO observations of UCHII regions not selected by outflows, which have a lower SiO detection rate, and that our SiO abundance increases with source – 8 – luminosity, we suggest that the SiO, in the 14 sources in which it is detected, is being generated in the outflow. Previous, high resolution observations of SiO also suggest that SiO can be enhanced in the outflows from high mass stars (i.e. Beuther et al. 2004, Beuther, Schilke & Gueth 2004) just as it is in the outflows from low mass stars. The nine non-detections in our sample could be caused by beam dilution if these sources are on average further away. However, if we compare the distances for sources with and without SiO detections, we find average distances of 6.3 ± 4.4 and 4.4 ± 2.5 kpc, respectively. Thus, the non-detections cannot be attributed to larger average source distances and so beam dilution can play only a minimal role in the non-detections of SiO. If these SiO non-detections are not due to distance effects, there must be some local phenomenon which can explain why SiO is not being detected in regions known to contain protostellar outflows. It is possible that the Si is evolving into different species and the SiO abundance is dropping back down to dark cloud values. Pineau des Forêts et al. (1997) suggest that a few×104 yr after the Si is liberated from dust grains and forms SiO, it can either freeze out back onto dust grains or oxidize and form SiO2. Thus, the lack of SiO may be due to silicon moving into other species if it was liberated more than 104 yr ago. This interpretation implies that we did not detect SiO in some of our sources because the outflow generating mechanism shut off more than 104 years ago, and the outflow observed in HCO+ (or in other molecules by other authors) is a remnant of previous accretion. The kinematic ages of nine of our sources are listed in the Wu et al (2004) catalog of high velocity outflows. Of these nine sources, five were also included in the P97 and S03 studies. This results in five sources for which we have both the kinematic age of the outflow, and the abundance of SiO. The relationship between outflow age and SiO abundance is shown in Figure 6 along with the model predictions of Pineau des Forêts et al. (1997). With only five points, it is difficult to draw conclusions about the relationship between outflow age and SiO abundance, especially given the uncertain beam filling factor. At higher resolution, these points would likely move upwards to higher abundances. A larger beam filling factor would move the points for the two young outflows (G5.89 and Cep A) towards the model predictions. As for the other three outflows, this would move them further from the model predictions. We suggest that the outflow generating mechanism is continuing to shock these regions, replenishing the SiO. The oldest of these sources (G192.58) shows an infall signature in HCO+, which also suggests that the outflow is still being powered. – 9 – 3.2. Source properties derived from HCO+ observations Given the large average distance to our sources (5.7 kpc), our 15′′ beam subtends an average linear distance of 0.4 pc. Thus, it is quite likely that the HCO+ emitting region does not entirely fill our single JCMT beam. In general we can determine the beam filling factor, f , for each source using TL = fsourceTs(1− e−τ ) (2) where TL is the line brightness temperature measured at the telescope (corrected for telescope efficiencies) and Ts is an approximation to the ambient temperature (44 K) which is valid at densities greater than ∼ 103 cm−3 (Rholfs & Wilson, 1994). In a number of our sources, the HCO+ is asymmetric, and thus cannot be consistently used to determine a beam filling factor. Instead, we can use the optically thin H13CO+ line, and we can simplify the above equation to f = TL/(Tsτ). These values are shown in Table 3. HCO+ becomes optically thick quite quickly due to its relatively high abundance with respect to H2 ([HCO +]/[H2] ∼ 10−8), and as such, can be used to roughly trace outflow and to trace infall (i.e. Myers et al. 1996) if the line profile shows a double peak. We determined the optical depth of HCO+ towards each source using Equation 1 of Choi et al. (1993), and found that in all but one case (G139.9), it is optically thick. In all cases, the optical depth of HCO+ is less than 77 (the abundance ratio between HCO+ and H13CO+, Wilson & Rood, 1994), resulting in optically thin H13CO+ towards all sources. Because HCO+ is optically thick in its line center, the line wings can be used to detect outflows, and so, if there is an outflow, it should be detectable in HCO+ even if it goes undetected in SiO. Gaussian profiles were fit to our HCO+ spectra (either single or double Gaussians, depending on the observed line shape), and the fits were subsequently subtracted from the spectra to leave only the residual outflowing gas. When using two Gaussians to fit the self absorbed spectra, we employed a method similar to the single Gaussian fitting of Purcell et al. (2006) because we used the sides of the detected lines to fit our profiles (see their Figure 3). Comparing the two Gaussian fits to single Gaussian fits showed no significant differences in distinguishing line wing intensity. Because of the possibility of contamination from SO2 emission in the blue shifted outflow wing (at -17 km s−1), the peak brightness of the residual emission was determined using only the red shifted wing emission. In all cases (except for G10.47 for which we could not find a linear baseline), we found a minimum of a 5σ peak brightness temperature in the residual line wing emission, with 19 of our sources having a minimum of a 10σ peak. This result suggests that we can detect outflow motions in all sources using our detections of HCO+, despite not detecting SiO towards every source. – 10 – Our observations show that for ten of our sources, the spectral line profile of the HCO+ emission has a double peak. This profile could either be due to self absorption of the optically thick HCO+ line or from multiple velocity components within our 15′′ beam. To break this degeneracy, we observed the optically thin H13CO+. If the H13CO+ line has a single peak at the same velocity as the HCO+ absorption feature, then it is likely that the HCO+ line is self absorbed. If, however, the H13CO+ also has two peaks, and they are at approximately the same velocities as the two HCO+ peaks, it would suggest that there are multiple components within the beam. Of our ten sources with double HCO+ peaks, only one shows a double peak profile in H13CO+ (G20.08). This results in nine sources with double peaked optically thick HCO+. In addition to the nine optically thick sources, similar line asymmetries appear in a number of other sources. However, in these sources, there is no clear emission gap producing a double peak profile, only an emission shoulder (i.e. De Vries & Myers 2005). If we take G75.78 as an example, the HCO+ line peak is red shifted from the rest velocity of the source, with a blue shifted emission shoulder. There are a number of different kinds of source dynamics that can lead to the double peaked line profiles seen in our spectra, such as infall, outflow and even rotation. However, infall is the only one of these processes which would produce line asymmetries which are consistently blue (i.e. the blue peak is higher than the red peak or shoulder). If these profiles were due to outflow or rotation, there would be no statistical reason to have more sources with higher blue peaks than red peaks. Many previous studies have investigated the statistical significance of using this type of optically thick blue line asymmetry to trace infall as opposed to other dynamical motions (i.e. Mardones et al. 1997 and Gregersen et al. 1997 for low mass star forming regions, and Fuller et al. 2005 for high mass star forming regions). Of the 10 sources in our survey which have double peaked HCO+ profiles, we suggest eight may be indicative of infall. The other two sources are G20.08 and G45.47. G20.08 has already been shown to have multiple components in the beam from the double peaked H13CO+ profile, and G45.47 has a brighter red peak than blue. There are two additional sources (G19.61, and G240.3) in which HCO+ has a strong red shifted shoulder, which we suggest may also be tracing infall. This analysis gives a total of ten infall candidates in our sample of 23 sources. A recent survey of HCO+ (J=1-0) towards sources with methanol masers shows an even distribution of sources with blue and red line asymmetries, and a higher percentage of self absorbed lines than in our study (Purcell et al. 2006). Of the six sources which overlap between our survey and that of Purcell et al, all six are self absorbed in HCO+ (J=1-0). Five of them have blue line asymmetries consistent with infall, while only one (G31.41) has – 11 – its red peak brighter than its blue peak. We only find self-absorption in HCO+ (J=4-3) for three of these six sources. In two of the sources for which we do not see a clear self-absorption feature, we do see evidence for a red shifted shoulder which may be showing unresolved infall. The sources in Purcell et al. (2006) have an even distribution of red and blue line asymmetries, while we have a clear bias towards detecting blue line asymmetries. This comparison could suggest that the higher energy J=4-3 transition of HCO+ is a better tracer of infalling gas because it does not self absorb as readily as the J=1-0 transition. For each of the 8 sources with blue, double peaked HCO+ profiles, we can determine an infall velocity (vin) using the two layer radiative transfer model of Myers et al. (1996). Using their equation 9, we find infall velocities for all eight double peaked infall sources (Table 4). The mass infall rate can then be determined using: ρV vin πnH2µmHr gmvin (3) where µ is the mean molecular weight (µ = 2.35), the geometric mean radius (rgm) is the unresolved circular radius of the HCO+ emitting region derived from the beam radius and the beam filling factor (rgm = frbeam), and nH2 is the ambient source density. For seven of the sources in Table 4, the ambient density was determined by either P97, Hofner et al. (2000), or Wang et al. (1993). We could not find the ambient density for the eighth source (G192.58). From this analysis, we determine mass infall rates ranging from 1×10−2 to 2×10−5 M⊙ yr−1. These values are slightly higher than those generally observed for low mass star forming regions, but are consistent with the accretion rates derived for high mass star forming regions by McKee & Tan (2003). Since outflow rates are orders of magnitude higher in high mass star forming regions (i.e. Beuther et al. 2002) it is not unreasonable to suggest that infall rates are also much higher in these regions. The mass outflow rate for only one of these sources (Cep A) can be determined from the Wu et al. (2004) survey of high velocity outflows by dividing the mass in the outflow by the kinematic age of the outflow. We find the ratio of the mass outflow rate to the mass infall rate to be Ṁout/Ṁin ≈ 16. This value is only slightly higher than values seen in other high mass star forming regions (i.e. Behrend & Maeder 2001). Also, models suggest that a mass equivalent to 20-30% of the mass accreted onto a protostar is ejected as a wind (i.e. Pelletier & Pudritz 1992, Shu et al. 1994), and that this wind entrains 5-20 times its mass in the outflow (Matzner & McKee 1999). – 12 – 3.3. Source properties derived from mapping G45.07 At a distance of 9.7 kpc, G45.07 is one of our furthest sources. This source was known from previous observations to have multiple continuum sources (De Buizer et al. 2003, 2005). The three continuum sources were observed in the mid-Infrared (MIR) and all three fall within 6′′ of our map center. A fairly young outflow has also been mapped at high resolution (< 3′′ synthesized beam) in CO and CS towards this region (Hunter et al. 1997). They observed a bipolar outflow with a position angle of -30◦ (east of north), as well as a red shifted absorption feature in their CS observations which they take to be indicative of infall. Due to the large distance to this source, we should be able to detect all of the emission associated with this source in a fairly small map. In the left panel of Figure 7 we present a map of the SiO (contours) and H13CO+ (halftone) emission in this region. The right panel of Figure 7 shows the HCO+ emission. In both figures, the 5σ (2.1 K km s−1) H13CO+ emission contour is plotted as a dashed line to help guide the eye. The first contour for SiO in the left panel and only contour of HCO+ in the right panel are also 5σ (2.2 and 3.5 K km s−1 respectively). The differences in the 5σ contour levels for each tracer come from the different single channel rms levels between the two tunings, and the width of each line as given in Table 2. Also plotted in both figures are the three MIR continuum sources observed by De Buizer et al (2003, 2005). If we did not have the added information provided by this map, there would be two main conclusions we could draw from our single pointing observations towards this source. The first is that the enhanced blue emission in all three tracers suggests that our pointing is observing more of the blue shifted outflow lobe than the red. Second, since we only have one peak in the spectrum of each tracer, there is only one source and we cannot classify it as infalling. The left panel of Figure 7 shows contours of SiO emission superimposed on the H13CO+ halftone. With beam spacings of 5′′, these maps are oversampled; however we note that much of the structure in the SiO emission is on scales comparable to the size of the JCMT beam. For instance, the structure at ∆α = −10′′, ∆δ = 0′′ is offset from the map center by more than the radius of our beam and could be independent from the emission at the map center. There is also SiO emission at ∆α = 5′′, ∆δ = 15′′, which is more than a full beam away from the map center, and suggests that the SiO emission is more extended that the primary beam of our observations. In fact, it appears as though there is a second SiO emission peak towards the upper left of the left panel of Figure 7. Interestingly, there does not appear to be as much H13CO+ emission at this northern position. This comparison shows that the SiO and H13CO+ lines are tracing different gas populations in this region. The excess SiO – 13 – emission is offset from the map center in the same direction as the CO emission shown in Hunter et al. (1997) at much higher resolution. The right panel of Figure 7 shows the HCO+ emission for this region. It appears that the HCO+ emission extends much further than the SiO emission, suggesting it is tracing the larger scale envelope material. The line through the middle of this plot indicates the cut taken for the position-velocity (PV) diagram along the outflow axis as described by Hunter et al. (1997) The two panels of Figure 8 show the PV diagrams for SiO and HCO+ in our maps both perpendicular and parallel to the outflow axis defined by Hunter et al. (1997). Our single pointing HCO+ spectrum (Figure 4) suggests we are observing more blue shifted outflow emission than red shifted emission; however, from our PV diagrams, we see that there is excess blue emission at all positions in our map. This excess blue emission cannot be due to outflow alone; instead, it could be due to an inherent velocity shift between the three continuum sources in our beam. We can, in fact, fit three Gaussian components to most of our HCO+ spectra. These Gaussians peak at velocities of 60, 52 and 44 km s−1, with the peak temperature for each component decreasing with velocity. The third component (at 44 km s−1) could not be fit at all positions because it was intrinsically weaker than the other two peaks, and was lost in the noise towards the edges of the map. It appears as though this third component might be contamination from SO2, which should occur at an apparent velocity of 41 km s−1 (or -17 km s−1 in Figure 4). Perpendicular to the outflow axis, the mean velocity of the HCO+ line peak appears to shift from ∼ 58 km s−1 at an offset of +15′′ from the source center to ∼ 61 km s−1 at an offset of −15′′ from the source (Figure 8). Given our velocity resolution (1.08 km s−1) and spatial resolution (15′′), is unclear whether this velocity shift is real. If it is, it could indicate large scale (∼ 1.4 pc) rotation within the core, on a much larger scale than would be expected for a rotating accretion disk. 3.4. Correlations between Datasets Previously, we discussed the reasons why we do not detect SiO towards a number of our sources, and have calculated the mass infall rates for the sources with double peaks in their HCO+ emission, but we have not yet discussed the correlations between the two species. In Figure 9 we plot the logarithm of the abundance of H13CO+ against the same quantity for SiO (open circles), as well as the column densities of both species (filled circles). The abundance of H13CO+ was calculated in the same manner as the abundance of SiO – 14 – described above (using the CS column density from P97 and the CS abundance from S03). The probability of obtaining these correlations if the data are, in fact, uncorrelated is 6×10−3 for the abundances, and 2×10−4 for the column densities. As stated earlier, SiO is a well known shock tracer, and as such, an increased abundance of SiO would suggest more shocked material within our 15′′ beam. The (generally) infall tracing HCO+ has been shown by some authors to be destroyed in strong shocks (i.e. Bergin et al. 1998, Jørgensen et al. 2004). However, Wolfire & Königl (1993) suggest that HCO+ can be enhanced in regions with high energy shocks, where electron abundances are much greater. This enhancement in the electron abundance increases the formation rate of ions, and we suggest that this is responsible for the H13CO+ abundance enhancement in our sources. This correlation between the abundances of H13CO+ and SiO suggests that HCO+ and H13CO+ are not only tracing infalling gas, but also the outflowing gas as well. This conclusion is supported by the strong, and broad, line wing emission detected in HCO+ (See Section 3.2). HCO+ over abundances have been seen in high mass star forming regions not included in this study like NGC 2071 (Girart et al. 1999) and Orion IRc2 (Vogel et al. 1984). In these two papers, the over abundances of H13CO+ are with respect to ambient cloud tracers such as CO and H2, rather than the high density or shock tracers like the CS and SiO with which we are comparing our H13CO+ abundances. However, Viti & Williams (1999a,b) show that HCO+ is indeed over abundant with respect to CS in the gas surrounding HH objects, and Jiménez-Serra et al. (2006) also show that the abundance of H13CO+ can be enhanced with respect to SiO by up to a factor of ten in the same regions ahead of HH objects. 4. Discussion and Conclusions Without maps of each region, it is impossible to tell how much of the HCO+ emission in the line center and in the line wings is due directly to infall and outflow motions; however, based on the arguments we have presented above, we suggest that ten sources show infall motions, and all 23 source show outflow motions based on the HCO+ line profiles. We have found evidence for recent outflow activity (SiO emission) in 14 out of our 23 sources. Seven of these outflow and infall sources overlap. M17S, G192.6 and G240 appear to show only infall signatures and no SiO outflow signatures. They do, however, appear to have HCO+ outflow signatures of a minimum of 8σ. Detection of line wing emission in HCO+ and the relationship between H13CO+ and SiO abundances described in the previous section suggest that while SiO is tracing outflow – 15 – in most sources and HCO+ is tracing infall in some sources, HCO+ is also observable in the outflowing gas for all regions. We find that the non-detection of SiO in nine of our sources is not due to beam dilution or larger average distances to the source, but possibly to older outflows for which the Si has likely either frozen back onto dust grains or evolved into SiO2. For these sources, it appears that the accretion may have ceased, and the observed outflowing gas is a remnant of previous accretion. We have found seven sources with SiO outflow signatures but no infall signatures in HCO+. This result could be due to a number of factors such as beam dilution of the infalling gas which masks the spectral line profile we would expect for large scale infall. It is possible that, as the outflow ages and widens, it may impinge on the region in which we could detect infalling gas. For outflow cones oriented along the line of sight, the younger, narrower outflows would have infalling gas with large line of sight velocities and be likely to produce an observable infall signature. However, for the older outflows which have widened, the largest infall velocities will be in the plane of the sky, and unobservable at the resolution of the JCMT. However, this effect would not be as pronounced for outflows in the plane of the sky (such as G5.89, for which we do not see an infall signature). It is difficult to assess the importance of this effect without detailed information on the outflow orientation in each source. Thus, we suggest that some of these sources may have finished accreting, and what we observe are remnant outflows from a previous phase of accretion. This scenario was suggested by Klaassen et al. (2006) to explain the large scale outflow in G5.89, and this source is one of these seven sources with an SiO outflow and no apparent infall signature. The seven sources which show recent outflow activity (those with SiO emission) and which appear to be undergoing infall are suggestive of ongoing accretion beyond the onset of the HII region. 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S., & Churchwell, E. 1989, ApJS, 69, 831 Wu, Y., Wei, Y., Zhao, M., Shi, Y., Yu, W., Qin, S., & Huang, M. 2004, A&A, 426, 503 This preprint was prepared with the AAS LATEX macros v5.2. – 20 – Table 1. Observed Sample of Massive Star Forming Regions. Name Position (J2000) RMS noise limit (K) VLSR Distance P92 Name RA DEC 347 GHz 356 GHz (km s−1) (kpc) ref G5.89 18 00 30.3 -24 03 58 0.060 0.111 9 2 1 W28A2 (1) G5.97 18 03 40.4 -24 22 44 0.044 0.069 10 2.7 5 · · · G8.67 18 06 19.0 -21 37 32 0.068 0.074 36 8.5 2 8.67-0.36 G10.47 18 08 38.4 -19 51 52 · · · b 0.100 67 12 2 W31 (1) G12.21 18 12 39.7 -18 24 21 0.042 0.076 24 16.3 2 12.21-0.1 M17S 18 20 24.8 -16 11 35 0.044 0.077 20 2.3 5 M17 (2) G19.61 18 27 38.1 -11 56 40 0.048 0.065 43 4.5 1 19.61-0.23 G20.08 18 28 10.4 -11 28 49 0.044 0.068 42 4.1 1 20.08-0.13 G29.96 18 46 03.9 -02 39 22 0.044 0.079 98 9 1 W43S G31.41 18 47 33.0 -01 12 36 0.044 0.061 97 8.5 1 31.41+0.31 G34.26 18 53 18.5 01 14 58 0.047 0.131 58 3.7 1 W44 G45.07c 19 13 22.1 10 50 53 0.044 0.073 59 9.7 1 45.07+0.13 G45.47 19 14 25.6 11 09 26 0.037 0.063 58 8.3 6 · · · G61.48 19 46 49.2 25 12 48 0.037 0.048 12 2 1 S88 B K3-50A 20 01 45.6 33 32 42 0.035 0.077 -24 8.6 3 K3-50 G75.78 20 21 44.1 37 26 40 0.038 0.068 0 5.6 1 ON 2N Cep A 22 56 17.9 62 01 49 0.031 0.076 -10 0.7 1 CEP A W3(OH) 02 27 03.8 61 52 25 0.026 0.079 -48 2.4 2 W3 (OH) G138.3 03 01 29.2 60 29 12 0.066 0.073 -38 3.8 1 S201 G139.9 03 07 23.9 58 30 53 0.058 0.071 -39 4.2 1 · · · G192.58 06 12 53.6 17 59 27 0.037 0.053 9 2.5 3 S255/7 G192.6 06 12 53.6 18 00 26 0.074 0.098 9 2.5 3 S255/7 G240.3 07 44 51.9 -24 07 40 0.035 0.048 68 6.4 4 · · · aName given to source in Plume et al. (1992). When included in P97 and S03, these were the source names used. bToo much chemistry in spectrum to determine a reliable rms noise level. – 21 – cThe rms noise limits for the two maps of this source are 0.1 K and 0.14 K for the 347 GHz and 356 GHz pointings respectively. Note. — The source velocity here is the reference velocity of each source, which has been removed from each spectrum. RMS noise levels are given in TMB. References:(1) Hanson et al. 2002, (2) S03, (3) Kurtz et al. 1994, (4) Kumar et al. 2003., (5) Churchwell et al. 1990, and (6) Hofner et al. 2000 – 22 – Table 2. Integrated Intensities for SiO, HCO+ and H13CO+. Name SiO Properties HCO+ Properties H13CO+ Properties TMBdv dv TMB TMBdv dv TMB TMBdv dv G5.89 2.3 75.3± 0.7 120 39.2 690.0±0.8 105 8.9 65.2± 0.4 35 G5.97 <0.1 <0.3± 0.1 8 12.4 50.5 ±0.2 15 0.6 1.6 ± 0.1 8 G8.67 0.2 1.6 ± 0.3 13 13.8 80.0 ±0.3 38 5.6 20.9± 0.2 10 G10.47 1.0 9.9 ± · · · 22 7.7 79.3 ±0.4 35 1.2 9.2 ± · · · 15 G12.21 0.2 2.5 ± 0.2 25 10.0 93.8 ±0.3 30 0.8 6.5 ± 0.2 15 M17S <0.1 <0.7± 0.2 15 16.2 79.0 ±0.2 15 2.9 9.4 ± 0.2 11 G19.61 0.8 15.9± 0.3 45 10.8 162.2±0.4 55 1.8 16.1± 0.2 20 G20.08 0.4 6.9 ± 0.3 40 6.5 75.0 ±0.3 40 1.1 9.7 ± 0.2 25 G29.96 0.5 7.9 ± 0.3 40 23.6 160.4±0.3 30 3.8 13.2± 0.1 10 G31.41 <0.1 <0.7± 0.2 15 1.6 6.8 ±0.2 13 <0.1 <0.3± 0.1 3 G34.26 1.3 26.0± 0.4 55 27.8 228.2±0.6 35 4.3 33.2± 0.2 25 G45.07 0.7 10.3± 0.3 37 15.0 169.4±0.4 45 1.5 11.2± 0.3 33 G45.47 <0.1 <0.3± 0.1 12 10.1 59.4 ±0.2 20 1.3 6.1 ± 0.1 12 G61.48 <0.1 <0.3± 0.1 8 10.8 58.5 ±0.1 18 1.5 4.7 ± 0.1 8 K3-50A 0.3 2.1 ± 0.1 16 18.6 154.8±0.3 25 1.8 10.6± 0.2 18 G75.78 0.3 2.2 ± 0.1 13 12.5 136.8±0.3 45 2.5 10.3± 0.2 17 Cep A 0.5 7.3 ± 0.2 45 23.2 234.7±0.4 60 4.5 23.4± 0.2 28 W3(OH) 1.4 14.5± 0.2 50 18.5 148.2±0.3 30 2.4 11.6± 0.1 15 G138.3 <0.2 <0.3± 0.1 2 5.0 10.6 ±0.2 8 0.3 0.5 ± 0.1 4 G139.9 <0.2 <0.3± 0.1 11 8.6 18.8 ±0.1 6 <0.2 <0.3± 0.1 2 G192.58 0.3 1.3 ± 0.1 8 14.5 84.2 ±0.2 25 1.1 5.5 ± 0.1 15 G192.6 <0.2 <0.6± 0.2 30 12.0 54.1 ±0.3 15 0.8 2.2 ± 0.2 8 G240.3 <0.1 <0.4± 0.2 10 7.3 59.1 ±0.2 30 0.4 2.5 ± 0.1 14 Note. — For the sources in which we did not detect SiO or H13CO+, 3σ upper limits on the brightness temperature and integrated intensities are given. – 23 – Table 3. Observed and Derived Source Parameters. Name Detection f a Column density (×1012) [SiO]/[H2]b LFIR toutflowc SO2 Infall H 13CO+ SiO Log(L⊙) (10 4 yr) G5.89 Y N 0.79 39.7 421.0 -8.90 5.25d 0.2 G5.97 N N 0.28 1.0 <1.7 · · · 5.23d · · · G8.67 Y Y 0.24 12.7 9.0 · · · 5.70d · · · G10.47 Y Y 0.16 5.6 55.4 -9.53 6.26d · · · G12.21 Y N 0.22 4.0 14.0 -10.15 6.17d · · · M17S N Y 0.33 5.7 <3.9 <-10.73 5.72d · · · G19.61 Y Y 0.22 9.8 89.0 -9.72 5.42d · · · G20.08 Y N 0.13 5.9 38.6 -10.48 4.86d · · · G29.96 Y N 0.49 8.0 44.2 -9.12 6.30d · · · G31.41 N N · · · <0.2 <3.9 <-11.89 5.45d · · · G34.26 Y Y 0.58 20.2 145.0 -10.43 5.77d · · · G45.07 Y N 0.32 6.8 57.6 -9.54 6.15d 4 G45.47 Y N 0.21 3.7 <1.7 · · · 6.04d · · · G61.48 N N 0.23 2.9 <1.7 <-10.80 5.01d 7 K3-50A Y N 0.40 6.5 11.7 -11.60 6.35e · · · G75.78 Y N 0.25 6.3 12.3 · · · 5.65d 3.7 Cep A Y Y 0.47 14.2 40.8 -9.78 4.40f 0.2 W3(OH) Y Y 0.39 7.1 81.1 -11.16 5.12e · · · G138.3 N N 0.11 0.3 <1.7 · · · 4.57e 17 G139.9 N N · · · <0.2 <1.7 · · · 4.82e 6 G192.58 Y Y 0.32 3.4 7.3 -10.94 4.79e 50 G192.6 Y Y 0.26 1.3 <3.4 · · · · · · · · · G240.3 Y Y 0.16 1.5 <2.2 · · · · · · 2.3 aHCO+ beam filling factor. bSiO abundance relative to H2 for sources with CS observations (P97) and abundance calculations (S03). cKinematic age of outflow from Wu et al. (2004). – 24 – dFar Infrared Luminosities modified from Wood & Churchwell (1989). eFar Infrared Luminosities modified from Kurtz et al. (1994). fFar Infrared Luminosities taken from Evans et al. (1981). – 25 – Table 4. Infall Velocities and Mass Infall Rates. Name Vin a nH2 b Ṁin (km s−1) (105 cm−1) (10−4 M⊙ yr G8.67 0.4 ± 0.1 1.8 4 ± 2 G10.47 1.8 ± 0.3 7.2 100± 80 M17S 1.4 ± 0.5 5.0 4 ± 2 Cep A 0.23±0.07 10.0 0.17± 0.07 W3(OH) 0.06±0.02 60.0 3 ± 1 G192.58 0.8 ± 0.3 · · · · · · G192.6 0.9 ± 0.4 4.0 2 ± 1 G34.26 1.5 ± 0.3 3.6 14 ± 4 aInfall velocities for sources with double peaked HCO+ profiles. bAmbient densities taken from P97 except for: G34.26 (Hofner et al. 2000), M17S (Wang et al. 1993). cMass infall rates. – 26 – Fig. 1.— Nine sources in which SiO was not detected (to 4 σ limits). The name of each source is given in the top left hand corner of each panel. For each source, the source rest velocity is plotted as ∆v = 0 km s−1. – 27 – Fig. 2.— HCO+ and H13CO+ sources with no SiO detections (the same sources as in Figure 1). Solid lines show HCO+ emission, while dashed lines show H13CO+ emission scaled up by a factor of four. The temperature scale on the left hand side of the panels is the scale used for the HCO+ spectra, and the temperature scale on the right hand side is that used for the H13CO+ spectra. For sources with SO2 detections, we have re-plotted the HCO + spectra on a larger intensity scale ( from -1 K to 3 K, in gray) to show the low lying SO2 emission. The SO2 emission, when present, is centered at -17 km s −1, and is indicated by an arrow. G31.41 is slightly offset to emphasize that the temperature scale has been magnified to show the emission feature. – 28 – Fig. 3.— SiO spectra for 14 source in which SiO was detected (above 4 σ limits). For this figure only, the range of temperatures plotted increases by row. The plotted temperature ranges for each row increase by 0.5 K per row. The temperature scale in the top row extends from 0.2 to 1 K (in the TMB scale), while the temperature scale in the bottom row extends from 0.2 to 2.5 K. Note that the velocity scale for G5.89 is much wider (a 78 km s−1 window as opposed to a 38 km s−1 window for the other sources) in order to show the full width of the emission line, but that the temperature scale is the same as that for W3(OH). – 29 – Fig. 4.— HCO+ and H13CO+ emission from sources with SiO detections (the same sources as in Figure 3). The solid, dashed and gray lines are the same as shown in Figure 2, as is the placement of the temperature scales. As in Figure 3, G5.89 has been plotted separately from the rest of the sources to stress that the velocity scale is larger for this source. – 30 – Fig. 5.— SiO abundance plotted as a function of source luminosity. The line of best fit shown is given by (Log([SiO]/[H2])=0.30±0.06Log(L/L⊙)-11.7±0.3), and was calculated only for sources detected in SiO. This shows that SiO abundance increases with source luminosity. This is contrary to what is expected for SiO produced in PDRs, suggesting the observed SiO is produced in outflows. – 31 – Fig. 6.— SiO abundance plotted as a function of outflow age. The outflow ages were taken from Wu et al. (2004). The solid line shows the model predictions of Pineau des Foréts et al. (1997) for the SiO abundance as a function of age (assuming Si returns to the dust grains), and the dashed line represents the canonical dark cloud abundance of SiO. The source shown with a downwards arrow represents G61.48, a source in which we did not detect SiO, and the value given is an upper limit to the SiO abundance. The error bars represent 30% calibration uncertainty between our observations and those of P97 and S03 – 32 – Fig. 7.— Left: SiO and H13CO+ towards G45.07. The halftone scale represents the in- tegrated H13CO+ emission, while the contours show the integrated intensity of SiO. The velocity range used to determine the integrated intensities is the same as the single pointing velocity range (dv), however the rms noise limits are those listed at the end of Section 2. The first SiO contour is 5σ (3.2 K km s−1), incrementing in steps of 2σ, with the same scale continuing into the white contours near the center. Right: HCO+ integrated intensity towards G45.07. The solid contour represents the 5σ emission for the HCO+ emission (3.5 K km s−1), and the solid line represents the cut used for the PV diagram along the outflow axis (PA = -30◦) presented in Figure 8. For both panels, the dashed line represents the 5σ H13CO+ emission contour (3.0 K km s−1). The three triangles represent the Mid IR sources detected by De Buizer et al. (2005), with the two points in the center corresponding to 230 GHz continuum sources detected at the Submillimeter Array (Klaassen et al. in prep). – 33 – Fig. 8.— Position-Velocity (PV) diagrams for SiO (halftone scale) and HCO+ emission (contours) in G45.07 both perpendicular to (top panel) and along the outflow axis (bottom panel) as defined in Hunter et al. (1997). These PV diagrams were taken at positions angles of 60◦ and -30◦ east of north respectively, with the cut for the bottom panel of this figure shown in the right panel of Figure 7. The first contour for HCO+ is 5σ, or 0.7 K since the rms noise limit for this map is 0.14 K, and the contours increase in increments of 5σ. The three dashed lines in the bottom panel show the peak velocities of the Gaussian fits to the HCO+ spectra. – 34 – Fig. 9.— The abundance of H13CO+ with respect to the abundance of SiO (open circles, dashed line of best fit) appears to increase faster than the respective column densities of these two species (filled circles, solid line of best fit). The equations of the two lines of best fit are: Log[X(H13CO+)]=(0.84±0.09)Log[X(SiO)]-(2.4±0.9), and Log[N(H13CO+)]=(0.5±0.1)Log[N(SiO)]+(5±1). This relationship could be due to HCO+ being enhanced (similarly to SiO), as discussed in the text. Introduction Observations Results Source Properties derived from SiO observations Source properties derived from HCO+ observations Source properties derived from mapping G45.07 Correlations between Datasets Discussion and Conclusions

0704.1246

Invariants of Welded Virtual Knots Via Crossed Module Invariants of Knotted Surfaces

Invariants of Welded Virtual Knots Via Crossed Module Invariants of Knotted Surfaces Louis H. Kauffman Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan St., Chicago, IL 60607-7045, USA [email protected] João Faria Martins∗ Departamento de Matemática, Instituto Superior Técnico (Universidade Técnica de Lisboa) Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected] September 8, 2021 Abstract We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non-trivial by calculating explicit examples. We define welded virtual graphs and consider invariants of them defined in a similar way. 2000 Mathematics Subject Classification: 57M25 (primary), 57Q45 (secondary). Keywords: welded virtual knots, knotted surfaces, crossed module, quandle invariants, Alexander module. 1 Introduction Welded virtual knots were defined in [K1], by allowing one extra move in addition to the moves appearing in the definition of a virtual knot. This extra move preserves the (combinatorial) fun- damental group of the complement, which is therefore an invariant of welded virtual knots (the knot group). Given a finite group G, one can therefore define a welded virtual knot invariant HG, by considering the number of morphisms from the fundamental group of the complement into G. The Wirtinger presentation of knot groups enables a quandle type calculation of this “Counting Invariant” HG. Not a lot of welded virtual knot invariants are known. The aim of this article is to introduce a new one, the “Crossed Module Invariant” HG , which depends on a finite automorphic crossed module G = (E,G, ⊲), in other words on a pair of groups E and G, with E abelian, and a left action of G on E by automorphisms. The Crossed Module InvariantHG reduces to the Counting Invariant HG when E = 0. However, the Crossed Module Invariant distinguishes, in some cases, between welded virtual links with the Also at Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologia, Av. do Campo Grande, 376, 1749-024, Lisboa, Portugal. http://arxiv.org/abs/0704.1246v2 same knot group, and therefore it is strictly stronger than the Counting Invariant. We will assert this fact by calculating explicit examples. Let G = (E,G, ⊲) be an automorphic crossed module. Note that the Counting Invariant HG is trivial whenever G is abelian. However, taking G to be abelian and E to be non-trivial, yields a non-trivial invariant HG , which is, as a rule, much easier to calculate than the Counting In- variant HG where G is generic group, and it is strong enough to tell apart some pairs of links with the same knot group. Suppose that the welded virtual link K has n-components. Let κn = Z[X1,X 1 , . . . ,Xn,X n ]. We will define a kn-module CM(K), depending only on K, up to isomorphism and permutations of the variables X1, . . . ,Xn. If G is abelian, then HG simply counts the number of crossed module morphisms CM(K) → G. We prove in this article that if K is classical then CM(K) coincides with the Alexander module Alex(K) of K. However, this is not the case if K is not classical. We will give examples of pairs of welded virtual links (K,K ′) with the same knot group (thus the same Alexander module) but with CM(K) ≇ CM(K ′). This will happen when K and K ′ have the same knot group, but are distinguished by their crossed module invariants for G abelian. Let us explain the construction of the Crossed Module Invariant HG . Extending a previous construction due to T. Yagima, Shin Satoh defined in [S] a map which associates an oriented knotted torus T (K), the “tube of K”, to each oriented welded virtual knot K. The map K 7→ T (K) preserves knot groups. In the case when K is a classical knot, then T (K) coincides with the torus spun of K, obtained by spinning K 4-dimensionally, in order to obtain an embedding of the torus S1 × S1 into S4. The existence of the tube map K 7→ T (K) makes it natural to define invariants of welded virtual knots by considering invariants of knotted surfaces. We will consider this construction for the case of the crossed module invariants IG(Σ) of knotted surfaces Σ, defined in [FM1, FM2]. Here G = −→ G, ⊲ is a finite crossed module. Note that the invariant IG on a knotted surface coincides with Yetter’s Invariant (see [Y2, P1, FMP]) of the complement of it. We can thus define a welded virtual knot invariant by considering HG(K) = IG(T (K)), where K is a welded virtual knot. A straightforward analysis of the crossed module invariant of the tube T (K) of the welded virtual knot K permits the evaluation of HG(K) in a quandle type way, albeit the biquandle we define is sensitive to maximal and minimal points, so it should probably be called a “Morse biquandle”. A proof of the existence of the invariant HG , where G is a finite crossed module, can be done directly, from the Morse biquandle obtained. In fact all the results of this article are fully inde- pendent of the 4-dimensional picture, and can be given a direct proof. Moreover, they confirm the results obtained previously for the crossed module invariants IG of knotted surfaces in S As we have referred to above, the tube map K 7→ T (K) preserves the fundamental group of the complements. We prove that HG is powerful enough to distinguish between distinct welded virtual links with the same knot group. For example, we will construct an infinite set of pairs (Pi, c1(P i )), where i is an odd integer, of welded virtual links with the following properties: 1. Pi and c1(P i ) each have two components for all i. 2. Pi and c1(P i ) have isomorphic knot groups for each i. 3. Pi and c1(P i ) can be distinguished by their crossed module invariant for each i. In fact Pi and c1(P i ) will be distinguished by their crossed module invariant HG with G = (E,G, ⊲) being an automorphic crossed module with G abelian. This in particular proves that the Crossed Module Invariant of knotted surfaces IG defined in [FM1, FM2, FM3] sees beyond the fundamental group of their complement, in an infinite number of cases. Figure 1: Classical and virtual crossings. ↔ ↔ ↔ ↔ ↔ Reidemeister-I Move Reidemeister-II Move Reidemeister-III Move Figure 2: Reidemeister Moves I, II and III. We will also give examples of pairs of 1-component welded virtual knots with the same knot group, but separated by their crossed module invariants. However, we will need to make use of computer based calculations in this case. In this article we will propose a definition of Welded Virtual Graphs. The Crossed Module Invariant of welded virtual links extends naturally to them. 2 An Invariant of Welded Virtual Knots 2.1 Welded virtual knots Recall that a virtual knot diagram is, by definition, an immersion of a disjoint union of circles into the plane R2, where the 4-valent vertices of the immersion can represent either classical or virtual crossing; see figure 1. The definition of an oriented virtual knot diagram is the obvious one. We say that two virtual knot diagrams are equivalent if they can be related by the moves of figures 2 and 3, as well as planar isotopy. It is important to note that in the oriented case we will need to consider all the possible orientations of the strands. A virtual knot is an equivalence class of virtual knot diagrams under the equivalence relation just described; see [K1]. Observe that, as far as virtual knots are concerned, we do not allow the moves shown in figure 4, called respectively the forbidden moves F1 and F2. Considering the first forbidden move F1 in addition to the ones appearing in the definition of a virtual knot, one obtains the notion of a “welded virtual knot”, due to the first author; see [K1]. 2.1.1 The fundamental group of the complement The (combinatorial) fundamental group of the complement of a virtual knot diagram (the knot group) is, by definition, generated by all the arcs of a diagram of it, considering the relations (called Wirtinger Relations) of figure 5 at each crossing. It is understood that in each calculation of a knot group from a virtual knot diagram we will use either the “Left Handed” or the “Right Handed” Wirtinger Relation. The final result will not depend on this choice. ↔ ↔ ↔ ↔ ↔ Figure 3: Virtual Reidemeister Moves. ↔ ↔F1 F2 Figure 4: The forbidden moves F1 and F2. Y Y −1XY Y Y XY Figure 5: Wirtinger Relations. The first two are called “Left Handed” and “Right Handed” Wirtinger Relations, respectively. In the case of classical knots or links, this does coincide with the fundamental group of the complement, so we can drop the prefix “combinatorial”. This combinatorial fundamental group is in fact an invariant of welded virtual knots. This can be proved easily. 2.2 Virtual knot presentations of knotted surfaces By definition, a torus link1 in S4 is an embedding of a disjoint union of tori S1 × S1 into S4, considered up to ambient isotopy. A knotted torus is an embedding of a torus S1 × S1 into S4, considered up to ambient isotopy. The definition of an oriented knotted torus or torus link is the obvious one. As proved in [S, Ya, CKS], it is possible to associate an oriented torus link T (K) ⊂ S4, the “tube of K”, to each oriented welded virtual link K. This correspondence was defined first in [Ya], for the case of classical knots. The extension to welded virtual knots was completed in [S]. The tube map is very easy to define. Given a virtual link diagram, we define the tube of it by considering the broken surface diagram obtained by doing the transition of figures 6 and 7. For the representation of knotted surfaces in S4 in the form of broken surface diagrams, we refer the reader to [CKS]. The tube of a virtual knot diagram has a natural orientation determined by the orientation of a ball in S3. It is proved in [S] that if K and L are diagrams of the same welded virtual knot then it follows that T (K) and T (L) are isotopic knotted surfaces in S4. This defines the tube of a welded virtual knot. For calculation purposes, however, it is important to have a definition of the “Tube Map” in terms of movies. Let D ⊂ R2 be an oriented virtual knot diagram. We can suppose, apart from planar isotopy, that the projection on the second variable is a Morse function on D. Define a movie of a knotted surface by using the correspondence of figures 8, 9 and 10. Note our convention of Not to be confused with the 3-dimensional notion of a torus link. Figure 6: The tube of a virtual knot at the vicinity of a classical crossing. Figure 7: The tube of a virtual knot at the vicinity of a virtual crossing. death of a circle saddle point saddle point birth of a circle Figure 8: Associating a knotted torus to a virtual knot: edges, minimal and maximal points and virtual crossings. All circles are oriented counterclockwise. Note that the movies should be read from bottom to top. reading movies of knotted surfaces from the bottom to the top. This yields an alternative way for describing the tube T (K) of the virtual knot K, if we are provided a diagram of it. It was proved in [S, Ya] that the correspondence K 7→ T (K), where K is a welded virtual knot, preserves the fundamental groups of the complement (the knot groups). Given a (classical) link K with n components sitting in the interior of the semiplane {(x, y, z) ∈ R3 : z ≥ 0}, we define the torus spun of K by rotating K 4-dimensionally around the plane {z = 0}. Therefore, we obtain an embedding of the disjoint union of n tori S1 × S1 into S4. It was shown in [S] that the torus spun of K is in fact isotopic to the tube T (K) of K. The correspondence K 7→ T (K) actually sends welded virtual links to ribbon torus links. In fact, any ribbon torus link is of the form T (K) for some welded virtual knot K. However, it is an open problem whether the map K 7→ T (K) is faithful; see [CKS, problems (1) and (2) of 2.2.2]. 2.2.1 Welded virtual arcs A virtual arc diagram is, by definition, an immersion of a disjoint union of intervals [0, 1] into the plane R2, where the 4-valent vertices of the immersion can represent either classical of virtual crossings. The definition of a welded virtual arc is similar to the definition of a welded virtual knot, but considering in addition the moves of figure 11; see [S]. A sphere link is, by definition, an embedding of a disjoint union of spheres S2 into S4, considered up to ambient isotopy. Similarly to ribbon torus links in S4, any ribbon sphere link admits a Figure 9: Associating a knotted torus to a virtual knot: classical crossing points, first case. All circles are oriented counterclockwise. Figure 10: Associating a knotted torus to a virtual knot: classical crossing points, second case. All circles are oriented counterclockwise. Figure 11: Moves on welded virtual arc diagrams. Figure 12: The tube of a welded virtual arc close to the endpoints. Y Y −1XY XY X−1 e ∂(e)X Figure 13: Definition of a colouring of a dotted knot diagram. presentation as the tube T (A), where A is a welded virtual arc. Here T (A) is defined in the same way as the tube of a welded virtual knot, considering additionally the movies of figure 12 at the end-points of the arcs of A . Therefore T (A) is an embedding of a disjoint union of spheres S2 into Suppose that the arc A is classical, and that it sits inside the semiplane {z ≥ 0} of R3, inter- secting the plane {z = 0} at the end-points of A, transversally. Then in fact T (A) is the spun knot of A; see [R, S, CKS]. We can define the knot group of a welded virtual arc exactly in the same way as we defined the combinatorial fundamental group of the complement of a welded virtual knot. As in the case of welded virtual knots, the map A 7→ T (A) preserves knot groups; see [S]. Suppose that A is a classical arc (with one component) sitting in the semiplane {z ≥ 0} of R3, intersecting the plane {z = 0} at the end-points of A. Let K be the obvious closure of A. Then it is easy to see that A and K have the same knot groups. Note that the fact that A is classical is essential for this to hold. This is also true if A may have some S1 components, even though it is strictly necessary that A have only one component homeomorphic to [0, 1]. 2.3 Crossed module invariants of knotted surfaces A crossed module (see [B]) G = −→ G, ⊲ is given by a group morphism ∂ : E → G together with a left action ⊲ of G on E by automorphisms. The conditions on ∂ and ⊲ are: 1. ∂(X ⊲ e) = X∂(e)X−1,∀X ∈ G,∀e ∈ E, 2. ∂(e) ⊲ f = efe−1,∀e, f ∈ E. Note that the second condition implies that the subgroup ker ∂ of E is central in E, whereas the first implies that ker ∂ is G-invariant. A dotted knot diagram is, by definition, a regular projection of a bivalent graph, in other words of a link, possibly with some extra bivalent vertices inserted. Let D be a dotted knot diagram, which we suppose to be oriented. Let also G = −→ G, ⊲ be a finite crossed module. Definition 1 A colouring of D is an assignment of an element of G to each arc of D and of an element of E to each bivalent vertex of D satisfying the conditions of figure 13. Definition 2 Let D be a knot diagram (without vertices). A dotting of D is an insertion of bivalent vertices in D, considered up to a planar isotopy sending D to D, setwise. If D is an oriented knot diagram, let V (D) be the free Q-vector space on the set of all colourings of all dottings of D. X X X ∂(e)X ∂(fe)X X ∂(fe)X e f fe Figure 14: Relations on colourings. Y −1⊲e eXY X−1⊲e−1 Y −1⊲e−1Y −1X⊲e Figure 15: Relations on colourings. Consider now the relations of figures 14 and 15. It is straightforward to see that they are local on the knot diagrams and that they transform colourings into colourings. Definition 3 Let D be an oriented knot diagram (without vertices). The vector space V(D) is defined as the vector space obtained from V (D) by modding out by the relations R1 to R6. Let D and D′ be oriented knot diagrams. If D and D′ differ by planar isotopy, then there exists an obvious map V(D)→ V(D′). In fact, if D and D′ differ by a Reidemeister move or a Morse move (in other words a birth/death of a circle or a saddle point), then there also exists a well defined map V(D)→ V(D′). All this is explained in [FM1]. In figures 16, 17, 18, 19 and 20 we display the definition of these maps for the case of the Reidemeister-II move and the Morse moves, which we are going to need in this article. The remaining cases of these moves can be dealt with by doing the transition shown in figure 21, and using the relations R1 to R6. In figure 18, δ is a Kronecker delta. Therefore, any movie of an oriented knotted surface Σ can be evaluated to give an element IG(Σ) ∈ Q. Theorem 4 The evaluation IG of a movie of an oriented knotted surface defines an isotopy invari- ant of oriented knotted surfaces. This is shown in [FM1]. The homotopy theoretical interpretation of the isotopy invariant IG is discussed in [FM2, FM3, FMP]. The construction of the invariant IG was initially inspired by Yetter’s Invariant of manifolds; see [Y2, P1, P2]. Actually IG defines an embedded TQFT, in other words, an invariant of link cobordisms con- sidered up to ambient isotopy fixing both ends. X XY Y 7−→e f eX⊲fXY X −1⊲e−1 e Figure 16: Map assigned to positive Reidemeister-II move. b X−1⊲b−1X−1⊲aY X−1⊲b7−→ Figure 17: Map assigned to negative Reidemeister-II move. X Y X ∂(e)X X ∂(e)X δ(Y, ∂(e)X) Figure 18: Map associated to saddle point moves. 1 7−→ Figure 19: Map associated with births of a circle. x2xn−1 . . . #Eδ(x1x2...xn−1xn, 1E) Figure 20: Map associated with deaths of a circle. ←→X ∂(e)X X−1 X−1∂(e)−1e X Figure 21: Inversion of strands. (∂(g)Y,f) (Y,f−1) (X,e) (X,e−1) Figure 22: Relations at maximal and minimal points. (XY X−1,X⊲f) (X,efX⊲f (X,e) (Y,f) (X−1Y X,X−1⊲f) (X,X−1⊲f−1ef) (X,e) (Y,f) (Y,Y −1⊲e−1ef) (Y −1XY,Y −1⊲e) (X,e) (Y,f) (Y,feY ⊲e−1) (Y XY −1,Y ⊲e) (X,e) (Y,f) (Y,f) (X,e) (X,e) (Y,f) Figure 23: Relations at crossings. 2.3.1 The case of ribbon knotted torus As we have seen, if Σ is a ribbon knotted surface, which topologically is the disjoint union of tori S1 × S1 or spheres S2, then we can represent it as the tube T (K) of welded virtual knot K, in the first case, or the tube T (A) of a welded virtual arc A, in the second case. We want to find an algorithm for calculating IG(T (K)), where K is a welded virtual knot, directly from a diagram of K itself, and analogously for a welded virtual arc A. A careful look at the definition of the invariant IG together with the definition of the tube map in 2.2 leads to the following definition: Definition 5 Let G = −→ G, ⊲ be a crossed module. Let also D be a welded virtual knot diagram. Suppose that the projection on the second variable defines a Morse function on D. A G-colouring2 of D is an assignment of a pair (X, f), where X ∈ G and f ∈ ker ∂, to each connected component of D minus its set of crossings and extreme points; of an element e ∈ ker ∂ to each minimal point; and an element g ∈ E to each maximal point, satisfying the conditions shown in figures 22 and 23. This should not be confused with the notion of a colouring which was considered in the definition of the invariant IG , above. X∈G,e∈ker ∂ X X ∂(g)Y=X ∂(g)Y =X δ(f,e−1) Figure 24: Calculation of IG of the tube of a welded virtual knot: minimal and maximal points. The reason for considering these relation is obvious from figure 24, and figure 25, and its counterparts for different types of crossings. Note that ker ∂ ⊂ E is central in E. However, for this calculus to approximate the definition of IG(T (D)), for D a virtual knot diagram, the relation of figure 26 still needs to be incorporated into the calculations. To avoid needing to involve this relation, we consider the following restriction on the crossed modules with which we work. Definition 6 (Automorphic Crossed Module) A crossed module G = −→ G, ⊲ is called automorphic if ∂(e) = 1,∀e ∈ E. Therefore, an automorphic crossed module is given simply by two groups G and E, with E abelian, and a left action ⊲ of G on E by automorphisms. Definition 7 (Reduced G-Colourings) Let G = (E,G, ⊲) be an automorphic crossed module. Let also D ⊂ R2 be a virtual knot diagram, such that the projection on the second variable is a Morse function on D. A reduced G-colouring of D is given by an assignment of a pair (X, e) ∈ G × E to each connected component of D minus its set of crossings and extreme points, satisfying the relations of figures 23 and 27. The following result is easy to prove by using all the information we provided, and the fact that, for any knot diagram, the number of minimal points of it equals the number of maximal points. Theorem 8 Let D be a virtual knot diagram, such that the projection on the second variable is a Morse function on D. Let also G = (E,G, ⊲) be a finite automorphic crossed module. Consider the quantity: HG(D) = #{reduced G-colourings of D}. Then HG(D) is an invariant of welded virtual knots. In fact: HG(D) = IG(T (D)). Here IG is the Crossed Module Invariant of oriented knotted surfaces defined in [FM1]. Exercise 1 Check directly that HG (where G is an automorphic finite crossed module) is an in- variant of welded virtual knots. Note that together with the moves defining welded virtual knots, we still need to check invariance under planar isotopy, thus enforcing us to check invariance under the moves of the type depicted in figure 28, usually called Yetter’s Moves; see [Y1, FY]. It is important to note that we need to consider all the possible different crossing informations, and, since we are working in the oriented case, all the possible orientations of the strands. X−1⊲f X−1Y X X−1⊲f X−1⊲f X−1Y X eX−1⊲f−1f eX−1⊲f−1f X−1Y X X−1⊲f Figure 25: Calculation of IG of the tube T (D) of a welded virtual knot D: the type of crossings relative to figure 9. ∂(g)X ∂(g)X ∂(g)X = = = Figure 26: An identity. Here e ∈ ker ∂. (Y,e) (Y,e−1) (X,e) (X,e−1) Figure 27: Reduced G-colouring at extreme points. ↔ ↔↔ ↔ Figure 28: Sample of Yetter’s moves capturing planar isotopy. (X,1E) (Y,1E) Figure 29: Reduced G-colourings of welded virtual arcs at end-points. Here X,Y ∈ G. Let G = −→ G, ⊲ be a crossed module. Define π1(G) = coker(∂) and π2(G) = ker ∂, which is an abelian group. Then π1(G) has a natural left action ⊲ ′ on π2(G) by automorphisms. In particular Π(G) = (π2(G), π1(G), ⊲ ′) is an automorphic crossed module. In fact G also determines a cohomology class k3 ∈ H3(π1(G), π2(G)), called the k-invariant of G. It is not difficult to extend the invariant HG(D), where D is a welded virtual knot, to handle non-automorphic crossed modules G, so that HG(D) = IG(T (D)). We do this by incorporating the relation in figure 26 into the notion of a G-colouring of a virtual knot diagram. However, it is possible to prove that for any welded virtual knot D and any finite crossed module G we have that IG(T (D)) equals IΠ(G)(T (D)), apart from normalisation factors. This can be proved by using the graphical framework presented in this article. Hence, we do not lose generality if we restrict our attention only to automorphic crossed modules. Problem 1 Let G = (E,G, ⊲) be an automorphic crossed module. Find a ribbon Hopf algebra AG acting on the vector space freely generated by G × E such that HG is the Reshetikhin-Turaev invariant of knots associated to it (see [RT]), and so that the case of welded virtual knots also follow from this Hopf algebra framework in a natural way. Note that in the case when E = 0, we can take AG to be the quantum double of the function algebra on G. The solution to this problem would be somehow the quantum double of a finite categorical group, and therefore would be of considerable importance. 2.3.2 The case of welded virtual arcs Let G = (E,G, ⊲) be a finite automorphic crossed module. Let also A be a virtual arc diagram. The notion of a reduced G-colouring of A is totally analogous to the concept of a reduced G-colouring of a virtual knot diagram, considering that if an arc of A has a free end then it must be coloured by (X, 1E), where X ∈ G; see figure 29. One can see this from figure 12. We have: Theorem 9 Let A be a virtual arc diagram. The quantity: HG(A) = #{reduced G-colourings of A} #E#{cups}−#{caps}−#{pointing upwards ends of A} is an invariant of A as a welded virtual arc. In fact HG(A) = IG(T (A)). Therefore, the graphical framework presented in this article is also a calculational device for calcu- lating the crossed module invariant of spun knots, accordingly to 2.2.1. The invariant HG of Theorem 9 actually is an invariant of virtual arcs of which some components may be circles. In fact, it also naturally extends to an invariant of welded virtual graphs, to be defined in 3.5.2. Figure 30: Classical and Virtual Hopf links. (X,e−1) (X,e) (Y,f) (Y,f−1) (Y,f) (X,e) (Y −1XY,Y −1⊲e) (Y,Y −1⊲e−1ef) Y −1⊲e=e Y −1XY=X Y −1⊲e−1ef=f Figure 31: Calculation of the crossed module invariant of the Virtual Hopf Link L. 3 Examples 3.1 Virtual and Classical Hopf Link 3.1.1 Virtual Hopf Link The simplest non-trivial welded virtual link is the Virtual Hopf Link L, depicted in figure 30. Note that L is linked since its knot group is {X,Y : XY = Y X} ∼= Z2. Let G = (E,G, ⊲) be a finite automorphic crossed module. Let us calculate the crossed module invariant HG of the Virtual Hopf Link L. This calculation appears in figure 31. From this we can conclude that: HG(L) = #{X,Y ∈ G; e, f ∈ E|XY = Y X, Y −1 ⊲ e = e} (1) = #E#{X,Y ∈ G; e ∈ E|XY = Y X, Y −1 ⊲ e = e}. (2) Note that the previous equation simplifies to HG(L) = #E#G#{Y ∈ G; e ∈ E|Y −1 ⊲ e = e}, when the group G is abelian. On the other hand it is easy to see that if O2 is a pair of unlinked unknots then we have: 2) = #G2#E2. (3) From equations (2) and (3), it thus follows that any finite automorphic crossed module (E,G, ⊲) with G abelian sees the knotting of the Virtual Hopf Link if there exists Y ∈ G and e ∈ E such that Y −1 ⊲ e 6= e. This is verified in any automorphic crossed module (E,G, ⊲) with ⊲ being a non-trivial action of G on E. Consider the automorphic crossed module A = (E = Z3, G = Z2, ⊲) such that 1 ⊲ a = a and −1 ⊲ a = −a, where a ∈ Z3 and Z2 = ({1,−1},×); see [BM]. Then this crossed module detects the knottedness of the Virtual Hopf Link L. If fact HA(L) = 6#{Y ∈ Z2; e ∈ Z3|Y −1 ⊲ e = e} = 24, whereas HA(O 2) = 36. 3.1.2 The Hopf Link The Hopf Link H is depicted in figure 30. Note that the fundamental group of the complement of it is, similarly with the Virtual Hopf Link L, isomorphic with Z2. (X,e−1) (X,e) (Y,f) (Y,f−1) (X,X−1⊲f−1ef) (X−1Y X,X−1⊲f) (X−1Y −1XY X,X−1Y −1⊲f−1X−1Y −1X⊲(ef)) (X−1Y X,X−1Y −1⊲fX−1Y −1X⊲(e−1f−1)ef) X−1Y −1XY X=X X−1Y X=Y X−1Y −1⊲f−1X−1Y −1X⊲(ef)=e X−1Y −1⊲fX−1Y −1X⊲(e−1f−1)ef=f Figure 32: Calculation of the crossed module invariant of the Hopf Link. Let us calculate the crossed module invariant of the Hopf Link H. To this end, let G = (E,G, ⊲) be a finite automorphic crossed module. We display the calculation of HG(H) in figure 32. This permits us to conclude that: HG(H) = # X,Y ∈ G; e, f ∈ E XY = Y X X−1Y −1 ⊲ f−1X−1Y −1X ⊲ (ef) = e which particularises to HG(H) = # X,Y ∈ G; e, f ∈ E : X−1Y −1 ⊲ f−1Y −1 ⊲ (ef) = e , (4) in the case when G is abelian. This is in agreement with the calculation in [FM2]. Let us see that the Hopf Link H is not equivalent to the Virtual Hopf Link L as a welded virtual link. Consider the automorphic crossed module A = (E = Z3, G = Z2, ⊲) defined above. We have (note that we switched to additive notation, more adapted to this example): HA(H) = # {X,Y ∈ Z2; e, f ∈ Z3 : −XY ⊲ f + Y ⊲ (e+ f) = e} = # {X,Y ∈ Z2; e, f ∈ Z3 : −XY ⊲ f + Y ⊲ f = e− Y ⊲ e} . In the case Y = 1, we are led to the equation −X ⊲ f + f = 0, which has 4 × 3 solutions in Z2×Z3×Z3. In the case Y = −1, we get the equation e = 2 −1(X⊲f−f), which has 3×2 solutions in Z2 × Z3 × Z3. Therefore, we obtain HA(H) = 18. Therefore, we have proved that the Virtual Hopf Link is not equivalent to the Hopf Link as a welded virtual link, and also that the Hopf Link is knotted, by using the crossed module invariant. As we have referred to before, the knot groups of the Hopf Link and the Virtual Hopf Link are both isomorphic with Z2. Therefore, we have proved that the crossed module invariant HG sees beyond the fundamental group of the complement of a welded virtual knot. Since the correspondence K 7→ T (K), where K is a welded virtual link, preserves the funda- mental groups of the complement we have also proved: Theorem 10 The Crossed Module Invariant IG of knotted surfaces defined in [FM1, FM2] is pow- erful enough to distinguish between knotted surfaces Σ,Σ′ ⊂ S4, with Σ diffeomorphic with Σ′, whose complements have isomorphic fundamental groups, at least in a particular case. Therefore, one of the main open problems about the Crossed Module Invariant IG of knotted sur- faces that prevails is whether the invariant IG can distinguish between knotted surfaces whose com- plements have isomorphic fundamental groups and second homotopy groups, seen as π1-modules, but have distinct Postnikov invariants k3 ∈ H3(π1, π2). This problem was referred to in [FM2]. Examples of pairs of knotted surfaces like this do exist; see [PS]. Figure 33: The Hopf Arc HA, the Trefoil Knot 31 and the Trefoil Arc 31 (X,e−1) (X,e) (Y,f) (Y,f−1) (X,X−1⊲f−1ef)(C,X−1⊲f) (C,C−1X−1⊲fC−1⊲(ef)−1ef) (B,C−1X−1⊲f−1C−1⊲(ef)) (A,B−1C−1X−1⊲fB−1C−1⊲(ef)−1B−1⊲(ef)) (B,B−1C−1X−1⊲f−1B−1C−1⊲(ef)B−1⊲(ef)−1ef) C=X−1Y X B=X−1Y −1XY X A=Y −1X−1Y XY =Y −1CY ⊲(ef)−1B−1⊲(ef)=e ⊲(ef)B−1⊲(ef)−1ef=f Figure 34: Calculation of the crossed module invariant of the Trefoil Knot 31. Exercise 2 Consider the Hopf Arc HA depicted in figure 33. Prove that HG(HA) = HG(L), where L is the Virtual Hopf Link. Here G = (E,G, ⊲) is any finite automorphic crossed module. In fact, cf. 3.5.1, T (L) is obtained from T (HA) by adding a trivial 1-handle, which explains this identity. We will go back to this later in 3.5.2. 3.2 Trefoil Knot and Trefoil Arc The Trefoil Knot 31 and the Trefoil Arc 31 ′ are depicted in figure 33. Let us calculate the crossed module invariant of the Trefoil Knot 31. Let G = (E,G, ⊲) be a finite automorphic crossed module. The calculation of HG(31) appears in figure 34. This permits us to conclude that: HG(31) = # X,Y ∈ G; e, f ∈ E X−1Y −1X−1=Y −1X−1Y −1 Y −1X−1Y −1⊲fY −1X−1Y −1X⊲(ef)−1Y −1⊲(ef)=e X,Y ∈ G; e, f ∈ E X−1Y −1X−1=Y −1X−1Y −1 Y −1X−1Y −1⊲fX−1Y −1⊲(ef)−1Y −1⊲(ef)=e . (6) This simplifies to: HG(31) = #{X ∈ G; e, f ∈ E|X −3 ⊲ fX−2 ⊲ (ef)−1X−1 ⊲ (ef) = e}, (7) when G = (E,G, ⊲) is an automorphic crossed module with G abelian; see 3.3.2. Note that the crossed module invariant of the Trefoil Arc 31 ′ can also be obtained from this calculation, by making f = 1E , and inserting the necessary normalisation factors; see 2.3.2. This yields: HG(31 ′) = #E# X,Y ∈ G; e ∈ E X−1Y −1X−1=Y −1X−1Y −1 X−1Y −1⊲e−1Y −1⊲e=e , (8) Figure 35: A non trivial welded virtual arc whose closure is trivial. which simplifies to: HG(31 ′) = #E#{X ∈ G; e ∈ E : X−2 ⊲ e−1X−1 ⊲ e = e}, (9) whenever G is abelian. This is coherent with the calculation in [FM1, FM2]. Observe that from equations (7) and (9) it follows that (we switch to additive notation): HG(31) = #{X ∈ G; e, f ∈ E : X −3 ⊲ f −X−2 ⊲ (e+ f) +X−1 ⊲ (e+ f) = e} = #{X ∈ G; e, f ∈ E : X−2 ⊲ X−1 ⊲ f − e −X−1 ⊲ X−1 ⊲ f − e X−1 ⊲ f − e = HG(31 Thus: HG(31) = HG(31 ′), (10) whenever G = (E,G, ⊲) is an automorphic crossed module with G abelian. An analogous identity holds for any classical 1-component knot, see 3.3.2. We will consider the crossed module invariants of the Trefoil Knot and the Trefoil Arc for the case when G = (E,G, ⊲) is an automorphic crossed module with G being a non-abelian group in 3.5.5. In this case the previous identity does not hold. Let us see that HG detects the knottedness of the Trefoil Knot 31. The crossed module A = (Z3,Z2, ⊲) defined previously detects it. In fact it is easy to see that HA(31) = 12. On the other hand, if O is the unknot, we have that HG(O) = #E#G, for any automorphic crossed module G = (E,G, ⊲). Thus 31 is knotted. Analogously we can prove that the Trefoil Arc 31 ′ is knotted. Exercise 3 Consider the virtual arc A of figure 35. Prove that if G = (E,G, ⊲) is an automorphic finite crossed module with G abelian then: HG(A) = #E#{X ∈ G; e ∈ E|X −2 ⊲ e−1X−1 ⊲ ee−1 = 1}. Thus the crossed module A = (Z3,Z2, ⊲) defined previously detects that it is knotted. However, it is easy to show that the closure of A is the trivial welded virtual knot, a fact confirmed by the crossed module invariant. 3.3 Universal module constructions Let G be an abelian group. Suppose that G = (E,G, ⊲) is an automorphic crossed module, where E is an abelian group. Consider a welded virtual link K. Suppose that K has n-components S1, where n is a positive integer. Let κn = Z[X1,X 1 , . . . ,Xn,X n ] be the ring of Laurent polynomials on the formal variables X1, . . . Xn. We can assign to K a κn-module, so that HG(K) will satisfy: HG(K) = #Hom(CM(K),G), where Hom(CM(K),G) denotes the set of all crossed module morphism CM(K)→ G. (Y,X⊲f) (X,e+f−X⊲f) (X,e) (Y,f) (Y,X−1⊲f) (X,−X−1⊲f+e+f) (X,e) (Y,f) (Y,−Y −1⊲e+e+f) (X,Y −1⊲e) (X,e) (Y,f) (Y,f+e−Y ⊲e) (X,Y ⊲e) (X,e) (Y,f) (Y,f) (X,e) (X,e) (Y,f)(X,e) (X,−e) (X,e) (X,−e) Figure 36: Defining relations for the module CM(K). 3.3.1 The definition of the module CM(K) Definition 11 Let K be a welded virtual link diagram. Suppose that K is an immersion of a disjoint union of n circles S1 into the plane, each of which is assigned a variable Xi, where i ∈ {1, . . . , n}; in other words, suppose that we have a total order on the set of all S1-components of K. The module CM(K) is defined as the κn-module generated by all the connected components of K minus the set of crossings of K and extreme points of K, modding out by the relations of figure 36. It is understood that any connected component is assigned a pair (X, e), where e ∈ CM(K) is the module element that the connected components defines, whereas X ∈ {X1, . . . ,Xn} is the labelling of the S1-component of K in which the connected component is included. By using the same technique as in Exercise 1 we can prove: Theorem 12 Let K be a welded virtual link diagram with n S1-components. The isomorphism class of the κn-module CM(K) depends only on the welded virtual link determined by K, up to reordering of the S1-components of K. In addition, if G = (E,G, ⊲) is an automorphic finite crossed module with G abelian we have: HG(K) = #Hom(CM(K),G). 3.3.2 Relation with the Alexander Module Let K be a welded virtual link diagram with n S1-components, each labelled with an Xi ∈ {X1, ...,Xn}. We can define the Alexander module Alex(K) of K, defined as the module over κn with a generator for each connected component of K minus its set of crossings, modulo the re- lations of figure 37, obtained from the right handed Wirtinger relations of figure 5 by applying Fox derivatives; see [BZ, Chapter 9], [K2, Chapter XI] or [F]. Therefore, if K is a classical 1-component knot, then Alex(K) ∼= Z[X,X−1]/ 〈∆(K) = 0〉 ⊕ Z[X,X−1], where ∆(K) denotes the Alexander polynomial of K; see for example [BZ, 9 C]. Let K be a welded virtual link diagram. The Alexander module Alex(K) depends only on the knot group of the welded virtual link defined by K, up to isomorphism and reordering of the S1-components of K. (X,e) (Y,f) (Y,f) (X,−Y −1⊲f+Y −1⊲e+Y −1X⊲f)) (Y,f) (X,e) (X,e) (Y,f) Figure 37: Relations at crossings for the Alexander Module Alex(K). (X,e) (Y,f) (Y,f) (X,−Y −1⊲f+Y −1⊲e+Y −1X⊲f) (Y,X⊲f) (X,e) (X,Y −1⊲e) (Y,f) Figure 38: Relations at crossings for the module Alex′(K). The module Alex(K) admits a variant Alex′(K) whose defining relations appear in figure 38. Note that the κn-module Alex(K) is isomorphic to Alex ′(K) whenever K is a classical link diagram. The module Alex′(K) is invariant under virtual and classical Reidemeister moves. However, Alex′(K) is not invariant under the first forbidden move F1; rather it is invariant under the second forbidden move F2; see subsection 2.1. Given a virtual link diagram K, we can define the mirror image K∗ of it by switching positive to negative crossings, and vice-versa, and leaving virtual crossings unchanged. Therefore, the module Alex′(K∗) depends only on the welded virtual knot defined by K, up to isomorphism and reordering of the components of K. Theorem 13 Let K be a welded virtual link diagram. There exists an isomorphism φ : CM(K)→ Alex′(K∗). Proof. We can suppose that K is the closure of a virtual braid B; see [KL, Ka]. This avoids needing to deal with the defining relations of CM(K) at maximal and minimal points. Let b be a connected component of the braid B minus its set of crossings, defining therefore an element b ∈ CM(K). The isomorphism φ : CM(K)→ Alex′(K∗) sends b to Z−1 ⊲ b, where Z is the product of all the elements Xi assigned to the strands of B on the left of b (each belonging to a certain S1-component of K). The remaining details are left to the reader. The Alexander module of the Trefoil Knot 31 is the module over Z[X,X −1] with generators e and f and the relation X2 ⊲ (e + f) − X ⊲ (e + f) + (e + f) = 0, thus we have Alex(31) = Z[X,X−1]/ X2 −X + 1 = 0 ⊕ Z[X,X−1]. In particular, it follows equation (7). Let K be a classical 1-component knot. By using Theorem 13, we can prove that for any automorphic crossed module G = (E,G, ⊲), with G abelian, the invariant HG(K) is determined by the Alexander module Alex(K) of K, and thus from the Alexander polynomial ∆(K) of K. This is not the case for non classical links, since the crossed module invariants of the virtual and classical Hopf links L and H; see subsection 3.1 are different, even though they have isomorphic Alexander modules. In fact we have: Alex(H),CM(H),Alex(L) = Z[X,X−1, Y, Y −1] ⊲ e⊕ Z[X,X−1, Y, Y −1] ⊲ f 〈(X − 1) ⊲ f = (Y − 1) ⊲ e〉 Figure 39: Shin Satoh’s Knot S. the module over the ring Z[X,X−1, Y, Y −1] with two generators e and f , and the relation (X − 1) ⊲ f = (Y − 1) ⊲ e, whereas CM(L) = Z[X,X−1, Y, Y −1] ⊲ e⊕ Z[X,X−1, Y, Y −1] ⊲ f 〈Y ⊲ f = f〉 These last two modules are not isomorphic, as the calculations in subsection 3.1 certify. 3.3.3 Welded virtual arcs Let A be a welded virtual arc with a single component. The Z[X,X−1]-modules Alex(A),Alex′(A) and CM(A) defined above can still be assigned to A, considering the analogue of the relations in figure 29 at the end-points of A, so that the elements of Alex(A),Alex′(A) and CM(A) assigned to the edges of A incident to its end-points are zero. Any welded virtual arc A can be obtained as the (incomplete) closure of some braid. Therefore the proof of Theorem 13 gives an isomorphism φ : CM(A)→ Alex′(A). Suppose that A is a classical arc sitting in the semiplane {z ≥ 0} of R3, intersecting the plane {z = 0} at the end-points of A, only. Since A is classical we have Alex(A) = Alex′(A). Let K be the obvious closure of A. Then Alex(K) = Z[X,X−1]/ 〈∆(K) = 0〉 ⊕ Z[X,X−1], where ∆(K) is the Alexander polynomial of K. Choosing a connected component of K minus its set of crossings, and sending the generator of Alex(K) it defines to zero yields a presentation of Z[X,X−1]/ 〈∆(K) = 0〉; see [BZ, Theorem 9.10]. Comparing with the definition of Alex(A), proves that Alex(A) = Z[X,X−1]/ 〈∆(K) = 0〉. Therefore it follows that CM(A) ∼= Z[X,X−1]/ 〈∆(K) = 0〉 if A is a classical arc and K is the closure of A. The discussion above also implies that if G = (E,G, ⊲) is an automorphic crossed module with G abelian then HG(K) = HG(A) whenever A is a classical 1-component arc and K is the 1-component knot obtained by closing A. This is not the case if G is not abelian. Problem 2 Let K be a welded virtual link. What is the algebraic topology interpretation of the module CM(K) in terms of the tube T (K) ⊂ S4 of K. 3.4 Shin Satoh’s Knot In [S], Shin Satoh considered the welded virtual link S displayed in figure 39. It is a welded virtual knot whose knot group is isomorphic with the knot group of the Trefoil Knot 31. It is possible to prove that S is not equivalent to any classical knot as a welded virtual knot, see [S], thus the Shin Satoh’s Knot S is not equivalent to the Trefoil. See also 3.5.5. Let us calculate the crossed module invariant of the Shin Satoh’s Knot S. Let G = (E,G, ⊲) be a finite automorphic crossed module. We consider in this case that G is an abelian group, (X,c−1) (X,c−1) (X,a) (X,a) (X,a−1) (X,a−1) (X,b) (X,c) (X,b−1) (X,b−1) (X,c) (X,d) (X,d−1) (X,d−1) (X,X−1⊲a−1) (X,X−1⊲a−1) (X,X−1⊲aa−1d) (X,X−1⊲b−1) (X,X−1⊲(ab)a−1db−1) (X,X−1⊲d−1) (X,X−1⊲dX−1⊲b−1d−1) (X,X−1⊲c−1) (X,X Figure 40: Calculation of the crossed module invariant of the Shin Satoh’s Knot S for G abelian. which makes the calculations much easier, since we simply need to calculate the Z[X,X−1]-module CM(S). The case when G is non-abelian is considered in 3.5.5. Figure 40 permits us to conclude that: HG(S) = # X ∈ G; a, b, c, d ∈ E X−1 ⊲ (ab)a−1db−1 = c−1 X−1 ⊲ d−1 = b−1 X−1 ⊲ c−1 = a−1 X−1 ⊲ cc−1X−1 ⊲ dX−1 ⊲ b−1d−1 = X−1 ⊲ a X ∈ G; a, d ∈ E X−1 ⊲ aX−2 ⊲ da−1dX−1 ⊲ d−1 = X ⊲ a−1 aX ⊲ a−1X−1 ⊲ dX−2 ⊲ d−1d−1 = X−1 ⊲ a The two equations in the final expression are equivalent. We obtain, switching to additive notation: HG(S) = # X ∈ G; a, d ∈ E|X−1 ⊲ a− a+X ⊲ a = X−1 ⊲ d− d−X−2 ⊲ d . (11) This should be compared with the crossed module invariant of the Trefoil Knot 31, for G abelian: HG(31) = #{X ∈ G; e, f ∈ E|X −3 ⊲ f −X−2 ⊲ (e+ f) +X−1 ⊲ (e+ f) = e} = #{X ∈ G; e, f ∈ E|X−3 ⊲ f −X−2 ⊲ f +X−1 ⊲ f = e−X−1 ⊲ e+X−2 ⊲ e} = #{X ∈ G; e, f ∈ E|X−2 ⊲ f −X−1 ⊲ f + f = X ⊲ e− e+X−1 ⊲ e}. Therefore it follows that if G = (E,G, ⊲) is an automorphic crossed module with G abelian then: HG(31) = HG(S). (12) Figure 41: Adding a trivial 1-handle to a knotted surface. On the top we display the original movie, and on the bottom the new movie, both read from left to right. A concise description of this modification is fission saddle, fusion saddle. We present in the following subsection (see 3.5.1) an alternative proof of this fact, which should reassure the reader that the calculations in this article are correct, despite this being somehow a negative example. We will also see below (see 3.5.5) that if we take G to be non-abelian, then we can prove that the Trefoil Knot is not equivalent to the Shin Satoh’s Knot, by using the crossed module invariant. 3.5 Welded Virtual Graphs 3.5.1 Crossed module invariants of knotted surfaces obtained by adding trivial 1- handles Let Σ ⊂ S4 be a knotted surface which we suppose to be connected. The knotted surface Σ′ obtained from Σ by adding a trivial 1-handle is defined simply as the connected sum Σ′ = Σ#T 2, where T 2 is a torus S1×S1, trivially embedded in S4. The non-connected case is totally analogous, but a connected component of Σ must be chosen. A movie of Σ′ is obtained from a movie of Σ by choosing a strand of the movie of Σ belonging to the chosen component of Σ, and making the modification shown in figure 41. The straightforward proof of the following theorem is left to the reader. Theorem 14 Let G = −→ G, ⊲ be a finite crossed module. If the oriented knotted surface Σ′ is obtained from the oriented knotted surface Σ by adding a trivial 1-handle then: (#ker ∂)2 (#E)2 IG(Σ), thus in particular IG(Σ) = IG(Σ ′) whenever G is automorphic. The tube T (S) of the Shin Satoh’s Knot S is obtained from the Spun Trefoil (the tube T (3′1) of the Trefoil Arc 3′1) by adding a trivial 1-handle; see [S] or 3.5.2. This fact together with equation (10) proves that HG(31) = HG(S), whenever G = (E,G, ⊲) is a finite automorphic crossed module with G abelian, as already proved by other means; see subsection 3.4. Here 31 is the Trefoil Knot. 3.5.2 Definition of welded virtual graphs Let K be an oriented virtual graph diagram. Note that K may have some bivalent vertices where the orientation of an edge of K may change; however, there cannot be a change of orientation of a strand at a crossing; see figure 42. Figure 42: A welded virtual graph. saddle point Figure 43: The tube of a virtual graph at a 3-valent vertex (movie version). As usual, all circles are oriented counterclockwise. Given a virtual graph diagram K, we can define the tube T (K) of it exactly in the same way as the tube of a virtual link or arc is defined. We consider the type of movie of figure 43 at the 3-valent vertices. For the broken surface diagram version of this see figure 44. We proceed analogously for n-valent vertices if n > 3. The 2-valent vertices do not affect the calculation of T (K). On the other hand 1-valent vertices were already considered in the case of virtual arcs. It is easy to see that the tube T (K) of a virtual graph is invariant under the moves defining welded virtual knots and arcs; see subsection 2.1 and 2.2.1. In addition, T (K) is invariant under the moves shown in figure 45. Note that if a strand in figure 45 is drawn without orientation, then this means that the corresponding identity is valid for any choice of orientation. The invariance under the first, second and fifth moves is immediate. The invariance under the third and forth moves follows from figures 6 and 44, by sliding the cylinder that goes inside the other cylinder towards the end strand, in the obvious way, as shown in figure 46. It is strictly necessary that the edges incident to the vertex in cause have compatible orientations in the sense shown in figure 45. Note that otherwise the crossing informations in the corresponding initial and final broken surface diagrams in figure 46 would not be compatible. The invariance of T (K) under the penultimate moves of figure 45 follows from the same argu- ment that proves invariance under the classical and virtual Reidemeister-I moves. Definition 15 (Welded Virtual Graph) The moves on oriented virtual graph diagrams of fig- ure 45, together with the ones defining welded virtual knots and welded virtual arcs define what we called a “welded virtual graph”. Note that the moves of figure 47 are not allowed. If K is a welded virtual graph, then a welded virtual graph K ′ for which the tube T (K ′) of K ′ is obtained from T (K) by adding a trivial 1-handle is obtained from K by choosing a string of K Figure 44: The tube of a virtual graph at a 3-valent vertex; broken surface diagram version of the movie of figure 43. ↔ ↔ ↔ Figure 45: Moves defining Welded Virtual Graphs. Notice that the third and forth moves have a variant for which the direction of each strand is reversed. However, these moves are a consequence of the remaining. Figure 46: An identity between broken surface diagrams of knotted surfaces (reverse orientation of the fourth move of figure 45.) Figure 47: Fordidden Moves. K K ′ Figure 48: Adding a trivial 1-handle to a welded virtual graph. On the left we display the original graph. Figure 49: Adding a trivial 1-handle to the Hopf Arc yields the Virtual Hopf Link. (in the correct component) and doing the transition shown in figure 48 (adding a trivial 1-handle to a welded virtual graph). For example, consider the Hopf Arc HA defined in Exercise 2. Then adding a trivial 1-handle to the unclosed component of it yields the Virtual Hopf Link L; see figure 49. Note the usage of the moves of figure 45. Let G1 be a welded virtual graph such that, topologically, G1 is the union of circles S 1 and intervals I = [0, 1]. Suppose that G′1 is obtained from G1 by adding a trivial 1-handle to an I- component of it. Then we can always use the moves of figure 45 to find a graph G2, equivalent to G′1 as a welded virtual graph, but so that, topologically, G2 is the union of circles S 1 and intervals I. This was exemplified above for the case of the Hopf Arc HA, and should be compared with the method indicated in [S, page 541]. It is a good exercise to verify that adding a trivial 1-handle to the Trefoil Arc yields the Shin Satoh’s Knot. 3.5.3 The fundamental group of the complement The (combinatorial) fundamental group of a welded virtual graph complement (the knot group) is defined in the same way as the knot group of a virtual knot or arc. However, we consider the relations of figure 50 at the vertices of a graph (the edges incident to a vertex may carry any orientation). Note that this is in sharp contrast with the classical fundamental group of graph complements. In fact, we can easily find examples of welded virtual graphs for which the classical and virtual knot groups are different. The θ-graph which appears in figure 42 is such an example. It is not difficult to see that the knot group is an invariant of welded virtual graphs. Moreover the tube map K 7→ T (K) preserves knot groups. Suppose that the graph K ′ is obtained from K by adding a trivial 1-handle. We can see that the X X X X X X X X . . . . . . Figure 50: The relations satisfied by the knot group of a welded virtual graph at a vertex. (X,e1) (X,e2) (X,en) (X,f1) (X,f2) (X,fm) . . . . . . e1e2...en=f1f2...fm Figure 51: Reduced G-colourings of a welded virtual graph diagram at a vertex. knot groups of K and K ′ are isomorphic, thus also that the fundamental groups of the complements of the tubes T (K) and T (K ′) in S4 are isomorphic. This can easily be proved directly. Given an arc A embedded in the upper semiplane {z ≥ 0} of R3, intersecting the plane {z = 0} at the end points of A, only, there exist two knotted tori naturally associated to A. The first one is obtained from the tube T (A) of A by adding a trivial 1-handle, and a virtual knot c1(A) representing it can be easily determined from A using the method indicated in [S] and 3.5.2. In the second one, one simply closes A in the obvious way, obtaining c2(A), before taking the tube of it. If A is a classical arc, with only one component, then we have that the fundamental groups of the complements of the knotted surfaces T (A), T (c1(A)) and T (c2(A)) are all isomorphic. This also happens if we allow A to have more than one component, as long as all the other components are diffeomorphic to S1. However, it is necessary that A be classical. The pairs of welded virtual knots (c1(A), c2(A)), one for each classical 1-component arc A, provide a family of welded virtual knots with the same knot group. For example if 3′1 is the Trefoil Arc, then c2(3 1) is the Trefoil Knot 31, whereas c1(3 1) is the Shin Satoh Knot. These two can be proven to be non-equivalent by using the crossed module invariant; see 3.5.5. See also subsections 3.6 to 3.10 for other analogous examples. Problem 3 Under which circunstancies are the welded virtual knots c1(A) and c2(A) equivalent? What to say about their tubes in S4. 3.5.4 Crossed module invariants of welded virtual graphs Let G = (E,G, ⊲) be an automorphic finite crossed module. The invariant HG of welded virtual knots, or arcs, extends in a natural way to an invariant of welded virtual graphs K, by considering: HG(K) = IG(T (K)), where IG is the 4-dimensional invariant defined in subsection 2.3. As before, HG(K) can be calcu- lated directly from a diagram of K. Definition 16 Let G = (E,G, ⊲) be a finite automorphic crossed module. Let K be an oriented welded virtual graph diagram chosen so that the projection on the second variable is a Morse function in K. A reduced G-colouring of K is given by an assignment of a pair (X, e) ∈ G × E to each arc of G minus its set of critical points, crossings and vertices, satisfying the conditions already shown for virtual knot and arc diagrams, and the relation displayed in figure 51. We have: Theorem 17 Let G = (E,G, ⊲) be a finite automorphic crossed module. Let also K be an oriented Figure 52: One type of Yetter’s moves capturing planar isotopy of graph diagrams. welded virtual graph diagram. The quantity: HG(K) = #{reduced G colourings of K}#E #{caps}#E−#{cups} #E#{pointing upward 1-valent vertices of K} n-valent vertices v of K #E1−#{edges of K incident to v from above} (13) coincides with IG(T (K)), and therefore defines an invariant of welded virtual graphs. Exercise 4 Check directly that HG is a topological invariant of welded virtual graphs. Together with the moves of figure 45, as well as the moves defining welded virtual knots and arcs, one still needs to check invariance under planar isotopy. Planar isotopies of graph diagrams are captured by Yetter’s moves shown in figure 28, as well as figure 52; see [Y1] and [FY]. Exercise 5 Check directly that HG is invariant under addition of trivial 1-handles, as shown in figure 48; cf. Theorem 3.5.1. 3.5.5 The Trefoil Knot is not equivalent to the Shin-Satoh’s Knot We now use the extension of the Crossed Module Invariant to welded virtual graphs to prove that the Shin Satoh’s Knot S is not equivalent to the Trefoil Knot 31. Let 3 1 be the Trefoil Arc. Recall that S is obtained from 3′1 by adding a trivial 1-handle, in other words S = c1(3 1); see 3.5.3. Therefore, whenever G = (E,G, ⊲) is a finite automorphic crossed module we have: 1) = HG(S). In particular, from equation (8): HG(S) = HG(3 1) = #E# X,Y ∈G;e∈E XY X=Y XY −XY ⊲e+Y ⊲e=e , (14) note that we switched to additive notation. Also, from equation (6): HG(31) = # X,Y ∈G;e,f∈E XY X=Y XY Y XY ⊲f−XY ⊲(e+f)+Y ⊲(e+f)=e . (15) A natural example of a finite automorphic crossed module G = (E,G, ⊲) with G non abelian is constructed by taking G = GLn(Zp) and E = (Zp) n. Here GLn(Zp) denotes the group of n × n matrices in Zp with invertible determinant, where p is a positive integer. The action of GLn(Zp) in (Zp) n is taken to be the obvious one. Denote these crossed modules by G(n, p). Computations with Mathematica prove that HG(n,p)(31) 6= HG(n,p)(S) for example for p = 3, 4, 5, 7 and n = 2; see the following table. This proves that the crossed module invariant dis- tinguishes the Trefoil Knot from the Shin Satoh’s Knot, even though they have the same knot group. Table 1: knot HG2,2 HG2,3 HG2,4 HG2,5 HG2,7 31 96 4320 24576 132000 2272032 S 96 4752 27648 168000 2765952 (X,e−1) (X,e) (Y,f) (Y,f−1) (X−1Y X,X−1⊲f) (X,X−1⊲f−1ef) (X−1Y XY −1X,X−1Y ⊲f−1X−1Y X⊲(ef)) (X−1Y X,X−1Y ⊲fX−1Y X⊲(ef)−1ef) (X−1Y XY −1XY X−1Y −1X,X−1Y XY −1X⊲e−1) (X−1Y XY −1X,X−1Y ⊲f−1X−1Y X⊲(ef)e−1X−1Y XY −1X⊲e) (X−1Y XY −1XY X−1Y −1X,X−1Y XY −1X⊲e−1X−1Y ⊲fX−1Y X⊲(ef)−1efY X−1Y ⊲f−1Y X−1Y X⊲(ef)Y ⊲(ef)−1) (Y X−1Y XY−1,Y X−1Y ⊲fY X−1Y X⊲(ef)−1Y ⊲(ef)) Figure 53: Calculation of the crossed module invariant of the Figure of Eight Knot 41. 3.6 Figure of Eight Knot Let G = (E,G, ⊲) be a finite automorphic crossed module. Let us calculate the crossed module invariant HG(41) of the Figure of Eight Knot 41. This calculation appears in Figure 53. This permits us to conclude that, if G = (E,G, ⊲) is an automorphic crossed module, then: HG(41) = # X,Y ∈G;e,f∈E X−1Y XY −1XY X−1Y −1X=Y X−1Y XY −1X⊲e−1X−1Y ⊲fX−1Y X⊲(ef)−1efY X−1Y ⊲f−1Y X−1Y X⊲(ef)Y ⊲(ef)−1=f X−1Y XY −1X=Y X−1Y XY −1 X−1Y ⊲f−1X−1Y X⊲(ef)e−1X−1Y XY −1X⊲e=Y X−1Y ⊲f−1Y X−1Y X⊲(ef)Y ⊲(ef)−1 Note that the first pair of equations which appear in the previous formula is equivalent to the second one. In the case when G is abelian, the previous formula simplifies to (passing to additive notation): HG(41) = # X ∈ G; e, f ∈ E|(X2 − 3X + 1) ⊲ e = (−X2 + 3X − 1) ⊲ f = #E# X ∈ G; e ∈ E|(X2 − 3X + 1) ⊲ e = 0 as it should, since the Alexander polynomial of the Figure of Eight Knot is ∆(41) = X 2 − 3X + 1; see 3.3.2. The value of the crossed module invariant for the Figure of Eight Arc 4′1 (figure 54), for G not necessarily abelian, can be obtained from equation (16) by making f = 1, and inserting the relevant normalisation factors. This yields: 1) = #E# X,Y ∈G;e∈E X−1Y XY −1XY X−1Y −1X=Y X−1Y XY −1X⊲e−1X−1Y X⊲e−1eY X−1Y X⊲eY ⊲e−1=1 Figure 54: The Figure of Eight Arc 4′1 and the welded virtual knot c1(4 1) obtained from it by adding a trivial 1-handle Consider the welded virtual knot c1(4 1) obtained from the Figure of Eight Arc 4 1 by adding a trivial 1-handle to it; see 3.5.2. This welded virtual knot appears in figure 54. By using Theorem 14, it thus follows that for any finite automorphic crossed module G we have HG(c1(4 1)) = HG(4 see also 3.5.4. Recall that by the discussion in 3.5.3, the knot groups of the welded virtual knots 41 = c2(4 1) and c1(4 1) are isomorphic. Consider the crossed modules G(n,p), where p and n are positive integers, obtained from GLn(Zp) acting on (Zp) n, defined in 3.5.5. Computations with Mathematica prove that HG(n,p)(c1(4 1)) 6= HG(n,p)(41) for p = 3 or p = 7; see the following table. This proves that the welded virtual knots 41 = c2(4 1) and c1(4 1) are not equivalent, even though they have the same knot groups. Table 2: knot HG2,2 HG2,3 HG2,4 HG2,5 HG2,7 41 48 3024 15360 228000 1876896 1) 48 3456 15360 228000 2272032 3.7 The Solomon Seal Knot Let G = (E,G, ⊲) be an automorphic finite crossed module. The crossed module invariant of the (5, 2)-torus knot 51 (the Solomon Seal Knot) is calculated in figure 55. This permits us to conclude that: HG(51) = # X,Y ∈G;e,f∈E XY XY X=Y XYXY Y XYXY ⊲fXY XY ⊲(ef)−1Y XY ⊲(ef)XY ⊲(ef)−1Y ⊲(ef)=e Note that if the crossed module G = (E,G, ⊲) is such that G is abelian, then the previous expression simplifies to: HG(51) = #E# X ∈ G; e ∈ E ∣X4 ⊲ e−X3 ⊲ e+X2 ⊲ e−X ⊲ e+ e = 0 as it should, since the Alexander polynomial of the knot 51 is ∆(51) = X 4 −X3 +X2 −X + 1. The crossed module invariant of the Solomon Seal Arc 5′1, and the welded virtual knot c1(5 obtained from it by adding a trivial 1 handle, each presented in figure 56, can be obtained from this calculation by making f = 1, and inserting the remaining normalisation factors. Therefore it follows that: 1) = #E# X,Y ∈G;e∈E XY XYX=Y XY XY XY XY ⊲e−1Y XY ⊲eXY ⊲e−1Y ⊲e=e Computations with Mathematica show that HG(n,p)(c1(5 1)) 6= HG(n,p)(51) for n = 2 and p = 5; see the following table. Therefore the pair (51, c1(51)) is a pair of welded virtual knots with the same knot group, but distinguished by their crossed module invariant. (X,e) (Y,f) (XY X−1,X⊲f) (X,X⊲f−1ef) (XY XY −1X−1,XY ⊲f−1XY X−1⊲(ef)) (XY X−1,XY ⊲fXY X−1⊲(ef)−1ef) (XY XY X−1Y −1X−1,XY X⊲fXY ⊲(ef)−1XY XY −1X−1⊲(ef)) (XY XY −1X−1,XY X⊲f−1XY ⊲(ef)XY XY −1X−1⊲(ef)−1ef) (XY XY XY −1X−1Y −1X−1,XY XY ⊲f−1XY XY X−1⊲(ef)XY ⊲(ef)−1XY XY X−1Y −1X−1⊲(ef)) (XY XY X−1Y −1X−1,XY XY ⊲fXY XY X−1⊲(ef)−1XY ⊲(ef)XY XY X−1Y −1X−1⊲(ef)−1ef) (Y XY XY XY −1X−1Y −1X−1Y −1,Y XY XY ⊲fXY XY ⊲(ef)−1Y XY ⊲(ef)XY ⊲(ef)−1Y ⊲(ef)) (XY XY XY −1X−1Y −1X−1,Y XY XY ⊲f−1XY XY ⊲(ef)Y XY ⊲(ef)−1XY ⊲(ef)Y ⊲(ef)−1ef) Figure 55: Calculation of the crossed module invariant of the torus knot 51. In the top two colourings, we are using the fact that XYXYX = Y XY XY . Figure 56: The Solomon Seal arc 5′1 and the welded virtual knot c1(5 1) obtained by adding a trivial 1 handle to it. Table 3: knot HG2,2 HG2,3 HG2,4 HG2,5 HG2,7 51 24 432 1536 168000 98784 1) 24 432 1536 204000 98784 Figure 57: The 2-bridge knot 52 and the 2-bridge arc 5 3.8 The 2-bridge knot 52 (Stevedore) We now consider the 2-bridge knot 52 and the 2-bridge arc 5 2, depicted in figure 57. Let us calculate their crossed module invariant. Suppose that G = (E,G, ⊲) is a finite automorphic crossed module. The calculation of HG(52) appears in figure 58. From this it follows that: HG(52) = # X,Y ∈G;e,f,g∈E Y X−1Y XY −1XY −1=X−1Y X−1Y XY −1X Y X−1Y X−1⊲e−1Y X−1Y ⊲(ef)Y X−1Y X−1Y −1X⊲(ef)−1Y ⊲(ef)=g−1 Z=Y X−1Y XY −1XY −1 X−1Y X−1⊲eX−1Y ⊲(ef)−1X−1Y X−1Y −1X⊲(ef)=gfZ−1⊲f−1 Y=Z−1XZ Z−1⊲f=e−1gY −1⊲g−1 . (17) Note that the last two equations in the previous formula follow from the remaining. When G is abelian, the previous expression reduces to: HG(52) = #E#{X ∈ G; e ∈ E|2X 2 ⊲ e− 3e+ 2X−1 ⊲ e = 0}, as it should since the Alexander polynomial of the 52 knot is ∆(52) = 2X 2−3+2X−2. The formula for the crossed module invariant of the arc 5′2 is: HG(52) = #E# X,Y ∈G;e,g∈E Y X−1Y XY −1XY −1=X−1Y X−1Y XY −1X YX−1Y X−1⊲e−1Y X−1Y ⊲eY X−1Y X−1Y −1X⊲e−1Y ⊲e=g−1 Z=Y X−1Y XY −1XY −1 X−1Y X−1⊲eX−1Y ⊲e−1X−1Y X−1Y −1X⊲e=g . (18) Below there is a table comparing the value of HG(n,p)(52) and HG(n,p)(c1(5 2)), for n = 2 and p = 2, 3, 4, 5, 7. Here as usual, c1(5 2) is obtained from the welded virtual arc 5 1 by adding a trivial 1-handle to it. In particular it follows that the welded virtual knots 52 = c2(5 2) and c1(5 2) are not equivalent, even though they have the same knot groups. Table 4: knot HG2,2 HG2,3 HG2,4 HG2,5 HG2,7 52 24 864 1536 72000 987840 2) 24 864 1536 84000 1481760 (X,f) (Y,e) (Y,e (X,f−1) (Z,g) (Z,g (X−1Y X,X−1⊲e) (X,X−1⊲e−1ef) (X−1Y XY −1X,X−1Y ⊲e−1X−1Y X⊲(ef)) (X−1Y X,X−1Y ⊲eX−1Y X⊲(ef)−1ef) (X−1Y X−1Y XY −1X,X−1Y X−1⊲eX−1Y ⊲(ef)−1X−1Y X−1Y −1X⊲(ef)) (X−1Y XY −1X,X−1Y X−1⊲e−1X−1Y ⊲(ef)X−1Y X−1Y −1X⊲(ef)−1ef) (Y −1ZY,Y −1⊲g) (Y,e−1gY −1⊲g−1) (Z−1XZ,Z−1⊲f−1) (Z,g−1f−1Z−1⊲f) Figure 58: Calculation of the crossed module invariant of the 2-bridge knot 52. 3.9 The (n, 2)-torus knot Let n be an odd integer. An analogous calculation as in the case of the Trefoil Knot and the Solomon Seal Knot proves that the crossed module invariant of the (n, 2)-torus knot Kn has the following expression (in additive notation): HG(Kn) = # X,Y ∈G;e,f∈E ⊲ (e+ f)− f = 0 Here Si = X if i is even and Si = Y if i is odd. On the other hand, the crossed module invariant of the arc An, obtained from Kn in the obvious way (see subsection 3.7 for the case n = 5) is: HG(An) = #E# X,Y ∈G;f∈E ⊲ f − f = 0 In the following table, we compare the value, for each positive odd integer n ≤ 17, of the crossed module invariants HG(2,3) and HG(2,5) for the pair of welded virtual knots (Kn, c1(An)), where c1(An) is obtained from An by adding a trivial 1-handle to it. Since the knot groups of c1(An) and of c2(An) = Kn are isomorphic, this gives some more examples of pairs of 1-component welded virtual knots with the same knot group, but distinguished by their crossed module invariant. Table 5: K3 K5 K7 K9 K11 K13 K15 K17 HG(2,3) 4320 432 432 4320 432 432 4320 432 HG(2,5) 132000 168000 12000 132000 12000 12000 288000 12000 c1(A3) c1(A5) c1(A7) c1(A9) c1(A11) c1(A13) c1(A15) c1(A17) HG(2,3) 4752 432 432 4752 432 432 4752 432 HG(2,5) 168000 204000 12000 168000 12000 12000 360000 12000 Figure 59: The link P and the associated arc P ′. 3.10 Final examples Let m be a positive integer. We can define an automorphic crossed module Am = (Zm,Z2, ⊲), where Z2 = {−1, 1,×}, and the action of Z2 on Zm is 1 ⊲ a = a and (−1) ⊲ a = −a, where a ∈ Zm. This generalises the crossed module A = A3 defined in subsection 3.1. Consider the link P , as well as the associated arc P ′, shown in figure 59. Let G = (E,G, ⊲) be a finite automorphic crossed module with G abelian. An easy calculation shows that: HG(P ) = #{X,Y ∈ G; e, f ∈ E| − Y −3X−3 ⊲ f + Y −3X−2 ⊲ (e+ f)− Y −2X−2 ⊲ (e+ f) + Y −2X−1 ⊲ (e+ f) − Y −1X−1 ⊲ (e+ f) + Y −1 ⊲ (e+ f) = e}. ′) = #E#{X,Y ∈ G; e ∈ E| Y −3X−2 ⊲ e− Y −2X−2 ⊲ e+ Y −2X−1 ⊲ e − Y −1X−1 ⊲ e+ Y −1 ⊲ e = e}. In the case of the automorphic crossed modules Am defined above, the previous formulae simplify to (for each positive integer m): HAm(P ) = m 2 + 2m#{a ∈ Zm|2a = 0}+m#{a ∈ Zm|6a = 0}, HAm(P ′) = m (m+#{a ∈ Zm|2a = 0}+m+#{a ∈ Zm|6a = 0}) , thus HA(P ) = 24 and HA(P ′) = 30. Here as usual A = (Z3,Z2, ⊲). Let c1(P ′) be the welded virtual link obtained by adding a trivial 1-handle to the unclosed component of P ′ (see 3.5.3), thus P ′ and c1(P ′) have the same crossed module invariant; see 3.5.4. Hence (P = c2(P ′), c1(P ′)) is a pair of welded virtual links with the same knot group (see Figure 60: Two virtual links, P = c2(P ′) and c1(P ′), with the same knot group but distinguished by their crossed module invariant. 3.5.3), but distinguished by their crossed module invariant HG , where G = (E,G, ⊲) is a finite automorphic crossed module, which can be chosen so that G is abelian. In particular we have Alex(P ) ∼= Alex(c1(P ′)), but CM(P ) ≇ CM(c1(P ′)); see subsection 3.3. Exercise 6 Prove directly that P and c1(P ′) have the same knot group and are distinguished by their crossed module invariant. Exercise 7 The previous example can be generalised. For each positive odd integer n, let Pn be the 3-dimensional torus link in S3 with 2n crossings, similar to the link P in figure 59; in other words, Pn is the (2, 2n)-torus link. Let also P n be the associated arc, and let c1(P n) be the welded virtual link obtained by adding a trivial 1-handle to the unclosed component of P ′n; see figures 59 and 60 for the case n = 6. Prove that for any automorphic crossed module G = (E,G, ⊲), with G abelian, we have that: HG(Pn) = #{X,Y ∈ G; e, f ∈ E| −X−nY −n ⊲ f + (XY )−k ⊲ Y −1 ⊲ (e+ f)− (e+ f) + Y −1 ⊲ (e+ f) = e}, n) = # X,Y ∈ G; e ∈ E (XY )−k ⊲ Y −1 ⊲ e− e + Y −1 ⊲ e = e Thus if n is odd then we have: HAm(Pn) = m 2 + 2m#{a ∈ Zm|2a = 0}+m#{a ∈ Zm|2na = 0}, HAm(P n) = m m+#{a ∈ Zm|2a = 0}+m+#{a ∈ Zm|2na = 0} where as usual Am = (Zm,Z2, ⊲) and m is a positive integer. In particular it follows that HAn(Pn) = 2n 2 + 2n HAn(c1(P n)) = HAn(P n) = 3n 2 + n. This provides an infinite sequence (Pn, c1(P n)), where n is an odd integer, of pairs of 2- component welded virtual links with the same knot group, but distinguished by their crossed module invariant. This sequence includes not only the previous example, but also the case of the Hopf Link and the Virtual Hopf Link in subsection 3.1. = Q1 = Q2 = Q3 Figure61: ThreeVirtualLinkswiththesameknotgroupbutdistinguishedbytheircrossedmodule invariant. Note that, taking tubes, the previous example gives an infinite set of pairs of non-isotopic em- beddings of a disjoint union of two tori S1 × S1 into S4 with the same fundamental group of the complement, but distinguished by their Crossed Module Invariant IG of [FM1]. Another interesting example is provided by the virtual links Q1, Q2 and Q3 shown in figure 61. The knot groups of Q1, Q2 and Q3 are all isomorphic to {X,Y,Z : XY = Y X,ZY = Y Z}. Let G = (E,G, ⊲) be an automorphic crossed module. A simple calculation shows that: HG(Q1) = # X,Y,Z ∈ G; e, f, g ∈ E −Y −1X−1⊲f+Y −1⊲(e+f)=e −Z−1Y ⊲g+Z−1⊲(−f+g)=−f HG(Q2) = # X,Y,Z ∈ G; e, f, g ∈ E −Y −1X−1⊲f+Y −1⊲(e+f)=e Z−1⊲f=f HG(Q3) = # X,Y,Z ∈ G; e, f, g ∈ E Y −1⊲e=e −Z−1Y ⊲g+Z−1⊲g=0 Therefore the crossed module invariant HA, where as usual A = (Z3,Z2, ⊲), separates these Q1, Q2 and Q3. Acknowledgements JFM was financed by Fundação para a Ciência e Tecnologia (Portugal), post-doctoral grant number SFRH/BPD/17552/2004, part of the research project POCI/MAT/60352/2004 (“Quantum Topol- ogy”), also financed by FCT, cofinanced by the European Community fund FEDER. LK thanks the National Science Foundation for support under NSF Grant DMS-0245588. References [BM] Barrett J.W; Mackaay M.: Categorical Representations of Categorical Groups, Theory Appl. Categ. 16 (2006), No. 20, 529–557 (electronic). [B] Brown R.: Groupoids and Crossed Objects in Algebraic Topology, Homology Homotopy Appl. 1 (1999), 1–78 (electronic). 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[F] Fox R.H.: A Quick Trip through Knot Theory. 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 120–167 Prentice-Hall, Englewood Cliffs, N.J. [FY] Freyd P.J.; Yetter D.N.: Braided Compact Closed Categories with Applications to Low- Dimensional Topology. Adv. Math. 77 (1989), no. 2, 156–182. [Ka] Kamada S.: Braid Presentation of Virtual Knots and Welded Knots. Osaka J. Math. Volume 44, Number 2 (2007), 441-458. [K1] Kauffman L.H.: Virtual Knot Theory. European J. Combin. 20 (1999), no. 7, 663–690. [K2] Kauffman L.H.: On Knots. Annals of Mathematics Studies, 115. Princeton University Press, Princeton, NJ, 1987. [KL] Kauffman L.H.; Lambropoulou S.: The L-Move and Virtual Braids. J. Knot Theory Rami- fications 15 (2006), no. 6, 773–811. [P1] Porter T.: Interpretations of Yetter’s Notion of G-Coloring: Simplicial Fibre Bundles and Non-Abelian Cohomology, J. Knot Theory Ramifications 5 (1996), no. 5, 687–720. 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Introduction An Invariant of Welded Virtual Knots Welded virtual knots The fundamental group of the complement Virtual knot presentations of knotted surfaces Welded virtual arcs Crossed module invariants of knotted surfaces The case of ribbon knotted torus The case of welded virtual arcs Examples Virtual and Classical Hopf Link Virtual Hopf Link The Hopf Link Trefoil Knot and Trefoil Arc Universal module constructions The definition of the module CM(K) Relation with the Alexander Module Welded virtual arcs Shin Satoh's Knot Welded Virtual Graphs Crossed module invariants of knotted surfaces obtained by adding trivial 1-handles Definition of welded virtual graphs The fundamental group of the complement Crossed module invariants of welded virtual graphs The Trefoil Knot is not equivalent to the Shin-Satoh's Knot Figure of Eight Knot The Solomon Seal Knot The 2-bridge knot 52 (Stevedore) The (n,2)-torus knot Final examples

0704.1247

A Rational Approach to Resonance Saturation in large-Nc QCD

A Rational Approach to Resonance Saturation in large-Nc QCD Pere Masjuan and Santiago Peris Grup de F́ısica Teòrica and IFAE Universitat Autònoma de Barcelona, 08193 Barcelona, Spain. Abstract We point out that resonance saturation in QCD can be understood in the large-Nc limit from the mathematical theory of Pade Approximants to meromor- phic functions. These approximants are rational functions which encompass any saturation with a finite number of resonances as a particular example, explaining several results which have appeared in the literature. We review the main prop- erties of Pade Approximants with the help of a toy model for the 〈V V − AA〉 two-point correlator, paying particular attention to the relationship among the Chiral Expansion, the Operator Product Expansion and the resonance spectrum. In passing, we also comment on an old proposal made by Migdal in 1977 which has recently attracted much attention in the context of AdS/QCD models. Fi- nally, we apply the simplest Pade Approximant to the 〈V V − AA〉 correlator in the real case of QCD. The general conclusion is that a rational approximant may reliably describe a Green’s function in the Euclidean, but the same is not true in the Minkowski regime due to the appearance of unphysical poles and/or residues. http://arxiv.org/abs/0704.1247v1 1 Introduction The strong Chiral Lagrangian is a systematic organization of the physics in powers of momenta and quark masses, but requires knowledge of the low-energy constants (LEC) to make reliable phenomenological predictions. As with any other effective field theory, these LECs play the role of coupling constants and contain the information which comes from the integration of the heavy degrees of freedom not explicitly present in the Chiral Lagrangian (e.g. meson resonances). At O(p4) there are 10 of these constants [1]. Although at this order there is enough independent information to extract the values for these constants from experiment, this will hardly ever be possible at the next order, O(p6), because the number of constants becomes more than a hundred [2]. In the electroweak sector the proliferation of constants appears already at O(p4) [3]. Although in principle these low-energy constants may be computed on the lattice, in practice this has only been accomplished in a few cases for the strong Chiral Lagrangian at O(p4), and only recently [4]. The large Nc expansion [5] stands out as a very promising analytic approach capa- ble of dealing with the complexities of nonperturbative QCD while, at the same time, offering a relatively simple and manageable description of the physics. For instance, mesons are qq states with no width, the OZI rule is exact and there is even a proof of spontaneous chiral symmetry breaking [6]. Furthermore, its interest has recently received a renewed boost indirectly through the connection of some highly supersym- metric gauge theories to gravity [7], although the real relevance of this connection for QCD still remains to be seen. However, in spite of all this, the fact that no solution to large-Nc QCD has been found keeps posing a serious limitation to doing phenomenol- ogy. For instance, in order to reproduce the parton model logarithm which is present in QCD Green’s functions in perturbation theory, an infinity of resonances is necessary whose masses and decay constants are in principle unknown. On the other hand, QCD Green’s functions seem to be approximately saturated by just a few resonances; a property which has a long-standing phenomenological support going all the way back to vector meson dominance ideas [8], although it has never been properly understood. In a modern incarnation, this fact translated into the very successful observation [9] that the strong LECs at O(p4) seem to be well saturated by the lowest meson in the relevant channel,1 after certain constraints are imposed on some amplitudes at high-energy in order to match the expected behavior in QCD [11, 12]. It was then realized that all these successful results could be encompassed at once as an approximation to large-Nc QCD consisting in keeping only a finite (as opposed to the original infinite) set of resonances in Green’s functions. This approximation to large-Nc QCD has been termed Minimal Hadronic Approximation (MHA) [13] because it implements the minimal constraints which are necessary to secure the leading non- trivial behavior at large energy of certain Green’s functions through the marriage of the old resonance saturation and the large-Nc approximation of QCD. In recent years, a large amount of work has been dedicated to studying the consequences of these ideas [14]. 1This is less clear in the scalar channel, however. See Ref. [10] However, the high-energy matching with a finite set of resonances, first suggested in [11], makes it clear that the treatment is not amenable to the methods of a conventional effective field theory. An effective field theory is an approximation for energies smaller than a heavy particle’s mass and, therefore, cannot deal with momentum expansions at infinity as in the case of the Operator product Expansion (OPE). In other words, the fact that the set of resonances in each channel is really infinite precludes the naive expansion at large momentum because there is always a mass in the spectrum which is even larger. The sum over an infinite set of resonances and the expansion for large momentum are operations which do not commute [15]. In those Green’s functions containing a contribution from the parton model logarithm, this is made self-evident since a naive expansion at large momentum can only produce powers and not a logarithm, which is why large-Nc QCD requires an infinity of resonances in the first place. The problem can be delayed one power of αs if one requires the use of the resonance Lagrangian [9] to be limited only to Green’s functions which are order parameters of spontaneous chiral symmetry breaking. These order parameters vanish to all orders in αs in the chiral limit 2 and, therefore, avoid the presence of the parton model logarithm which, otherwise, would preclude from the outset any matching to a finite number of resonances. However, the concept of a Lagrangian whose validity is restricted only to a certain class of Green’s functions has never been totally clear; and even if the resonance Lagrangian is restricted by definition to order parameters, the problem surfaces again in the presence of logarithmic corrections from nontrivial anomalous dimensions, which make the exact matching at infinite momentum impossible. In a slightly different context, a somewhat similar observation was also made in Ref. [16]. In this paper it was observed that it is impossible to satisfy the large momentum fall-off expected in large-Nc QCD for the form factors which can be defined through a three-point Green’s function, if the sum over resonances in the Green’s function is restricted to a finite set. Interestingly, this again pointed to an incompatibility of the QCD short-distance behavior with an approximation to large Nc which only kept a finite number of resonances. A further piece of interesting evidence results from the comparison between the analysis in Refs. [17] and [11]. After imposing some good high-energy behavior in several Green’s functions and form factors including, in particular, the axial form factor governing the decay π → γeν, Ref. [11] obtains, keeping only one vector state V and one axial-vector state A, that their two masses must be constrained by the relation 2MV . The work in Ref. [17], on the contrary, does not use the axial form factor and obtains, after performing a very good fit within the same set of approximations, the precise values MV = 775.9 ± 0.5 MeV and MA = 938.7 ± 1.4 MeV. These values for the masses, although close, are not entirely compatible with the previous relation. In other words, the short-distance constraint from the axial form factor is not fully compatible with the short-distance constraints used in [17] if restricted to only one vector and one axial-vector states3. 2E.g., the two-point correlator 〈V V −AA〉. 3Adding one further state does not change the conclusion [17]. In this paper we would like to suggest that all the above properties can be un- derstood if the approximation to large Nc QCD with a finite number of resonances is reinterpreted within the mathematical Theory of Pade Approximants (PA) to mero- morphic functions [18]. QCD Green’s functions in the large Nc and chiral limits have an analytic structure in the complex momentum plane which consists of an infinity of isolated poles but no cut, i.e. they become meromorphic functions [19]. As such, they have a well-defined series expansion in powers of momentum around the origin with a finite radius of convergence given by the first resonance mass4. This is all that is needed to construct a Pade Approximant. A theorem by Pommerenke [20] assures then convergence of any near diagonal PA to the true function for any finite momen- tum, over the whole complex plane, except perhaps in a zero-area set. The poles of the original Green’s function (i.e. the resonance masses) belong to this zero-area set because not even the original function is defined there, but there are also extra poles. These extra poles are called “defects” in the mathematical literature [18]. When the Green’s function being approximated is of the Stieltjes type5, the poles of the PA are always real and located on the Minkowski region Re(q2) = Re(−Q2) > 0, approaching the physical poles as the order of the PA is increased [21]. However, this takes place in a hierarchical way and, while the poles in the PA which are closest to the origin are also very close to the physical masses, the agreement quickly deteriorates and one may find that the last poles are several times bigger than their physical counterparts [22]. The same is true of the residues. In section 3, we will see with the help of a model that the same properties are met in a meromorphic function whose spectral function is not positive definite, except that some of the poles in the PA may even be complex. This means that Minkowskian properties, such as masses and decay constants, cannot be reliably determined from a PA except, perhaps, from the first poles which are closest to the origin. If not all the residues and/or masses are physical, then there is no reason why they should be the same in the form factor governing π → γeν and in the Green’s function 〈V V − AA〉, explaining the different results found in [11] and [17] we alluded to above. Furthermore, the form factors of all but the lightest mesons, defined through the residues of the corresponding 3-point Green’s functions, will not be reliably determined from a PA to that Green’s function, again in agreement with the findings in [16]. The situation in the Euclidean is different. In general, PAs cannot be expanded at infinite momentum to generate an OPE type expansion for the true function. Never- theless, Pommerenke’s theorem assures a good approximation at any finite momentum, no matter how large. Of course, the order of the PA will have to increase, the larger the momentum region one wishes to approximate. For instance, in Ref. [21] it was shown with the help of a simple model how, even in the case of the 〈V V 〉 correlator which contains a logQ2 at large values of Q2 > 0, the PAs are capable of approximat- ing the true function at any arbitrarily large (but finite) value of Q2 > 0, without the need for a perturbative continuum. In section 3 we will show, again with the help of a 4The pion pole can always be eliminated multiplying by enough powers of momentum. We are assuming here the existence of a nonvanishing gap in large-Nc QCD. 5Roughly this means that the associated spectral function is positive definite, like in the case of the two-point correlator 〈V V 〉. See Ref. [18] (chapter 5) for a more precise definition. model, how this is also true in the more general case of a non-positive definite spectral function such as 〈V V −AA〉. This means that PAs are a reliable way to approximate the original Green’s function in the Euclidean but not in the Minkowskian regime. In 1977, A.A. Migdal [23] suggested PAs as a method to extract the spectrum of large-Nc QCD from the leading term in the OPE of the 〈V V 〉 correlator, i.e. from the parton model logarithm. However, nowadays this proposal should be considered unsatisfactory for a number of reasons [24], the most simple of them being that different spectra may lead to the same parton model logarithm [25]. In fact, the full OPE series is expected to be only an asymptotic expansion at Q2 = ∞ (i.e. with zero radius of convergence), and PAs constructed from this type of expansions cannot in general reproduce the position of the physical poles [26]. For instance, we show this explicitly with the help of a model for 〈V V − AA〉 in the Appendix. Migdal’s approach has been recently adopted (in disguise) in some models exploiting the so-called AdS/QCD correspondence [27] and, consequently, the same criticism also applies to them. In Ref. [28] a model for the 〈V V − AA〉 two-point correlator with a spectrum consisting of an infinity of resonances was suggested as a theoretical laboratory for studying the relationship between the spectrum and the coefficients of the OPE. In this paper several conventional methods usually employed in the literature were tested against the exact result from the model. These included: Finite Energy Sum Rules as in Ref. [33], pinched weights as in Ref. [34], Laplace transforms as in Ref. [35] and, finally, also resonance saturation as in the MHA method. The bottom line was that no method was able to produce very accurate predictions for the OPE coefficients. In all the methods but the last one, the reason for this lack of accuracy was basically due to the fact that the OPE requires an integral over the whole spectrum, whereas the integral is actually cut off at an upper limit (in the real case, the upper limit is mτ ). This is why even if one uses the real spectrum the result may be inaccurate [29]. In the case of the MHA the reason was, as we will comment upon below, that the poles were not allowed to be complex. In section 3 we will revisit this 〈V V − AA〉 model, now from the point of view of PAs. The model reproduces the power behavior of QCD at large Q2 > 0 except that the model is simple enough not to have any logQ2 and, therefore, it cannot reproduce the nonvanishing anomalous dimensions which exist in QCD. We do not think this is a major drawback, however, because in QCD these logarithms are always screened by at least one power of αs and, hence, in an approximate sense, it may be licit to ignore them. In the model such an approximation becomes exact6. Will the PAs be able to reproduce the large Q2 expansion of the 〈V V −AA〉 model? We will see that the answer is affirmative. Therefore, the reason why the MHA method was not able to predict accurately the OPE coefficients in Ref. [28] is because the lowest PA has complex poles which were not allowed in [28]. When these complex poles are considered, the accuracy achieved is better and, most importantly, improves for a higher PA. Since the model allows the construction of PAs of a very high order, we have checked this convergence up to the Pade P 5052 , which is able to reproduce the first non vanishing coefficient of the OPE in the model with an accuracy of 52 decimal figures. Together 6For a model with a logQ2, the reader may consult Ref. [21]. with other numerical examples which will be discussed in section 3, we take this as a clear evidence of the convergence of the method. This renders some confidence that PAs may also do a good job in the real case of QCD. The rest of the paper is organized as follows. In section 2 we review some generali- ties of rational approximants, in section 3 we describe the 〈V V −AA〉 model and apply different rational approximants to learn about the possible advantages and disadvan- tages of them. In section 4 we apply the simplest PA to the case of the real 〈V V −AA〉 two-point function in QCD. Finally, we close with some conclusions. 2 Rational approximations: generalities Let a function f(z) have an expansion around the origin of the complex plane of the f(z) = n , z → 0 . (1) One defines a Pade Approximant (PA) to f(z) , denoted by PMN (z), as a ratio of two polynomials QM(z), RN (z) 7, of order M and N (respectively) in the variable z, with a contact of order M + N with the expansion of f(z) around z = 0. This means that, when expanding PMN (z) around z = 0, one reproduces exactly the first M + N coefficients of the expansion for f(z) in Eq. (1): PMN (z) = QM(z) RN (z) ≈ f0 + f1 z + f2 z2 + ...+ fM+N zM+N +O(zN+M+1) . (2) At finite z, the rational function PMN (z) constitutes a resummation of the series (1). Of special interest for us will be the case when N =M +k, for a fixed k, because then the function behaves like 1/zk at z = ∞. The corresponding PAs PMM+k(z) belong to what is called the near-diagonal sequence for k 6= 0, with the case k = 0 being the diagonal sequence. The convergence properties of the PAs to a given function are much more difficult than those of normal power series and this is an active field of research in Applied Mathematics. In particular, those which concern meromorphic functions8 are rather well-known and will be of particular interest for this work. The main result which we will use is Pommerenke’s Theorem [20] which asserts that the sequence of (near) diagonal PA’s to a meromorphic function is convergent everywhere in any compact set of the complex plane except, perhaps, in a set of zero area. This set obviously includes the set of poles where the original function f(z) is clearly ill-defined but there may be some other extraneous poles as well. For a given compact region in the complex plane, the previous theorem of convergence requires that, either these extraneous poles move very far away from the region as the order of the Pade increases, or they pair up with a close-by zero becoming what is called a defect in the mathematical jargon 7Without loss of generality we define, as it is usually done, RN (0) = 1. 8A function is said to be meromorphic when its singularities are only isolated poles. [30]. These are to be considered artifacts of the approximation. Near the location of these extraneous poles the PA approximation clearly breaks down but, away from these poles, the approximation is safe. In the physical case the original function f(z) will be a Green’s function G(Q2) of the momentum variable Q2. In QCD in the large Nc limit this Green’s function is meromorphic with all its poles located on the negative real axis in the complex Q2 plane. These poles are identified with the meson masses. On the other hand, the region to be approximated by the PAs will be that of euclidean values for the momentum, i.e. Q2 > 0. The expansion of G(Q2) for Q2 large and positive coincides with the Operator Product Expansion. In general a meromorphic function does not obey any positivity constraints and, as we will see, this has as a consequence that some of the poles and residues of the PAs may become complex 9. This clearly precludes any possibility that these poles and residues may have anything to do with the physical meson masses and decay constants. However, and this is very important to realize, this does not spoil the validity of the rational approximation provided the poles, complex or not, are not in the region of Q2 one is interested in. It is to be considered rather as the price to pay for using a rational function, which has only a finite number of poles, as an approximation to a meromorphic function with an infinite set of poles. When the position of the poles in the original Green’s function is known, at least for the lowest lying states, it is interesting to devise a rational approximation which has this information already built in. The corresponding approximants are called Partial Pade Approximants (PPAs) in the mathematical literature [31] and are given by a rational function PMN,K(Q N,K(Q RN(Q2) TK(Q2) , (3) where QM(Q 2), RN(Q 2) and TK(Q 2) are polynomials of order M,N and K (respec- tively) in the variable Q2. The polynomial TK(Q 2) is defined by having K zeros precisely at the location of the lowest lying poles of the original Green’s function10 2) = (Q2 +M21 ) (Q 2 +M22 ) ... (Q 2 +M2K) . (4) As before the polynomial RN (Q 2) is chosen so that RN (0) = 1 and, together with 2), they are defined so that the ratio PMN,K(Q 2) matches exactly the first M +N terms in the expansion of the original function around Q2 = 0, i.e. : N,K(Q 2) ≈ f0 + f1 Q2 + f2 Q4 + ... + fM+N Q2M+2N +O(Q2N+2M+2) . (5) At infinity, the PPA in Eq. (3) obviously falls off like 1/Q2N+2K−2M . Exactly as it happens in the case of PAs, also the PPAs will have complex poles for a general 9A special case which does obey positivity constraints is when the function is Stieltjes. In this case the poles and residues of the PAs are purely real and with the same sign as those of the original function [21]. 10For simplicity, we will assume that all the poles are simple. meromorphic function, which prevents it from any interpretation in terms of meson states. Finally, another rational approximant defined in mathematics is the so-called Pade Type Approximant (PTA) [31] TMN (Q QM (Q TN (Q2) , (6) where TN (Q 2) is also given by the polynomial (4), now with N preassigned zeros at the corresponding position of the poles of the original Green’s function, G(Q2). The polynomial QM(Q 2) is defined so that the expansion of the PTA around Q2 = 0 agrees with that of the original function up to and including terms of order M + 1, i.e. 2) ≈ f0 + f1 Q2 + f2 Q4 + ... + fM Q2M +O(Q2M+2) . (7) At large values of Q2, one has that TMN (Q 2) falls off like 1/Q2N−2M . Clearly the PTAs are a particular case of the PPAs, i.e. TMN (Q 2) = PM0,N(Q 2) and coincide with what has been called the Hadronic Approximation to large-Nc QCD in the literature [13]. Let us summarize the mathematical jargon. A Pade Type Approximant (PTA) is a rational function with all the poles chosen in advance precisely at the physical masses. A Pade Approximant (PA) is when all the poles are left free. The intermediate situation, with some poles fixed at the physical masses and some left free, corresponds to what is called a Partial Pade Approximant (PPA). 3 Testing rational approximations: a model Let us consider the two-point functions of vector and axial-vector currents in the chiral limit ΠV,Aµν (q) = i d4x eiqx〈JV,Aµ (x)J† V,Aν (0)〉 = qµqν − gµνq2 ΠV,A(q 2) , (8) with J V (x) = d(x)γ µu(x) and J A(x) = d(x)γ µγ5u(x). As it is known, the difference ΠV (q 2)− ΠA(q2) satisfies an unsubtracted dispersion relation11 ΠV−A(q t− q2 − iǫ ImΠV−A(t) . (9) Following Refs. [32, 28], we define our model by giving the spectrum as ImΠV (t) = 2F ρ δ(t−M2ρ ) + 2 F 2V (n)δ(t−M2V (n)) , ImΠA(t) = 2F 0 δ(t) + 2 F 2A(n)δ(t−M2A(n)) . (10) 11The upper cutoff which is needed to render the dispersive integrals mathematically well defined can be sent to infinity provided it respects chiral symmetry [15]. Here Fρ,Mρ are the electromagnetic decay constant and mass of the ρ meson and FV,A(n) are the electromagnetic decay constants of the n− th resonance in the vector (resp. axial) channels, while MV,A(n) are the corresponding masses. F0 is the pion decay constant in the chiral limit. The dependence on the resonance excitation number n is the following: F 2V,A(n) = F 2 = constant , M2V,A(n) = m V,A + n Λ 2 , (11) in accord with known properties of the large-Nc limit of QCD [5] as well as alleged properties of the associated Regge theory [37]. The combination ΠLR(q (ΠV (q 2)−ΠA(q2)) (12) thus reads ΠLR(q −q2 +M2ρ −q2 +M2V (n) −q2 +M2A(n) . (13) This two-point function can be expressed in terms of the Digamma function ψ(z) = log Γ(z) as [28] ΠLR(q −q2 +M2ρ −q2 +m2A −q2 +m2V . (14) To resemble the case of QCD, we will demand that the usual parton-model logarithm is reproduced in both vector and axial-vector channels and that the difference (9) has an operator product expansion which starts at dimension six. A set of parameters satisfying these conditions is given by12 F0 = 85.8 MeV , Fρ = 133.884 MeV , F = 143.758 MeV , (15) Mρ = 0.767 GeV, mA = 1.182 GeV, mV = 1.49371 GeV , Λ = 1.2774 GeV , and is the one we will use in this section. This set of parameters has been chosen to resemble those of the real world, while keeping the model at a manageable level. For instance, the values of Fρ and Mρ in (15) are chosen so that the function ΠLR in (14) has vanishing 1/Q2 and 1/Q4 in the OPE at large Q2 > 0, as in real QCD. In fact, the model admits the introduction of finite widths (which is a 1/Nc effect) in the manner described in Ref. [32], after which the spectral function looks reasonably similar to the experimental spectral function. This comparison can be found in Fig. 5 of Ref. [28]. But this model is also interesting for a very different reason. In Ref. [28] several attempts were made at determining the coefficients of the OPE by using the methods which have become common practice in the literature. Among those we may list Finite Energy Sum Rules [33], with pinched weights [34], Laplace sum rules [35] and Minimal Hadronic Approximation [13]. As it turned out, when these methods were tested on 12These numbers have been rounded off for the purpose of presentation. Some of the exercises which will follow require much more precision than the one shown here. C0 C2 C4 C6 C−4 C−6 C−8 −7.362 21.01 −43.92 81.81 −2.592 1.674 −0.577 Table 1: Values of the coefficients C2k from the high- and low-Q 2 expansions of Q2 ΠLR(−Q2) in Eq. (16) in units of 10−3 GeV 2−2k. Notice that C−2 = 0 and C0 = −F 20 (the pion decay constant in the chiral limit), see text. the model, none of them was able to produce very accurate results. We think that this makes the model very interesting (and challenging !) as a way to assess systematic errors [36]. Defining the expansion of the Green’s function (9) in Q2 = −q2 around Q2 = 0,∞ Q2 ΠLR(−Q2) ≈ C2k Q 2k , with k = 0,±1,±2,±3, . . . (16) one obtains that the coefficients accompanying inverse powers of momentum, akin to the Operator Product Expansion at large Q2 > 0, are given by (p = 1, 2, 3, ... with k = 1− p): C2k = −F 20 δp,1 + (−1)p+1 F 2ρM F 2Λ2p−2 , (17) where Bp(x) are the Bernoulli polynomials [10]. As stated above, Fρ andMρ are defined by the condition that the above expression (17) vanishes for k = 0,−1 enforcing that Q2 ΠLR(−Q2) ∼ Q−4 at large momentum, as in QCD. We emphasize that the above coefficients of the OPE in Eq. (17) can not be calculated by a naive expansion at large Q2 of the Green’s function in Eq. (13). In other words, physical masses and decay constants do not satisfy the Weinberg sum rules [15]. On the other hand, for the coefficients accompanying nonnegative powers of mo- mentum, akin to the chiral expansion at small Q2, one has (k = 1, 2, 3, ...): C0 = −F 20 , C2k = (−1)k+1 (k − 1)! ψ(k−1) − ψ(k−1) where ψ(k−1)(z) = dk−1ψ(z)/dzk−1. In Table 1 we collect the values for the first few of these coefficients C2k. Let us start with the construction of the rational approximants to the function Q2 ΠLR(−Q2). Since our original function (14) falls off at large Q2 as Q−4, this is a constraint we will impose on all our approximants. The simplest PA satisfying the right falloff at large momentum is P 02 (Q 2), so we will begin with this case. In order to simplify the results, and unless explicitly stated otherwise, we will assume that dimensionful quantities are expressed in units of GeV to the appropriate power. Fixing the three unknowns with the first three coefficients from the chiral expansion of (14) (i.e. C0,2,4) one gets the following rational function P 02 (Q − r2R (Q2 + zR)(Q2 + z , r2R = 3.379×10−3 , zR = 0.6550+ i 0.1732 . (19) 5 10 15 20 25 Im (q ) Re (q ) Figure 1: Location of the poles (dots) and zeros (squares) of the Pade Approximant P 5052 (−q2) in the complex q2 plane. We recall that Q2 = −q2. Notice how zeros and poles approximately coincide in the region which is farthest away from the origin. When the order of the Pade is increased, the overall shape of the figure does not change but the two branches of complex poles move towards the right, i.e. away from the origin. We can hardly overemphasize the striking appearance of a pair of complex-conjugate poles on the Minkowski side of the complex Q2 plane. Obviously, this means that these poles cannot be interpreted in any way as the meson states appearing in the physical spectrum (10,13). In spite of this, if one expands (19) for large values of Q2 > 0, one finds C−4 = −r2R = −3.379 × 10−3 which is not such a bad approximation for this coefficient of the OPE, see Table 1. Even better is the prediction of the fourth term in the chiral expansion, which is C6 = 79.58× 10−3. This agreement is not a numerical coincidence and the approximation can be sys- tematically improved if more terms of the chiral expansion are known. In order to exemplify this, we have amused ourselves by constructing the high-order PA P 5052 (Q This rational approximant correctly determines the values for C−4,−6,−8 with (respec- tively) 52, 48 and 45 decimal figures. In the case of C103, which is the first predictable term from the chiral expansion for this Pade, the accuracy reaches some staggering 192 decimal figures. This is all in agreement with Pommerenke’s theorem [20]. As it happens for the PA (19), also higher-order PAs may develop some artificial poles. In particular, Figure 1 shows the location of the 52 poles of the PA P 5052 (Q 2) in the complex q2 plane. Of these, the first 25 are purely real and the rest are complex- conjugate pairs. A detailed numerical analysis reveals that the poles and residues reproduce very well the value of the meson masses and decay constants for the lowest part of the physical spectrum of the model given in (13-15), but the agreement deteri- orates very quickly as one gets farther away from the origin, eventually becoming the complex numbers seen in Fig. 1. It is by creating these analytic defects that rational functions can effectively mimic with a finite number of poles the infinite tower of poles present in the original function (14). For instance the values of the first pole and residue in P 5052 (Q 2) reproduce those of the ρ in (15) within 193 astonishing decimal places for both. However, in the case of the 25th pole, which is the last one still purely real, its location agrees with the physical mass only with 3 decimal figures. This is not to be considered as a success, however, because after the previous accuracy, this is quite a dramatic drop. In fact, the residue associated with this 25th pole comes out to be 29 times the true value. The lesson we would like to draw from this exercise should be clear: the determination of decay constants and masses extracted as the residues and poles of a PA deteriorate very quickly as one moves away from the origin. There is no reason why the last poles and residues in the PA are to be anywhere near their physical counterparts and their identification with the particle’s mass and decay constant should be considered unreliable. Clearly, this particularly affects low-order PAs. A very good accuracy can also be obtained in the determination of global euclidean observables such as integrals of the Green’s function over the interval 0 ≤ Q2 < ∞. Notice that the region where one approximates the true function is far away from the artificial poles in the PA. For instance, one may consider the value for the integral Iπ = (−1) dQ2 Q2ΠLR(Q 2) = 4.78719× 10−3, (20) which, up to a constant, would yield the electromagetic pion mass difference in the chiral limit [38] in the model (14). The PA P 5052 (Q 2) reproduces the value for this integral with more than 42 decimal figures. This suggests that one may use the integral (20) as a further input to construct a PA. For example if we fix the three unknowns in the PA P 02 (Q 2) by matching the first two terms from the chiral expansion but now we complete it with the pion mass dif- ference (20) instead of a third term from the chiral expansion as we did in (19),13 the approximant results to be P̃ 02 (Q − r2R (Q2 + zR)(Q2 + z , with r2R = 2.898× 10−3 , zR = 0.5618 + i 0.2795 . This determines C−4 = −2.898×10−3 and C4 = −41.26×10−3, which shows that using the pion mass difference is not a bad idea. Notice how the position of the artificial pole has changed with respect to (19). Artificial poles and analytic defects are transient in nature, i.e. they appear and disappear from a point in the complex plane when the order of the Pade is changed. On the contrary, the typical sign that a pole in a Pade is associated with a truly physical pole is its stability under these changes in the order of the Pade. Of course, when the order in the Pade increases there have to be new poles by definition, and it is natural to expect that some of them will be defects. Pade Approximants place some effective poles and residues in the complex Q2 plane in order to mimic the behavior of the true Green’s function, but it can mimic the function only away from the poles, e.g. in the Euclidean region. Obviously, PAs cannot converge at the poles, in agreement with Pommerenke’s theorem [20], since not even the true function is well defined there. The point is that what may look like a small correction in the Euclidean region may turn out to be a large number in the Minknowski region. To exemplify this in simple terms, 13We remark that this procedure, although reasonable from the phenomenological point of view, strictly speaking lies outside the standard mathematical theory of rational approximants [18, 31]. let us consider a very small parameter ǫ and imagine that a given Pade P (Q2) produces the rational approximant to the true Green’s function G(Q2) given by G(Q2) ≈ P (Q2) ≡ R(Q2) + ǫ Q2 +M2 , (22) where R(Q2) is the part of the Pade which is independent of ǫ. Although for Q2 > 0 there is a sense in which the last term is a small correction precisely because of the smallness of ǫ, for Q2 < 0 this is no longer true because of the pole at Q2 = −M2. This pole is in general a defect and may not represent any physical mass. In fact, associated with this pole, there is a very close-by zero of the Pade P (Q2) at Q2 = −M2 − ǫ R(−M2)−1, as can be immediately checked in (22). This is another way of saying that a defect is characterized by having an abnormally small residue and is the origin of the pairs of zeros and poles in the y-shaped branches of Fig. 1. Therefore, not only are defects unavoidable but one could say they are even necessary for a Pade Approximant to approximate a meromorphic function with an infinite set of poles. Similarly to masses, also decay constants may be unreliable. To see this, imagine now that our Pade is given by P (Q2) = Q2 +M2 (Q2 +M2) (Q2 +M2 + ǫ2) , (23) again for a very small ǫ. As before, the term proportional to ǫ may be considered a small correction for Q2 > 0. However, at the pole Q2 = −M2 the decay constant becomes F + ǫ−1 which, for ǫ small, may represent a huge correction. When the poles are preassigned at the physical masses, like in the case of PTAs, it is the value of the residues that compensates for the fact that the rational approximant lacks the infinite tower of resonances. As we saw before, the residues of the poles in the Pade which lie farthest away from the origin are the ones which get the largest distortion relative to their physical counterparts. In real life, the number of available terms from the chiral expansion for the con- struction of a PA is very limited. Since the masses and decay constants of the first few vector and axial-vector resonances are known, one may envisage the construction of a rational approximant having some of its poles at the prescribed values given by the known masses of these resonances. If all the poles in the approximant are prescribed this way (as in the MHA), we have a PTA. On the contrary, when some of the poles are prescribed but some are also left free, then we have a PPA (see the previous section). Assuming that the first masses are known, let us proceed to constructing the PTAs (6). The lowest such PTA is T02(Q 2), which contains two poles at the physical masses of the ρ and the first A in the tower. Fixing the residue through the chiral expansion to be C0 = −F 20 , one obtains − F 20M2ρM2A (Q2 +M2ρ )(Q 2 +M2A) . (24) Even though it has the same number of inputs (C0 and the two masses), this rational approximant does not do such a good job as the PAs (19) or (21). For instance, C−4 is 2.3 times larger than the true value in Table 1. As we have already stated, one way to intuitively understand this result is the following. The OPE is an expansion at Q2 = ∞ and therefore knows about the whole spectrum because no resonance is heavy enough with respect to Q2 to become negligible in the expansion, i.e., the infinite tower of resonances does not decouple in the OPE. Chopping an infinite set of poles down to a finite set may be a good approximation, but only at the expense of some changes. These changes amount to the appearance of poles and residues in the PA which the original function does not have. This is how the PA (19) manages to approximate the true function (14). However, by construction, the PTA (24) does not allow the presence of any artificial pole because, unlike in a PA, all its poles are fixed at the physical values. Consequently, it only has its residues as a means to compensate for the infinite tower of poles present in the true function and, hence, does a poorer job than the PA (19), particularly in determining large-Q2 observables like C−4. Indeed, the role played by the residues in the approximation can be appreciated by comparing the true values of the decay constants to those extracted from (24). Although the one of the ρ is within 30% of the true value, that of the A is off by 100%. A different matter is the prediction of low-energy observables such as, e.g., the chiral coefficients. In this case heavy resonances make a small contribution and this means that the infinite tower of resonances does decouple.14 Truncating the infinite tower down to a finite set of poles is not such a severe simplification in this case, which helps understand why a PTA may do a good job predicting unknown chiral coefficients. Indeed, (24) reproduces the value of C2 within an accuracy of 15%, growing to 22% in the case of C4. A global observable like Iπ averages the low and the high Q 2 behaviors and ends up differing from the true value (20) by 35%. This gives some confidence that observables which are integrals over Euclidean momentum may be reasonably estimated with MHA as, e.g., in the BK calculation of Ref. [39]. Improving on the PTA (24) by adding in the first resonance mass from the vector tower produces the following approximant a + b Q2 (Q2 +M2ρ )(Q 2 +M2A)(Q 2 +M2V ) , with a = −13.5× 10−3, b = +1.33× 10−4 , (25) where the values of the chiral coefficients C0 and C2 have been used to determine the parameters a and b. The prediction for C4 is much better now (only 2% off), in agreement with our previous comments. The prediction for C−4 is still very bad, becoming now 19 times smaller than the exact value. Nevertheless, it eventually gets much better if PTAs of very high order are constructed. For instance, we have found C−4 = −2.58 × 10−3 for the approximant T79 with 9 poles. Similarly, we have also checked that the prediction of the chiral coefficients and the integral (20) improve with higher-order PTAs. However, another matter is the prediction of the residues. For instance, the predic- tion for the decay constant of the state with mass MV in (25) is smaller than the exact value in the model (15) by a factor of 2. In general, we have seen that the residues 14This is because the residues F 2 in the Green’s function (14) stay constant as the masses grow. This behavior does not hold in the case of the scalar and pseudoscalar two-point functions [10]. of the poles always deteriorate very quickly so that the residue corresponding to the pole which is at the greatest distance from the origin is nowhere near the exact value. We again explicitly checked this up to the approximant T79, in which case the decay constant for this pole is almost 5 times smaller than the exact value. The conclusion, therefore, is that PTAs are able to approximate the exact function only at the expense of changing the residues of the poles from their physical values. Identifying residues with physical decay constants may be completely wrong in a PTA for the poles which are farthest away from the origin. As an intermediate approach between PAs and PTAs, there are the PPAs (3) where some poles are fixed at their physical values while some others are left free. The simplest of such rational approximants is P01,1(Q 2) (see the previous section for notation). Fixing its 3 unknowns with M2ρ , C0 and C2, one obtains 1,1(Q − r2R (Q2 +M2ρ )(Q 2 + zR) , with r2R = 3.75× 10−3 , zR = 0.8665 . (26) As can be seen, the mass (squared) of the first A resonance is predicted to be at zR which is sensibly smaller than the true value in (15)15. The rational function (26) predicts C−4 = −r2R = −3.75 × 10−3 which is a better determination than that of the PTA (24) with the same number of inputs, and C4 = −45.52× 10−3 which is not bad either. Concerning the pion mass difference, one gets Iπ = 5.22 × 10−3. However, as compared to the PAs (19) or (21), the PPA (26) does not represent a clear improvement. In order to improve on accuracy of the PPA, one may try to use the mass and decay constant of the first resonance, Mρ and Fρ, in addition to the pion mass difference and the chiral coefficients C0, C2 and build the P 2,1(Q 2), which can be written as: 2,1(Q F 2ρM Q2 +M2ρ a− F 2ρM2ρ Q2 (Q2 + zc) (Q2 + z∗c ) a = 17.43× 10−3, zc = 1.24 + i 0.34 . This PPA, upon expansion at large and small Q2, determines C−4 = −2.47× 10−3 and C4 = −44.0× 10−3 to be compared with the corresponding coefficient in Table 1. The accuracy obtained is better than that of (21), but this is probably to be expected since (27) has more inputs. Based on the previous numerical experiments done on the model in Eq. (14,15) (and many others), we now summarize the following conclusions. Although, in principle, the PAs have the advantage of reaching the best precision by carefully adjusting the polynomial in the denominator to have some effective poles which simulate the infinite tower present in (14), they have the disadvantage that some of the terms in the low- Q2 expansion are required precisely to construct this denominator. This hampers the construction of high-order PAs and consequently limits the possible accuracy. When the locations of the first poles in the true function are known, there is the possibility to construct PTAs (with all the poles fixed at the true values) and PPA (with some of the poles fixed and some left free). As we have seen, although the PTA may approximate low-Q2 properties of the true function reasonably well, the large-Q2 15Intriguinly enough, this is also what happens in the real case of QCD [9, 17]. properties tend to be much worse, at least as long as they are not of unrealistically high order. The PPAs, on the other hand, interpolate smoothly between the PAs (only free poles) and the PTAs (no free pole). Depending on the case, one may choose one or several of these rational approximants. However, common to all the rational approximants constructed is the fact that the residues and/or poles which are farthest away from the origin are in general unrelated to their physical counterparts. 4 The QCD case Let us now discuss the real case of large-Nc QCD in the chiral limit. In contrast to the case of the previous model, any analysis in this case is limited by two obvious facts. First, any input value will have an error (from experiment and because of the chiral and large-Nc limits), and this error will propagate through the rational approximant. And second, it is not possible to go to high orders in the construction of rational approxi- mants due to the rather sparse set of input data. In spite of these difficulties one may feel encouraged by the phenomenological fact that resonance saturation approximates meson physics rather well. The simplest PA to the function Q2ΠLR(−Q2) with the right fall-off as Q−4 at large Q2 is P 02 (Q P 02 (Q 1 + A Q2 +B Q4 . (28) The values of the three unknowns a, A and B may be fixed by requiring that this PA reproduces the correct values for F0, L10 and Iπ 16 given by F0 = 0.086± 0.001 GeV , δmπ = 4.5936± 0.0005 MeV =⇒ Iπ = (5.480± 0.006)× 10−3GeV4 , L10(0.5 GeV) ≤ L10 ≤ L10(1.1 GeV) =⇒ L10 = (−5.13± 0.6)× 10−3 . (29) The low-energy constant L10 is related to the chiral coefficient C2, in the notation of Eq. (16), by C2 = −4L10. Since L10 does not run in the large-Nc limit, it is not clear at what scale to evaluate L10(µ) [40]. In Eq. (29) we have varied µ in the range 0.5 GeV ≤ µ ≤ 1.1 GeV as a way to estimate 1/Nc systematic effects. The central value corresponds to the result for L10(Mρ) found in Ref. [42]. The other results in (29) are extracted from Refs. [1, 41]. Obviously, the PA (28) can also be rewritten as P 02 (Q (Q2 + zV )(Q2 + zA) , (30) in terms of two poles zV,A. In order to discuss the nature of these poles, we will define the dimensionless parameter ζ by the combination ζ ≡ −4L10 = 2.06± 0.25 , (31) 16Recall that Iπ is, up to a constant, the electromagnetic pion mass difference δmπ [17] and is defined in terms of ΠLR as in Eq. (20). where the values in (29) above have been used in the last step. Imposing the constraints (29) on the PA (30) one finds two types of solutions depending on the value of ζ : for ζ > 2 the two poles zV,A are real, whereas for ζ < 2 the two poles are complex. At ζ = 2, the two solutions coincide. To see this, let us write the set of equations satisfied by the PA (30) as: F 20 = zV zA −4L10 = F 20 Iπ = F zV zA zA − zV . (32) The first of these equations can be used to determine the value of the residue r2 in terms of zV zA. In order to analyze the other two, let us first assume that both poles zV,A are real. In this case, they also have to be positive or else the integral Iπ will not exist because it runs over all positive values of Q2. Let us now make the change of variables zV = R (1− x) , zA = R (1 + x) . (33) The condition zV,A > 0 translates into R > 0, |x| < 1. In terms of these new variables, the second and third equations in (32) can be combined into 1 + x , (34) where the definition (31) for ζ has been used. With the help of the identity log(1 + x/1− x) = 2 th−1x (for |x| < 1), one can finally rewrite this expression as th−1x , (x real) (35) which is an equation with a solution for x only if ζ ≥ 2. Once this value of x is found, the value of R can always be obtained from one of the last two equations (32) and this determines the two real poles zV,A from (33). On the other hand, when ζ < 2, Eq. (35) does not have a solution. However, according to (31), ζ can also be smaller than 2. In order to study this case, we may use the identity th−1(i y) = i tan−1(y) to rewrite the above equation (35) in terms of the variable x = i y (y real) as tan−1y , (y real). (36) One now finds that this equation has a solution for y when ζ ≤ 2. In this case the poles of the PA (28) are complex-conjugate to each other and can be obtained as zV,A = R(1±i y). These poles, obviously, cannot be associated with any resonance mass and this is why this solution has been discarded in all resonance saturation schemes up to now. However, from the point of view of the rational approximant (28) there is C0 C2 C4 C6 C8 C−4 C−6 C−8 −F 20 −4L10 −43± 13 81± 53 −145± 120 −4.1± 0.5 6± 2 −7± 6 Table 2: Values of the coefficients C2k in the high- and low-Q 2 expansions of Q2 ΠLR(−Q2) in Eq. (16) in units of 10−3 GeV 2−2k. Recall that C−2 = 0. nothing wrong with this complex solution, as the approximant is real and well behaved. From the lessons learned in the previous section with the model, there is no reason to discard this solution since, as we saw, rational approximants may use complex poles to produce accurate approximations. Therefore, we propose to use both the complex as well as the real solution for the poles zV,A, at least insofar as the value for ζ ≷ 2. In this case we obtain, using the values given in Eqs. (29), (ζ ≥ 2) , r2 = −(4.1± 0.5)× 10−3 , zV = (0.77)2 ± 0.15 , zA = (0.96)2 ± 0.41 (37) (ζ ≤ 2) , r2 = −(3.9± 0.1)× 10−3 , zV = z∗A = (0.66± 0.06) + i (0.25± 0.25) , (38) in units of GeV6 for r2, and GeV2 for zV,A. The two solutions in Eqs. (37,38) have been separated for illustrative purposes only. It is clear that they are continuously connected through the boundary at ζ = 2, at which value the two poles coincide and zV = zA ≃ 0.72. The errors quoted are the result of scanning the spread of values in (29) through the equations (32). With both set of values in (37,38), one can get to a prediction for the chiral and OPE coefficients by expansion in Q2 and 1/Q2, respectively. These expansions of the PA can be done entirely in the Euclidean region Q2 > 0, away from the position of the poles zV,A, whether real or complex. Recalling the notation in Eq. (16), the above P produces the coefficients for these expansions collected in Table 2. The values for the OPE coefficients C−4,−6,−8 in this table are compatible with those of Ref. [17], after multiplying by a factor of two in order to agree with the normalization used by these authors. However, the spectrum in our case is different because of the complex solution in (38). As we saw in the previous section with a model, this again shows that Euclidean properties of a given Green’s function, such as the OPE and chiral expansions, or integrals over Q2 > 0 are safer to approximate with a rational approximant than Minkowskian quantities, such as resonance masses and decay constants. 5 Conclusions In this article we pointed out that approximating large-Nc QCD with a finite number of resonances may be reinterpreted within the mathematical Theory of Pade Approxi- mants to meromorphic functions [18]. The main results of this theory may be summarized as follows. One may expect convergence of a sequence of Pade Approximants to a QCD Green’s function in the large-Nc limit in any compact region of the complex Q 2 plane except at most in a zero-area set [20]. This set without convergence comprises the poles of the original Green’s function together with some other artificial poles generated by the approximant which the original function does not have. As the order of the PA grows, the previous convergence property implies that any given artificial pole either goes to infinity, away from the relevant region, or is almost compensated by a nearby zero. This symbiosis between a pole and a zero is called a defect. Although close to a pole the rational approximation breaks down, in a region which is far away from it the approximation should work well. We have reviewed the main results of this theory with the help of a model for the two-point Green’s function 〈V V −AA〉. The simpler case of a Green’s function of the Stieltjes type, such as the two-point correlator 〈V V 〉, was previously considered in Ref. [21]. We have seen in the case of this particular model how rational approximants create the expected artificial poles (and the corresponding residues) in the Minkowski region Re(q2) > 0 while, at the same time, yielding an accurate description of the Green’s function in the Euclidean region Re(q2) < 0. This happens in a hierarchical way: although the first poles/residues in a PA may be used to describe the physical masses/decay constants reasonably well, the last ones give only a very poor description. Therefore, it is in general unreliable to extract properties of individual mesons, such as masses and decay constants, from an approximation to large-Nc QCD with only a finite number of states. Since a form factor, like a decay constant, is obtained as the residue of a Green’s function at the corresponding pole(s), this also means that one may not extract a meson form factor from a rational approximant to a 3-point Green’s function, in agreement with [16]. This observation may explain why the analysis of Ref. [43], which is based on an extraction of matrix elements such as 〈π|S|P 〉 and 〈π|P |S〉 from the 3-point function 〈SPP 〉, finds values for the Kℓ3 form factor which are different from those obtained in other analyses [44]. In spite of all the above problems related to the Minkowski region, our model shows how Pade Approximants may nevertheless be a useful tool in other regions of momentum space. We think that this is also true in the real case of QCD in the large- Nc limit. In this case one may use the first few terms of the chiral and operator product expansions of a given Green’s function to construct a Pade Approximant which should yield a reasonable description of this function in those regions of momentum space which are free of poles. In this construction, Pade Approximants containing complex poles, if they appear, should not be dismissed. We have also reanalyzed the simplest approximation to the 〈V V − AA〉 Green’s function in real QCD which consists of keeping only two poles, and we have found that, depending on the value of the combination ζ in Eq. (31), these two poles may actually be complex. However, if not all the residues and masses in a rational approximant are physical, this poses a challenge to any attempt to use a Lagrangian with a finite number of resonances such as, for example, the ones in Ref. [9, 11], for describing Green’s functions in the large-Nc limit of QCD. Even if these Lagrangians are interpreted in terms of PTAs, with the poles fixed at the physical value of the meson masses, we have seen how the residues then get very large corrections with respect to their physical counterparts. Can these residues be efficiently incorporated in a Lagrangian framework? We hope to be able to devote some work to answering this and related questions in the future. Acknowledgements S.P. is indebted to M. Golterman, M. Knecht and E. de Rafael for innumerable discussions during the last years which have become crucial to shape his understanding on these issues. He is also very grateful to C. Diaz-Mendoza, P. Gonzalez-Vera and R. Orive, from the Dept. of Mathematical Analysis at La Laguna Univ., for invaluable conversations on the properties of Pade Approximants as well as for hospitality. We thank S. Friot, M. Golterman, M. Jamin, R. Kaiser, J. Portoles and E. de Rafael for comments on the manuscript. This work has been supported by CICYT-FEDER-FPA2005-02211, SGR2005-00916 and by the EU Contract No. MRTN-CT-2006-035482, “FLAVIAnet”. APPENDIX Here we will show how the PAs constructed from the OPE do not in general re- produce even the first resonances in the spectrum, unlike those constructed from the chiral expansion. Again, we will use the model of section 3 as an example. Recalling the definition of the OPE given in Eq. (16), with the corresponding coefficients (17), it is straightforward to construct a PA in 1/Q2 around infinity, i.e. by matching powers of the OPE in 1/Q2. The construction parallels that in Eq. (2) but with the replace- ment z = 1/Q2. 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0704.1248

Unification and Fermion Mass Structure

Unification and fermion mass structure. Graham G. Ross ∗and Mario Serna † Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP November 18, 2021 Abstract Grand Unified Theories predict relationships between the GUT-scale quark and lepton masses. Using new data in the context of the MSSM, we update the values and uncertainties of the masses and mixing angles for the three generations at the GUT scale. We also update fits to hierarchical patterns in the GUT-scale Yukawa matrices. The new data shows not all the classic GUT-scale mass relationships remain in quantitative agreement at small to moderate tan β. However, at large tan β, these discrepancies can be eliminated by finite, tan β-enhanced, radiative, threshold corrections if the gluino mass has the opposite sign to the wino mass. Explaining the origin of fermion masses and mixings remains one of the most important goals in our attempts to go beyond the Standard Model. In this, one very promising possibility is that there is an underlying stage of unification relating the couplings responsible for the fermion masses. However we are hindered by the fact that the measured masses and mixings do not directly give the structure of the underlying Lagrangian both because the data is insufficient unambiguously to reconstruct the full fermion mass matrices and because radiative corrections can obscure the underlying structure. In this letter we will address both these points in the context of the MSSM. We first present an analysis of the measured mass and mixing angles continued to the GUT scale. The analysis updates previous work, using the precise measurements of fermion masses and mixing angles from the b-factories and the updated top-quark mass from CDF and D0. The resulting data at the GUT scale allows us to look for underlying patterns which may suggest a unified origin. We also explore the sensitivity of these patterns to tanβ-enhanced, radiative threshold corrections. We next proceed to extract the underlying Yukawa coupling matrices for the quarks and leptons. There are two difficulties in this. The first is that the data cannot, without some assumptions, determine all elements of these matrices. The second is that the Yukawa coupling matrices are basis dependent. We choose to work in a basis in which the mass matrices are hierarchical in structure with the off-diagonal elements small relative to the appropriate combinations of on-diagonal matrix elements. This is the basis we think is most likely to display the structure of the underlying theory, for example that of a spontaneously broken family symmetry in which the hierarchical structure is ordered by the (small) order parameter breaking the symmetry. With this structure to leading order the observed masses and mixing angles determine the mass matrix elements on and above the diagonal, and our analysis determines these entries, again allowing for significant tanβ enhanced radiative corrections. The resulting form of the mass matrices provides the “data” for developing models of fermion masses such as those based on a broken family symmetry. The data set used is summarized in Table 1. Since the fit of reference [4] (RRRV) to the Yukawa texture was done, the measurement of the Standard-Model parameters has improved considerably. We highlight a few of the changes in the data since 2000: The top-quark mass has gone from Mt = 174.3 ± 5 GeV to Mt = 170.9 ± 1.9 GeV. In 2000 the Particle Data Book reported mb(mb) = 4.2 ± 0.2 GeV [5] which has improved to mb(mb) = 4.2 ± 0.07 GeV today. In addition each higher order QCD correction pushes down the value of mb(MZ) at the scale of the Z bosons mass. In 1998 mb(MZ) = 3.0 ± 0.2 GeV [6] and today it is mb(MZ) = 2.87 ± 0.06 GeV [7]. The most significant shift in the data relevant to the RRRV fit is a downward revision to the strange-quark mass at the scale µL = 2 GeV from ms(µL) ≈ 120± 50 MeV [5] to ∗[email protected] †[email protected] http://arxiv.org/abs/0704.1248v2 Low-Energy Parameter Value(Uncertainty in last digit(s)) Notes and Reference mu(µL)/md(µL) 0.45(15) PDB Estimation [1] ms(µL)/md(µL) 19.5(1.5) PDB Estimation [1] mu(µL) +md(µL) [8.8(3.0), 7.6(1.6)] MeV PDB, Quark Masses, pg 15 [1]. ( Non-lattice, Lattice ) −(md+mu)2/4 22.8(4) Martemyanov and Sopov [2] ms(µL) [103(20) , 95(20)] MeV PDB, Quark Masses, pg 15 [1]. [Non-lattice, lattice] mu(µL) 3(1) MeV PDB, Quark Masses, pg 15 [1]. Non-lattice. md(µL) 6.0(1.5) MeV PDB, Quark Masses, pg 15 [1]. Non-lattice. mc(mc) 1.24(09) GeV PDB, Quark Masses, pg 16 [1]. Non-lattice. mb(mb) 4.20(07) GeV PDB, Quark Masses, pg 16,19 [1]. Non-lattice. Mt 170.9 (1.9)GeV CDF & D0 [3] Pole Mass (Me,Mµ,Mτ ) (0.511(15), 105.6(3.1), 1777(53) ) MeV 3% uncertainty from neglect- ing Y e thresholds. A Wolfenstein parameter 0.818(17) PDB Ch 11 Eq. 11.25 [1] ρ Wolfenstein parameter 0.221(64) PDB Ch 11 Eq. 11.25 [1] λ Wolfenstein parameter 0.2272(10) PDB Ch 11 Eq. 11.25 [1] η Wolfenstein parameter 0.340(45) PDB Ch 11 Eq. 11.25 [1] |VCKM | 0.97383(24) 0.2272(10) 0.00396(09) 0.2271(10) 0.97296(24) 0.04221(80) 0.00814(64) 0.04161(78) 0.999100(34)  PDB Ch 11 Eq. 11.26 [1] sin 2β from CKM 0.687(32) PDB Ch 11 Eq. 11.19 [1] Jarlskog Invariant 3.08(18)× 10−5 PDB Ch 11 Eq. 11.26 [1] vHiggs(MZ) 246.221(20) GeV Uncertainty expanded. [1] ( α−1EM (MZ), αs(MZ), sin 2 θW (MZ) ) ( 127.904(19), 0.1216(17), 0.23122(15)) PDB Sec 10.6 [1] Table 1: Low-energy observables. Masses in lower-case m are MS running masses. Capital M indicates pole mass. The light quark’s (u,d,s) mass are specified at a scale µL = 2 GeV. VCKM are the Standard Model’s best fit values. today’s value ms(µL) = 103± 20 MeV. We also know the CKM unitarity triangle parameters better today than six years ago. For example, in 2000 the Particle Data book reported sin 2β = 0.79 ± 0.4 [5] which is improved to sin 2β = 0.69± 0.032 in 2006 [1]. The sin 2β value is about 1.2 σ off from a global fit to all the CKM data [8], our fits generally lock onto the global-fit data and exhibit a 1 σ tension for sin 2β. Together, the improved CKM matrix observations add stronger constraints to the textures compared to data from several years ago. We first consider the determination of the fundamental mass parameters at the GUT scale in order simply to compare to GUT predictions. The starting point for the light-quark masses at low scale is given by the χ2 fit to the data of Table 1 mu(µL) = 2.7± 0.5 MeV md(µL) = 5.3± 0.5 MeV ms(µL) = 103± 12 MeV. (1) Using these as input we determine the values of the mass parameters at the GUT scale for various choices of tanβ but not including possible tanβ enhanced threshold corrections. We do this using numerical solutions to the RG equations. The one-loop and two-loop RG equations for the gauge couplings and the Yukawa couplings in the Standard Model and in the MSSM that we use in this study come from a number of sources [6] [9][10] [11]. The results are given in the first five columns of Table 2. These can readily be compared to expectations in various Grand Unified models. The classic prediction of SU(5) with third generation down- quark and charged-lepton masses given by the coupling B 5f .10f .5H 1 is mb(MX)/mτ (MX) = 1 [12]. This ratio is given in Table 2 where it may be seen that the value agrees at a special low tanβ value but for large tanβ it is some 25% smaller than the GUT prediction2. A similar relation between the strange quark and the muon is untenable and to describe the masses consistently in SU(5) Georgi and Jarlskog [14] proposed that the second generation masses should come instead from the coupling C 5f .10f .45H leading instead to the relation 3ms(MX)/mµ(MX) = 1. As may be seen from Table 2 in all cases this ratio is approximately 0.69(8). The prediction of Georgi and Jarlskog for the lightest generation masses follows from the relation Det(Md)/Det(M l) = 1. This results from the form of their mass matrix which is given by3  , M l = A −3C  (2) in which there is a (1, 1) texture zero4 and the determinant is given by the product of the (3, 3), (1, 2) and (2, 1) elements. If the (1, 2) and (2, 1) elements are also given by 5f .10f .5H couplings they will be the same in the down-quark and charged-lepton mass matrices giving rise to the equality of the determinants. The form of eq(2) may be arranged by imposing additional continuous or discrete symmetries. One may see from Table 2 that the actual value of the ratio of the determinants is quite far from unity disagreeing with the Georgi Jarlskog relation. In summary the latest data on fermion masses, while qualitatively in agreement with the simple GUT relations, has significant quantitative discrepancies. However the analysis has not, so far, included the SUSY threshold corrections which substantially affect the GUT mass relations at large tanβ [15]. A catalog of the full SUSY threshold corrections is given in [16]. The particular finite SUSY thresholds discussed in this letter do not decouple as the super partners become massive. We follow the approximation described in Blazek, Raby, and Pokorski (BRP) for threshold corrections to the CKM elements and down-like mass eigenstates [17]. The finite threshold corrections to Y e and Y u and are generally about 3% or smaller δY u, δY d . 0.03 (3) and will be neglected in our study. The logarithmic threshold corrections are approximated by using the Standard-Model RG equations from MZ to an effective SUSY scale MS. The finite, tanβ-enhanced Y d SUSY threshold corrections are dominated by the a sbottom-gluino loop, a stop-higgsino loop, and a stop-chargino loop. Integrating out the SUSY particles at a scale MS leaves the matching condition at that scale for the Standard-Model Yukawa couplings: δmsch Y uSM = sinβ Y u (4) δmsch Y d SM = cosβ U 1 + Γd + V CKM Γ u VCKM Y ddiag U R (5) Y e SM = cosβ Y e. (6) All the parameters on the right-hand side take on their MSSM values in the DR scheme. The factor δmsch converts the quark running masses from MS to DR scheme. The β corresponds to the ratio of the two Higgs VEVs vu/vd = tanβ. The U matrices decompose the MSSM Yukawa couplings at the scale MS : Y u = U diagU R and Y d = U diagU R. The matrices Y diag and Y diag are diagonal and correspond to the mass eigenstates divided by the appropriate VEV at the scale MS . The CKM matrix is given by VCKM = U L . The left-hand side involves the Standard-Model Yukawa couplings. The matrices Γ u and Γd encode the SUSY threshold corrections. If the squarks are diagonalized in flavor space by the same rotations that diagonalize the quarks, the matrices Γu and Γd are diagonal: Γd = diag(γd, γd, γb), Γ u = diag(γu, γu, γt). In general the squarks are 15f , 10f refer to the SU(5) representations making up a family of quarks and leptons while 5H is a five dimensional representation of Higgs scalars. 2We’d like to thank Ilja Dorsner for pointing out that the tan β dependence of mb/mτ (MX) is more flat than in previous studies (e.g. ref. [13]). This change is mostly due to the higher effective SUSY scale MS , the higher value of αs(MZ) found in global standard model fits, and smaller top-quark mass Mt. 3The remaining mass matrix elements may be non-zero provided they do not contribute significantly to the deteminant 4Below we discuss an independent reason for having a (1, 1) texture zero. Parameters Input SUSY Parameters tanβ 1.3 10 38 50 38 38 γb 0 0 0 0 −0.22 +0.22 γd 0 0 0 0 −0.21 +0.21 γt 0 0 0 0 0 −0.44 Parameters Corresponding GUT-Scale Parameters with Propagated Uncertainty yt(MX) 6 −5 0.48(2) 0.49(2) 0.51(3) 0.51(2) 0.51(2) yb(MX) 0.0113 +0.0002 −0.01 0.051(2) 0.23(1) 0.37(2) 0.34(3) 0.34(3) yτ (MX) 0.0114(3) 0.070(3) 0.32(2) 0.51(4) 0.34(2) 0.34(2) (mu/mc)(MX) 0.0027(6) 0.0027(6) 0.0027(6) 0.0027(6) 0.0026(6) 0.0026(6) (md/ms)(MX) 0.051(7) 0.051(7) 0.051(7) 0.051(7) 0.051(7) 0.051(7) (me/mµ)(MX) 0.0048(2) 0.0048(2) 0.0048(2) 0.0048(2) 0.0048(2) 0.0048(2) (mc/mt)(MX) 0.0009 +0.001 −0.00006 0.0025(2) 0.0024(2) 0.0023(2) 0.0023(2) 0.0023(2) (ms/mb)(MX) 0.014(4) 0.019(2) 0.017(2) 0.016(2) 0.018(2) 0.010(2) (mµ/mτ )(MX) 0.059(2) 0.059(2) 0.054(2) 0.050(2) 0.054(2) 0.054(2) A(MX) 0.56 +0.34 −0.01 0.77(2) 0.75(2) 0.72(2) 0.73(3) 0.46(3) λ(MX) 0.227(1) 0.227(1) 0.227(1) 0.227(1) 0.227(1) 0.227(1) ρ̄(MX) 0.22(6) 0.22(6) 0.22(6) 0.22(6) 0.22(6) 0.22(6) η̄(MX) 0.33(4) 0.33(4) 0.33(4) 0.33(4) 0.33(4) 0.33(4) J(MX) × 10 −5 1.4+2.2−0.2 2.6(4) 2.5(4) 2.3(4) 2.3(4) 1.0(2) Parameters Comparison with GUT Mass Ratios (mb/mτ )(MX) 1.00 +0.04 −0.4 0.73(3) 0.73(3) 0.73(4) 1.00(4) 1.00(4) (3ms/mµ)(MX) 0.70 −0.05 0.69(8) 0.69(8) 0.69(8) 0.9(1) 0.6(1) (md/3me)(MX) 0.82(7) 0.83(7) 0.83(7) 0.83(7) 1.05(8) 0.68(6) (detY detY e )(MX) 0.57 +0.08 −0.26 0.42(7) 0.42(7) 0.42(7) 0.92(14) 0.39(7) Table 2: The mass parameters continued to the GUT-scale MX for various values of tanβ and threshold corrections γt,b,d. These are calculated with the 2-loop gauge coupling and 2-loop Yukawa coupling RG equations assuming an effective SUSY scale MS = 500 GeV. not diagonalized by the same rotations as the quarks but provided the relative mixing angles are reasonably small the corrections to flavour conserving masses, which are our primary concern here, will be second order in these mixing angles. We will assume Γu and Γd are diagonal in what follows. Approximations for Γu and Γd based on the mass insertion approximation are found in [18][19][20]: γt ≈ y t tanβ , µ2) ∼ y2t γu ≈ −g 2 M2 µ ,m2χ2 ,m ũ) ∼ 0 (8) M3 µ I3(m M3 µ I3(m where I3 is given by 2, b2, c2) = a2b2 log a + b2c2 log b + c2a2 log c (a2 − b2)(b2 − c2)(a2 − c2) . (11) In these expressions q̃ refers to superpartner of q. χj indicate chargino mass eigenstates. µ is the coefficient to the Hu Hd interaction in the superpotential. M1,M2,M3 are the gaugino soft breaking terms. A t refers to the soft top-quark trilinear coupling. The mass insertion approximation breaks down if there is large mixing between the mass eigenstates of the stop or the sbottom. The right-most expressions in eqs(7,9,10) assume the relevant squark mass eigenstates are nearly degenerate and heavier than M3 and µ. These expressions ( eqs 7 - 10) provide an approximate mapping from a supersymmetric spectra to the γi parameters through which we parameterize the threshold corrections; however, with the exception of Column A of Table 4, we do not specify a SUSY spectra but directly parameterize the thresholds corrections through γi. The separation between γb and γd is set by the lack of degeneracy of the down-like squarks. If the squark masses for the first two generations are not degenerate, then there will be a corresponding separation between the (1,1) and (2,2) entries of Γd and Γu. If the sparticle spectra is designed to have a large At and a light stop, γt can be enhanced and dominate over γb. Because the charm Yukawa coupling is so small, the scharm-higgsino loop is negligible, and γu follows from a chargino squark loop and is also generally small with values around 0.02 because of the smaller g2 coupling. In our work, we approximate Γ 22 ∼ Γ 11 ∼ 0. The only substantial correction to the first and second generations is given by γd [15]. As described in BRP, the threshold corrections leave |Vus| and |Vub/Vcb| unchanged to a good approxi- mation. Threshold corrections in Γu do affect the Vub and Vcb at the scale MS giving V SMub − V V MSSMub V SMcb − V V MSSMcb ⋍ − (γt − γu) . (12) The threshold corrections for the down-quark masses are given approximately by md ⋍ m d (1 + γd + γu) ms ⋍ m s (1 + γd + γu) mb ⋍ m b (1 + γb + γt) where the superscript 0 denotes the mass without threshold corrections. Not shown are the nonlinear effects which arise through the RG equations when the bottom Yukawa coupling is changed by threshold effects. These are properly included in our final results obtained by numerically solving the RG equations. Due to our assumption that the squark masses for the first two generations are degenerate, the combina- tion of the GUT relations given by detM l/ detMd (3ms/mµ) (mb/mτ ) = 1 is unaffected up to nonlinear effects. Thus we cannot simultaneously fit all three GUT relations through the threshold corrections. A best fit requires the threshold effects given by γb + γt ≈ −0.22± 0.02 (13) γd + γu ≈ −0.21± 0.02. (14) giving the results shown in the penultimate column of Table 2, just consistent with the GUT predictions. The question is whether these threshold effects are of a reasonable magnitude and, if so, what are the implications for the SUSY spectra which determine the γi? From eqs(9,10), at tanβ = 38 we have ∼ −0.5, ∼ 1.0 The current observation of the muon’s (g − 2)µ is 3.4 σ [21] away from the Standard-Model prediction. If SUSY is to explain the observed deviation, one needs tanβ > 8 [22] and µM2 > 0 [23]. With this sign we must have µM3 negative and the d̃, s̃ squarks only lightly split from the b̃ squarks. M3 negative is characteristic of anomaly mediated SUSY breaking[24] and is discussed in [25][26][20][27]. Although we have deduced M3 < 0 from the approximate eqs(9,10), the the correlation persists in the near exact expression found in eq(23) of ref [17]. Adjusting to different squark splitting can occur in various schemes[28]. However the squark splitting can readily be adjusted without spoiling the fit because, up to nonlinear effects, the solution only requires the constraints implied by eq(13), so we may make γb > γd and hence make m by allowing for a small positive value for γt. In this case A t must be positive. It is of interest also to consider the threshold effects in the case that µM3 is positive. This is illustrated in the last column of Table 2 in which we have reversed the sign of γd, consistent with positive µM3 , and chosen γb ≃ γd as is expected for similar down squark masses. The value of γt is chosen to keep the equality between mb and mτ . One may see that the other GUT relations are not satisfied, being driven further away by the threshold corrections. Reducing the magnitude of γb and γd reduces the discrepancy somewhat but still limited by the deviation found in the no-threshold case (the fourth column of Table 2). Parameter 2001 RRRV Fit A0 Fit B0 Fit A1 Fit B1 Fit A2 Fit B2 tanβ Small 1.3 1.3 38 38 38 38 a′ O(1) 0 0 0 0 −2.0 −2.0 ǫu 0.05 0.030(1) 0.030(1) 0.0491(16) 0.0491(15) 0.0493(16) 0.0493(14) ǫd 0.15(1) 0.117(4) 0.117(4) 0.134(7) 0.134(7) 0.132(7) 0.132(7) |b′| 1.0 1.75(20) 1.75(21) 1.05(12) 1.05(13) 1.04(12) 1.04(13) arg(b′) 90o +93(16)o − 93(13)o +91(16)o − 91(13)o +93(16)o − 93(13)o a 1.31(14) 2.05(14) 2.05(14) 2.16(23) 2.16(24) 1.92(21) 1.92(22) b 1.50(10) 1.92(14) 1.92(15) 1.66(13) 1.66(13) 1.70(13) 1.70(13) |c| 0.40(2) 0.85(13) 2.30(20) 0.78(15) 2.12(36) 0.83(17) 2.19(38) arg(c) − 24(3)o − 39(18)o − 61(14)o − 43(14)o − 59(13)o − 37(25)o − 60(13)o Table 3: Results of a χ2 fit of eqs(15,16) to to the data in Table 2 in the absence of threshold corrections. We set a′ as indicated and set c′ = d′ = d = 0 and f = f ′ = 1 at fixed values. At tanβ near 50 the non-linear effects are large and b − τ unification requires γb + γt ∼ −0.1 to −0.15. In this case it is possible to have t − b − τ unification of the Yukawa couplings. For µ > 0,M3 > 0, the “Just-so” Split-Higgs solution of references [29, 30, 31, 32] can achieve this while satisfying both b → s γ and (g − 2)µ constraints but only with large γb and γt and a large cancellation in γb + γt. In this case, as in the example given above, the threshold corrections drive the masses further from the mass relations for the first and second generations because µM3 > 0. It is possble to have t−b−τ unification with µM3 < 0, satisfying the b → s γ and (g − 2)µ constraints in which the GUT predictions for the first and second generation of quarks is acceptable. Examples include Non-Universal Gaugino Mediation [33] and AMSB; both have some very heavy sparticle masses ( & 4 TeV) [20]. Minimal AMSB with a light sparticle spectra( . 1 TeV), while satisfying (g − 2)µ and b → s γ constraints, requires tanβ less than about 30 [23]. We turn now to the second part of our study in which we update previous fits to the Yukawa matrices responsible for quark and lepton masses. As discussed above we choose to work in a basis in which the mass matrices are hierarchical with the off-diagonal elements small relative to the appropriate combinations of on-diagonal matrix elements. This is the basis we think is most likely to display the structure of the underlying theory, for example that of a spontaneously broken family symmetry, in which the hierarchical structure is ordered by the (small) order parameter breaking the symmetry. With this structure to leading order in the ratio of light to heavy quarks the observed masses and mixing angles determine the mass matrix elements on and above the diagonal provided the elements below the diagonal are not anomalously large. This is the case for matrices that are nearly symmetrical or for nearly Hermitian as is the case in models based on an SO(10) GUT. For convenience we fit to symmetric Yukawa coupling matrices but, as stressed above, this is not a critical assumption as the data is insensitive to the off-diagonal elements below the diagonal and the quality of the fit is not changed if, for example, we use Hermitian forms. We parameterize a set of general, symmetric Yukawa matrices as: Y u(MX) = y d′ǫ4u b ′ ǫ3u c ′ ǫ3u b′ ǫ3u f ′ ǫ2u a ′ ǫ2u c′ ǫ3u a ′ ǫ2u 1  , (15) Y d(MX) = y d ǫ4d b ǫ d c ǫ b ǫ3d f ǫ d a ǫ c ǫ3d a ǫ  . (16) Although not shown, we always choose lepton Yukawa couplings at MX consistent with the low-energy lepton masses. Notice that the f coefficient and ǫd are redundant (likewise in Y u). We include f to be able to discuss the phase of the (2,2) term. We write all the entries in terms of ǫ so that our coefficients will be O(1). We will always select our best ǫ parameters such that |f | = 1. RRRV noted that all solutions, to leading order in the small expansion parameters, only depend on two Parameter A B C B2 C2 tanβ 30 38 38 38 38 γb 0.20 −0.22 +0.22 −0.22 +0.22 γt −0.03 0 −0.44 0 −0.44 γd 0.20 −0.21 +0.21 −0.21 +0.21 a′ 0 0 0 −2 −2 ǫu 0.0495(17) 0.0483(16) 0.0483(18) 0.0485(17) 0.0485(18) ǫd 0.131(7) 0.128(7) 0.102(9) 0.127(7) 0.101(9) |b′| 1.04(12) 1.07(12) 1.07(11) 1.05(12) 1.06(10) arg(b′) 90(12)o 91(12)o 93(12)o 95(12)o 95(12)o a 2.17(24) 2.27(26) 2.30(42) 2.03(24) 1.89(35) b 1.69(13) 1.73(13) 2.21(18) 1.74(10) 2.26(20) |c| 0.80(16) 0.86(17) 1.09(33) 0.81(17) 1.10(35) arg(c) − 41(18)o − 42(19)o − 41(14)o − 53(10)o − 41(12)o Y u33 0.48(2) 0.51(2) 0.51(2) 0.51(2) 0.51(2) Y d33 0.15(1) 0.34(3) 0.34(3) 0.34(3) 0.34(3) Y e33 0.23(1) 0.34(2) 0.34(2) 0.34(2) 0.34(2) (mb/mτ )(MX) 0.67(4) 1.00(4) 1.00(4) 1.00(4) 1.00(4) (3ms/mµ)(MX) 0.60(3) 0.9(1) 0.6(1) 0.9(1) 0.6(1) (md/3me)(MX) 0.71(7) 1.04(8) 0.68(6) 1.04(8) 0.68(6)∣∣∣detY d(MX) detY e(MX ) ∣∣∣ 0.3(1) 0.92(14) 0.4(1) 0.92(14) 0.4(1) Table 4: A χ2 fit of eqs(15,16) including the SUSY threshold effects parameterized by the specified γi. phases φ1 and φ2 given by φ1 = (φ b − φ f )− (φb − φf ) (17) φ2 = (φc − φa)− (φb − φf ). (18) where φx is the phase of parameter x. For this reason it is sufficient to consider only b ′ and c as complex with all other parameters real. As mentioned above the data favours a texture zero in the (1, 1) position. With a symmetric form for the mass matrix for the first two families, this leads to the phenomenologically successful Gatto Sartori Tonin [34] relation Vus(MX) ≈ ∣∣bǫd − |b′|ei φb′ ǫu ∣∣∣∣ . (19) This relation gives an excellent fit to Vus with φ1 ≈ ± 90 o, and to preserve it we take d, d′ to be zero in our fits. As discussed above, in SU(5) this texture zero leads to the GUT relation Det(Md)/Det(M l) = 1 which, with threshold corrections, is in good agreement with experiment. In the case that c is small it was shown in RRRV that φ1 is to a good approximation the CP violating phase δ in the Wolfenstein parameterization. A non-zero c is necessary to avoid the relation Vub/Vcb = mu/mc and with the improvement in the data, it is now necessary to have c larger than was found in RRRV 5. As a result the contribution to CP violation coming from φ2 is at least 30%. The sign ambiguity in φ1 gives rise to an ambiguity in c with the positive sign corresponding to the larger value of c seen in Tables 3 and 4. Table 3 shows results from a χ2 fit of eqs(15,16) to to the data in Table 2 in the absence of threshold corrections. The error, indicated by the term in brackets, represent the widest axis of the 1σ error ellipse in parameter space. The fits labeled ‘A’ have phases such that we have the smaller magnitude solution of |c|, and fits labeled ‘B’ have phases such that we have the larger magnitude solution of |c|. As discussed above, it is not possible unambiguously to determine the relative contributions of the off-diagonal elements of the up and down Yukawa matrices to the mixing angles. In the fit A2 and B2 we illustrate the uncertainty 5As shown in ref. [35], it is possible, in a basis with large off-diagonal entries, to have an Hermitian pattern with the (1,1) and (1,3) zero provided one carefully orchestrates cancelations among Y u and Y d parameters. We find this approach requires a strange-quark mass near its upper limit. associated with this ambiguity, allowing for O(1) coefficients a′. In all the examples in Table 3, the mass ratios, and Wolfenstein parameters are essentially the same as in Table 2. The effects of the large tanβ threshold corrections are shown in Table 4. The threshold corrections depend on the details of the SUSY spectrum, and we have displayed the effects corresponding to a variety of choices for this spectrum. Column A corresponds to a “standard” SUGRA fit - the benchmark Snowmass Points and Slopes (SPS) spectra 1b of ref([36]). Because the spectra SPS 1b has large stop and sbottom squark mixing angles, the approximations given in eqns(7-10) break down, and the value for the correction γi in Column A need to be calculated with the more complete expressions in BRP [17]. In the column A fit and the next two fits in columns B and C, we set a′ and c′ to zero. Column B corresponds to the fit given in the penultimate column of Table 2 which agrees very well with the simple GUT predictions. It is characterized by the “anomaly-like” spectrum with M3 negative. Column C examines the M3 positive case while maintaining the GUT prediction for the third generation mb = mτ . It corresponds to the “Just-so” Split-Higgs solution. In the fits A, B and C the value of the parameter a is significantly larger than that found in RRRV. This causes problems for models based on non-Abelian family symmetries, and it is of interest to try to reduce a by allowing a′, b′ and c′ to vary while remaining O(1) parameters. Doing this for the fits B and C leads to the fits B2 and C2 given in Table 4 where it may be seen that the extent to which a can be reduced is quite limited. Adjusting to this is a challenge for the broken family-symmetry models. Although we have included the finite corrections to match the MSSM theory onto the standard model at an effective SUSY scale MS = 500 GeV, we have not included finite corrections from matching onto a specific GUT model. Precise threshold corrections cannot be rigorously calculated without a specific GUT model. Here we only estimate the order of magnitude of corrections to the mass relations in Table 2 from matching the MSSM values onto a GUT model at the GUT scale. The tanβ enhanced corrections in eq(7-10) arise from soft SUSY breaking interactions and are suppressed by factors of MSUSY /MGUT in the high-scale matching. Allowing for O(1) splitting of the mass ratios of the heavy states, one obtains corrections to yb/yτ (likewise for the lighter generations) of O( g (4π)2 ) from the X and Y gauge bosons and O( (4π)2 ) from colored Higgs states. Because we have a different Higgs representations for different generations, these threshold correction will be different for correcting the 3ms/mµ relation than the mb/mτ relation. These factors can be enhanced in the case there are multiple Higgs representation. For an SU(5) SUSY GUT these corrections are of the order of 2%. Plank scale suppressed operators can also induce corrections to both the unification scale [37] and may have significant effects on the masses of the lighter generations [38]. In the case that the Yukawa texture is given by a broken family symmetry in terms of an expansion parameter ǫ, one expects model dependent corrections of order ǫ which may be significant. In summary, in the light of the significant improvement in the measurement of fermion mass parameters, we have analyzed the possibility that the fermion mass structure results from an underlying supersymmetric GUT at a very high-scale mirroring the unification found for the gauge couplings. Use of the RG equations to continue the mass parameters to the GUT scale shows that, although qualitatively in agreement with the GUT predictions coming from simple Higgs structures, there is a small quantitative discrepancy. We have shown that these discrepancies may be eliminated by finite radiative threshold corrections involving the supersymmetric partners of the Standard-Model states. The required magnitude of these corrections is what is expected at large tanβ, and the form needed corresponds to a supersymmetric spectrum in which the gluino mass is negative with the opposite sign to the Wino mass. We have also performed a fit to the recent data to extract the underlying Yukawa coupling matrices for the quarks and leptons. This is done in the basis in which the mass matrices are hierarchical in structure with the off-diagonal elements small relative to the appropriate combinations of on-diagonal matrix elements, the basis most likely to be relevant if the fermion mass structure is due to a spontaneously broken family symmetry. We have explored the effect of SUSY threshold corrections for a variety of SUSY spectra. The resulting structure has significant differences from previous fits, and we hope will provide the “data” for developing models of fermion masses such as those based on a broken family symmetry. M.S. acknowledges support from the United States Air Force Institute of Technology. The views expressed in this letter are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the US Government. 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0704.1249

Cluster tilting for one-dimensional hypersurface singularities

CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN Abstract. In this article we study Cohen-Macaulay modules over one-dimensional hypersurface sin- gularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological methods, using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfy τ2 = id. In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve singularities. Introduction Motivated by the Fomin-Zelevinsky theory of cluster algebras [FZ1, FZ2, FZ3], a tilting theory in cluster categories was initiated in [BMRRT]. For a finite dimensional hereditary algebra H over a field k, the associated cluster category CH is the orbit category D b(H)/F , where Db(H) is the bounded derived category of finite dimensional H-modules and the functor F : Db(H) → Db(H) is τ−1[1] = S−1[2]. Here τ denotes the translation associated with almost split sequences/triangles and S the Serre functor [BK] on Db(H). (See [CCS] for an independent definition of a category equivalent to the cluster category when H is of Dynkin type An). An object T in a cluster category CH was defined to be a (cluster) tilting object if Ext (T, T ) = 0, and if Ext1CH (X,X ⊕ T ) = 0, then X is in addT . The corresponding endomorphism algebras, called cluster tilted algebras, were investigated in [BMR1] and subsequent papers. A useful additional property of a cluster tilting object was that even the weaker condition Ext1CH (X,T ) = 0 implies that X is in addT , called Ext-configuration in [BMRRT]. Such a property also appears naturally in the work of the second author on a higher theory of almost split sequences in module categories [I1, I2] and the notion corresponding to the above definition was called maximal 1-orthogonal. For the category mod(Λ) of finite dimensional modules over a preprojective algebra of Dynkin type Λ over an algebraically closed field k, the concept corresponding to the above definition of cluster tilting object in a cluster category was called maximal rigid [GLSc]. Also in this setting it was shown that being maximal 1-orthogonal was a consequence of being maximal rigid. The same result holds for the stable category mod(Λ). The categories CH and mod(Λ) are both triangulated categories [Ke, H], with finite dimensional ho- momorphism spaces, and they have Calabi-Yau dimension 2 (2-CY for short) (see [BMRRT, Ke][AR, 3.1,1.2][C][Ke, 8.5]). The last fact means that there is a Serre functor S = Σ2, where Σ is the shift functor in the triangulated category. For an arbitrary 2-CY triangulated category C with finite dimensional homomorphism spaces over a field k, a cluster tilting object T in C was defined to be an object satisfying the stronger property discussed above, corresponding to the property of being maximal 1-orthogonal/Ext-configuration [KR]. The corresponding class of algebras, containing the cluster tilted ones, have been called 2-CY tilted. With this concept many results have been generalised from cluster categories, and from the stable categories mod(Λ), to this more general setting in [KR], which moreover contains several results which are new also in the first two cases. One of the important applications of classical tilting theory has been the construction of derived equivalences: Given a tilting bundle T on a smooth projective variety X , the total right derived functor of Hom(T, ) is an equivalence from the bounded derived category of coherent sheaves onX to the bounded The first author was supported by the DFG project Bu 1866/1-1, the second and last author by a Storforsk grant 167130 from the Norwegian Research Council. http://arxiv.org/abs/0704.1249v3 2 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN derived category of finite dimensional modules over the endomorphism algebra of T . Analogously, cluster tilting theory allows one to establish equivalences between very large factor categories appearing in the local situation of Cohen-Macaulay modules and categories of modules over finite dimensional algebras. Namely, if CM(R) is the stable category of maximal Cohen-Macaulay modules over an odd-dimensional isolated hypersurface singularity, then CM(R) is 2-CY. If it contains a cluster tilting object T , then the functor Hom(T, ) induces an equivalence between the quotient of CM(R) by the ideal of morphisms factoring through τT and the category of finite dimensional modules over the endomorphism algebra B = End(T ). It is then not hard to see that B is symmetric and the indecomposable nonprojective B-modules are τ -periodic of τ -period at most 2. In this article, we study examples of this setup arising from finite, tame and wild CM-type isolated hypersurface singularities R. The endomorphism algebras of the cluster tilting objects in the tame case occur in lists in [BS, Er, Sk]. We also obtain a large class of symmetric finite dimensional algebras where the stable AR-quiver consists only of tubes of rank one or two. Examples of (wild) selfinjective algebras whose stable AR-quiver consists only of tubes of rank one or three were known previously [AR]. In the process we investigate the relationship between cluster tilting and maximal rigid objects. It is of interest to know if the first property implies the second one in general. In this paper we provide interesting examples where this is not the case. The setting we deal with are the simple isolated hypersurface singularities R in dimension one over an algebraically closed field k, with the stable category CM(R) of maximal Cohen-Macaulay R-modules being our 2-CY category. These singularities are indexed by the Dynkin diagrams, and in the cases Dn for odd n and E7 we give examples of maximal rigid objects which are not cluster tilting. We also deal with cluster tilting and (maximal) rigid objects in the category CM(R), defined in an analogous way. We also investigate the other Dynkin diagrams, and it is interesting to notice that there are cases with no nonzero rigid objects (An, n even, E6, E8), and cases where the maximal rigid objects coincide with the cluster tilting objects (An, n odd and Dn, n even). In the last case we see that both loops and 2- cycles can occur for the associated 2-CY tilted algebras, whereas this never happens for the cases CH and mod(Λ) [BMRRT, BMR2, GLSc]. The results are also valid for any odd-dimensional simple hypersurface singularity, since the stable categories of Cohen-Macaulay modules are all triangle equivalent [Kn, So]. We shall construct a large class of one-dimensional hypersurface singularities R, where CM(R) or CM(R) has a cluster tilting object, including examples coming from simple singularities and minimally elliptic singularities. We classify all rigid objects in CM(R) for these R, in particular, we give a bijection between cluster tilting objects in CM(R) and elements in a symmetric group. Our method is based on a higher theory of almost split sequences [I1, I2], and a crucial role is played by the endomorphism algebras EndR(T ) (called ‘three-dimensional Auslander algebras’) of cluster tilting objects T in CM(R). These algebras have global dimension three, and have 2-CY tilted algebras as stable factors. The functor HomR(T, ) : CM(R) → mod(EndR(T )) sends cluster tilting objects in CM(R) to tilting modules over EndR(T ). By comparing cluster tilting mutations in CM(R) and tilting mutation in CM(EndR(T )), we can apply results on tilting mutation due to Riedtmann-Schofield [RS] and Happel-Unger [HU1, HU2] to get information on cluster tilting objects in CM(R). We focus on the interplay between cluster tilting theory and birational geometry (see section 5 for definitions). In [V1, V2], Van den Bergh established a relationship between crepant resolutions of sin- gularities and certain algebras called non-commutative crepant resolutions, via derived equivalence. It is known that endomorphism algebras of cluster tilting objects of three-dimensional normal Gorenstein sin- gularities are 3-CY in the sense that the bounded derived category of finite length modules is 3-CY, and they form a class of non-commutative crepant resolutions [I2, IR]. Thus we have a connection between cluster tilting theory and birational geometry. We translate Katz’s criterion [Kat] for three-dimensional cAn–singularities for existence of crepant resolutions to a criterion for one-dimensional hypersurface sin- gularities for existence of cluster tilting objects. Consequently the class of hypersurface singularities, which are shown to have cluster tilting objects by using higher almost split sequences, are exactly the class having non-commutative crepant resolutions. However we do not know whether the number of cluster tilting objects has a meaning in birational geometry. CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 3 In section 2 we investigate maximal rigid objects and cluster tilting objects in CM(R) for simple one-dimensional hypersurface singularities. We decide whether extension spaces are zero or not by using covering techniques. In section 3 we point out that we could also use the computer program Singular [GP] to accomplish the same thing. In section 4 we construct cluster tilting objects for a large class of isolated hypersurface singularities, where the associated 2-CY tilted algebras can be of finite, tame or wild representation type. We also classify cluster tilting and indecomposable rigid objects for this class. In section 5 we establish a connection between existence of cluster tilting objects and existence of small resolutions. In section 6 we give a geometric approach to some of the results in section 4. Section 7 is devoted to computing some concrete examples of 2-CY tilted algebras. In section 8 we generalize results from section 2 to 2-CY triangulated categories with only a finite number of indecomposable objects. We refer to [Y] as a general reference for representation theory of Cohen-Macaulay rings, and [AGV, GLSh] for classification of singularities. Our modules are usually right modules, and composition of maps fg means first g, then f . We call a module basic if it is a direct sum of mutually non-isomorphic indecomposable modules. Acknowledgment The first author would like to thank Duco van Straten and the second author would like to thank Atsushi Takahashi and Hokuto Uehara for stimulating discussions. 1. Main results Let (R,m) be a local complete d-dimensional commutative noetherian Gorenstein isolated singularity and R/m = k ⊂ R, where k is an algebraically closed field of characteristic zero. We denote by CM(R) the category of maximal Cohen-Macaulay modules over R. Then CM(R) is a Frobenius category (i.e. an exact category with enough projectives and injectives which coincide), and so the stable category CM(R) is a Hom-finite triangulated category with shift functor Σ = Ω−1 [H]. For an integer n, we say that CM(R) or CM(R) is n-CY if there exists a functorial isomorphism HomR(X,Y ) ≃ DHomR(Y,Σ for any X,Y ∈ CM(R). We collect some fundamental results. • We have AR-duality HomR(X,Y ) ≃ DExt R(Y, τX) with τ ≃ Ω2−d [Au]. In particular, CM(R) is (d− 1)-CY. • If R is a hypersurface singularity, then Σ2 = id [Ei]. Consequently, if d is odd, then τ = Ω and CM(R) is 2-CY. If d is even, then τ = id and CM(R) is 1-CY, hence any non-free Cohen-Macaulay R-module M satisfies Ext1R(M,M) 6= 0. • (Knörrer periodicity) CM(k[[x0, · · · , xd, y, z]]/(f + yz)) ≃ CM(k[[x0, · · · , xd]]/(f)) for any f ∈ k[[x0, · · · , xd]] [Kn] ([So] in characteristic two). We state some of the definitions, valid more generally, in the context of CM(R) and CM(R). Definition 1.1. Let C = CM(R) or CM(R). We call an object M ∈ C • rigid if Ext1R(M,M) = 0, • maximal rigid if it is rigid and any rigid N ∈ C satisfying M ∈ addN satisfies N ∈ addM , • cluster tilting if addM = {X ∈ C | Ext1R(M,X) = 0} = {X ∈ C | Ext R(X,M) = 0}. Cluster tilting objects are maximal rigid, but we show that the converse does not necessarily hold for 2-CY triangulated categories CM(R). If C is 2-CY, then M ∈ C is cluster tilting if and only if addM = {X ∈ C | Ext1R(M,X) = 0}. 4 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN Definition 1.2. Let C = CM(R) (or CM(R)) be 2-CY and M ∈ C a basic cluster tilting object. Take an indecomposable summand X of M = X⊕N . Then there exist short exact sequences (or triangles) (called exchange sequences) → Y and Y such that Ni ∈ addN and f0 is a minimal right (addN)-approximation. Then Y ⊕N is a basic cluster tilting object again called cluster tilting mutation of M [BMRRT, GLSc][IY, Def. 2.5, Th. 5.3]. In this case f1 is a minimal right (addN)-approximation and gi is a minimal left (addN)-approximation automatically, so X ⊕N is a cluster tilting mutation of Y ⊕N . It is known that there are no more basic cluster tilting objects containing N [IY, Th. 5.3]. Let R = k[[x, y, z2, · · · , zd]]/(f) be a simple hypersurface singularity so that in characteristic zero f is one of the following polynomials, (An) x 2 + yn+1 + z22 + z 3 + · · · + z d (n ≥ 1) (Dn) x 2y + yn−1 + z22 + z 3 + · · · + z d (n ≥ 4) (E6) x 3 + y4 + z22 + z 3 + · · · + z (E7) x 3 + xy3 + z22 + z 3 + · · · + z (E8) x 3 + y5 + z22 + z 3 + · · · + z Then R is of finite Cohen-Macaulay representation type [Ar, GK, Kn, So]. We shall show the following result in section 2 using additive functions on the AR quiver. We shall explain another proof using Singular in section 3. Theorem 1.3. Let R be a simple hypersurface singularity of dimension d ≥ 1 over an algebraically closed field k of characteristic zero. (1) Assume that d is even. Then CM(R) does not have non-zero rigid objects. (2) Assume that d is odd. Then the number of indecomposable rigid objects, basic cluster tilting objects, basic maximal rigid objects, and indecomposable summands of basic maximal rigid objects in CM(R) are as follows: f indec. rigid cluster tilting max. rigid summands of max. rigid (An) n : odd 2 2 2 1 (An) n : even 0 0 1 0 (Dn) n : odd 2 0 2 1 (Dn) n : even 6 6 6 2 (E6) 0 0 1 0 (E7) 2 0 2 1 (E8) 0 0 1 0 We also consider a minimally elliptic curve singularity Tp,q(λ) (p ≤ q). Assume for simplicity that our base field k is algebraically closed of characteristic zero. Then these singularities are given by the equations xp + yq + λx2y2 = 0, where 1 and certain values of λ ∈ k have to be excluded. They are of tame Cohen-Macaulay representation type [D, Kah, DG]. We divide into two cases. (i) Assume 1 . This case occurs if and only if (p, q) = (3, 6) or (4, 4), and Tp,q(λ) is called simply elliptic. The corresponding coordinate rings can be written in the form T3,6(λ) = k[[x, y]]/(y(y − x 2)(y − λx2)) T4,4(λ) = k[[x, y]]/(xy(x − y)(x− λy)), where in both cases λ ∈ k \ {0, 1}. CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 5 (ii) Assume 1 . Then Tp,q(λ) does not depend on the continuous parameter λ, and is called a cusp singularity. In this case the corresponding coordinate rings can be written in the form Tp,q = k[[x, y]]/((x p−2 − y2)(x2 − yq−2)). We shall show the following result in section 6 by applying a result in birational geometry. Theorem 1.4. Let R be a minimally elliptic curve singularity Tp,q(λ) over an algebraically closed field k of characteristic zero. (a) CM(R) has a cluster tilting object if and only if p = 3 and q is even or if both p and q are even. (b) The number of indecomposable rigid objects, basic cluster tilting objects, and indecomposable summands of basic cluster tilting objects in CM(R) are as follows: p, q indec. rigid cluster tilting summands of cluster tilting p = 3, q : even 6 6 2 p, q : even 14 24 3 We also prove the following general theorem, which includes both Theorem 1.3 (except the assertion on maximal rigid objects) and Theorem 1.4. The ‘if’ part in (a) and the assertion (b) are proved in section 4 by a purely homological method. The proof of (a), including another proof of the ‘if’ part, is given in section 6 by applying Katz’s criterion in birational geometry. Theorem 1.5. Let R = k[[x, y]]/(f) (f ∈ (x, y)) be a one-dimensional reduced hypersurface singularity over an algebraically closed field k of characteristic zero. (a) CM(R) has a cluster tilting object if and only if f is a product f = f1 · · · fn with fi /∈ (x, y) (b) The number of indecomposable rigid objects, basic cluster tilting objects, and indecomposable summands of basic cluster tilting objects in CM(R) are as follows: indec. rigid cluster tilting summands of cluster tilting 2n − 2 n! n− 1 The following result gives a bridge between cluster tilting theory and birational geometry. The termi- nologies are explained in section 5. Theorem 1.6. Let (R,m) be a three-dimensional isolated cAn–singularity over an algebraically closed field k of characteristic zero defined by the equation g(x, y) + zt and R′ a one-dimensional singularity defined by g(x, y). Then the following conditions are equivalent. (a) Spec(R) has a small resolution. (b) Spec(R) has a crepant resolution. (c) (R,m) has a non-commutative crepant resolution. (d) CM(R) has a cluster tilting object. (e) CM(R′) has a cluster tilting object. (f) The number of irreducible power series in the prime decomposition of g(x, y) is n+ 1. We end this section by giving an application to finite dimensional algebras. A 2-CY tilted algebra is an endomorphism ring EndC(M) of a cluster tilting object T in a 2-CY triangulated category C. In section 7, we shall show the following result and compute 2-CY tilted algebras associated with minimally elliptic curve singularities. Theorem 1.7. Let (R,m) be an odd-dimensional isolated hypersurface singularity and Γ a 2-CY tilted algebra coming from CM(R). Then we have the following. (a) Γ is a symmetric algebra. (b) All components in the stable AR-quiver of infinite type Γ are tubes of rank 1 or 2. 6 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN For example, put R = k[[x, y]]/((x− λ1y) · · · (x− λny)) and M = k[[x, y]]/((x− λ1y) · · · (x− λiy)) for distinct elements λi ∈ k. Then M is a cluster tilting object in CM(R) by Theorem 4.1, so Γ = EndR(M) satisfies the conditions in Theorem 1.7. Since CM(R) has wild Cohen-Macaulay representation type if n > 4 [DG, Th. 3], we should get a family of examples of finite dimensional symmetric k-algebras whose stable AR-quiver consists only of tubes of rank 1 or 2, and are of wild representation type. 2. Simple hypersurface singularities Let R be a one-dimensional simple hypersurface singularity. In this case the AR-quivers are known for CM(R) [DW], and so also for CM(R). We use the notation from [Y]. In order to locate the indecomposable rigid modules M , that is, the modules M with Ext1(M,M) = 0, the following lemmas are useful, where part (a) of the first one is proved in [HKR], and the second one is a direct consequence of [KR] (generalizing [BMR1]). Lemma 2.1. (a) Let C be an abelian or triangulated k-category with finite dimensional homomor- phism spaces. Let A −−−→ B1 ⊕ B2 (g1,g2) −−−−→ C be a short exact sequence or a triangle, where A is indecomposable, B1 and B2 nonzero, and (g1, g2) has no nonzero indecomposable summand which is an isomorphism. Then Hom(A,C) 6= 0 . (b) Let 0 → A −→ C → 0 be an almost split sequence in CM(R), where R is an isolated hypersurface singularity, and B has at least two indecomposable nonprojective summands in a decomposition of B into a direct sum of indecomposable modules. Then Ext1(C,C) 6= 0. Proof. (a) See [HKR, Lem. 6.5]. (b) Using (a) together with the above AR-formula and τ2 = id, we obtain DExt1(C,C) ≃ Hom(τ−1C,C) = Hom(τC,C) ≃ Hom(A,C) 6= 0, where D = Homk( , k). � Lemma 2.2. Let T be a cluster tilting object in the Hom-finite connected 2-CY category C, and Γ = EndC(T ). (a) The functor G = HomC(T, ) : C → mod(Γ) induces an equivalence of categories G : C/add(τT ) → mod(Γ). (b) The AR-quiver for Γ is as a translation quiver obtained from the AR-quiver for C by removing the vertices corresponding to the indecomposable summands of τT . (c) Assume τ2 = id. Then we have the following. (i) Γ is a symmetric algebra. (ii) The indecomposable nonprojective Γ-modules have τ-period one or two. (iii) If C has an infinite number of nonisomorphic indecomposable objects, then all components in the stable AR-quiver of Γ are tubes of rank one or two. (d) If C has only a finite number n of nonisomorphic indecomposable objects, and T has t nonisomor- phic indecomposable summands, then there are n− t nonisomorphic indecomposable Γ-modules. Proof. For (a) and (b) see [BMR1, KR]. Since C is 2-CY, we have τ = Σ, and a functorial isomorphism DHomC(T, T ) ≃ HomC(T,Σ 2T ) = HomC(T, τ 2T ) ≃ HomC(T, T ). This shows that Γ is symmetric. Let C be an indecomposable nonprojective Γ-module. Viewing C as an object in C we have τ2CC ≃ C, and τC is not a projective Γ-module since C is not removed. Hence we have τ2ΓC ≃ C. If C has an infinite number of nonisomorphic indecomposable objects, then Γ is of infinite type. Then each component of the AR-quiver is infinite, and hence is a tube of rank one or two. Finally, (d) is a direct consequence of (a). � CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 7 We also use that in our cases we have a covering functor Π: k(ZQ) → CM(R), where Q is the appropriate Dynkin quiver and k(ZQ) is the mesh category of the translation quiver ZQ [Rie, Am], (see also [I1, Section 4.4] for another explanation using functorial methods). For the one-dimensional simple hypersurface singularities we have the cases An (n even or odd), Dn (n odd or even), E6, E7 and E8. We now investigate them case by case. Proposition 2.3. In the case An (with n even) there are no indecomposable rigid objects. Proof. We have the stable AR-quiver oo // · · · //oo In/2 Here, and later, a dotted line between two indecomposable modules means that they are connected via τ . Since τIj ≃ Ij for each j, Ext 1(Ij , Ij) 6= 0 for j = 1, · · · , n/2. Hence no Ij is rigid. � Proposition 2.4. In the case An (with n odd) the maximal rigid objects coincide with the cluster tilting objects. There are two indecomposable ones, and the corresponding 2-CY tilted algebras are k[x]/(x (n+1) Proof. For simplicity, we write l = (n− 1)/2. We have the stable AR-quiver //oo · · · //oo Ml ==zzzzz aaDDDDD Since τMi ≃Mi for i = 1, · · · , l, we have Ext1(Mi,Mi) ≃ Hom(Mi, τMi) ≃ Hom(Mi,Mi) 6= 0. So only the indecomposable objects N− and N+ could be rigid. We use covering techniques and additive functions to compute the support of Hom(N−, ), where we refer to [BG] for the meaning of the diagrams below. >>}}} <<yyy ==zzz · · · >>}}} >>}}} ==zzz ??��� // N− //// Ml >>~~~ // N+ // Ml >>~~~ // N− · · · ??��� >>~~~ >>~~~ BB��� BB��� BB��� BB��� BB��� BB��� // 1 // 1 BB��� // 0 // 1 BB��� // 1 1 // 0 // 0 BB��� BB��� BB��� BB��� We see that Hom(N−, N+) = 0, so Ext 1(N+, N+) = Ext 1(N+, τN−) = 0, and Ext 1(N−, N−) = 0. Since Ext1(N+, N−) 6= 0, we see that N+ and N− are exactly the maximal rigid objects. Further Hom(N−,Mi) 6= 0 for all i, so Ext 1(N+,Mi) 6= 0 and Ext 1(N−,Mi) 6= 0 for all i. This shows that N+ and N− are also cluster tilting objects. The description of the cluster tilted algebras follows directly from the above picture. � Proposition 2.5. In the case Dn with n odd we have two maximal rigid objects, which both are inde- composable, and neither one is cluster tilting. 8 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN Proof. We have the AR-quiver // Y1 // Y2 // M2 // // · · · // M(n−3)/2 **UUU X(n−1)/2 jjUUU ttiiii A // X1 \\9999999 // N1 ]];;;;;;; // X2 ]];;;;;;; // N2 ]];;;;;;; // · · · // N(n−3)/2 bbDDDDDDDD 44iiii Using Lemma 2.1, the only candidates for being indecomposable rigid are A and B. We compute the support of Hom(A, ) AA��� BB��� · · · AA��� ??��� ??��� BB��� Y1 · · · CC��� BB��� where B = τA and l = (n− 3)/2. We see that Hom(A,B) = 0, so that Ext1(A,A) = 0. Then A is clearly maximal rigid. Since Hom(A,M1) = 0, we have Ext 1(A,N1) = 0, so A is not cluster tilting. Alternatively, we could use that we see that End(A)op ≃ k[x]/(x2), which has two indecomposable modules, whereas CM(R) has 2n− 3 indecomposable objects. If A was cluster tilting, End(A)op would have had 2n− 3 − 1 = 2n− 4 indecomposable modules, by Lemma 2.2. � Proposition 2.6. In the case D2n with n a positive integer we have that the maximal rigid objects coincide with the cluster tilting ones. There are 6 of them, and each is a direct sum of two nonisomorphic indecomposable objects. The corresponding 2-CY-tilted algebras are given by the quiver with relations · α // · oo αβα = 0 = βαβ in the case D4, and by ·γ ;; α // · oo with γn−1 = βα, γβ = 0 = αγ and · α // · oo with (αβ)n−1α = 0 = (βα)n−1β for 2n > 4. Proof. We have the AR-quiver vvnnn // Y1 · · · // Yn−1 ``BBBBBBB A // X1 // XX1111111111 YY3333333333 // X2 YY3333333333 // N2 YY3333333333 // · · · // Xn−1 ZZ66666666666 FF 66nnnn hhQQQQ By Lemma 2.1, the only possible indecomposable rigid objects are: A, B, C+, C−, D+, D−. CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 9 We compute the support of Hom(C+, ): C+ // Yl AA��� // D+ // Xl ??��� // C+ // Yl ??��� // D+ // Xl ??��� // C+ // Yl CC��� · · · @@��� ??��� Nl−1 Ml−1 >>~~~ @@��� @@��� ??��� AA��� >>~~~ >>~~~ >>~~~ BB��� >>}}} · · · AA��� >>}}} where l = n− 1 1 // 1 BB��� // 0 // 1 BB��� // 1 // 1 BB��� // 0 // 1 BB��� // 1 // 0 BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� We see that Hom(C+, D+) = 0, so Ext 1(C+, C+) = 0 = Ext 1(D+, D+). Further, Hom(C+, C−) = 0, so Ext1(C+, D−) = 0. By symmetry Ext 1(D−, D−) = 0 = Ext 1(C−, C−) and Ext 1(D+, C−) = 0. Also Ext1(C+, A) = 0, Ext 1(C+, B) 6= 0, so Ext 1(D+, B) = 0, Ext 1(D+, A) 6= 0. Further Ext 1(C+, X) 6= 0 for X 6= A,D−, C+. We now compute the support of Hom(A, ) >>~~~ C+ // Yl ??��� D+ // Xl · · · ??��� >>~~~ @@��� M1 M1 @@��� >>~~~ ??��� >>}}} · · · ??~~~ >>~~~ >>}}} ??~~~ 10 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN where l = n− 1 and we have an odd number of columns and rows. BB��� 1 // 1 BB��� 0 // 0 BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� We see that Hom(A,B) = 0, so Ext1(A,A) = 0, hence also Ext1(B,B) = 0. Since Hom(A,D−) = 0, we have Ext1(A,C−) = 0, hence Ext 1(B,D−) = 0. Since Hom(A,C−) 6= 0, we have Ext 1(A,D−) 6= 0, so Ext1(B,D+) 6= 0. It follows that C+ ⊕D−, C− ⊕D+, C+ ⊕A, D+ ⊕B, A⊕ C− and B ⊕D− are maximal rigid. These are also cluster tilting: We have Hom(A,Xi) 6= 0, Hom(A,Ni) 6= 0, so Ext 1(B,Xi) 6= 0, Ext1(B,Ni) 6= 0. Similarly, Ext 1(A, Yi) 6= 0, Ext 1(A,Mi) 6= 0. Also Hom(C+, Yi) 6= 0, Hom(C+, Ni) 6= 0, so Ext1(D+, Yi) 6= 0, Ext 1(D+, Ni) 6= 0. Hence Ext 1(C+, Xi) 6= 0, Ext1(C+,Mi) 6= 0. So Ext 1(D−, Yi) 6= 0, Ext 1(D−, Ni) 6= 0, Ext 1(C−, Xi) 6= 0, Ext 1(C−,Mi) 6= 0. We see that each indecomposable rigid object can be extended to a cluster tilting object in exactly two ways, which we would know from a general result in [IY, Th. 5.3]. The exchange graph is as follows: {C+, D−} {B,D−} rrrrrr {A,C+} {B,D+} {A,C−} {C−, D+} rrrrrr Considering the above pictures, we get the desired description of the corresponding 2-CY tilted algebras in terms of quivers with relations. � Proposition 2.7. In the case E6 there are no indecomposable rigid objects. Proof. We have the AR-quiver // M1 ::uuuu ddIIII // N1 \\8888888 The only candidates for indecomposable rigid objects according to Lemma 2.1 are M1 and N1. We compute the support of Hom(M1, ). CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 11 >>~~~ ??��� · · · ??��� // M2 // X >>~~~ // M2 // X AA��� >>~~~ ??��� · · · AA��� ??��� >>~~~ BB��� BB��� · · · BB��� // 1 // 1 BB��� 0 // 1 BB��� BB��� BB��� · · · BB��� BB��� BB��� We see that Hom(M1, N1) 6= 0, so that Ext 1(M1,M1) 6= 0 and Ext 1(N1, N1) 6= 0. � Proposition 2.8. In the case E7 there are two maximal rigid objects, which both are indecomposable, and neither of them is cluster tilting. Proof. We have the AR-quiver ___ D // M2 // // Y3 ^^>>>>>>>> wwppp // Y1 xxppp B // N2 ^^======== // X2 __???????? // X3 ��������������� __>>>>>>>> // X1 ^^======== // N1 __???????? Using Lemma 2.1, we see that the only candidates for indecomposable rigid objects are A, B, M1, N1, C and D. We first compute the support of Hom(A, ). ??��� X1 · · · @@��� // C // X3 ??��� AA��� ??��� @@��� @@��� ??��� · · · AA��� @@��� ??��� // 1 // 1 // 0 // 1 // 1 // 1 // 0 // 1 // 1 // 1 // 0 // 0 We see that Ext1(A,A) = 0, and so also Ext1(B,B) = 0, so A and B are rigid. Next we compute the support of Hom(M1, ). @@��� ??��� @@��� · · · @@��� // C // X3 ??��� // D // Y3 ??��� @@��� ??��� N2 · · · ??��� BB��� BB��� BB��� · · · BB��� 1 // 1 BB��� // 0 // 1 BB��� BB��� BB��� 1 · · · BB��� We see that Ext1(M1,M1) 6= 0 and Ext 1(N1, N1) 6= 0, so that M1 and N1 are not rigid. Then we compute the support of Hom(C, ). 12 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN @@��� ??��� X1 · · · C // X3 @@��� D // Y3 @@��� C // X3 ??��� @@��� ??��� @@��� ??��� · · · ??��� BB��� BB��� 1 · · · 1 // 1 BB��� 0 // 1 BB��� 1 // 2 // BB��� BB��� BB��� BB��� BB��� · · · BB��� We see that Ext1(C,C) 6= 0 and Ext1(D,D) 6= 0, so that C and D are not rigid. Hence A and B are the rigid indecomposable objects, and they are maximal rigid. Since Ext1(A,C) = 0, we see that A and hence B is not cluster tilting. � Proposition 2.9. In the case E8 there are no indecomposable rigid objects. Proof. We have the AR-quiver __ B2 N2 // // X1 ]];;;;;;; xxqqq xxqqq // C1 // B1 // N1 M2 // C2 ^^<<<<<<< // Y1 ]];;;;;;; �������������� // Y2 ]];;;;;;; // D1 ]];;;;;;; // A1 ]];;;;;;; // M1 ]]<<<<<<< The only candidates for indecomposable rigid objects are M1, N1, M2, N2, A2 and B2, by Lemma 2.1. We first compute the support of Hom(M1, ): ??��� ??��� · · · ??��� B2 // X1 ??��� // A2 // Y1 ??��� ??��� ??��� @@��� ??��� ??��� @@��� ??��� ??��� ??��� · · · @@��� @@��� ??��� ??��� BB��� BB��� 0 · · · BB��� // 1 // 1 BB��� // 0 // 1 BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� BB��� · · · BB��� BB��� BB��� BB��� BB��� We see that Ext1(M1,M1) 6= 0, and hence Ext 1(N1, N1) 6= 0. CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 13 Next we compute the support of Hom(M2, ): @@��� ??��� ??��� · · · @@��� // B2 // X1 ??��� // A2 // Y1 ??��� ??��� ??��� ??��� · · · BB��� BB��� BB��� · · · BB��� 1 // 1 BB��� 0 // 1 BB��� BB��� BB��� BB��� · · · We see that Ext1(M2,M2) 6= 0, and hence Ext 1(N2, N2) 6= 0. Finally we compute the support of Hom(A2, ): ??��� ??��� D2 · · · A2 // Y1 AA��� // B2 // X1 ??��� // A2 // Y1 // ??��� ??��� ??��� ??��� ??��� ??��� B1 · · · ??��� BB��� BB��� 1 · · · 1 // 1 BB��� // 0 // 1 BB��� // 1 // 2 BB��� BB��� BB��� BB��� BB��� BB��� 1 · · · BB��� It follows that Ext1(A2, A2) 6= 0, and similarly Ext 1(B2, B2) 6= 0. Hence there are no indecomposable rigid objects. � 3. Computation with Singular An alternative way to carry out computations of Ext1–spaces in the stable category of maximal Cohen- Macaulay modules is to use the computer algebra system Singular, see [GP]. Let R = k[x1, x2, . . . , xn]〈x1,x2,...,xn〉/I be a Cohen-Macaulay local ring which is an isolated singularity, and M and N two maximal Cohen- Macaulay modules. Denote by R̂ the completion of R. Since all the spaces ExtiR(M,N) (i ≥ 1) are finite- dimensional over k and the functor mod(R) → mod(R̂) is exact, maps the maximal Cohen-Macaulay modules to maximal Cohen-Macaulay modules and the finite length modules to finite length modules, we can conclude that dimk(Ext R(M,N)) = dimk(Ext bR(M̂, N̂)). As an illustration we show how to do this for the case E7. Proposition 3.1. In the case E7 there are two maximal rigid objects, which both are indecomposable and neither of them is cluster tilting. 14 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN By [Y] the AR-quiver of CM(R) has the form // M2 // // Y3 aaCCCCC vvmmm // Y1 vvmmm B // N2 ``BBBBB // X2 aaDDDDD // X3 aaCCCCC FF aaBBBBB // N1 aaDDDDD By Lemma 2.1 only the modules A,B,C,D,M1, N1 can be rigid. Since B = τ(A), D = τ(C), N1 = τ(M1), the pairs of modules (A,B), (C,D) and (M1, N1) are rigid or not rigid simultaneously. By [Y] we have the following presentations: x2+y3 −−−−→ R −→ R −→ A −→ 0, ( x y y2 −x) −−−−−→ R2 x( x y y2 −x) −−−−−−→ R2 −→ C −→ 0, xy2 −x) −−−−−−→ R2 xy2 −x2) −−−−−−→ R2 −→M1 −→ 0, so we can use the computer algebra system Singular in order to compute the Ext1–spaces between these modules. > Singular (call the program ‘‘Singular’’) > LIB ‘‘homolog.lib’’; (call the library of homological algebra) > ring S = 0,(x,y),ds; (defines the ring S = Q[x, y]〈x,y〉) > ideal I = x3 + xy3; (defines the ideal x3 + xy3 in S) > qring R = std(I); (defines the ring Q[x, y]〈x,y〉/I) > module A = [x]; > module C = [x2, xy2], [xy, -x2]; > module M1 = [x2, xy2], [y, -x2]; (define modules A,C,M1) > list l = Ext(1,A,A,1); // dimension of Ext1: -1 (Output: Ext1R(A,A) = 0) > list l = Ext(1,C,C,1); // ** redefining l ** // dimension of Ext1: 0 (the Krull dimension of Ext1R(C,C) is 0) // vdim of Ext1: 2 (dimk(Ext R(C,C)) = 2) > list l = Ext(1,M1,M1,1); // ** redefining l ** // dimension of Ext1: 0 // vdim of Ext1: 10 > list l = Ext(1,A,C,1); // ** redefining l ** // dimension of Ext1: -1 This computation shows that the modules A and B are rigid, C,D,M1 and N1 are not rigid and since Ext1R(A,C) = 0, there are no cluster tilting objects in the stable category CM(R). 4. One-dimensional hypersurface singularities We shall construct a large class of one-dimensional hypersurface singularities having a cluster tilting object, then classify all cluster tilting objects. Our method is based on the higher theory of almost split sequences and Auslander algebras studied in [I1, I2]. We also use a relationship between cluster tilting objects in CM(R) and tilting modules over the endomorphism algebra of a cluster tilting object [I2]. Then we shall compare cluster tilting mutation given in Definition 1.2 with tilting mutation by using results due to Riedtmann-Schofield [RS] and Happel-Unger [HU1, HU2]. In this section, we usually consider cluster tilting objects in CM(R) instead of CM(R). CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 15 Let k be an infinite field, S := k[[x, y]] and m := (x, y). We fix f ∈ m and write f = f1 · · · fn for irreducible formal power series fi ∈ m (1 ≤ i ≤ n). Put Si := S/(f1 · · · fi) and R := Sn = S/(f). We assume that R is reduced, so we have (fi) 6= (fj) for any i 6= j, and R is then an isolated singularity. Our main results in this section are the following, where the part (a) remains true in any dimension. Theorem 4.1. (a) i=1 Si is a rigid object in CM(R). i=1 Si is a cluster tilting object in CM(R) if the following condition (A) is satisfied. (A) fi /∈ m 2 for any 1 ≤ i ≤ n. Let Sn be the symmetric group of degree n. For w ∈ Sn and I ⊆ {1, · · · , n}, we put Swi := S/(fw(1) · · · fw(i)), Mw := Swi and SI := S/( Theorem 4.2. Assume that (A) is satisfied. (a) There are exactly n! basic cluster tilting objects Mw (w ∈ Sn) and exactly 2 n− 1 indecomposable rigid objects SI (∅ 6= I ⊆ {1, · · · , n}) in CM(R). (b) For any w ∈ Sn, there are exactly n! basic Cohen-Macaulay tilting EndR(Mw)-modules HomR(Mw,Mw′) (w ′ ∈ Sn) of projective dimension at most one. Moreover, all algebras EndR(Mw) (w ∈ Sn) are derived equivalent. It is interesting to compare with results in [IR], where two-dimensional (2-Calabi-Yau) algebras Γ are treated and a bijection between elements in an affine Weyl group and tilting Γ-modules of projective dimension at most one is given. Here the algebra is one-dimensional, and Weyl groups appear. Here we consider three examples. (a) Let R be a curve singularity of type A2n−1 or D2n+2, so R = S/((x− yn)(x + yn)) or R = S/(y(x− yn)(x+ yn)). By our theorems, there are exactly 2 or 6 cluster tilting objects and exactly 3 or 7 indecomposable rigid objects in CM(R), which fits with our computations in section 1. (b) Let R be a curve singularity of type T3,2q+2(λ) or T2p+2,2q+2(λ), so R = S/((x− y2)(x − yq)(x + yq)) (R = S/(y(y − x2)(y − λx2)) for q = 2), R = S/((xp − y)(xp + y)(x− yq)(x + yq)) (R = S/(xy(x− y)(x− λy)) for p = q = 1). By our theorems, there are exactly 6 or 24 cluster tilting objects and exactly 7 or 15 indecom- posable rigid objects in CM(R). (c) Let λi ∈ k (1 ≤ i ≤ n) be mutually distinct elements in k. Put R := S/((x− λ1y) · · · (x− λny)). By our theorems, there are exactly n! cluster tilting objects and exactly 2n − 1 indecomposable rigid objects in CM(R). First of all, Theorem 4.1(a) follows immediately from the following observation. Proposition 4.3. For g1, g2 ∈ m and g3 ∈ S, put R := S/(g1g2g3). If g1 and g2 have no common factor, then Ext1R(S/(g1g3), S/(g1)) = 0 = Ext R(S/(g1), S/(g1g3)). Proof. We have a projective resolution → R → S/(g1g3) → 0. Applying HomR( , S/(g1)), we have a complex S/(g1) g1g3=0 −→ S/(g1) → S/(g1). This is exact since g1 and g2 have no common factor. Thus we have the former equation, and the other one can be proved similarly. � 16 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN Our plan of proof of Theorem 4.1(b) is the following. (i) First we shall prove Theorem 4.1 under the following stronger assumption: (B) m = (f1, f2) = · · · = (fn−1, fn). (ii) Then we shall prove the general statement of Theorem 4.1. We need the following general result in [I1, I2]. Proposition 4.4. Let R be a complete local Gorenstein ring of dimension at most three and M a rigid Cohen-Macaulay R-module which is a generator (i.e. M contains R as a direct summand). Then the following conditions are equivalent. (a) M is a cluster tilting object in CM(R). (b) gl. dim EndR(M) ≤ 3. (c) For any X ∈ CM(R), there exists an exact sequence 0 →M1 →M0 → X → 0 with Mi ∈ addM . (d) For any indecomposable direct summand X of M , there exists an exact sequence 0 → M2 → X with Mi ∈ addM and a is a right almost split map in addM . Proof. (a)⇔(b) For d = dimR, take the d-cotilting module T = R and apply [I2, Th. 5.1(3)] for m = d and n = 2 there. (a)⇔(c) See [I1, Prop. 2.2.2]. (a)⇒(d) See [I1, Th. 3.3.1]. (d)⇒(b) For any simple EndR(M)-module S, there exists an indecomposable direct summand X of M such that S is the top of the projective HomR(M,X). Since Ext R(M,M2) = 0, the sequence in (d) gives a projective resolution 0 → HomR(M,M2) → HomR(M,M1) → HomR(M,M0) → HomR(M,X) → S → 0. Thus we have pdS ≤ 3 and gl. dim EndR(M) ≤ 3. � The sequence in (d) is called a 2-almost split sequence when X is non-projective and a and b are right minimal. In this case a is surjective, c is a left almost split map in addM , and b and c are left minimal. There is a close relationship between 2-almost split sequences and exchange sequences [IY]. We shall construct exact sequences satisfying the above condition (d) in Lemma 4.5 and Lemma 4.6 below. We use the isomorphism HomR(Si, Sj) ≃ (fi+1 · · · fj)/(f1 · · · fj) i < j S/(f1 · · · fj) i ≥ j. Lemma 4.5. Let R = S/(f) be a one-dimensional reduced hypersurface singularity, S0 := 0 and 1 ≤ i < (a) We have exchange sequences (see Definition 1.2) 0 → Si −−−−→ Si+1 ⊕ Si−1 (1 fi+1) −−−−−→ S/(f1 · · · fi−1fi+1) → 0, 0 → S/(f1 · · · fi−1fi+1) (fi1 ) −−→ Si+1 ⊕ Si−1 (−1 fi) −−−−−→ Si → 0. (b) If (fi, fi+1) = m, then we have a 2-almost split sequence 0 → Si −−−−→ Si+1 ⊕ Si−1 fi fifi+1 1 fi+1 −−−−−−−→ Si+1 ⊕ Si−1 (−1 fi) −−−−−→ Si → 0 in add i=1 Si. Proof. (a) Consider the map a := (−1 fi) : Si+1 ⊕ Si−1 → Si. Any morphism from Sj to Si factors through 1 : Si+1 → Si (respectively, fi : Si−1 → Si) if j > i (respectively, j < i). Thus a is a minimal right (add j 6=i Sj)-approximation. It is easily checked that Ker a = {s ∈ Si+1 | s ∈ fiSi} = (fi)/(f1 · · · fi+1) ≃ S/(f1 · · · fi−1fi+1), where we denote by s the image of s via the natural surjection Si+1 → Si. Consider the surjective map b := (1 fi+1) : Si+1 ⊕Si−1 → S/(f1 · · · fi−1fi+1). It is easily checked that Ker b = {s ∈ Si+1 | s ∈ (fi+1)/(f1 · · · fi−1fi+1)} = (fi+1)/(f1 · · · fi+1) ≃ Si, where we denote by s the image of s via the natural surjection Si+1 → S/(f1 · · · fi−1fi+1). CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 17 (b) This sequence is exact by (a). Any non-isomorphic endomorphism of Si is multiplication with an element in m, which is equal to (fi, fi+1) by our assumption. Since fi+1 (respectively, fi) : Si → Si factors through 1 : Si+1 → Si (respectively, fi : Si−1 → Si), we have that a is a right almost split map. � Now we choose fn+1 ∈ m such that m = (fn, fn+1), and fn+1 and f1 · · · fn have no common factor. This is possible by our assumption (A). Lemma 4.6. We have an exact sequence 0 → Sn−1 −fn+1 −−−−−→ Sn ⊕ Sn−1 (fn+1 fn) −−−−−−→ Sn with a minimal right almost split map (fn+1 fn) in add i=1 Si. Proof. Consider the map a := (fn+1 fn) : Sn⊕Sn−1 → Sn. Any morphism from Sj (j < n) to Sn factors through fn : Sn−1 → Sn. Any non-isomorphic endomorphism of Sn is multiplication with an element in m = (fn+1, fn). Since fn : Sn → Sn factors through fn : Sn−1 → Sn, we have that a is a right almost split map. It is easily checked that Ker a = {s ∈ Sn−1 | fns ∈ fn+1Sn} = (fn+1, f1 · · · fn−1)/(f1 · · · fn−1), which is isomorphic to Sn−1 by the choice of fn+1. In particular, a is right minimal. � Thus we finished the proof of Theorem 4.1 under the stronger assumption (B). To show the general statement of Theorem 4.1, we need some preliminary observations. Let us consider cluster tilting mutation in CM(R). We use the notation introduced at the beginning of this section. Lemma 4.7. For w ∈ Sn, we assume that Mw is a cluster tilting object in CM(R). Then, for 1 ≤ i < n and si = (i i+ 1), we have exchange sequences 0 → Swi → S i+1 ⊕ S i−1 → S i → 0 and 0 → S i → S i+1 ⊕ S i−1 → S i → 0. Proof. Without loss of generality, we can assume w = 1. Then the assertion follows from Lemma 4.5(a). � Immediately, we have the following. Proposition 4.8. Assume that Mw is a cluster tilting object in CM(R) for some w ∈ Sn. (a) The cluster tilting mutations of Mw are Mwsi (1 ≤ i < n). (b) Mw′ is a cluster tilting object in CM(R) for any w ′ ∈ Sn. Proof. (a) This follows from Lemma 4.7. (b) This follows from (a) since Sn is generated by si (1 ≤ i < n). � The following result is also useful. Lemma 4.9. Let R and R′ be complete local Gorenstein rings with dimR = dimR′ and M a rigid object in CM(R) which is a generator. Assume that there exists a surjection R′ → R, and we regard CM(R) as a full subcategory of CM(R′). If R′ ⊕M is a cluster tilting object in CM(R′), then M is a cluster tilting object in CM(R). Proof. We use the equivalence (a)⇔(c) in Proposition 4.4, which remains true in any dimension [I1, Prop. 2.2.2]. For any X ∈ CM(R), take a right (addM)-approximation f : M0 → X of X . Since M is a generator of R, we have an exact sequence 0 → Y → M0 → X → 0 with Y ∈ CM(R). Since R′ is a projective R′-module, f is a right add(R′ ⊕M)-approximation of X . Since R′ ⊕M is a cluster tilting object in CM(R′), we have Y ∈ add(R′ ⊕M). Since Y ∈ CM(R), we have Y ∈ addM . Thus M satisfies condition (c) in Proposition 4.4. � Now we shall prove Theorem 4.1. Since k is an infinite field and the assumption (A) is satisfied, we can take irreducible formal power series gi ∈ m (1 ≤ i < n) such that h2i−1 := fi and h2i := gi satisfy the following conditions: • (hi) 6= (hj) for any i 6= j. 18 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN • m = (h1, h2) = (h2, h3) = · · · = (h2n−2, h2n−1). Put R′ := S/(h1 · · ·h2n−1). This is reduced by the first condition. Since we have already proved Theorem 4.1 under the assumption (B), we have that⊕2n−1 i=1 S/(h1 · · ·hi) is a cluster tilting object in CM(R ′). By Proposition 4.8,⊕2n−1 i=1 S/(hw(1) · · ·hw(i)) is a cluster tilting object in CM(R ′) for any w ∈ S2n−1. In particular, S/(f1 · · · fi)) ⊕ ( S/(f1 · · · fng1 · · · gi)) is a cluster tilting object in CM(R′). Moreover we have surjections R′ → · · · → S/(f1 · · · fng1g2) → S/(f1 · · · fng1) → R. Using Lemma 4.9 repeatedly, we have that i=1(S/(f1 · · · fi)) is a cluster tilting object in CM(R). Thus we have proved Theorem 4.1. � Before proving Theorem 4.2, we give the following description of the quiver of the endomorphism algebras. Proposition 4.10. Assume that (A) is satisfied. (a) The quiver of EndR( i=1 Si) is // · · ·oo // Sn−1oo where in addition there is a loop at Si (1 ≤ i < n) if and only if (fi, fi+1) 6= m. (b) We have the quiver of EndR( i=1 Si) by removing the vertex Sn from the quiver in (a). Proof. We only have to show (a). We only have to calculate minimal right almost split maps in i=1 Si. We have a minimal right almost split map Sn⊕Sn−1 → Sn by Lemma 4.6. If (fi, fi+1) = m (1 ≤ i < n), then we have a minimal right almost split map Si+1 ⊕ Si−1 → Si by Lemma 4.5. We only have to consider the case (fi, fi+1) 6= m (1 ≤ i < n). Take g ∈ m such that (fi, fi+1, g) = m. It is easily check (cf proof of Lemma 4.5) that we have a right almost split map c := (−1 g fi) : Si+1 ⊕ Si ⊕ Si−1 → Si. Assume that c is not right minimal. Then there exists a right almost split map of the form c′ : Si+1 ⊕ Si → Si (i > 1), Si ⊕ Si−1 → Si or Si+1 ⊕ Si−1 → Si. For the first case, it is easily checked that fi : Si−1 → Si does not factor through c ′, a contradiction. Similarly we have the contradiction for the remaining cases. Thus c is the minimal right almost split map. � In the rest we shall show Theorem 4.2. We recall results on tilting mutation due to Riedtmann- Schofield [RS] and Happel-Unger [HU1, HU2]. For simplicity, a tilting module means a tilting module of projective dimension at most one. Let Γ be a module-finite algebra with n simple modules over a complete local ring with n simple modules. Their results remain valid in this setting. Recall that, for basic tilting Γ-modules T and U , we write T ≥ U if Ext1Γ(T, U) = 0. By tilting theory, we have FacT = {X ∈ mod(Γ) | Ext Γ(T,X) = 0}. Thus Ext1Γ(T, U) = 0 is equivalent to FacT ⊃ FacU , and ≥ gives a partial order. On the other hand, we call a Γ-module T almost complete tilting if pd ΓT ≤ 1, Ext Γ(T, T ) = 0 and T has exactly (n−1) non-isomorphic indecomposable direct summands. We collect some basic results. Proposition 4.11. (a) Any almost complete tilting Γ-module has at most two complements. (b) T and U are neighbors in the partial order if and only if there exists an almost complete tilting Γ-module which is a common direct summand of T and U . CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 19 (c) Assume T ≥ U . Then there exists a sequence T = T0 > T1 > T2 > · · · > U satisfying the following conditions. (i) Ti and Ti+1 are neighbors. (ii) Either Ti = U for some i or the sequence is infinite. (d) T ≥ U if and only if there exists an exact sequence 0 → T → U0 → U1 → 0 with Ui ∈ addU . If the conditions in (b) above are satisfied, we call T a tilting mutation of U . We also need the following easy observation on Cohen-Macaulay tilting modules. For a module-finite R-algebra Γ, we call a Γ-module Cohen-Macaulay if it is a Cohen-Macaulay R-module. As usual, we denote by CM(Γ) the category of Cohen-Macaulay Γ-modules. Lemma 4.12. Let Γ be a module-finite algebra over a complete local Gorenstein ring R such that Γ ∈ CM(R), and T and U tilting Γ-modules. Assume U ∈ CM(Γ). (a) If T ≥ U , then T ∈ CM(Γ). (b) Let P be a projective Γ-module such that HomR(P,R) is a projective Γ op-module. Then P ∈ addU . Proof. (a) By Proposition 4.11(d), there exists an exact sequence 0 → T → U0 → U1 → 0 with Ui ∈ addU . Thus the assertion holds. (b) We have Ext1Γop(HomR(P,R),HomR(U,R)) = 0. Since we have a duality HomR( , R) : CM(Γ) ↔ CM(Γop), it holds Ext1Γ(U, P ) = 0. There exists an exact sequence 0 → P → U0 → U1 → 0 with Ui ∈ addU [H, Lem. III.2.3], which must split since Ext Γ(U, P ) = 0. Thus we have P ∈ addU . � Finally, let us recall the following relation between cluster tilting and tilting (see [I2, Th. 5.3.2] for (a), and (b) is clear). Proposition 4.13. Let R = S/(f) be a one-dimensional reduced hypersurface singularity and M , N and N ′ cluster tilting objects in CM(R). (a) HomR(M,N) is a tilting EndR(M)-module of projective dimension at most one. (b) If N ′ is a cluster tilting mutation of N , then HomR(M,N ′) is a tilting mutation of HomR(M,N). Now we shall prove Theorem 4.2. Fix w ∈ Sn and put Γ := EndR(Mw). Since Mw is a generator of R, the functor HomR(Mw, ) : CM(R) → CM(Γ) is fully faithful. By Theorem 4.1, Mw is a cluster tilting object in CM(R). By Proposition 4.13(a), HomR(Mw,Mw′) (w ′ ∈ Sn) is a Cohen-Macaulay tilting Γ-module. (b) Take any Cohen-Macaulay tilting Γ-module U . Since P := HomR(Mw, R) is a projective Γ-module such that HomR(P,R) = Mw = HomR(R,Mw) is a projective Γ-module, we have P ∈ addU by Lemma 4.12(b). In particular, by Proposition 4.11(a)(b), • any Cohen-Macaulay tilting Γ-module has at most (n − 1) tilting mutations which are Cohen- Macaulay. Conversely, by Proposition 4.8 and Proposition 4.13(b), • any Cohen-Macaulay tilting Γ-module of the form HomR(Mw,Mw′) (w ′ ∈ Sn) has precisely (n− 1) tilting mutations HomR(Mw,Mw′si) (1 ≤ i < n) which are Cohen-Macaulay. Consequently, any successive tilting mutation of Γ = HomR(Mw,Mw) has the form HomR(Mw,Mw′) for some w′ ∈ Sn if each step is Cohen-Macaulay. Using this observation, we shall show that U is isomorphic to HomR(Mw,Mw′) for some w ′. Since Γ ≥ U , there exists a sequence Γ = T0 > T1 > T2 > · · · > U satisfying the conditions in Proposition 4.11(c). By Lemma 4.12(a), each Ti is Cohen-Macaulay. Thus the above observation implies that each Ti has the form HomR(Mw,Mwi) for some wi ∈ Sn. Moreover, wi 6= wj for i 6= j. Since Sn is a finite group, the above sequence must be finite. Thus U = Ti holds for some i, hence the proof is completed. 20 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN (a) Let U be a cluster tilting object in CM(R). Again by Proposition 4.13(a), HomR(Mw, U) is a Cohen-Macaulay tilting Γ-module. By part (b) which we already proved, HomR(Mw, U) is isomorphic to HomR(Mw,Mw′) for some w ′ ∈ Sn. Since the functor HomR(Mw, ) : CM(R) → CM(Γ) is fully faithful, U is isomorphic to Mw′, and the former assertion is proved. For the latter assertion, we only have to show that any rigid object in CM(R) is a direct summand of some cluster tilting object in CM(R). This is valid by the following general result in [BIRS, Th. 1.9]. � Proposition 4.14. Let C be a 2-CY Frobenius category with a cluster tilting object. Then any rigid object in C is a direct summand of some cluster tilting object in C. We end this section with the following application to dimension three. Now let S′′ := k[[x, y, u, v]], fi ∈ m = (x, y) (1 ≤ i ≤ n) and R ′′ := S′′/(f1 · · · fn + uv). For w ∈ Sn and I ⊆ {1, · · · , n}, we put Uwi := (u, fw(1) · · · fw(i)) ⊂ R ′′, Mw := Uwi and UI := (u, fi) ⊂ S We have the following result (see 5.2 for definition). Corollary 4.15. Under the assumption (A), we have the following. (a) There are exactly n! basic cluster tilting objects Mw (w ∈ Sn) and exactly 2 n− 1 indecomposable rigid objects UI (∅ 6= I ⊂ {1, · · · , n}) in CM(R (b) There are non-commutative crepant resolutions EndR′′(Mw) (w ∈ Sn) of R ′′, which are derived equivalent. Proof. (a) We only have to apply Knörrer periodicity CM(R) → CM(R′′) [Kn, So] as follows: Since Swi ∈ CM(R) has a projective resolution −→ R → Swi → 0 for a := fw(1) · · · fw(i) and b := fw(i+1) · · · fw(n), the corresponding object X ∈ CM(R ′′) has a projective resolution (u ab −v) −→ R′′2 (v ab −u) −→ R′′2 → X → 0. It is easily checked that X is isomorphic to (u, a) = Uwi . (b) Any cluster tilting object gives a non-commutative crepant resolution by [I2, Th. 5.2.1]. They are derived equivalent by [I2, Cor. 5.3.3]. � For example, k[[x, y, u, v]]/((x− λ1y) · · · (x− λny) + uv) has a non-commutative crepant resolution for distinct elements λ1, · · · , λn ∈ k. 5. Link with birational geometry There is another approach to the investigation of cluster tilting objects for maximal Cohen-Macaulay modules, using birational geometry. More specifically there is a close connection between resolutions of three-dimensional Gorenstein singularities and cluster tilting theory, provided by the so-called non- commutative crepant resolutions of Van den Bergh. This gives at the same time alternative proofs for geometric results, using cluster tilting objects. The aim of this section is to establish a link with small resolutions. We give relevant criteria for having small resolutions, and apply them to give an alternative approach to most of the results in the previous sections. Let (R,m) be a three-dimensional complete normal Gorenstein singularity over an algebraically closed field k of characteristic zero, and let X = Spec(R). A resolution of singularities Y −→ X is called • crepant, if ωY ∼= π ∗ωX for canonical sheaves ωX and ωY of X and Y respectively. • small, if the fibre of the closed point has dimension at most one. CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 21 A small resolution is automatically crepant, but the converse is in general not true. However, both types of resolutions coincide for certain important classes of three-dimensional singularities. A cDV (compound Du Val) singularity is a three-dimensional singularity given by the equation f(x, y, z) + tg(x, y, z, t) = 0, where f(x, y, z) defines a simple surface singularity and g(x, y, z, t) is arbitrary. A cDV singularity is called cAn if the intersection of f(x, y, z) + tg(x, y, z, t) = 0 with a generic hyperplane ax+ by + cz + dt = 0 in k4 is an An surface singularity. Generic means that the coefficients (a, b, c, d) belong to a non-empty Zariski open subset in k4. Theorem 5.1. [Re, Cor. 1.12, Th. 1.14] Let X be a three-dimensional Gorenstein singularity. (a) If X has a small resolution, then it is cDV. (b) If X is an isolated cDV singularity, then any crepant resolution of X is small. Since any isolated cDV singularity is terminal [Re], we can apply Van den Bergh’s results on non- commutative crepant resolutions defined as follows. Definition 5.2. [V2, Def. 4.1] Let (R,m) be a three-dimensional normal Gorenstein domain. An R- module M gives rise to a non-commutative crepant resolution if (i) M is reflexive, (ii) A = EndR(M) is Cohen-Macaulay as an R–module, (iii) gl.dim(A) = 3. The following result establishes a useful connection. Theorem 5.3. [V1, Cor. 3.2.11][V2, Th. 6.6.3] Let (R,m) be an isolated cDV singularity. Then there exists a crepant resolution of X = Spec(R) if and only if there exists a non-commutative one in the sense of Definition 5.2. The existence of a non-commutative crepant resolution turns out to be equivalent to the existence of a cluster tilting object in the triangulated category CM(R). Theorem 5.4. [I2, Th. 5.2.1][IR, Cor. 8.13] Let (R,m) be a three-dimensional normal Gorenstein domain which is an isolated singularity. Then the existence of a non-commutative crepant resolution is equivalent to the existence of a cluster tilting object in the stable category of maximal Cohen-Macaulay modules CM(R). Proof. For convenience of the reader, we give an outline of the proof (see also Proposition 4.4). Let us first assume that M is a cluster tilting object in CM(R). Then M is automatically reflexive. From the exact sequence 0 −→ Ω(M) −→ F −→M −→ 0 we obtain (1) 0 −→ EndR(M) −→ HomR(F,M) −→ HomR(Ω(M),M) −→ Ext R(M,M) −→ 0. Since M is rigid, Ext1R(M,M) = 0. Moreover, depth(HomR(F,M)) = depth(M) = 3 and depth(HomR(Ω(M),M) ≥ 2, and hence depth(EndR(M)) = 3 and A = End(M) is maximal Cohen- Macaulay over R. For the difficult part of this implication, claiming that gl.dim(A) = 3, we refer to [I1, Th. 3.6.2]. For the other direction, let M be a module giving rise to a non-commutative crepant resolution. Then by [IR, Th. 8.9] there exists another module M ′ giving rise to a non-commutative crepant resolution, which is maximal Cohen-Macaulay and contains R as a direct summand. By the assumption, depth(EndR(M ′)) = 3 and we can apply [IR, Lem. 8.5] to the exact sequence (1) to deduce that Ext1R(M ′,M ′) = 0, so that M ′ is rigid. The difficult part saying that M ′ is cluster tilting is proven in [I2, Th. 5.2.1]. � We now summarize the results of this section. 22 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN Theorem 5.5. Let (R,m) be an isolated cDV singularity. Then the following are equivalent. (a) Spec(R) has a small resolution. (b) Spec(R) has a crepant resolution. (c) (R,m) has a non-commutative crepant resolution. (d) CM(R) has a cluster tilting object. We have an efficient criterion for existence of a small resolution of a cAn–singularity. Theorem 5.6. [Kat, Th. 1.1] Let X = Spec(R) be an isolated cAn–singularity. (a) Let Y −→ X be a small resolution. Then the exceptional curve in Y is a chain of n projective lines and X has the form g(x, y) + uv, where the curve singularity g(x, y) has n + 1 distinct branches at the origin. (b) If X has the form g(x, y) + uv, where the curve singularity g(x, y) has n+ 1 distinct branches at the origin, then X has a small resolution. Using the criterion of Katz together with Knörrer periodicity, we get additional equivalent conditions in a special case. Theorem 5.7. Let (R,m) be an isolated cAn–singularity defined by the equation g(x, y) + zt. Then the following conditions are equivalent in addition to (a)-(d) in Theorem 5.5. (e) Let R′ be a one-dimensional singularity defined by g(x, y). Then CM(R′) has a cluster tilting object. (f) The number of irreducible power series in the prime decomposition of g(x, y) is n+ 1. Proof. (a)⇔(f) This follows from Theorem 5.6. (d)⇔(e) By the Knörrer periodicity there is an equivalence of triangulated categories between the stable categories CM(R) ∼= CM(R′). For, the equivalence of these stable categories given in [Kn, So] is induced by an exact functor taking projectives to projectives. � Theorem 5.8. Assume that the equivalent conditions in Theorem 5.7 are satisfied. Then the following numbers are equal. (a) One plus the number of irreducible components of the exceptional curve of a small resolution of Spec(R). (b) The number of irreducible power series in the prime decomposition of g(z, t). (c) The number of simple modules of non-commutative crepant resolutions of (R,m). (d) One plus the number of non-isomorphic indecomposable summands of basic cluster tilting objects in CM(R). Proof. (a) and (b) are equal by Theorem 5.6. (a) and (c) are equal by [V1, Th. 3.5.6]. (c) and (d) are equal by [IR, Cor. 8.8]. � 6. Application to curve singularities In this section we apply results in the previous section to some curve singularities to investigate whether they have some cluster tilting object or not. In addition to simple singularities, we study some other nice singularities. In what follows we refer to [AGV] as a general reference for classification of singularities. To apply results in previous sections to minimally elliptic singularities, we also consider a three- dimensional hypersurface singularity Tp,q,2,2(λ) = k[[x, y, u, v]]/(x p + yq + λx2y2 + uv). To apply Theorem 5.7 to a curve singularity, we have to know that the corresponding three-dimensional singularity is cAn. It is given by the following result, where we denote by ord(g) the degree of the lowest term of a power series g. Proposition 6.1. We have the following properties of three-dimensional hypersurface singularities: CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 23 (a) An (1 ≤ n) is a cA1–singularity, (b) Dn (4 ≤ n) and En (n = 6, 7, 8) are cA2–singularities, (c) T3,q,2,2(λ) (6 ≤ q) is a cA2–singularity, (d) Tp,q,2,2(λ) (4 ≤ p ≤ q) is a cA3–singularity, (e) k[[x, y, z, t]]/(x2 + y2 + g(z, t)) (g ∈ k[[z, t]]) is a cAm–singularity if m = ord(g) − 1 ≥ 1. We shall give a detailed proof at the end of this section. In view of Theorem 5.7 and Proposition 6.1, we have the following main result in this section. Theorem 6.2. (a) A simple three-dimensional singularity satisfies the equivalent conditions in The- orem 5.7 if and only if it is of type An (n is odd) or Dn (n is even). (b) A Tp,q,2,2(λ)–singularity satisfies the equivalent conditions in Theorem 5.7 if and only if p = 3 and q is even or if both p and q are even. (c) A singularity k[[x, y, u, v]]/(uv + f1 · · · fn) with irreducible and mutually prime fi ∈ (x, y) ⊂ k[[x, y]] (1 ≤ i ≤ n) satisfies the equivalent conditions in Theorem 5.7 if and only if fi /∈ (x, y) for any i. Proof. Each singularity is cAm by Proposition 6.1, and defined by an equation of the form g(x, y) + uv. By Theorem 5.7, we only have to check whether the number of irreducible power series factors of g(x, y) is m+ 1 or not. (a) For an An–singularity, we have m = 1 and g(x, y) = x 2 + yn+1. So g has two factors if and only if n is odd. For a Dn–singularity, we have m = 2 and g(x, y) = (x 2 + yn−2)y. So g has three factors if and only if n is even. For an En–singularity, we have m = 2 and g(x, y) = x 3 + y4, x(x2 + y3) or x3 + y5. In each case, g does not have three factors. (b) First we consider the simply elliptic case. We have m = 2 and g(x, y) = y(y − x2)(y − λx2) for (p, q) = (3, 6), and m = 3 and g(x, y) = xy(x − y)(x− λy) for (p, q) = (4, 4). In both cases, g has m+ 1 factors. Now we consider the cusp case. We have m = 2 for p = 3 and m = 3 for p > 3, and g(x, y) = (xp−2 − y2)(x2 − yq−2). So g has m+ 1 factors if and only if p = 3 and q is even or if both p and q are even. (c) We have m = i=1 ord(fi) − 1 and g = f1 · · · fn. So g has m+ 1 factors if and only if ord(fi) = 1 for any i. � Immediately we have the following conclusion. Corollary 6.3. (a) A simple curve singularity R has a cluster tilting object if and only if it is of type An (n is odd) or Dn (n is even). The number of non-isomorphic indecomposable summands of basic cluster tilting objects in CM(R) is 1 for type An (n is odd) and 2 for type Dn (n is even). (b) A Tp,q(λ)-singularity R has a cluster tilting object if and only if p = 3 and q is even or if both p and q are even. The number of non-isomorphic indecomposable summands of basic cluster tilting objects in CM(R) is 2 if p = 3 and q is even, and 3 if both p and p are even. (c) A singularity R = k[[x, y]]/(f1 · · · fn) with irreducible and mutually prime fi ∈ (x, y) ⊂ k[[x, y]] (1 ≤ i ≤ n) has a cluster tilting object if and only if fi /∈ (x, y) 2 for any i. In this case, the number of non-isomorphic indecomposable summands of basic cluster tilting objects in CM(R) is n− 1. In view of Theorem 4.2, we have completed the proof of Theorem 1.5. In the rest of this section, we shall prove Proposition 6.1. Let k be an algebraically closed field of characteristic zero, R = k[[x1, x2, . . . , xn]] the local ring of formal power series and m its maximal ideal. We shall need the following standard notions. Definition 6.4. For f ∈ m2 we denote by J(f) = 〈 ∂f , . . . , ∂f 〉 its Jacobi ideal. The Milnor number µ(f) is defined as µ(f) := dimk(R/J(f)). 24 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN The following lemma is standard (see for example [AGV, GLSh]): Lemma 6.5. A hypersurface singularity f = 0 is isolated if and only if µ(f) <∞. Definition 6.6 ([AGV]). Two hypersurface singularities f = 0 and g = 0 are called right equivalent ∼ g) if there exists an algebra automorphism ϕ ∈ Aut(R) such that g = ϕ(f). Note that f ∼ g implies an isomorphism of k–algebras R/(f) ∼= R/(g). The following lemma is straightforward, see for example [GLSh, Lem. 2.10]. Lemma 6.7. Assume f ∼ g, then µ(f) = µ(g). In what follows, we shall need the next standard result on classification of singularities, see for example [GLSh, Cor. 2.24]. Theorem 6.8. Let f ∈ m2 be an isolated singularity with Milnor number µ. Then ∼ f + g for any g ∈ mµ+2. We shall need the following easy lemma. Lemma 6.9. Let f = x2 + y2 + p(x, y, z), where p(x, y, z) = zn + p1(x, y)z n−1 + · · · + pn(x, y) is a homogeneous form of degree n ≥ 3. Then ∼ x2 + y2 + zn. Proof. Write p(x, y, z) = zn + xu + yv for some homogeneous forms u and v of degree n− 1. Then x2 + y2 + zn + xu+ yv = (x+ u/2)2 + (y + v/2)2 + zn − (u2 + v2)/4. After a change of variables x 7→ x+ u/2, y 7→ y + v/2 and z 7→ z we reduce f to the form f = x2 + y2 + zn + h, where h ∈ m2(n−1) ⊂ mn+1. Note that µ(x2 + y2 + zn) = n− 1, hence by Theorem 6.8 we have ∼ x2 + y2 + zn. Now we are ready to give a proof of Proposition 6.1. We only have to show the assertion (e) since the other cases are special cases of this. We denote by H the hyperplane in a four-dimensional space defined by the equation t = αx+ βy + γz, α, β, γ ∈ k. We put g(z, t) = a0z m+1 + a1z mt+ · · · + am+1t m+1 + (higher terms). Then the intersection of H with the singularity defined by the equation x2 + y2 + g(z, t) is given by the equation f = h+ (higher terms), where h = x2 + y2 + a0z m+1 + a1z m(αx+ βy + γz) + · · · + am+1(αx+ βy + γz) Now we consider the case m = 1. We have h ∼ x2 + y2 + z2 since any quadratic form can be diagonalized using linear transformations. By Lemma 6.7, we have µ(h) = µ(x2 + y2 + z2) = 1. Hence ∼ x2 + y2 + z2 by Theorem 6.8. Next we consider the case m ≥ 2. Assume α ∈ k satisfies a0 + a1α+ · · ·+ am+1α m+1 6= 0. By Lemma 6.9, we have h ∼ x2 + y2 + zm+1. By Lemma 6.7, we have µ(h) = µ(x2 + y2 + zm+1) = m. Hence ∼ x2 + y2 + zm+1 by Theorem 6.8. Consequently, x2 + y2 + g(z, t) is cAm. � CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 25 7. Examples of 2-CY tilted algebras Since the 2-CY tilted algebras coming from maximal Cohen-Macaulay modules over hypersurfaces have some nice properties, it is of interest to have more explicit information about such algebras. This section is devoted to some such computations for algebras coming from minimally elliptic singularities. We obtain algebras appearing in classification lists for some classes of tame self-injective algebras [Er, BS, Sk]. We start with giving some general properties which are direct consequences of Lemma 2.2. Theorem 7.1. Let (R,m) be an odd-dimensional isolated hypersurface singularity and Γ a 2-CY tilted algebra coming from CM(R). Then we have the following. (a) Γ is a symmetric algebra. (b) All components in the stable AR-quiver of infinite type Γ are tubes of rank 1 or 2. We now start with our computations of 2-CY tilted algebras coming from minimally elliptic singular- ities. We first introduce and investigate two classes of algebras, and then show that they are isomorphic to 2-CY tilted algebras coming from minimally elliptic singularities. For a quiver Q with finitely many vertices and arrows we define the radical completion k̂Q of the path algebra kQ by the formula k̂Q = lim kQ/ radn(kQ). The reason we deal with completion is the following: Let Q be a finite quiver, J the ideal of k̂Q generated by the arrows and I ⊆ J2 a complete ideal such that Λ = k̂Q/I is finite-dimensional. Lemma 7.2. The ideal I is generated in k̂Q by a minimal system of relations, that is, a set of elements ρ1, · · · , ρn of I whose images form a k-basis of I/IJ + JI. The lemma is shown by a standard argument (cf [BMR3, Section 3]). Its analogue for the non complete path algebra is not always true. For example, for the algebra Λ = B2,2(λ) defined below, the elements ρ1, · · · , ρn listed as generators for I form a minimal system of relations. So they generate I in k̂Q. They also yield a k-basis of I ′/I ′J + JI ′ −→ I/IJ + JI, where I ′ = I ∩ kQ and J ′ = J ∩ kQ. But they do not generate the ideal I ′ of kQ since, as one can show, the quotient kQ/〈ρ1, · · · , ρn〉 is infinite-dimensional. On the other hand, the ideal I ′ is generated by the preimage ρ1, · · · , ρn of a basis of I ′/I ′J ′ + J ′I ′ if the quotient kQ/〈ρ1, · · · , ρn〉 is finite-dimensional, since then the ideal 〈ρ1, · · · , ρn〉 contains a power of J ′. This happens for example for the algebra A2(λ) as defined below, cf. also [Sk, 5.9] and [BS, Th. 1]. We know that for all vertices i, j of Q, we have dimk ei(I/IJ + JI)ej = dimk Ext Λ(Si, Sj) where Si and Sj denote the simple Λ-modules corresponding to the vertices i and j [B]. When Λ is 2-CY tilted, then dim Ext1Λ(Sj , Si) ≥ dim Ext Λ(Si, Sj) (see [BMR3, KR]). Thus the number of arrows in Q is an upper bound on the number of elements in a minimal system of relations. Definition 7.3. (1) For q ≥ 2 and λ ∈ k∗ we write Aq(λ) = k̂Q/I, where Q = ·ϕ ## α // · I = 〈ψα− αϕ, βψ − ϕβ, ϕ2 − βα, ψq − λαβ〉. If q = 2, then we additionally assume λ 6= 1. (It can be shown that for q ≥ 3 we have Aq(λ) ∼= Aq(1), so we drop the parameter λ in this case.) (2) For p, q ≥ 1 and λ ∈ k∗ we write Bp,q(λ) = k̂Q/I, where Q = ·ϕ ## α // · γ // · 26 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN I = 〈βα− ϕp, γδ − λψq, αϕ− δγα, ϕβ − βδγ, δψ − αβδ, ψγ − γαβ〉. For p = q = 1 we additionally assume λ 6= 1. When p = q = 1, the generators ϕ and ψ can be excluded and B1,1(λ) is given by the completion of the path algebra of the quiver Q = · α // · γ // · modulo the relations I = 〈αβα − δγα, αβδ − λδγδ, γαβ − λγδγ, βδγ − βαβ〉. For (p, q) 6= (1, 1) we have Bp,q(λ) ∼= Bp,q(1). In particular, for p = 1 and q ≥ 2 the algebra is isomorphic to k̂Q/I, where Q = · α // · γ // · I = 〈γδ − ψq, αβα − δγα, βαβ − βδγ, δψ − αβδ, ψγ − γαβ〉. It turns out that the algebras Aq(λ) and Bp,q(λ) are finite dimensional. In order to show this it suffices to check that all oriented cycles in k̂Q/I are nilpotent. Lemma 7.4. In the algebra Aq(λ) the following zero relations hold: αβα = 0, βαβ = 0, αϕ2 = ψ2α = 0, ϕ2β = βψ2 = 0, ϕ4 = 0, ψq+2 = 0. Proof. We have to consider separately the cases q = 2 and q ≥ 3. Let q = 2, then we assumed λ 6= 1. We have αβα = αϕ2 = ψ2α = λ−1αβα, hence αβα = 0. In a similar way we obtain βαβ = 0. Then αϕ2 = αβα = 0, ϕ2 = βαβα = 0 and the remaining zero relations follow analogously. Let q ≥ 3. Then ψqα = αβα = αϕ2 = ψ2α, so (1 − ψq−2)ψ2α = 0 and hence ψ2α = αβα = 0 in k̂Q/I. The remaining zero relations follow similarly. � Lemma 7.5. We have the following relations in Bp,q(λ): ϕp+2 = 0, ψq+2 = 0, γαϕ = ψγα = 0, ϕβδ = βδψ = 0. Moreover, αβ · δγ = δγ · αβ. For q ≥ p ≥ 2 we have (αβ)2 = (δγ)2 = 0, for q > p = 1 we have (αβ)3 = 0, (δγ)2 = 0, (αβ)2 · (δγ) = 0 and for p = q = 1 (αβ)3 = (γδ)3 = 0, (αβ)2 = αβ · δγ = λ(δγ)2. The proof is completely parallel to the proof of the previous lemma and is therefore skipped. � The main result of this section is the following CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 27 Theorem 7.6. (a) Let R be a T3,2q+2(λ)–singularity, where q ≥ 2 and λ ∈ k ∗. Then in the triangulated category CM(R) there exists a cluster tilting object with the corresponding 2-CY-tilted algebra isomorphic to Aq(λ). (b)For R = T2p+2,2q+2(λ) the category CM(R) has a cluster tilting object with endomorphism algebra isomorphic to Bp,q(λ). Proof. (a) We consider first the case of T3,2q+2(λ). The coordinate ring of T3,6(λ) is isomorphic to R = k[[x, y]]/(y(y − x2)(y − λx2)), where λ 6= 0, 1. Consider Cohen-Macaulay modules M and N given by the two-periodic free resolutions M = (R −−−→ R y(y−λx2) −−−−−−→ R), N = (R y(y−x2) −−−−−→ R y−λx2 −−−−→ R). Then M ⊕N is cluster tilting by Theorem 4.1 or Corollary 6.3. In order to compute the endomorphism algebra End(M ⊕N), note that End(M) ∼= k[ϕ]/〈ϕ where ϕ = (x, x) is an endomorphism of M viewed as a two-periodic map of a free resolution. In End(M) we have (y, y) = (x, x)2 = ϕ2. Similarly, End(N) ∼= k[ψ]/〈ψ 4〉, ψ = (x, x), (y, y) = λ(x, x)2 = λψ2, Hom(M,N) = k2 = 〈(1, y), (x, xy)〉, Hom(N,M) = k2 = 〈(y, 1), (xy, x)〉. The isomorphism A2(λ) −→ End(M ⊕N) is given by ϕ 7→ (x, x), ψ 7→ (x, x), α 7→ (1, y), β 7→ (y, 1). Assume now q ≥ 3 and R = T3,2q+2. By [AGV] we may write R = k[[x, y]]/((x− y2)(x2 − y2q)). Consider the Cohen-Macaulay module M ⊕N , where M = (R −−−→ R x2−y2q −−−−−→ R), N = (R (x−y2)(x+yq) −−−−−−−−−→ R −−−→ R). Again, by a straightforward calculation End(M) ∼= k[ϕ]/〈ϕ4〉, ϕ = (y, y), End(N) ∼= k[ψ]/〈ψq+2〉, ψ = (y, y) Hom(M,N) = k2 = 〈(1, x+ yq), (y, y(x+ yq))〉, Hom(M,N) = k2 = 〈(x+ yq, 1), (y(x+ yq), y)〉. If q ≥ 4 then End(M ⊕N) is isomorphic to k̂Q/I, where Q = ·ϕ ## α // · and the relations are βα = ϕ2, αβ = 2ψq, αϕ = ψα, ϕβ = βψ ϕ = (y, y), ψ = (y, y), α = (1, x+ yq), β = (x+ yq, 1). By rescaling all generators α 7→ 2aα, β 7→ 2bβ, ϕ 7→ 2fϕ, ψ 7→ 2gψ for properly chosen a, b, f, g ∈ Q one can easily show End(M ⊕N) ∼= Aq. 28 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN The case q = 3 has to be considered separately, since this time the relations are βα = ϕ2 + ϕ3, αβ = 2ψq, αϕ = ψα, ϕβ = βψ. We claim that there exist invertible power series u(t), v(t), w(t), z(t) ∈ k[[t]] such that the new generators ϕ′ = u(ϕ)ϕ, ψ′ = v(ψ)ψ, α′ = αw(ϕ) = w(ψ)α, β′ = βz(ψ) = z(ϕ)β satisfy precisely the relations of the algebra A3. This is fulfilled provided we have the following equations in k[[t]]:  zw = u2(1 + tu) zw = 2v3 uw = vw uz = vz. This system is equivalent to u(t) = v(t) = (2 − t)−1 = )2 + . . . ) and hence the statement is proven. The case of T2p+2,2q+2(λ) is essentially similar. For p = q = 1 we have R = k[[x, y]]/(xy(x − y)(x− λy)). Take  M = (R −−−→ R xy(x−λy) −−−−−−→ R), N = (R x(x−y) −−−−−→ R y(x−λy) −−−−−→ R), K = (R xy(x−y) −−−−−→ R −−−→ R). By Theorem 4.1 or Corollary 6.3, M ⊕N ⊕K is cluster tilting. Moreover, B1,1(λ) ≃ End(M ⊕N ⊕K). Let now R = k[[x, y]]/((xp − y)(xp + y)(yq − x)(yq + x)), where (p, q) 6= (1, 1) and  M = (R −−−→ R (yq+x)(yq−x)(xp+y) −−−−−−−−−−−−−−→ R), N = (R (xp−y)(xp+y) −−−−−−−−−→ R (yq−x)(yq+x) −−−−−−−−−→ R), K = (R (xp−y)(xp+y)(yq+x) −−−−−−−−−−−−−−→ R −−−→ R). By Theorem 4.1 or Corollary 6.3, M ⊕N ⊕K is cluster tilting, and by a similar case-by-case analysis it can be verified that End(M ⊕N ⊕K) ∼= Bp,q. � We have seen that the algebras Aq(λ) and Bp,q(λ) are symmetric, and the indecomposable nonpro- jective modules have τ -period at most 2, hence Ω-period dividing 4 since τ = Ω2 in this case. A direct computation shows that the Cartan matrix is nonsingular. Note that these algebras appear in Erd- mann’s list of algebras of quaternion type [Er], see also [Sk], that is, in addition to the above properties, the algebras are tame. Note that for the corresponding algebras, more relations are given in Erdmann’s list. This has to do with the fact that we are working with the completion, as discussed earlier. In our case all relations correspond to different arrows in the quiver. The simply elliptic ones also appear in Bia lkowski-Skowroński’s list of weakly symmetric tubular algebras with a nonsingular Cartan matrix. This provides a link between some stable categories of maximal Cohen-Macaulay modules over isolated hypersurface singularities, and some classes of finite dimensional algebras, obtained via cluster tilting theory. Previously a link between maximal Cohen-Macaulay modules and finite dimensional algebras was given with the canonical algebras of Ringel, via the categories Coh(X) of coherent sheaves on weighted projective lines in the sense of Geigle-Lenzing [GL]. Here the category of vector bundles is equivalent to the category of graded maximal Cohen-Macaulay modules with degree zero maps, over some isolated CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 29 singularity. And the canonical algebras are obtained as endomorphism algebras of certain tilting objects in Coh(X) which are vector bundles. Note that it is known from work of Dieterich [D], Kahn [Kah], Drozd and Greuel [DG] that mini- mally elliptic curve singularities have tame Cohen-Macaulay representation type. Vice versa, any Cohen- Macaulay tame reduced hypersurface curve singularity is isomorphic to one of the Tp,q(λ), see [DG]. Moreover, simply elliptic singularities are tame of polynomial growth and cusp singularities are tame of exponential growth. Furthermore, the Auslander-Reiten quiver of the corresponding stable categories of maximal Cohen-Macaulay modules consists of tubes of rank one or two, see [Kah, Th. 3.1] and [DGK, Cor. 7.2]. It should follow from the tameness of CM(T3,p(λ)) and CM(Tp,q(λ)) that the associated 2-CY tilted algebras are tame. We point out that in the wild case we can obtain symmetric 2-CY tilted algebras where the stable AR-quiver consists of tubes of rank one and two, and most of them should be wild. It was previously known that there are examples of wild selfinjective algebras whose AR-quivers consist of tubes of rank one or three [AR]. 8. Appendix: 2-CY triangulated categories of finite type In this section, we consider a more general situation than in section 2. Let k be an algebraically closed field and C a k-linear connected 2-Calabi-Yau triangulated category with only finitely many indecompos- able objects. We show that it follows from the shape of the AR quiver of C whether cluster tilting objects (respectively, non-zero rigid objects) exist in C or not. Let us start with giving the possible shapes of the AR quiver of C. Recall that a subgroup G of Aut(Z∆) is called weakly admissible if x and gx do not have a common direct successor for any vertex x in Z∆ and g ∈ G\{1} [XZ, Am]. Proposition 8.1. The AR quiver of C is Z∆/G for a Dynkin diagram ∆ and a weakly admissible subgroup G of Aut(Z∆) which contains F ∈ Aut(Z∆) defined by the list below. Moreover, G is generated by a single element g ∈ Aut(Z∆) in the list below. ∆ Aut(Z∆) F g (An) n : odd Z× Z/2Z ( , 1) (k, 1) (k|n+3 , n+3 is odd) (An) n : even Z n+ 3 k (k|n+ 3) (Dn) n : odd Z× Z/2Z (n, 1) (k, 1) (k|n) (D4) Z× S3 (4, 0) (k, σ) (k|4, σ k = 1) (Dn) n : even, n > 4 Z× Z/2Z (n, 0) (k, 0) (k|n) or (k, 1) (k|n, is even) (E6) Z× Z/2Z (7, 1) (1, 1) or (7, 1) (E7) Z 10 1, 2, 5 or 10 (E8) Z 16 1, 2, 4, 8 or 16 In each case, elements in the torsion part of Aut(Z∆) are induced by the automorphism of ∆. The torsionfree part of Aut(Z∆) is generated by τ except the case (An) with even n, in which case it is generated by the square root of τ . Proof. By [XZ] (see also [Am, 4.0.4]), the AR quiver of C is Z∆/G for a Dynkin diagram ∆ and a weakly admissible subgroup G of Aut(Z∆). Since C is 2-Calabi-Yau, G contains F . By [Am, 2.2.1], G is generated by a single element g. By the condition F ∈ 〈g〉, we have the above list. � Note that, by a result of Keller [Ke], the translation quiver Z∆/G for any Dynkin diagram ∆ and any weakly admissible group G of Aut(Z∆) is realized as the AR quiver of a triangulated orbit category Db(H)/g for a hereditary algebra H of type ∆ and some autofunctor g of Db(H). 30 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN Theorem 8.2. (1) C has a cluster tilting object if and only if the AR quiver of C is Z∆/g for a Dynkin diagram ∆ and g ∈ Aut(Z∆) in the list below. ∆ Aut(Z∆) g (An) n : odd Z× Z/2Z ( , 1) (3|n) or (n+3 (An) n : even Z (3|n) or n+ 3 (Dn) n : odd Z× Z/2Z (k, 1) (k|n) (D4) Z× S3 (k, σ) (k|4, σ k = 1, (k, σ) 6= (1, 1)) (Dn) n : even, n > 4 Z× Z/2Z (k, k) (k|n) (E6) Z× Z/2Z (7, 1) (E7) Z 10 (E8) Z 8 or 16 (2) C does not have a non-zero rigid object if and only if the AR quiver of C is Z∆/g for a Dynkin diagram ∆ and g ∈ Aut(Z∆) in the list below. ∆ Aut(Z∆) g (An) n : odd Z× Z/2Z − (An) n : even Z 1 (Dn) n : odd Z× Z/2Z − (D4) Z× S3 (1, 1) (Dn) n : even, n > 4 Z× Z/2Z (1, 0) (E6) Z× Z/2Z (1, 1) (E7) Z 1 (E8) Z 1 or 2 Proof. Our method is based on the computation of additive functions in section 2. We refer to [I1, Section 4.4] for detailed explanation. (1) Assume that g is on the list. Then one can check that C has a cluster tilting object. For example, consider the (Dn) case here. Fix a vertex x ∈ Z∆ corresponding to an end point of ∆ which is adjacent to the branch vertex of ∆. Then the subset {(1, 1)lx | l ∈ Z} of Z∆ is stable under the action of g, and gives a cluster tilting object of C. Conversely, assume that C has a cluster tilting object. Then one can check that g is on the list. For example, consider the (An) case with even n here. By [CCS, I1], cluster tilting objects correspond to dissections of a regular (n + 3)-polygon into triangles by non-crossing diagonals. The action of g shows that it is invariant under the rotation of 2kπ -radian. Since the center of the regular (n + 3)-polygon is contained in some triangle or its edge, we have 2kπ = 2π, 4π , π or 2π . Since k|n+ 3 and n is even, we have k = n+ 3 or n+3 (2) If g is on the list above, then one can easily check that C does not have non-zero rigid objects. 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Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, LMS 146, 1990. Johannes-Gutenberg Universität Mainz, Fachbereich Physik, Mathematik und Informatik, Institut für Math- ematik, 55099 Mainz, Germany E-mail address: [email protected] Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan E-mail address: [email protected] UFR de Mathématiques, UMR 7586 du CNRS, Case 7012, Université Paris 7, 2 place Jussieu, 75251 Paris Cedex 05, France E-mail address: [email protected] Institutt for matematiske fag, Norges Teknisk-naturvitenskapelige universitet, N-7491, Trondheim, Norway E-mail address: [email protected] Introduction Acknowledgment 1. Main results 2. Simple hypersurface singularities 3. Computation with Singular 4. One-dimensional hypersurface singularities 5. Link with birational geometry 6. Application to curve singularities 7. Examples of 2-CY tilted algebras 8. Appendix: 2-CY triangulated categories of finite type References

0704.1250

Gemini Mid-IR Polarimetry of NGC1068: Polarized Structures Around the Nucleus

Gemini Mid-IR Polarimetry of NGC1068: Polarized Structures Around the Nucleus C. Packham1, S. Young2, S. Fisher3, K. Volk3, R. Mason3, J. H. Hough2 P. F. Roche4 M. Elitzur5, J. Radomski6, and E. Perlman7 [email protected] ABSTRACT We present diffraction limited, 10µm imaging polarimetry data for the cen- tral regions of the archetypal Seyfert AGN, NGC1068. The position angle of polarization is consistent with three dominant polarizing mechanisms. We iden- tify three distinct regions of polarization: (a) north of the nucleus, arising from aligned dust in the NLR, (b) south, east and west of the nucleus, consistent with dust being channeled toward the central engine and (c) a central minimum of polarization consistent with a compact (≤22pc) torus. These observations pro- vide continuity between the geometrically and optically thick torus and the host galaxy’s nuclear environments. These images represent the first published mid- IR polarimetry from an 8-m class telescope and illustrate the potential of such observations. Subject headings: galaxies: nuclei — galaxies: Seyfert — galaxies: structure galaxies: individual(NGC 1068) — infrared: galaxies — polarization: galaxies 1University of Florida, Department of Astronomy, 211 Bryant Space Science Center, P.O. Box 112055, Gainesville, FL, 32611-2055, USA 2Center for Astrophysics Research, University of Hertfordshire, Hatfield, AL10 9AB, UK 3Gemini Observatory, Northern Operations Center, 670 N. A’ohuku Place, Hilo, Hawaii,96720, USA 4University of Oxford, Department of Astrophysics, Keble Road, Oxford, OX1 3RH, UK 5University of Kentucky, Department of Physics and Astronomy, 600 Rose Street, Lexington, KY, 40506, 6Gemini Observatory, Southern Operations Center, c/o AURA, INC., Casilla 603, La Serena, Chile 7Physics and Space Sciences Department, Florida Institute of Technology, 150 West University Boulevard, FL, USA http://arxiv.org/abs/0704.1250v1 – 2 – 1. Introduction The unified theory of Seyfert (Sy) type active galactic nuclei (AGN) holds that all types of Sy AGN are essentially the same object, viewed from different lines of sight (LOS). Surrounding the central engine is a geometrically and optically thick, dusty, molecular torus, obscuring the broad emission line region from some LOS. In this scheme, the Sy classification depends solely on the LOS and exact torus properties. Such theories received a major boost through the detection of scattered, and hence polarized, broad emission lines in the spectrum of NGC1068 (Antonucci & Miller 1985), revealing an obscured Sy 1 central engine in the previously classified Sy 2 AGN, entirely consistent with unified theories. Whilst fundamental to unified theories, the torus remains difficult to image directly at optical/IR wavelengths, with perhaps the most direct observation of the torus made by speckle interferometry in the near-IR (Weigelt et al. 2004). Strong evidence for signifi- cant amounts of obscuring material in the central 100pc-scale nuclear regions of NGC1068, possibly in the form of a torus, is provided by observations of CO and HCN emission (Planesas et al. 1991; Jackson et al. 1993; Schinnerer et al. 2000), and recent Chandra X- ray observations (e.g. Ogle et al. 2003). Mid-IR spectroscopy reveals a moderately deep (τ9.7 ≈ 0.4) silicate absorption feature at the nucleus (Roche et al. 1984; Siebenmorgen et al. 2004), whose strength is approximately constant up to ∼1′′ south of the brightest mid-IR point (Mason et al. 2006; Rhee & Larkin 2006). Applying the Nenkova et al. (2002) clumpy torus model, Mason et al. (2006) suggested the torus is compact (≤ 15pc), in good agreement with mid-IR interferometric observations (Jaffe et al. 2004). Further evidence for a compact torus was found through AO fed H2 1-0S(1) observations (Davies et al. 2006), finding a 15pc clump of H2 extending from the nucleus at the same PA as the line of masers. The observed extent of the torus, or nuclear obscuring material in NGC1068, is partly dependent on the wavelength and/or observational technique. Young et al. (1996) used imaging polarimetry to observe the silhouette of obscuring material against the southern ionization cone, which they attributed to the torus, with a derived diameter of ∼200pc in the H-band. The close proximity (1′′ ≡ 72pc) and high brightness of NGC1068 makes it an ideal target for polarimetry, a traditionally photon-starved application. Near-IR studies of NGC1068 by Packham et al. (1997), Lumsden et al. (1999) and Simpson et al. (2002) clearly detected the bi-conical ionization structure in scattered light. In the nuclear regions, there is a trend to a position angle (PA) of polarization being perpendicular to the radio jet with increasing wavelength. Modeling of the nuclear regions requires both an extended area of scattering particles as well as dichroic absorption of nuclear emission, possibly by dust in, or associated with, the torus (Young et al. (1995), Watanabe et al. (2003)). Bailey et al. (1988) found that the PA of polarization rotates by ∼70◦ between 4 and – 3 – 5µm. The 10µm spectropolarimetry of Aitken et al. (1984) showed a similar PA of polariza- tion to that at 5µm, and a constant degree of polarization through the silicate absorption feature. These results are entirely consistent with the predicted 90◦ change from dichroic absorption to dichroic emission from aligned dust grains. That the PA change was only ∼70◦ is attributable to dilution of the dichroic emission component by polarized flux in the ex- tended scattering cones, most likely from dichroic emission from dust in the narrow emission line region (NLR) (Bailey et al. 1988). To investigate the contributions of the various polarizing mechanisms and structures in the nucleus of NGC1068, Lumsden et al. (1999) performed imaging polarimetry using a broad-band 8-13 µm filter. These data represented the first and only published mid-IR imaging polarimetry of an AGN, but their interpretation was complicated by the ∼0.7′′ resolution of the data. To take advantage of the improved spatial resolution attainable from an 8 m-class telescope, we obtained new mid-IR imaging polarimetry of NGC1068 during commissioning of this mode at the Gemini North telescope. 2. Observations We obtained imaging polarimetry of NGC1068 during commissioning of the polarimetry unit of Michelle (Glasse et al. 1997) on UT 2005 December 19 on the Gemini North 8.1m telescope. These observations were primarily aimed at measuring the degree and position angle (PA) of polarization with NGC 1068 as a test object, and hence used a limited on-source time of 148 seconds. Michelle uses a Raytheon 320 x 240 pixel Si:As IBC array, providing a plate scale of 0.1′′ per pixel in imaging mode. Images were obtained in the 9.7µm (δλ = 1.0µm, 50% cut-on/off) filter only, using the standard chop-nod technique to remove time-variable sky background, telescope thermal emission and so-called “1/f” detector noise. The chop throw was 15′′ and the telescope was nodded every ∼90 s. The chop throw was fixed at 0◦ (N-S). Conditions were photometric and the observations were diffraction limited (∼0.30′′FWHM). Michelle employs a warm, rotatable half wave retarder (or half wave plate, HWP) to modulate the polarization signal, located upstream of the entrance window of the dewar. A cold wire grid polarizer is used as the polarimetric analyzer, located in a collimated beam. Images were taken at four HWP PAs in the following sequence: 0◦, 45◦, 45◦, 0◦, 22.5◦, 67.5◦, 67.5◦, 22.5◦ in the first nod position, and the sequence repeated in the second nod position. In this manner, the Stokes parameters can be computed as close in time as possible, reducing the effects of variations in sky transmission and emission. This sequence, however, requires many motions of the HWP, and is therefore under evaluation with a view to reducing the – 4 – number of HWP motions to increase observing efficiency. Data were reduced using the Gemini IRAF package in conjunction with Starlink POLPACK software (Berry & Gledhill 2003). The difference for each chop pair in a given nod position and HWP PA was calculated, and then differenced with the second nod position at the same HWP PA. Images were aligned through shifting by fractional pixel values to account for slight image drift between frames, and then the Stokes parameters I, Q and U computed using POLPACK. A total of 20 nod positions were recorded, and residual array/electronic noise was removed through use of a median-filter noise mask. The data were reduced through creation of four individual I, Q and U maps and also through coadding all frames at their respective HWP PA first and then producing a single I, Q and U map. The S/N in the latter method is slightly higher in the individual Q and U maps, presumably due to a ’smoothing’ of the Q and U during the co-addition; these are the data used in this paper. The efficiency and zero angle calibration of the polarimeter were measured through observations of two polarized sources and comparison with measurements published by Smith et al. (2000). The instrumental polarization was estimated to be ≤0.3% through observations of two stars that fulfilled the criteria of (a) high proper motion (hence nearby), (b) high galactic latitude (to minimize the presence of Galactic dust) and (c) an intermediate spectral type star (to minimize intrinsic stellar nebulosity). 3. Results Figure 1 shows the total flux image (color-scale and contours) with the polarization vectors overlaid. The polarization vectors are plotted where the S/N is ≥54 in total flux, and contours are linearly spaced in intensity, starting at a S/N of 27. Figure 2 shows the polarized intensity map, produced by multiplying the degree of polarization by the total intensity image. As in Figure 1, only where the S/N in the total intensity image is ≥54 are data plotted. The resultant polarization vectors are contained within an approximate N-S oriented ellipse, major/minor axes 1.7′′/1.2′′ respectively. The integrated degree of polarization within that ellipse is 2.48±0.57% at a PA of 26.7±15.3◦. The errors in the degree and PA of polarization are estimated through independent measurements of the four individual polarization maps and computing the standard deviation. It should be noted that the exposure time in those four individual maps was very short, where systematic effects could dominate, and hence the quoted errors may be an overestimation. The degree of polarization is higher than the 1.30% measured by Lumsden et al. (1999) in a 2′′ aperture, consistent with an increased observed polarization as often arises with improved spatial resolution. The PA of polarization is significantly different from Lumsden’s measurement of 49◦ in a 2′′ aperture. – 5 – However, our data shows a PA rotation of 94◦ from the K band data of Packham et al. (1997) and Lumsden et al. (1999), entirely as expected if the dominant polarizing mechanism changes from dichroic absorption to emission between the two wavebands, as described in §1. We speculate the Lumsden et al. (1999) ∼0.68′′ results suffered significantly greater contamination in their beam, possibly from surrounding extended polarization, as compared to our ∼0.30′′ results. Additionally, the wider bandwidth of Lumsden et al. (1999) would have been more affected by the different and competing polarizing components. The degree and PA of polarization suggests contributions from three components. The first extends ∼1′′ north of the mid-IR peak and is coincident with the inner regions of the radio jet, with a PA of polarization approximately N-S. The second region extends south, east and west of the nucleus, with a PA of polarization of ∼35◦. Finally, the degree of polarization drops to a minimum very close to the mid-IR total flux peak, which we believe is most likely to arise from an unresolved polarization contribution with a PA of polarization approximately orthogonal to that of the more extended emission, leading to a reduction in the measured polarization. The polarized intensity image reveals polarized emission extending north of the mid-IR peak, and two areas of enhanced flux east and west of the mid-IR peak, and a minimum close to the mid-IR total flux peak. Table 1 summarizes the locations and polarization components. 4. Discussion Polarization at mid-IR wavelengths most likely arises from either dichroic absorption or emission, both due to dust grains with a preferred alignment. The integrated PA of polar- ization in these data confirms and enhances the interpretation of the near-90◦ PA flip from near- to mid-IR wavelengths, with the factor ∼2.5 increase in spatial resolution providing a more accurate and consistent result. Galliano et al. (2003, 2005) suggested, based on spatial coincidence, the [OIII] clouds in the ionization cone are the dominant mid-IR sources away from the compact torus. The polarized flux image shows a similar spatial correspondence with the [OIII], and the PA of polarization north of the nucleus is consistent with the in- terpretation of Bailey et al. (1988) of dichroic emission in the NLR, possibly through dust alignment via jet streaming or a helical magnetic field associated with the jet. We discount directly observed synchrotron radiation from the radio jet accounting for the polarization, as an extrapolation of the radio emission to the mid-IR provides too little flux. Hence, this data confirms the extended mid-IR polarized emission north of the nucleus is dominantly from dust in the ionization cone. South, east and west of the nucleus, as the PA of polarization is perpendicular to that – 6 – in the near-IR where the polarization is thought to be produced by dichroic absorption, we suggest the dominant polarizing mechanism is dichroic emission by grains aligned to the same field direction as the absorbing grains, in agreement with other authors (i.e. Bailey et al. (1988)). Due to the Barnett effect (Lazarian (2003), and references therein), grains align with their short axes parallel to the local magnetic field, and the PA of polarization is parallel to the direction of the magnetic field for dichroic absorption and orthogonal for dichroic emission. The location of the polarized emission areas and PA of polarization is suggestive of warm aligned dust grains being channeled from the host galaxy toward the torus. Indeed, the PA of the polarized flux is coincident with the H2 material that Davies et al. (2006) associated with molecular material in a compact torus. Dichroic absorption in an unresolved optically thick central region could account for the minimum in polarization close to the peak of mid-IR flux, with a PA of polarization approximately orthogonal to the more extended dichroic emission to the east, west and south. Alternatively the mid-IR flux in the innermost regions could arise from a strong mid-IR, intrinsically unpolarized, source. However, there is tentative evidence of the central regions showing a twist in the PA of polarization, tending toward a similar PA found in the dichroic absorption at near-IR wavelengths (e.g. Packham et al. (1997)), which supports the dichroic absorption interpretation. In both possibilities, a compact (≤0.3′′ (≤22pc) diameter) torus could account for this result. If correct, the polarization minimum indicates the true position of the central engine, which is not coincident with the mid-IR total flux peak, but displaced by ∼0.2′′ to the west. CO (Schinnerer et al. 2000) and optical HST (Catchpole & Boksenberg 1997) obser- vations are interpreted as evidence of a warped molecular disk on 100pc scales, partially obscuring the nuclear regions of the host galaxy and ionization cone pointing away from Earth. Indeed, Schinnerer et al. (2000) speculate this material, rather than a compact torus, is responsible for obscuring the AGN. Elitzur & Shlosman (2006) suggest the 100pc molec- ular structure is the extension of the pc scale disk of masers (Greenhill & Gwinn (1997), Gallimore et al. (2001), Galliano et al. (2003)). We suggest that our data provide continuity between the geometrically thick torus (height/radius ∼1) to the flatter, larger galactic disk (height/radius ∼0.15). The western polarized feature is considerably larger than the compact (≤ few pc) torus suggested by several authors (e.g. Jaffe et al. 2004; Mason et al. 2006; Packham et al. 2005; Radomski et al. 2006) on the basis of mid-IR imaging and modeling, but much smaller than the suggested torus detected by Young et al. (1996). However, the feature is detected in polarized flux, a technique that increases contrast by removing the dominant, unpolarized, emission. Distinct from total flux, polarimetric observations are therefore potentially much – 7 – more sensitive to emission from the putative faint, outer regions of the torus where the interaction with the inner regions of the host galaxy must occur. We suggest that a way to reconcile the evidence for a compact torus with these observations and others, such as extended silicate absorption (Roche et al. (2006), Roche et al. (2007)) and 100 pc-scale CO discs, is that the compact, geometrically and optically thick torus is often surrounded by a larger, more diffuse structure, associated with the dusty central regions of the host galaxy. Where the torus ends and the host galaxy dust structure starts may be more of a question of semantics rather than a true physical boundary. These observations examine the interaction between the host galaxy and possible entrainment into the outer torus regions. Further multiple-wavelength polarimetric observations of both NGC1068 and other AGN are required to test this hypothesis. We are grateful to the Gemini, UKIRT and ATC science and engineering staff for their outstanding work on Michelle and the Gemini telescope, and wish to note especially Chris Carter. Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agree- ment with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Particle Physics and Astronomy Research Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Coun- cil (Australia), CNPq (Brazil) and CONICET (Argentina). REFERENCES Aitken, D. K., Briggs, G., Bailey, J. A., Roche, P. F., & Hough, J. H. 1984, Nature, 310, 660 Antonucci, R. R. J., & Miller, J. S. 1985, ApJ, 297, 621 Bailey, J., Axon, D. J., Hough, J. H., Ward, M. J., McLean, I., & Heathcote, S. R. 1988, MNRAS, 234, 899 Berry, D. S., & Gledhill, T. M. 2003, Starlink User Notes. Available from http://star-www.rl.ac.uk Catchpole, R. M., & Boksenberg, A. 1997, Ap&SS, 248, 79 Davies, R., Genzel, R., Tacconi, L., Mueller Sanchez, F., & Sternberg, A. 2006, ArXiv Astrophysics e-prints, arXiv:astro-ph/0612009 Elitzur, M., & Shlosman, I. 2006, ApJ, 648, L101 http://star-www.rl.ac.uk http://arxiv.org/abs/astro-ph/0612009 – 8 – Galliano, E., Alloin, D., Granato, G. L., & Villar-Mart́ın, M. 2003, A&A, 412, 615 Galliano, E., Pantin, E., Alloin, D., & Lagage, P. O. 2005, MNRAS, 363, L1 Gallimore, J. F., Henkel, C., Baum, S. A., Glass, I. S., Claussen, M. J., Prieto, M. A., & Von Kap-herr, A. 2001, ApJ, 556, 694 Glasse, A. C. H, Atad-Ettedgui, E. I., & Harris, J. W., SPIE 2871, 1197 Greenhill, L. J., & Gwinn, C. R. 1997, Ap&SS, 248, 261 Lumsden, S. L., Moore, T. J. T., Smith, C., Fujiyoshi, T., Bland-Hawthorn, J., & Ward, M. J. 1999, MNRAS, 303, 209 Jackson, J. M., Paglione, T. A. D., Ishizuki, S., & Nguyen-Q-Rieu 1993, ApJ, 418, L13 Jaffe, W., et al. 2004, Nature, 429, 47 Lazarian, A. 2003, Journal of Quantitative Spectroscopy and Radiative Transfer, 79, 881 Mason, R. E., Geballe, T. R., Packham, C., Levenson, N. A., Elitzur, M., Fisher, R. S., & Perlman, E. 2006, ApJ, 640, 612 Nenkova, M., Ivezić, Ž., & Elitzur, M. 2002, ApJ, 570, L9 Ogle, P. M., Brookings, T., Canizares, C. R., Lee, J. C., & Marshall, H. L. 2003, A&A, 402, Packham, C., Young, S., Hough, J. H., Axon, D. J., & Bailey, J. A. 1997, MNRAS, 288, 375 Packham, C., Radomski, J. T., Roche, P. F., Aitken, D. K., Perlman, E., Alonso-Herrero, A., Colina, L., & Telesco, C. M. 2005, ApJ, 618, L17 Planesas, P., Scoville, N., & Myers, S. T. 1991, ApJ, 369, 364 Radomski, J. T., et al. 2006, American Astronomical Society Meeting Abstracts, 209, #149.11 Rhee, J. H., & Larkin, J. E. 2006, ApJ, 640, 625 Roche, P. F., Whitmore, B., Aitken, D. K., & Phillips, M. M. 1984, MNRAS, 207, 35 Roche, P. F., Packham, C., Telesco, C. M., Radomski, J. T., Alonso-Herrero, A., Aitken, D. K., Colina, L., & Perlman, E. 2006, MNRAS, 367, 1689 – 9 – Roche, P. F., Packham, C., Aitken, D. K., & Mason, R. E. 2007, MNRAS, 375, 99 Schinnerer, E., Eckart, A., Tacconi, L. J., Genzel, R., & Downes, D. 2000, ApJ, 533, 850 Siebenmorgen, R., Krügel, E., & Spoon, H. W. W. 2004, A&A, 414, 123 Simpson, J. P., Colgan, S. W. J., Erickson, E. F., Hines, D. C., Schultz, A. S. B., & Trammell, S. R. 2002, ApJ, 574, 95 Smith, C. H., Wright, C. M., Aitken, D. K., Roche, P. F., & Hough, J. H. 2000, MNRAS, 312, 327 Watanabe, M., Nagata, T., Sato, S., Nakaya, H., & Hough, J. H. 2003, ApJ, 591, 714 Weigelt, G., Wittkowski, M., Balega, Y. Y., Beckert, T., Duschl, W. J., Hofmann, K.-H., Men’shchikov, A. B., & Schertl, D. 2004, A&A, 425, 77 Young, S., Hough, J. H., Axon, D. J., Bailey, J. A., & Ward, M. J. 1995, MNRAS, 272, 513 Young, S., Packham, C., Hough, J. H., & Efstathiou, A. 1996, MNRAS, 283, L1 This preprint was prepared with the AAS LATEX macros v5.2. – 10 – Table 1: Polarization component summary Locale, Aperture, λ Degree of Polarization PA of Polarization Emitting Component Nucleus, 1.7′′x1.2′′ 2.48±0.57% 26.7±15.3◦ Several 9.7µm North region ∼2% ∼8◦ Dichroic emission from dust 9.7µm aligned through jet interaction East, West, South ∼3.5% ∼35◦ Dichroic emission from regions, 9.7µm galactic dust or torus outer edge Innermost region ≤1% - Dichroic absorption or 9.7µm unpolarized source Nucleus, 2′′ 1.3±0.05% 49±3◦ Several, including dichroic 10µm emission Nucleus, 2′′ 4.11±0.46% 120.6±2.38◦ Several, including dichroic 2.2µm absorption a9.7µm data from this paper b10µm data from Lumsden et al. (1999) c2.2µm data from Packham et al. (1997) – 11 – Fig. 1.— Total flux image (color) with the polarization vectors for the central regions of NGC1068. The length of the vector is proportional to the degree of polarization, and the angle shows the PA of polarization. Each pixel is 0.1′′, and the 10% polarization scale bar is shown in the upper right. North is up, and east is to the left. – 12 – Starlink GAIA::Skycat Total Intensity all_coadded_pi_thresh20.sdf 160.0 121.5 packham Jan 17, 2007 at 13:07:23 Fig. 2.— Polarized flux image of the central regions of NGC1068. The “X” shows the position of the peak total flux. Each pixel is 0.1′′. Introduction Observations Results Discussion

0704.1251

Coupling between magnetic ordering and structural instabilities in perovskite biferroics: A first-principles study

Coupling between magnetic ordering and structural instabilities in perovskite biferroics: A first-principles study Nirat Ray and Umesh V. Waghmare Theoretical Sciences Unit Jawaharlal Nehru Centre for Advanced Scientific Research Jakkur PO, Bangalore 560 064, India (Dated: November 13, 2018) We use first-principles density functional theory-based calculations to investigate structural insta- bilities in the high symmetry cubic perovskite structure of rare-earth (R = La, Y, Lu) and Bi-based biferroic chromites, focusing on Γ and R point phonons of states with para-, ferro-, and antiferro- magnetic ordering. We find that (a) the structure with G-type antiferromagnetic ordering is most stable, (b) the most dominant structural instabilities in these oxides are the ones associated with ro- tations of oxygen octahedra, and (c) structural instabilities involving changes in Cr-O-Cr bond angle depend sensitively on the changes in magnetic ordering. The dependence of structural instabilities on magnetic ordering can be understood in terms of how super-exchange interactions depend on the Cr-O-Cr bond angles and Cr-O bond lengths. We demonstrate how adequate buckling of Cr-O-Cr chains can favour ferromagnetism. Born effective charges (BEC) calculated using the Berry phase expression are found to be anomalously large for the A-cations, indicating their chemical relevance to ferroelectric distortions. I. INTRODUCTION A ferroic is a material which exhibits spontaneous and switchable ordering of electric polarization or magnetiza- tion or elastic strain. Materials exhibiting more than one of such orderings termed ‘multiferroics’- have recently be- come the focus of much research1. Most of the biferroics investigated in recent years are ABO3 oxides with per- ovskite structure. The d0−ness or the zero occupancy of tranistion metal B cation is known chemically to fa- vor ferroelectricity2. Hence, the availability of transition metal d -electrons in the perovskite oxides necessary for magnetism, reduces the tendency for off-centering ferro- electric distortions2 making multiferroics relatively rare. How the ordering of d−electronic spins of the B cation in- fluence ferroelectric or other competing structural insta- bilities has not yet been explored and understood. The coupling between magnetic ordering and structural in- stabilities is expected to involve interesting physics and is of direct relevance to technological applications3 such as multiple state memory elements and novel memory media. There are at least three families of high tempera- ture biferroic materials. Bi- based perovskite oxides, like BiMnO3 4, BiCrO3 5, and BiFeO3 6,7 are known to be promising biferroics. Ferroelectricity in these materi- als arises from the stereochemical activity of the 6s lone pair electrons of Bi. Hexagonal rare earth manganates LnMnO3 and InMnO3 are biferroics which exhibit im- proper or geometric ferroelectricity8,9,10,11. Rare earth chromites LnCrO3 12,13 (with Ln = Ho, Er, Tm, Yb, Lu or Y) have been recently shown to be biferroic; YCrO3 has been shown12 to exhibit canted antiferromagnetic be- havior below 140 K and a ferroelectric transition around 473 K. Similarly, LuCrO3 becomes a canted antiferro- magnet below 115 K, and is ferroelectric below 488 K13. The absence of any ferroelctricity in LaCrO3, has been attributed to the large size of the La3+ ion, in comparison with Y3+. However, small values of polarization reported for these materials (about 2µC/cm2 for YCrO3 12 and 6µC/cm2 of BiFeO3 7) inspite of large A-cation off- centering distortions remain a puzzle. More recently14, a new concept of ‘local non-centrosymmetry’ in YCrO3 has been proposed to account for the small value of polarization observed. Perovskite oxides are known to have many competing structural instabilities15 and this competition is further enriched by the magnetic insta- bilities. The coupling and competition between various magnetic and structural ordering can be partly respon- sible for weak ferroelectricity or the possibility of local non-centrosymmetry. Our goal here is to investigate this issue through determination of various structural insta- bilities for different magnetic orderings, with a focus on rare earth chromites. We present results of detailed electronic structure and frozen phonon (at the Γ and R points) calculations for a set of five materials (LuCrO3, YCrO3, LaCrO3, BiCrO3 and YFeO3) in the cubic phase with three different mag- netic orderings (para-, ferro- and antiferromagnetic). In Section II, we briefly describe the methods used in cal- culations here. In section III, we report results for struc- tural energetics, the electronic density of states (DOS) and the Born Effective charges (BEC) for the cubic phase of these materials with different magnetic orderings. In Section IV, we report results for structural instabilities in chromites and compare them with those in a related compound YFeO3. Since the many-electron correlations are important in magnetic oxides, we estimate their ef- fects on the structural instabilities through use of the Hubbard parameter U16. Using our results for structural instabilities, we show how certain nonpolar structural instabilities can cooperatively stabilize ferromagnetism. Our work reveals how the structural instabilities of these http://arxiv.org/abs/0704.1251v1 biferroic oxides depend on the size of A-cation (with the same B-cation), and a change in B-cation. We interpret the results using arguments based on superexchange17, and the well-known Goodenough-Kanamori rules18. Fi- nally, we summarize in Section V. FIG. 1: Perovskite structure: Cell doubled along the 〈111〉 direction, to represent G-type antiferroamgnetic ordering. II. METHODOLOGY Our calculations are based on first-principles pseu- dopotential based Density Functional Theory within a generalized gradient approximation (GGA)19 as imple- mented in the PWSCF package20. The interaction be- tween ions and electrons is approximated with ultra- soft pseudopotentials21 treating explicitly 11 electrons [(n-1)s2 (n-1)p6 (n-1)d1 and ns2] in the valence shell of Lu(n=6), La(n=6) and Y(n=5). We consider 6 valence electrons for Oxygen [2s22p4] and 14 electrons for Cr [3s23p63d54s1]. We used a plane wave basis with kinetic energy cut off of 25 Ryd (150 Ryd) to represent wave functions (density). For cubic structures, we sample the Brillouin zone using a 5 × 5 × 5 Monkhorst Pack Mesh22, and a denser mesh (6 × 6 × 6) and higher energy cut-off (30 Ryd) for energy differences. Phonon frequencies cal- culated with these larger parameters do not differ much from those calculated with a lower cut-off. We perform spin polarised calculations by initializing different spins on neighbouring magnetic ions; For paramagnetic order- ing we initialise a zero spin on Cr ions. To represent anti- ferromagnetic ordering, the unit cell is doubled along the 〈111〉 direction (see Fig1). To investigate structural in- stabilities in the prototype cubic structure, we determine its dynamical matrix using frozen phonon calculations. We use a finite difference form of the first derivative to compute an element of the force constant matrix: Kiαjβ = − TABLE I: Lattice constants of various oxides in the cubic perovskite structure with experimental unit cell volumes. Stress(GPa) a (Å) PM FM AFM LuCrO3 3.77 -4.712 3.302 1.429 YCrO3 3.79 -3.875 3.883 2.240 LaCrO3 3.88 -7.043 -0.083 -1.564 YFeO3 3.83 a -15.8 -5.9 2.5 BiCrO3 3.85 b -15.4 -9.7 -11.2 aRef.25. bRef.4. = −Fiα(ujβ = ∆)− Fiα(ujβ = −∆) where Fiα is the Hellman-Feynman force acting on the ith atom in α direction, and, ujβ the displacement of the jth atom in β direction with respect to the equilibrium structure. We used ∆=0.04 Å, about 1% of the lattice constant. The dynamical matrix is then calculated from the force constant matrix, Diαjβ = Kiαjβ√ , (2) whose eigenvalues correspond to the square of the phonon frequencies (ω2). In periodic systems, the dynamical charge tensor or Born effective charge tensor can be defined23 as the co- efficient of proportionality between the macroscopic po- larization created in direction β and a rigid displacement of the sublattice of atoms j in direction α, Z∗j,αβ = Ωo ∂P el ∂uj,α , (3) Ωo being the unit cell volume. The polarization is deter- mined using the berry phase formalism24 as implemented in the PWSCF package. III. PROPERTIES OF THE CUBIC PEROVSKITE STRUCTURE We have determined electronic structure of RCrO3 compounds (R=Y, Lu, La) in the high symmetry cu- bic structure with different magnetic orderings. This is accomplished through calculations with different initial guesses for atomic spin polarization and optimizing with respect to spin density. All our calculations are for the ex- perimental unit cell volumes, as ferroelectricity is known to be sensitive to lattice constants or pressure (see lat- tice constants listed in Table I). In many magnetic com- pounds, a change of magnetic ordering causes a stress which induces a structural distortion26. This concept of ‘magnetic stress’ was introduced to describe structural phase transitions that are induced by magnetic order- ing, and applied to materials with degenerate (usual eg) orbitals27. In these materials as well, a change in mag- netic ordering (with fixed lattice parameters) produces a change in stress (see Table I). For LuCrO3 and YCrO3, the introduction of spin polarization produces a change from compressive stress in the paramagnetic phase to a tensile stress, with stress being minimum in the antifer- romagnetic structure (the lowest energy ordering). In LaCrO3, the stress remains compressive with all three magnetic orderings. A. Electronic structure of YCrO3, LuCrO3 and LaCrO3 1. Cubic Paramagnetic(PM) Structure FIG. 2: Density of States for cubic PM YCrO3, LuCrO3 and LaCrO3. The fermi level (indicated by a dashed line) has been set to zero in all the three cases. First, we present results for the highest symmetry cu- bic structure with no spin polarization. Although this state is experimentally inaccessible, it provides a useful reference for understanding the spin-polarized structures discussed later in the paper. The plotted energy range is from -10 to 4 eV, and the lower lying semicore states have been omitted for clarity. In the PM cubic YCrO3, LuCrO3 and LaCrO3, (see Fig 2) there is high density of electronic states at the Fermi level, driving the system towards a Stoner instability28. This suggests that this phase should be unstable with respect to spin polariza- tion and/or structural distortions. The contribution of various orbitals to the DOS can be understood better by examining the orbital projected density of states (see Fig 3) which show that, the contribution between -8 to -3 eV is mainly from Oxygen 2p orbitals. Cr d-orbitals con- tribute predominantly to the peaks at the Fermi level. In contrast, the contribution from the Lu d-orbitals is substantial only 2eV above the Fermi level. 2. Cubic Ferromagnetic(FM) Structure Ferromagnetic cubic structure is simulated by initial- izing spins on both Cr ions in the same direction. In FIG. 3: Orbital resolved density of states for cubic PM LuCrO3. The high density of states at the Fermi level hints that it is an unstable phase. FIG. 4: Total and orbital resolved Density of states for cubic ferromagnetic YCrO3, LuCrO3 and LaCrO3 with the Fermi level set to zero in all the three cases. all chromites studied here, the ferromagnetic structures have a magnetic moment of 3 µB in accordance with the Hund’s rule value expected for a d3 configuration. The majority spins are represented by the solid line on the positive Y axis, and the minority spins on the negative Y axis. The introduction of spin polarization reduces the energy by approximately 2 eV per unit cell. The source of stabilization is clear from the density of states (see Fig 4) which reveals opening of a gap at the Fermi level. The states corresponding to non-magnetic atoms are unchanged in comparison with PM ordering. The down-spin Cr 3d states are split off from the O 2p states creating a wide gap for the minority states. The up-spin Cr 3d states hybridize with the O 2p states and there is a very small gap for the majority carriers. The den- sity of electronic states at the Fermi level is still finite having a small contribution from the Cr d-orbitals. This hints that either the ferromagnetic phase may not be the most stable, and that either an antiferromagnetic (AFM) spin arrangement could lower the energy of the system, or that the cubic structure is unstable and a structural distortion will lower the energy of the system. Since Cr3+ is a d3 ion, it is Jahn-Teller inactive, and the structural distortions (if any) probably involve the A-cation (at the corners), or the oxygen anions. 3. Antiferromagnetic Structure We simulated antiferromagnetic structure by initializ- ing antiparallel spins on the two Cr ions in the super- cell. It is well known that, the superexchange between eg orbitals of adjacent ions connected through oxygen with a 180o metal-oxygen-metal bond angle, is much stronger than the interaction between the correspond- ing t2g orbitals, since the former is mediated by stronger dpσ bonds as compared to the weaker dpπ bonds in the latter17,18. So, we expect a superexchange interaction in which there is a weak coupling between the t2g orbitals of the adjacent Cr atoms giving rise to an antiferromag- netic interaction. Further, this coupling will be stronger in the cubic structure as the bond angle between Cr-O-Cr is 1800, as compared to a distorted structure. From su- FIG. 5: Density of states for antiferromagnetic YCrO3, LuCrO3 and LaCrO3. perexchange arguments applied to d3 configurations, the structure with G-type antiferromagnetic ordering hav- ing rhombohedral symmetry should be most stable. We consider collinear spins assuming that the canting of the spins would be small. We find that the AFM structure is lower in energy by about 0.4 eV than the FM phase. We note that this gain in energy by FM ordering with respect to PM ordering is much more (around 2eV) than the gain in energy in going from the FM to the AFM structure (see Table III). LuCrO3, like YCrO3, is also found to be insulating with the introduction of a gap at the Fermi level (see Fig. 5). Both spin channels have identical density of states consistent with an AFM spin arrangement. From the orbital projected density of states for AFM arrangement, we find that the t2g orbitals of Cr are fully occupied (and constitute the HOMO), whereas the eg orbitals are unoccupied. Although the LUMO consists of Cr d-orbitals (about 2eV above the Fermi level), the Lu d states also appear within the same energy range (see Fig. 6). FIG. 6: Orbital resolved density of states for antiferromag- netic LuCrO3. B. Born Effective Charge (BEC) The effective charge tensors have been calculated from polarization differences between the perfect and distorted structures in the AFM phase. The anomalous values of Z* so obtained indicate that a large force is felt by a given ion due to small macroscopic electric field, thus favoring a tendency for off-centering and toward a polarized ground state. The effective charge of the A-cation (see Table II) is about the same for the three cases. For LaCrO3, we find a larger BEC on Cr and one of the oxygen atom moving along the bond. This is possibly because of larger Cr-O bond length (arising from larger size of La cation, see Table III) and correspondingly greater contribution from the long-range charge transfer. We expect from this, that the superexchange interaction in LaCrO3 should be stronger as well. TABLE II: The XX component of Born effective charge tensor for AFM LaCrO3, LuCrO3 and YCrO3. Nominal charges are indicated in brackets. Z*A Z*B Z*Ox Z*Oy,z LuCrO3 4.42(3) 3.43(3) -2.56(-2) -2.62(-2)) YCrO3 4.45(3) 3.44(3) -2.62(-2) -2.66(-2)) LaCrO3 4.5(3) 3.76(3) -3.82(-2) -2.22(-2)) TABLE III: Relative Energies of different Magnetic phases, Cr-O bond lengths, and Neel’s temperatures for cubic LuCrO3, YCrO3 and LaCrO3 (Energy of the PM phase has been set to zero). PM FM G-AFM Cr-O Bond length TN LuCrO3 0.0 1.89 eV 2.3 eV 1.88(Å) 115 K YCrO3 0.0 2.02 eV 2.4 eV 1.90(Å) 140 K LaCrO3 0.0 2.04 eV 2.5 eV 1.94(Å) 282 K aNeel’s temperature taken from Ref. 12,13. bRef. 4. IV. STRUCTURAL INSTABILITIES A. Coupling with Magnetic Ordering In order to represent G-type antiferromagnetic order- ing which has been shown to be most favourable energet- ically, we use a unit cell doubled along the 〈111〉 direc- tion. We determine structural instabilities in this struc- ture with different magnetic orderings. A single unit cell has 10 atoms which results in 30 phonon branches: 3 ac- coustic (which have zero frequency at k=(0,0,0)) and 27 optical, some of which are triply degenerate. We are in- terested mainly in optical modes with imaginary phonon frequencies corresponding to instabilities in the structure. Doubling the unit cell along the 〈111〉 direction, gives us access to zone boundary phonon modes (R-point) which form the dominant structural instabilities in this struc- ture, along with the zone-center modes. In the paramagnetic phase, both YCrO3 and LuCrO3 exhibit a zone center instability at 116.5 and 144.8 cm−1 respectively, which is a polar mode (with Γ15 symme- try) involving mainly the off centering of A-cation. This instability, however, is absent in LaCrO3. We find two instabilities in the FM phase: Γ15 and Γ25 modes. The non-polar Γ25 mode involves oxygen displacements only, and is strongly unstable in the FM phase. The Γ15 mode involves the A-cation (rare earth ion) moving in a direc- tion opposite to that of the oxygen cage and Cr atom resulting in a ferroelectric polar structural distortion. Note that the Cr atom moves in the same direction as the oxygen ion, in contrast to the behavior of Ti ion in BaTiO3 15 and PbTiO3 29, but similar to the behavior of Mn in BiMnO3 11 and Cr in BiCrO3 With AFM spin arrangement, for LuCrO3, we find three triply degenerate instabilities, at 339.5 cm−1, 145 cm−1 and the weakest at around 60 cm−1. For YCrO3, the corresponding instabilities are at 309 cm−1, 140 cm−1 and 80 cm−1 respectively. The strongest instability (at around 300 cm−1 for the two materials) has R25 symme- try and corresponds to rotation of the corner connected oxygen octahedra. The next instability is the ferroelec- tric Γ15 mode (around 140 cm −1). The weakest insta- bility (60 cm−1 for LuCrO3 and 80 cm −1 for YCrO3) has R15 symmetry and involves displacement of the A- cations (Lu and Y for our case) and small oxygen dis- placements; these are antiparallel in neighbouring unit cells. In LaCrO3, we find only two triply degenerate in- stabilities. The first instability at around 220 cm−1 cor- responding to the oxygen rotations (the R25 mode) and the second close to 18 cm−1 having Γ15 symmetry (see Fig 7). FIG. 7: Eigenvectors of the unstable Γ point phonon modes: Γ15 and Γ25 modes For the rare earth chromites studied in this paper, un- stable R-point modes in the AFM phase do not change significantly with magnetic ordering. Γ point instabili- ties in contrast depend strongly on magnetic ordering. Only the high frequency phonons are affected in going from para- to antiferromagnetic phase and all Γ15 and R15 modes are softer in the FM phase. The Γ25 mode which brings about a significant change in the Cr-O-Cr as well as O-Cr-O bond angle shows a spectacular change with magnetic ordering. This mode, highly unstable in the ferromagnetic phase (at around 200 cm −1) becomes stable at around 50 cm−1 with para- and antiferromagnetic orderings. This behavior, although not as pronounced, is also seen for this mode in LaCrO3. The R25 instability, also involving a change in the Cr-O-Cr bond angle, is not affected by the change in magnetic ordering possibly because the O-Cr-O bond angle still remains unchanged. Another significant change is observed for the R’25 mode, which involves a movement of the two B-cations (Cr in our case) in opposite directions. After introduction of spin-polarization, the R25’ mode (close to 250 cm −1 for PM phase) becomes more stable at around 400 cm−1 for the FM and AFM phases. We compare our results with the Bi-based biferroic chromite, BiCrO3 and find a similar behavior of the Γ25 mode here as well. We thus attribute this behavior to the B cation (Cr ion for the chromites) and expect it to be their general behavior. To interpret the general trend in phonon frequencies (see Fig. 8), the following rules apply: 1. The modes which involve a change in the Cr-O- Cr bond angle (as well as O-Cr-O bond angle) are more stable in the AFM phases and are relatively less, or un- stable, in the FM phase. This behaviour is seen in the Γ25 and Γ15 modes for LuCrO3. 2. Secondly, the modes which involve a change in the Cr-O bond length tend to harden with the introduction of spin polarization, as observed for the R’25 and R’12 modes. FIG. 8: Changes in phonon frequencies of cubic LuCrO3 with different magnetic orderings. Insets show similar curves for LaCrO3 and BiCrO3. In order to study the effect of change in B-cation (see Fig 9), we compare instabilities in YFeO3 with YCrO3. We find that a change in magnetic ordering has an op- posite effect on the unstable modes involving a change in Fe-O-Fe bond angle. A spectacular change is the R’2 mode, an oxygen breathing mode, which softens in the FM phase as compared to the PM and AFM phases. The Γ25 mode also shows a different behaviour, showing a sta- bilization with FM ordering. These differences are due to the filled eg orbitals in Fe 3+, which are unoccupied in Cr3+, leading to a much stronger superexchange inter- action mediated by the eg orbitals, and have a different geometry dependence. B. Stabilization of Ferromagnetic Ordering The spectacular change in the frequency of the Γ25 mode with FM ordering, prompts us to discuss whether ferromagnetism can be stabilized in these chromites by varying the Cr-O-Cr bond angle. Since the Γ25 mode is unstable only in the FM phase, we want to study the effect of freezing in a distortion of this mode. We have calculated the total energy as a function of Γ25 displace- ments in the 〈111〉 and 〈100〉 directions, for the FM and AFM phases (see Fig 10). For LuCrO3, we observe a FIG. 9: Changes in phonon frequencies with changes in B- cation: YCrO3 and YFeO3. crossover, at a displacement of 0.407 Å, beyond which the FM phase is energetically favoured. The Cr-O-Cr bond angle, at the crossover point is found to be 153◦, which is significantly different from, the value suggested by Good- enough and Kanamori for ferromagnetic superexchange interaction (130◦) in 1951. On examining the Density of states beyond the crossover point (see Fig. 11), we find more significant hybridization between Cr and oxygens for the FM phase. Secondly, the FM phase so stabilized is found to be insulating. FIG. 10: Energy vs displacement corresponding to the Γ25 mode for FM and AFM orderings in LuCrO3. FM becomes more stable than the AFM state for rhombohedral distortions greater than 0.407 Å. C. Effect of Correlations As mentioned earlier in the paper, the LDA+U method has been successfully applied to describe the electronic structure of sytems containing localized d and f elec- trons where LDA sometimes leads to incorrect results16, and recently it has been applied to obtain structural pa- rameters that are in better agreement with experimental FIG. 11: Density of states for FM and AFM phases at Γ25 displacement of 0.407 Å (see Fig. 10, where the FM state is more stable). results than LDA or GGA.7,30. In this work, we use a value of U = 3.0 eV, adapted from work on full struc- tural optimization of BiCrO3 7. With the introduction of correlations through U parameter, we find that the modes in FM phase do not change significantly. Our re- sults for YCrO3 (see Fig 12) bear that only Γ25, R’2 and R’12 modes are noticeably affected. Correlations lead to softening of the R’2 and R’12 modes by 50 cm −1 , and tend to harden the Γ25 mode in the PM phase. In an AFM spin arrangement, these R-point modes are hard- ened by approximately 20 cm−1 and the Γ25 instability is softened. FIG. 12: Effect of correlations on the phonon modes of YCrO3 with different magnetic orderings. For each mode, data on the left of the vertical dashed line represents estimates with GGA and the data on the right represents estimates with GGA+U (U=3.0 eV). V. SUMMARY In conclusion, we have determined structural instabil- ities of LaCrO3, LuCrO3, YCrO3 and BiCrO3 in their cubic perovskite structures with different magnetic or- derings. Our finding that the G-type antiferromagnetic ordering is most stable can be explained with superex- change arguments. Ferroelectric structural instabilities in the cubic structures involve A-cation (Lu or Y) dis- placements, as indicated by the eigenvectors of the fer- roelectric Γ15 modes and an anomalous BEC of the A- cation. We find that certain phonon frequencies de- pend sensitively on magnetic ordering: the modes in- volving a change in bond-angle are stable (harder) with the antiferro- and paramagnetic ordering than in the FM state; on the other hand, the modes involving a change in Cr-O bond length are softer in the paramagnetic phase and comparable in the FM and AFM states. The Γ25 oxygen mode brings about a significant change in the Cr- O-Cr bond angle and is highly unstable in the FM phase, and corresponding structural distortion leads to stabiliza- tion of ferromagnetic ordering in these chromites. Among the competing structural instabilities the antiferrodis- tortive instability (R25 mode) is the strongest. Electron correlations are found to have little effect on the unstable phonon modes, but result in a slight change in a few of the stable phonon modes in the PM phase. We note that the effects of magnetic ordering on structural instabilities are quite different (in fact, opposite sometimes) in YFeO3 with respect to chromites, and a more detailed study is required to understand such couplings in ferrites. Origin of small polarization and/or local non-centrosymmetry14 is probably from the relatively weak ferroelectric instabil- ities and their competition with various structural mag- netic instabilities, and our work should be useful in for- mulating a phenomenological analysis of the same. VI. ACKNOWLEDGMENTS Nirat Ray thanks JNCASR for Summer Research fel- lowship Programme and Joydeep Bhattacharjee for dis- cussions. UVW is thankful to Professor C N R Rao for stimulating discussions and encouragement for this work and acknowledges use of central computing facility and financial support from the Centre for Computational Ma- terials Science at JNCASR. 1 N. A. Spaldin, Phys. World, April 2004; T. Kimura et al Nature 426, 55 (2003); N. A. Spaldin and M. Fiebig, Sci- ence 309, 391 (2005). 2 N.A. Hill, J. Phys. Chem. B 104, 6684-6709 (2000). 3 Wood, V. E.; Austin, A. E. Magnetoelectric Interaction Phenomena in Crystals; Freeman, A. J., Schmid, H., Eds.; Gordon and Breach: Langhorne, PA, 1975. 4 N.A. Hill and K.M. Rabe, Phys. Rev. B 59, 8759 (1999); A. Moreira dos Santos et al, Phys. Rev. B 66, 064425 (2002). 5 N.A. Hill, Pio Bättig and Claude Daul, J. Phy Chem B 106, 3383-3388 (2002). 6 J. Wang et al., Science 299, (5613), 1719 (2003). 7 J. B. Neaton et al, Phys. Rev. B 71, 014113 (2005). 8 B. B. VanAken, Thomas T.M. Palstra1, A. Filippetti and N. A. Spaldin, Nat. Mat. 3, 164 (2004). 9 C. R. Serrao et al., J. Appl. Phys. 100, 076104 (2006). 10 C. J. Fennie and K. M. Rabe, Phys. Rev. B 72, 100103(R) (2005). 11 M. Feibig, J. Magn. Magn. Mater. 290-291, 883 (2005). 12 C. R. Serrao et al., Phys Rev B 72, 220101 (2005). 13 J. R. Sahu et al., J. Mater. Chem. 17, 42-44 (2007). 14 K. Ramesha et al., J. Phys. Cond. Matter 19, 102202 doi:10.1088/0953-8984/19/10/102202 15 Ph Ghosez, E. Cockayne, U.V. Waghmare and K.M. Rabe, Phys. Rev. B 60, 836 (1999). 16 V. I. Anisimov, F. Aryasetiawan and A. I. Lichtenstein, J. Phys. Cond. Matter 9, 767 (1997). 17 P. W. Anderson, Phys. Rev. 79, 350 - 356 (1950). 18 J. B. Goodenough, Phys. Rev. 100 564 (1955). 19 Y. Wang and J. P. Perdew, Phys. Rev. B 44, 13298 (1991). 20 S. Baroni, A. Dal Corso, S. de Gironcoli, and P. Giannozzi, 2001, http://www.pwscf.org. 21 D. Vanderbilt, Phys. Rev. B 41, 7892 (1990). 22 H.J. Monkhorst and J.D. Pack, Phys. Rev. B 13, 5188 (1976); 16, 1748 (1977). 23 Ph. Ghosez, J. P. Michenaud, and X. Gonze, Phys. Rev. B 58, 6224 (1998). 24 R. Resta, Berry Phase in Electronic Wave functions, available at: http://ale2ts.ts.infm.it:6163/˜resta/publ/notes trois.ps.gz. 25 M. Eibschitz, Acta. Cryst. 19, 337 (1965). 26 Alessio Filippetti and Nicola A. Hill, Phys. Rev. Lett. 85, 5166 - 5169 (2000). 27 P. Gopal, N. A. Spaldin and U. V. Waghmare, Phys. Rev. B 70, 205104 (2004). 28 E.C. Stoner, Proc. R. Soc. 165, 372 (1938). 29 R. E. Cohen, Nature 358, 136138 (1992). 30 O. Bengone, M. Alouani, J. Hugel, and P. Blöchl, Comput. Mater. Sci. 24, 192 (2002). 31 A. Kokalji, XCrySDen – a new program for displaying crys- talline structures and electron densities, J. Mol. Graphics Modelling 17, 176 (1999). http://www.pwscf.org http://ale2ts.ts.infm.it:6163/\char 126resta/publ/notes_trois.ps.gz

0704.1252

LNRF-velocity hump-induced oscillations of a Keplerian disc orbiting near-extreme Kerr black hole: A possible explanation of high-frequency QPOs in GRS 1915+105

arXiv:0704.1252v2 [astro-ph] 23 May 2007 Astronomy & Astrophysics manuscript no. grs1915-corrected c© ESO 2018 October 30, 2018 LNRF-velocity hump-induced oscillations of a Keplerian disc orbiting near-extreme Kerr black hole: A possible explanation of high-frequency QPOs in GRS 1915+105 Zdeněk Stuchlı́k, Petr Slaný, and Gabriel Török Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-74601 Opava, Czech Republic Received / Accepted ABSTRACT Context. At least four high-frequency quasiperiodic oscillations (QPOs) at frequencies 41 Hz, 67 Hz, 113 Hz, and 167 Hz were re- ported in a binary system GRS 1915+105 hosting near-extreme Kerr black hole with a dimensionless spin a > 0.98. Aims. We attempt to explain all four observed frequencies by an extension of the standard resonant model of epicyclic oscillations. Methods. We use the idea of oscillations induced by the hump of the orbital velocity profile (related to locally non-rotating frames– LNRF) in discs orbiting near-extreme Kerr black holes, which are characterized by a “humpy frequency” νh, that could excite the radial and vertical epicyclic oscillations with frequencies νr, νv. Due to non-linear resonant phenomena, the combinational frequen- cies are allowed as well. Results. Assuming mass M = 14.8 M⊙ and spin a = 0.9998 for the GRS 1915+105 Kerr black hole, the model predicts frequencies νh = 41 Hz, νr = 67 Hz, νh + νr = 108 Hz, and νv − νr = 170 Hz corresponding quite well to the observed ones. Conclusions. For black-hole parameters being in good agreement with those given observationally, the forced resonant phenom- ena in non-linear oscillations, excited by the ”hump-induced” oscillations in a Keplerian disc, can explain high-frequency QPOs in near-extreme Kerr black-hole binary system GRS 1915+105 within the range of observational errors. Key words. Accretion, accretion discs – Black hole physics – Methods: data analysis 1. Introduction Detailed analysis of the variable X-ray black-hole binary system (microquasar) GRS 1915+105 reveals high-frequency QPOs ap- pearing at four frequencies, namely ν1 = (41 ± 1) Hz, ν2 = (67 ± 1) Hz (Morgan et al. 1997; Strohmayer 2001), and ν3 = (113±5) Hz, ν4 = (167±5) Hz (Remillard 2004). In this range of its errors, both pairs are close to the frequency ratio 3:2 suggest- ing the possible existence of resonant phenomena in the system. Observations of oscillations with these frequencies have differ- ent qualities, but in all four cases the data are quite convincing; see (McClintock & Remillard 2004; Remillard & McClintock 2006). Several models have been developed to explain the kHz QPO frequencies, and it is usually preferred that these oscillations are related to the orbital motion near the inner edge of an accretion disc. In particular, two ideas based on the strong-gravity prop- erties have been proposed. While Stella & Vietri (1998, 1999) introduced the “Relativistic Precession Model” considering that the kHz QPOs directly manifest the modes of a slightly per- turbed (and therefore epicyclic) relativistic motion of blobs in the inner parts of the accretion disc, Kluźniak & Abramowicz (2001) propose models based on non-linear oscillations of an accretion disc that assume resonant interaction between orbital and/or epicyclic modes. In a different context, the possibility of resonant coupling between the epicyclic modes of motion in the Send offprint requests to: Z. Stuchlı́k, e-mail: [email protected] Kerr spacetime was also mentioned in the early work of Aliev & Galtsov (1981). In the case of near-extreme Kerr black holes, it was sug- gested that the epicyclic oscillations in the disc could be excited by resonances with the so-called “hump-induced” oscillations, see papers of Aschenbach (2004, 2006) and Stuchlı́k et al. (2004, 2007). This idea was proposed so as to extend standard orbital (resonant) models meant to explain high-frequency QPOs ob- served in black-hole sources. Recently, careful and detailed analysis of the spectral contin- uum from GRS 1915+105 has put a strong limit on the black- hole spin,1 namely 0.98 < a < 1 (McClintock et al. 2006), indicating the presence of near-extreme Kerr black hole whose mass has been restricted observationally to M = (14.0±4.4) M⊙, see (McClintock & Remillard 2004; Remillard & McClintock 2006). Therefore, the microquasar GRS 1915+105 seems to be an appropriate candidate to test the extended resonant model with hump-induced oscillations.2 The idea of hump-induced oscillations and their possible res- onant coupling with the epicyclic ones is briefly discussed in Sect. 2. The related resonant model, assuming the excitation of epicyclic oscillations by the hump-induced oscillations through 1 Units c = G = M = 1 (M is the total mass of the Kerr black hole) and the Boyer-Lindquist (B-L) coordinates (t, r, θ, ϕ) are used hereafter. 2 However, Middleton et al. (2006) refer to a substantially lower, in- termediate value of black-hole spin, a ∼ 0.7, to which the model of hump-induced oscillations cannot be applied. http://arxiv.org/abs/0704.1252v2 2 Zdeněk Stuchlı́k et al.: LNRF-velocity hump induced oscillations in GRS 1915+105 non-linear resonant phenomena, is applied to GRS 1915+105 in Sect. 3, concluding remarks are presented in Sect. 4. 2. Hump-induced and epicyclic oscillations in Keplerian discs and possible resonant coupling In order to describe the local processes in an accretion disc, it is necessary to choose a local observer (characterized by its reference frame). In general relativity there is no preferred ob- server. On the other hand, if we want to study processes related to the orbital motion of matter in the disc, it is reasonable to choose the observers with zero angular momentum, so-called ZAMOs, as their reference frames do not rotate with respect to the spacetime, and thus ZAMOs should reveal local kinematic properties of the disc in the clearest way. (In rotating–stationary, axisymetric–spacetimes, they are dragged along with the space- time.) In the Kerr spacetime, ZAMOs are represented by locally non-rotating frames (LNRF); see Bardeen et al. (1972). Notice that in the Schwarzschild spacetime, LNRF correspond to the static observer frames. Aschenbach (2004) finds that for near-extreme Kerr black holes with the spin a > 0.9953, the test-particle orbital veloc- ity V(ϕ) related to LNRF reveals a hump in the equatorial plane (θ = π/2). This non-monotonicity is located in a small region inside the ergosphere of the black-hole spacetime close to, but above, the marginally stable orbit.3 Therefore, it can be relevant for thin accretion discs around near-extreme Kerr black holes, as the inner edge of the disc can extend down to the innermost stable circular orbit (ISCO). Moreover, Stuchlı́k et al. (2005) shows that for a > 0.99979 the similarly humpy behavior of the orbital velocity in LNRF also takes place for the non-geodesic motion of test perfect fluid in marginally stable barotropic tori characterized by the uni- form distribution of the specific angular momentum, ℓ(r, θ) ≡ −Uϕ/Ut = const, where the motion of fluid elements is given by the 4-velocity field Uµ = (U t(r, θ), 0, 0, Uϕ(r, θ)). Outside the equatorial plane, the non-monotonic behavior of V(ϕ) in marginally stable tori is represented by the topology change of the cylindrical equivelocity surfaces in the region of the hump, because the toroidal equivelocity surfaces centered around the circle corresponding to the local minimum of V(ϕ) in the equa- torial plane exist for a > 0.99979 (Stuchlı́k et al. 2005). This suggests a generation of possible instabilities in radial and ver- tical directions; see Stuchlı́k et al. (2004). In the following, we restrict our attention to the case of Keplerian discs. Heuristic connection between the positive part of the veloc- ity gradient, ∂V(ϕ)/∂r, and the excitation of epicyclic oscilla- tions in Keplerian discs was suggested by Aschenbach (2004, 2006), who defined the characteristic frequency of oscillations, induced by the humpy profile of V(ϕ), by the maximum posi- tive slope of the orbital velocity in terms of the coordinate ra- dius, νcrit ≡ (∂V(ϕ)/∂r)max. This coordinate-dependent defini- tion was corrected in Stuchlı́k et al. (2007), where the proper radial distance dr̃ = grr dr rather than the coordinate dis- tance dr was used to define the characteristic (critical) frequency νr̃crit ≡ (∂V (ϕ)/∂r̃)max. Such a locally defined critical frequency was further related to a stationary observer at infinity, obtaining the so-called “humpy frequency” −(gtt + 2ωgtϕ + ω2gϕϕ)r=rh ν 3 We stress that the Aschenbach effect is frame-dependent, as it is related to LNRF, but recall the arguments for relevance of the LNRF point of view at the beginning of the section. 0 0.001 0.002 0.003 0.004 0 0.001 0.002 0.003 0.004 1 − a a = (0.9998 ± 0.0001) ✉ GRS 1915+105 ✛ 607 ✛ 1.29 Fig. 1. Spin-dependence of the humpy frequency νh and the humpy ra- dius rh. For completeness, the B-L radius of the innermost stable circu- lar geodesic (ISCO) is plotted. (rh − 2) − rh(r2h + a 2) + 2a2 r5h + a 4(3rh + 2) − 2a3r h (3rh + 1) 2∆3/2h h + a) 2a2r2h(2rh − 5) − 2ar h (5rh − 9) 2∆3/2h h + a) , (1) where gµν are the metric coefficients of the Kerr geometry and ω = −gtϕ/gϕϕ is the angular velocity of the LNRF; see, e.g., Bardeen et al. (1972); ∆h = r h − 2rh + a 2. The analytic formula is given for the equatorial plane (θ = π/2). The B-L radius rh where the positive gradient of the velocity profile in terms of the proper radial distance reaches its maximum, so-called “humpy radius”, is given by the condition ∂V(ϕ) = 0 (2) leading to the equation 3a7(r + 2) + a6 r(21r2 + 18r − 4) − a5r(33r2 + 10r + 20) r(45r3 − 62r2 − 68r + 16) − a3r3(83r2 − 122r − 60) +a2r4 r(27r2 − 130r + 136) − 9ar5(7r2 − 26r + 24) r(3r − 2) = 0, (3) which must be solved numerically. The spin dependence of the humpy radius and the related humpy frequency is illustrated in Fig. 1. The humpy radius rh falls monotonically with increas- ing spin a, while the humpy frequency νh has a maximum for Zdeněk Stuchlı́k et al.: LNRF-velocity hump induced oscillations in GRS 1915+105 3 a = 0.9998, where νh (max) = 607 (M⊙/M) Hz, and it tends to νh (a→1) = 588 (M⊙/M) Hz. When particles following a Keplerian circular orbit are per- turbed, they begin to follow, in the first approximation, an epicyclic motion around the equilibrium Keplerian orbit, gen- erally characterized by the frequencies of the radial and vertical epicyclic oscillations νr, νv (Aliev & Galtsov 1981; Nowak & Lehr 1998): r = ν K(1 − 6r + 8ar−3/2 − 3a2r−2), (4) v ≡ ν θ = ν K(1 − 4ar + 3a2r−2), (5) where νK is the Keplerian orbital frequency 2π(r3/2 + a) . (6) The ratios of the humpy frequency and the epicyclic frequencies at the humpy radius were determined in Stuchlı́k et al. (2007) revealing almost spin-independent asymptotic behavior for a → 1 represented closely by the ratios of integer numbers, νv : νr : νh ∼ 11 : 3 : 2, which imply a possibility of resonant phenomena between the hump-induced and epicyclic oscillations predicted by Aschenbach (2004). The ratios of the epicyclic frequencies and the humpy frequency are given in the dependence on the black-hole spin in Fig. 2. 3. Application of the hump-induced resonance model to high-frequency QPOs in GRS 1915+105 Primarily concentrating on the lower pair of frequencies, we as- sume that the lowest frequency is directly the humpy frequency, νh ≡ ν1 = (41 ± 1) Hz, (7) while the second lowest frequency corresponds directly to the radial epicyclic frequency at the same radius rh, νr ≡ ν2 = (67 ± 1) Hz. (8) These frequencies are close to a 3:2 ratio, therefore the forced non-linear resonance can be relevant in such a situation. The ratio of νr/νh = (1.63 ± 0.06) gives the black hole spin a = (0.9998 ± 0.0001) (the uncertainty of the spin is implied by un- certainties of the lower pair of frequencies being ∼ 1 Hz); see Fig. 2. Notice that this spin corresponds to the maximal pos- sible value of the humpy frequency νh (max) (Fig. 1). Since the humpy frequency is 1/M-scaled, the absolute value of νh im- plies the black hole mass M = (14.8±0.4) M⊙. The correspond- ing humpy radius is rh = 1.29 +0.01 −0.02 (Fig. 1). At such a radius, the vertical epicyclic frequency of a particle orbiting the Kerr black hole with the mass and spin inferred above reaches the value νv = (0.23 ± 0.01) kHz. Then the upper pair of observed frequencies can be explained, within the range of observational errors ±5 Hz, by combinational frequencies at the humpy radius rh in the following way: ν3 ∼ (νr + νh) = (108 ± 2) Hz (9) ν4 ∼ (νv − νr) = (0.17 ± 0.01) kHz. (10) 4. Conclusions The idea of epicyclic oscillations induced by the LNRF-velocity hump in the region where the positive part of the velocity gra- dient reaches its maximum is able to address all four high- frequency QPOs observed in the X-ray source GRS 1915+105. -8 -7 -6 -5 -4 -3 log(1 − a) a = (0.9998 ± 0.0001) ✲ GRS 1915+105 ✲ ✻νr :νh (νv − νr) :(νh + νr) νv :νr νv :νh ∼ 3:2 ∼ 7:2 ∼ 6:1 Fig. 2. Spin dependence of frequency ratios including the radial (νr) and vertical (νv) epicyclic frequencies, and the humpy frequency (νh) evaluated at the same radius rh where the humpy frequency is defined. The range of the spin relevant for GRS 1915+105 Kerr black hole is shaded. For the mean value a = 0.9998, the frequency ratios are close to the ratios of integer numbers, suggesting a possibility of resonances between hump-induced and epicyclic oscillations in GRS 1915+105. The model implies a near-extreme spin of the central black hole (a ∼ 0.9998), which agrees well with results from the spec- tral continuum fits, and the black-hole mass M ∼ 14.8 M⊙ be- ing well inside the interval given by other observational meth- ods. Note that the orbital resonance model of Kluźniak & Abramowicz, assuming the parametric resonance between the vertical and radial epicyclic oscillations in frequency ratio 3:2 represented by the upper pair of observed frequencies, also gives the spin a > 0.99 but for M ≃ 18M⊙ (Török et al. 2005). On the other hand, the “Relativistic Precession Model” gives a substan- tially lower value for the spin: a ∼ 0.3 (Stella et al. 1999). In the presented model, we assume that all four observed frequencies arise due to forced non-linear oscillations of the Keplerian disc at the same radius rh, excited by the hump- induced oscillations characterized by the humpy frequency νh. The black-hole parameters a, M are fixed by the requirement that the lower pair of observed frequencies is identified with the humpy frequency and the radial epicyclic frequency, ν1 ≡ νh, ν2 ≡ νr. Assuming non-linear resonant phenomena enabling the existence of combinational frequencies and the possibility of observing them, the upper pair of observed frequencies can be explained as the combinational ones of the humpy frequency and both epicyclic frequencies, ν3 ∼ (νr + νh), ν4 ∼ (νv − νr). Moreover, both frequency ratios νr : νh, and (νv − νr) : (νr + νh) are close to 3:2 ratio (Fig. 2), in which the resonant phenomena can be strong enough. On the other hand, as 4νh = (164± 4) Hz, which is also close to the uppermost frequency, there is an- other possibility of explaining ν4 through a sub-harmonic reso- nance forced by the humpy oscillations as well. Finally, note that Strohmayer (2001) also reports another relatively weak QPO at frequency of (56 ± 2) Hz. If this is the case (which, according to our knowledge, has not been confirmed by other observations yet), it could be related to the second harmonic of the combina- tional frequency4 (νr − νh) = (26 ± 2) Hz. 4 Combinational frequency (νr − νh) corresponds to the same order of nonlinearity as (νr + νh). Note added in the manuscript: After the paper was accepted we ob- tained an information that a weak QPO at frequency 27 Hz is referenced in Belloni et al. (2001). 4 Zdeněk Stuchlı́k et al.: LNRF-velocity hump induced oscillations in GRS 1915+105 Generally, other harmonics and combinational frequencies may occur in a non-linear oscillating system corresponding to higher approximations, when the equation of motion describing the non-linear oscillations is solved by the method of successive approximations. The statement by Landau & Lifshitz (1976) that “As the degree of approximation increases, however, the strength of the resonances, and the widths of the frequency ranges in which they occur, decrease so rapidly that in practice only the resonances at frequencies5 ν ≈ pν0/q with small p and q can be observed” can explain why a QPO near the frequency 237 Hz, corresponding to the vertical epicyclic frequency νv at the same radius rh as the previously mentioned humpy and radial epicyclic frequencies νh, νr, is not directly observed, despite the commen- surability of these frequencies represented by the frequency ra- tios νv : νh ∼ 6 : 1 and νv : νr ∼ 7 : 2 (Fig. 2). Acknowledgements. The authors are supported by the Czech grant MSM 4781305903. References Aliev, A. N. & Galtsov, D. V. 1981, Gen. Relativity Gravitation, 13, 899 Aschenbach, B. 2004, Astronomy and Astrophysics, 425, 1075 Aschenbach, B. 2006, Chines J. Astronom. Astrophys., 6, 221, special issue for Frascati Workshop 2005, astro-ph/0603193 Bardeen, J. M., Press, W. H., & Teukolsky, S. A. 1972, Astrophys. J., 178, 347 Belloni, T., Méndez, M., & Sánchez-Fernández, C. 2001, Astronomy and Astrophysics, 372, 551 Kluźniak, W. & Abramowicz, M. A. 2001, Acta Phys. Polon. B, 32, 3605 Landau, L. D. & Lifshitz, E. M. 1976, Course of Theoretical Physics (Volume 1): Mechanics (Elsevier Butterworth-Heinemann), 3rd edition McClintock, J. E. & Remillard, R. A. 2004, in Compact Stellar X-Ray Sources, ed. W. H. G. Lewin & M. van der Klis (Cambridge: Cambridge University Press), astro-ph/0306213 McClintock, J. E., Shafee, R., Narayan, R., et al. 2006, Astrophys. J., 652, 518, astro-ph/0606076 Middleton, M., Done, C., Gierliński, M., & Davis, S. W. 2006, Monthly Notices Roy. Astronom. Soc., 373, 1004, astro-ph/0601540 Morgan, E. H., Remillard, R. A., & Greiner, J. 1997, Astrophys. J., 482, 993 Nowak, M. A. & Lehr, D. E. 1998, in Theory of Black Hole Accretion Disks, ed. M. A. Abramowicz, G. Björnsson, & J. E. Pringle (Cambridge: Cambridge University Press), 233–253 Remillard, R. A. 2004, AIPC, 714, 13 Remillard, R. A. & McClintock, J. E. 2006, Annual Review of Astronomy and Astrophysics, 44, 49, arXiv:astro-ph/0606352v1 Stella, L. & Vietri, M. 1998, Astrophys. J., 492, L59, astro-ph/9709085 Stella, L. & Vietri, M. 1999, Phys. Rev. Lett., 82, 17, astro-ph/9812124 Stella, L., Vietri, M., & Morsink, S. M. 1999, Astrophys. J., 524, L63, astro-ph/9907346 Strohmayer, T. E. 2001, Astrophys. J., 554, L169 Stuchlı́k, Z., Slaný, P., & Török, G. 2004, in Proceedings of RAGtime 4/5: Workshops on black holes and neutron stars, Opava, 14–16/13–15 Oct 2002/03, ed. S. Hledı́k & Z. Stuchlı́k (Opava: Silesian University in Opava) Stuchlı́k, Z., Slaný, P., & Török, G. 2007, Astronomy and Astrophysics, 463, 807, astro-ph/0612439 Stuchlı́k, Z., Slaný, P., Török, G., & Abramowicz, M. A. 2005, Phys. Rev. D, 71, 024037, gr-qc/0411091 Török, G., Abramowicz, M. A., Kluźniak, W., & Stuchlı́k, Z. 2005, Astronomy and Astrophysics, 436, 1, astro-ph/0401464 5 p, q are integers.

0704.1253

Compton thick AGN in the Suzaku era

arXiv:0704.1253v1 [astro-ph] 10 Apr 2007 Compton thick AGN in the Suzaku era Andrea Comastri1, Roberto Gilli1, Cristian Vignali2, Giorgio Matt3, Fabrizio Fiore4, Kazushi Iwasawa5 1 INAF – Osservatorio Astronomico di Bologna, Bologna, Italy 2 Dipartimento di Astronomia, Università di Bologna, Bologna, Italy 3 Dipartimento di Fisica, Università di Roma Tre, Roma, Italy 4 INAF – Osservatorio Astronomico di Roma, Monteporzio, Italy 5 MPE – Garching, Germany (Received ) Suzaku observations of two hard X–ray (> 10 keV) selected nearby Seyfert 2 galaxies are presented. Both sources were clearly detected with the pin Hard X–ray Detector up to several tens of keV, allowing for a fairly good characterization of the broad band X–ray continuum. Both sources are heavily obscured, one of which (NGC 5728) being Compton thick, while at lower energies the shape and intensity of the scattered/reflected continuum is very different. Strong iron Kα lines are detected in both sources. There are also hints for the presence of a broad relativistic iron line in NGC 4992. §1. Introduction A fraction as high as 50% of Seyfert 2 galaxies in the nearby Universe are ob- scured in the X–ray band by column densities of the order of, or larger than the inverse of the Thomson cross-section (NH ≥ σ ≃ 1.5× 1024 cm−2), hence dubbed Compton thick (CT). If the optical depth (τ = NHσT ) for Compton scattering does not exceed values of the order of “a few”, X–ray photons with energies higher than 10–15 keV are able to penetrate the obscuring material and reach the observer. For higher values of τ , the entire X–ray spectrum is depressed by Compton down scattering and the X–ray photons are effectively trapped by the obscuring material irrespective of their energy. The former class of sources (mildly CT) can be effi- ciently detected by X–ray instruments sensitive above 10 keV, while for the latter (heavily CT) their nature may be inferred through indirect arguments, such as the presence of a strong iron Kα line over a flat reflected continuum. The search for and the characterization of the physical properties of CT AGN is relevant to under- stand the evolution of accreting Supermassive Black Holes (SMBHs). In particular, mildly CT AGN are the most promising candidates to explain the residual (i.e. not yet resolved) spectrum of the X–ray background around its 30 keV peak (Comastri 2004a; Worsley et al. 2005) but only a handful of them are known beyond the local Universe (see Comastri 2004b for a review). If this were the case, we may be missing a not negligible fraction of the accretion power in the Universe and of the baryonic matter locked in SMBH (Marconi et al. 2004). An unbiased census of extremely obscured AGN would require to survey the hard X–ray sky above 10 keV with good sensitivity. Such an argument is one of the key scientific drivers of the SimbolX mission (Ferrando et al. 2006), which will be hopefully launched in the next decade. typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.1253v1 2 A. Comastri et al. For the time being one has to rely on the observations obtained by the high energy detectors on board BeppoSAX, INTEGRAL, Swift and, more recently, Suzaku. Though limited to bright and thus low redshift sources, they have proven to be quite successful in finding heavily obscured CT AGN. As a first step forward towards a census of CT AGN we have conceived a program with Suzaku to observe hard X– ray selected bright AGN from the INTEGRAL/IBIS (Beckmann et al. 2006) and Swift/BAT (Markwardt et al. 2005) catalogues. The goal of this program is to discover “new” CT AGN which are likely to be present among the already detected sources, but not recognized as such due to the poor counting statistics and/or the lack of information at lower energies. In order to select the most suitable candidates, we have considered the sources in the above mentioned AGN catalogues with a bright hard X–ray flux and tentative evidence of intrinsic absorption from observations at lower energies. For a few of them the column densities are estimated to be close to the CT threshold. Suzaku observations were obtained for NGC 5728 and NGC 4992. §2. The Suzaku observations The reprocessed (v1.2) data were reduced using standard calibration products available in November 2006. Source spectra are obtained from the Front Illuminated XIS chips with an extraction radius of ∼3′, while background spectra are extracted from nearby regions with a larger radius to guarantee good statistics. The effective exposure time for both sources is of the order of 30 ksec. The pin hard X–ray source spectra were obtained taking into account both the instrumental background appropriate for each observation and the cosmic X–ray background. The pin/XIS intercalibration constant was fixed at 1.16. In the following, we report the basic results obtained from the analysis of the X–ray spectra of the two sources and refer to Comastri et al. (2007, in preparation) for a more exhaustive description of the data analysis and interpretation. Energy (keV) 5.5 6 6.5 7 7.5 Energy (keV) iron Kα line iron Kβ line CT obscured power law unobscured reflection Fig. 1. Left panel: The unfolded broad band spectrum of NGC 5728 with the various components used to model the continuum and the iron lines. Right panel: A zoom on the ”iron band” showing a strong Kα line at ∼ 6.4 keV and a less prominent Kβ line (∼ 7 keV) on top of the underlying continuum (upper line) made by the sum of a CT obscured power law (middle line) and an unobscured reflected component (lower line). Compton Thick AGN 3 105 20 Energy (keV) 4.5 5 5.5 6 6.5 7 Energy (keV) Fig. 2. Left panel: The unfolded broad band spectrum of NGC 4992. The reflection dominated absorbed continuum, the relativistic disk line and a weak unobscured reflected component are reported. Right panel: The residuals vs. the best fit continuum in the 4.5–7.5 keV band. 2.1. NGC 5728 The Suzaku spectrum of NGC 5728 is shown in Fig. 1 (left panel). The source is clearly detected by the pin detector up to about 50 keV. The primary X–ray continuum is absorbed by Compton thick gas (NH ≃ 2.1 ± 0.2 × 10 24 cm−2). The power law slope has been fixed at Γ = 1.9 due to the narrow energy range (20–40 keV) over which the continuum is free from obscuration effects. At lower energy, the continuum can be represented by a two component model: a flat one responsible for most of the X–ray flux in the ∼ 2–6 keV energy range and a steep one taking over below 2 keV. The former may be ascribed to reflection of cold material presumably from the inner wall of the torus, while the latter has a power law shape and can be identified as primary emission scattered by off–nuclear gas into the line of sight, or unresolved soft X–ray emission lines, as commonly observed in Seyfert 2 galaxies (Guainazzi & Bianchi 2007). The scattered/reflected flux accounts for 1–2 % of the total unabsorbed flux (∼ 5× 10−11 erg s−1 cm−2) in the 2–10 keV band. The 2–50 keV unabsorbed luminosity is 2.3 × 1043 erg −1, typical of a bright Seyfert galaxy. It is interesting to note that the hard (> 10 keV) X–ray flux as measured by the pin detector is consistent within 20% with the Swift/BAT measurement in the overlapping energy range. A zoom of the ∼ 5–8 keV unfolded spectrum is shown in Fig. 1 (right panel). The iron line complex is best fitted with two gaussian lines: a strong (EW ≃ 1.0±0.3 keV) iron Kα line at ∼ 6.4 keV and a Kβ (EW ∼ 130±70 eV) at ∼ 7 keV. The relative ratio is consistent with that expected from cold neutral gas. The addition of a Compton shoulder parameterized by a Gaussian profile centered at 6.3 keV and σ = 40 eV (Matt 2002), though not statistically required, accounts for some 10% of the Kα line flux, in reasonably good agreement with the value expected for reflection from Compton thick matter. 2.2. NGC 4992 The Seyfert 2 galaxy NGC 4992 is detected by Suzaku up to about 30 keV with a flux consistent (within 10%) with that reported by INTEGRAL. The continuum (a power law with Γ=1.9) is heavily obscured (NH ∼ 4.5 ± 0.5 × 10 23 cm−2) but 4 A. Comastri et al. not Compton thick. The high energy spectrum is best fitted by adding a strong, absorbed, disk reflection component to the primary power law. The quality of the data is not such to tightly constrain the intensity of the reflection component. The 90% lower limit (R > 5) indicate a reflection dominated spectrum which is similar to that reported by Miniutti et al. (2007) from the analysis of the XMM-Newton data of IRAS 13197-1627. The source is extremely weak below 3–4 keV. The addition of an unabsorbed reflection spectrum only marginally improves the fit (Fig. 2, left panel). A zoom of the residuals in the 4.5–7.5 keV range, wrt the best fit continuum model, is shown in Fig. 2 (right panel). The shape of the residuals suggests the presence of a broad line. Indeed the best fit to the line emission is obtained with a diskline model. Leaving only the line flux and the disk inclination angle as free parameters, the line equivalent width is ∼ 750± 200 eV and the inclination angle is < 40 degrees (at 90% confidence). The best fit EW is consistent with a reflection dominated nature of the broad band spectrum. The absorption corrected 2–50 keV luminosity is ∼ 6× 1043 erg s−1. 2.3. Epilogue Relatively shallow Suzaku observations of two hard X–ray selected (with INTE- GRAL/IBIS and Swift/BAT) nearby Seyfert 2 galaxies have revelead a wealth of spectral complexity in their X–ray spectra. The good sensitivity over a broad X– ray energy range makes Suzaku very efficient to study the most obscured sources in the nearby Universe and will eventually allow us to establish the AGN absorption distribution at high column densities. Acknowledgements We thank G. Miniutti for extremely useful discussions. Support from the Italian Space Agency (ASI) under the contract ASI-INAF I/023/05/0 is acknowledged. References 1) V. Beckmann, et al. ApJ 638 (2006), 642. 2) A. Comastri, Multiwavelength AGN Surveys; R. Mujica and R. Maiolino eds. World Sci- entific Publishing Company, Singapore (2004a), p. 323 3) A. Comastri, Supermassive Black Holes in the Distant Universe, A.J. Barger eds., Kluwer academic publishers, (2004b), p. 245. 4) P. Ferrando, et al. Space Telescopes and Instrumentation II: Ultraviolet to Gamma Ray. M.J.L. Turner, G.Hasinger, eds.. Proc. of the SPIE, Volume 6266 (2006), p. 62660 5) M. Guainazzi, S. Bianchi, MNRAS 374 (2007), 1290. 6) A. Marconi, G. Risaliti, R. Gilli, et al., MNRAS 351 (2004), 69. 7) C.B. Markwardt, et al. ApJ 633 (2005), L77. 8) G. Matt, MNRAS 337 (2002), 147. 9) G. Miniutti, et al., MNRAS 375 (2007) 227 10) M.A. Worsley, A.C Fabian, F.E. Bauer, et al., MNRAS 357 (2005), 1281.

0704.1254

Spin-dependence of Ce $4f$ hybridization in magnetically ordered systems: A spin-resolved photoemission study of Ce/Fe(110)

Spin-dependence of Ce 4f hybridization in magnetically ordered systems: A spin-resolved photoemission study of Ce/Fe(110) Yu. S. Dedkov,1,∗ M. Fonin,2 Yu. Kucherenko,3 S. L. Molodtsov,1 U. Rüdiger,2 and C. Laubschat1 1Institut für Festkörperphysik, Technische Universität Dresden, 01062 Dresden, Germany 2Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany 3Institute for Metal Physics, National Academy of Sciences of Ukraine, 03142 Kiev, Ukraine (Dated: October 30, 2018) Abstract Spin- and angle-resolved resonant (Ce 4d → 4f) photoemission spectra of a monolayer Ce on Fe(110) reveal spin-dependent changes of the Fermi-level peak intensities. That indicate a spin- dependence of 4f hybridization and, thus, of 4f occupancy and local moment. The phenomenon is described in the framework of the periodic Anderson model by 4f electron hopping into the exchange split Fe 3d derived bands that form a spin-gap at the Fermi energy around the Γ point of the surface Brillouin zone. PACS numbers: 71.20.Eh, 75.30.Mb, 75.70.-i, 79.60.-i ∗ Corresponding author. E-mail: [email protected] http://arxiv.org/abs/0704.1254v1 As a function of chemical composition, the electronic properties of Ce 4f states in in- termetallic compounds vary from localized 4f 1 character over heavy-fermion behavior and mixed valence to the boarder of itinerant behavior [1]. This fascinating variety of characters is already reflected in Ce metal, where in the course of the famous isostructural γ → α transition a magnetic phase transforms into a nonmagnetic one depending on temperature and/or pressure accompanied by a volume collapse of 15% [2]. While in the promotion model this phenomenon was ascribed to a transition from a trivalent 4f 1(5d6s)3 to a tetravalent 4f 0(5d6s)4 configuration [3], later studies related the effect to a Mott-transition from local- ized to itinerant character of the 4f state [4] or to a Kondo collapse [5]. The promotion model is clearly ruled out by photoemission (PE) that reveals only weak intensity changes of the total 4f derived emission upon the γ → α transition [6]. Instead of a single 4f 0 PE final state at about 2 eV binding energy (BE) as expected from a localized 4f 1 ground state a second 4f -derived feature is observed at the Fermi energy, EF , that increases in intensity upon the γ → α transition [6]. An itinerant description based on the local density approximation (LDA) fails to explain this double-peak structure [7], it is, however, well reproduced in the framework of the single-impurity Anderson model (SIAM) considering electron hopping between localized 4f 1 and valence-band (VB) states [8]. A momentum dependence of the 4f signal as recently observed by angle-resolved PE experiments [9, 10, 11] could be explained considering the translational symmetry of the solid within a simple approach to the periodic Anderson model (PAM) [10, 11, 12]. From both SIAM and PAM the Fermi-peak intensity may be taken as a direct measure for the hopping probability. The latter should increase with the VB density of states at EF , and in fact huge Fermi-level peaks are typically observed in PE spectra of Ce transition-metal compounds reflecting α-like behavior of the Ce 4f states due to hybridization with transition- metal d-bands [13]. A spin-dependence of 4f hopping may be expected for magnetically ordered systems where the exchange splitting of the VB leads to strong variations of the density of states at EF for differently oriented VB spins. Respective spin-dependent γ → α transitions have not be observed so far, the effect, however, could be of high importance for the understanding of magnetic anomalies in these systems since the local magnetic properties of the Ce atoms may strongly vary as a function of 4f spin orientation. In this contribution we report for the first time on a spin-dependent γ → α−like transition observed by a spin- and angle-resolved resonant PE from an ordered Ce adlayer on Fe(110). Although hybridization is expected to be relatively weak in the outermost surface layer due to the low coordination of the Ce atoms [14], the quasi two-dimensional structure of the system allows for a proper determination of the position in k space probed in the experiment as necessary for a quantitative description within PAM applied here. For Ce/Fe(110), our local spin density approximation (LSDA) slab calculations reveal at the Γ point a strong reduction of majority-spin states around EF that should lead to a respective weakening of 4f hybridization for this spin orientation. In fact, our spin- and angle-resolved PE spectra show a lower Fermi-level peak intensity for the 4f majority- than minority-spin orientation. Simulations of the PE spectra within PAM reproduce this effect as well as a spin-dependent splitting of the ionization peak observed in the experimental data. Similar spin-dependencies are expected to be of high importance for the understanding of magnetic anomalies in a series of other RE systems, where hybridization phenomena were experimentally observed and successfully described within SIAM or PAM [12, 15]. A Fe(110) substrate was prepared by thermal deposition of Fe films with a thickness of 50 Å on W(110) and subsequent annealing at 450K. Low-energy electron diffraction (LEED) yielded in sharp patterns with two-fold symmetry as expected for a structurally ordered bcc Fe(110) surface [Fig. 1(a)]. Further deposition of 0.5 monolayer (close-packed atomic arrangement) of Ce metal at 300K led to a sharp overstructure in the LEED pattern [Fig. 1(b)] that could be reproduced by a kinematic LEED simulation [Fig. 1(d)] with the structural model shown in Fig. 1(c). Ce atoms are placed on hollow-sites of the bcc Fe(110) surface reproducing the arrangement of a (110) plane of fcc γ-Ce expanded by 11%. Spin- and angle-resolved resonant PE experiments at the Ce 4d → 4f absorption threshold were performed using a hemisherical PHOIBOS150 electron-energy analyzer (SPECS) equipped with a 25 kV mini-Mott spin-detector and synchrotron radiation from beamline U125/1- PGM of BESSY (Berlin). The energy and angle resolutions were set to 100meV and ±2◦, respectively. The light incidence angle was 30◦ with respect to the sample surface, and the photoelectrons were collected around the surface normal. Spin-resolved measurements were performed in normal emission geometry at 130K in magnetic remanence after having applied a magnetic field pulse of about 500Oe along the in-plane 〈11̄0〉 easy axis (perpendicular to electric field vector of the light) of the Fe(110) film. The experimental setup asymmetry was accounted for in the standard way by measuring spin-resolved spectra for two opposite directions of applied magnetic field [16, 17]. The base pressure in the experimental chamber was in the upper 10−11mbar range rising shortly to the upper 10−10 range during evaporation and annealing. Fig. 2 shows spin-resolved PE data of Ce/Fe(110) taken on- and off-resonance at 121 eV and 112 eV photon energies, respectively. The off-resonance spectra are dominated by emis- sions from Fe 3d-derived bands and are very similar to respective data of the pure Fe sub- strate (not shown here). The spectra reflect clearly the exchange splitting of the Fe 3d bands into a minority-spin component at EF (”spin down”: filled triangles) and a majority-spin component shifted to higher BE (”spin up”: open triangles). While the spectra of the pure substrate remain almost unchanged when going from 112 eV to 121 eV photon energy, the on-resonance spectra of Ce/Fe(110) reveal an additional feature around 2.2 eV BE that is ascribed to the resonantly enhanced 4f signal. In order to extract the Ce 4f contributions from these spectra, the off-resonance data were subtracted from the on-resonance spectra after proper normalization of the intensities with respect to the photon flux and the slowly varying Fe 3d photoionization cross section. The resulting spin-resolved 4f spectra are shown in the upper part of Fig. 3 together with the corresponding spin polarization P (inset) defined as P = (I↑ − I↓)/(I↑ + I↓), where I↑ and I↓ denote the intensities of the majority- and minority-spin channels, respectively. The spectra reveal the well-known double-peak structure of the Ce 4f emission consisting of a main maximum at 2.2 eV corresponding to the ionization peak expected for an unhybridized 4f 1 ground state and the hybridization peak at EF . From the weak intensity of the latter relative to the ionization-peak signal, a weak hybridization similar to the one in γ-Ce can be concluded as it is expected for a Ce surface layer [18]. The most important observation is, however, that the intensity of the hybridization peak is larger for the minority- than for the majority-spin component (Fig. 3) indicating larger 4f -hybridization of the former. The spin polarization of both, the ionization and the hybridization peaks, gives a negative sign indicating that the preferred orientation of the Ce 4f spins is opposite to the magnetization direction of the Fe layers. In addition to the double-peak structure another feature is visible around 1 eV BE (Fig. 3), that is weaker in intensity and shifts to lower BE when going from the minority- to the majority-spin component. In order to understand the ground-state magnetic properties of Ce/Fe(110), as a first step fully relativistic spin-polarized band-structure calculations were performed by means of the linear muffin-tin orbital (LMTO) method. A pure Fe surface and the Ce/Fe(110) system were considered using the structural model shown in Fig. 1(c). The Fe substrate was simulated by a five-layer slab of Fe atoms with (110) orientation of the surface. The results were compared to data calculated for the isostructural non-f system La/Fe(110). For the atoms in the middle layer of the Fe slab the calculations give a local electronic structure close to that obtained for Fe bulk [19]. The calculated Fe 3d spin moment value lies between 2.35µB and 2.40µB. At the surface it increases to 2.60µB. In all cases contributions of s and p electrons to magnetic moment are negligible. By the presence of a Ce overlayer the Fe 3d spin moments of the surface atoms are reduced to 2.14µB and 2.50µB, respectively, depending on whether the Fe atoms are nearest neighbors of Ce atoms or not. Replacing in the calculation Ce by La atoms give very similar results indicating that the electronic structure of the Fe atoms is perturbed by interactions with extended valence states (mainly 5d) of the overlayer. The calculations yield for a La atom on the Fe(110) surface a local spin moment of −0.24µB, determined mainly by the 5d electrons (−0.20µB). The negative sign stands for an antiparallel orientation with respect to the Fe 3d spin moment. For the Ce atom the local spin moment is equal to −1.12µB, with 5d and 4f contributions of −0.28µB and −0.82µB, respectively. Thus, like in other Ce-Fe systems [20, 21, 22], the Ce 4f electrons reveal a spin orientation opposite to Fe 3d majority spin in agreement with the PE experiment. Since the 4f electrons have additionally a large positive orbital momentum of 2.80µB due to their reduced atomic coordination at the surface the total moment equals to 1.70µB and corresponds, thus, to ferromagnetic coupling with respect to the Fe 3d spins. At finite temperatures magnetic disorder leads to the situation encountered in the experiment where a part of the 4f spins are flipped into the opposite direction. In order to describe the observed variation of 4f hybridization as a function of spin orientation, we used the simplified periodic Anderson model that was recently successfully applied to explain the angle-resolved PE spectra of CePd3 [10] and Ce/W(110) [11]. In this approach the double occupation of the 4f states is ignored (on-site f − f Coulomb interaction energy, Uff → ∞) and k vector conservation upon hybridization is assumed. In this case a simplified (without Uff term) Anderson Hamiltonian can be written as follows εσ(k)d+ εσf (k)f kσ + f where the VB states |kσ〉 have a dispersion εσ(k) and are described by creation (annihilation) operators d+ kσ (dkσ). The operator f kσ creates a f electron with momentum k, spin σ, and energy εσf (k). We assume that a non-hybridized f band has no dispersion: ε f (k) = εσf allowing, however, a possible small difference in the energy positions of 4f levels with different spin σ due to exchange interaction. The two electron subsystems (VB and 4f states) are coupled via a hybridization V σ (E) that leaves the electron spin unaffected, i. e. spin-flips upon electron hopping are excluded. E denotes the BE with respect to EF . This form of the Hamiltonian allows us to diagonalize it for each particular k point of the surface Brillouin zone (BZ) and for each spin state σ. For the hybridization matrix element V σ (E) we use calculated f -projected local expansion coefficients cσf (E,k) of the Bloch functions around the rare-earth sites: V (E) = ∆·cσf (E,k), where ∆ is a constant, adjustable parameter. Expansion coefficients cσf (E,k) that charac- terize the local f character of VB states were taken from the results of the band-structure calculations of the La/Fe(110) system, in order to exclude the contribution of localized Ce 4f orbitals. For normal emission of the photoelectrons we have to consider VB states at the Γ point of the surface BZ. The calculated values of ∣cσf (E,Γ) are shown in the bottom part of Fig. 2. The energy distributions of the VB states of local f character are quite different for majority- and minority-spin electrons. Since these states are formed by linear combination of wave functions of the neighboring atoms (mainly Fe 3d) penetrating into the La atomic spheres, they reflect to some extent the energy and spin distribution of the latter (see off-resonance spectra in Fig. 2). Their different amplitude and energy distribution for majority- and minority-spin states causes strong differences in the respective hybridiza- tion matrix elements and results in different shape of the 4f PE spectra for the two spin directions. The spectral functions of the Ce 4f emission were calculated using the parameters ε −1.9 eV, ε = −1.7 eV, and ∆ = 0.85 eV. These values deviate from those used in Ref. [11] for Ce/W(110) only by slightly higher BE of the non-hybridized 4f level resulting from the lower coordination of the Ce atoms. An energy-dependent life-time broadening of the form ΓL = 0.030 eV+0.085E was considered. The calculated spectral functions were additionally broadened with a Gaussian (ΓG = 100meV) to simulate finite instrumental resolution and an integral background was added to take into account inelastic scattering. The calculated spin-resolved Ce 4f PE spectra are presented in Fig. 3 (lower part). The energy distribution of the PE intensity agrees well with that of the experimental spectra (Fig. 3, upper part). The minority-spin spectrum reveals high intensity of the hybridization peak due to large density of the minority-spin VB states close to EF . A shoulder near 1 eV BE is formed by hybridization with VB peaks at 0.9 eV and 1.3 eV BE (Fig. 1). In accordance with the experiment, in the calculated majority-spin spectrum the ionization peak is split into three components (maxima at 0.9 eV, 2.1 eV, and shoulder at 3 eV BE) as a result of hybridization with the VB states (at 1.4 eV and between 2 eV and 3 eV BE). No majority- spin hybridization peak is obtained in the calculation due to the negligibly small density of VB states for this spin direction at the Fermi level. This theoretical result deviate from the experiment where a reduced but finite hybridization peak was observed. The latter may be ascribed to the finite angle resolution of the experiment that samples also regions in the k space where majority-spin bands cross EF . The calculated spin polarization (Fig. 3, inset in the lower part) reproduces qualitatively the energy dependence of the measured polarization. Particularly good agreement is obtained for the points where the spin polarization changes its sign. In summary, we have shown that the observed spin-dependence of the shape of the Ce 4f emission in Ce/Fe(110) system may be explained by a spin-dependence of 4f -hybridization. From this result 4f -occupancy as well as effective magnetic moment are generally expected to vary with spin-orientation, an effect that may be of crucial importance for the understanding of many-body effects and magnetic anomalies in RE systems. This work was funded by the Deutsche Forschungsgemeinschaft, SFB 463, Projects TP B4 and TP B16 as well as SFB513. We would like to acknowledge BESSY staff for technical support during experiment. [1] J.G. Sereni in Handbook on the Physics and Chemistry of Rare-Earths, ed. by K. A. Gschnei- dner, Jr. and L. R. Eyring, Vol. 15, p. 1 (North-Holland, 1991). [2] K. A. Gschneider, Jr. et al., J. Phys. Chem. Solids 23, 1191 (1962); D. G. Koskenmaki and K. A. Gschneider in Handbook on the Physics and Chemistry of Rare Earths, edited by K. A. Gschneider, Jr. and L. R. Eyring (North-Holland, Amsterdam, 1978). [3] R. Ramirez and L. M. Falicov, Phys. Rev. B 3, 2425 (1971). [4] B. Johansson, Philos. Mag. 30, 469 (1974); B. Johansson et al., Phys. Rev. Lett. 74, 2335 (1995). [5] J. W. Allen and R. M. Martin, Phys. Rev. Lett. 49, 1106 (1982); M. Lavagna et al., Phys. Lett. 99, 210 (1982); J. W. Allen and L. Z. Liu, Phys. Rev. B 46, 5047 (1992). [6] D. Wieliczka et al., Phys. Rev. B 26, 7056 (1982); E. Wouilloud et al., Phys. Rev. B 28, 7354 (1983); A. Fujimori and J.H. Weaver, Phys. Rev. B 32, 3422 (1985); F. Patthey et al., Phys. Rev. Lett 55, 1518 (1985); E. Weschke et al., Phys. Rev. B 44, 8304 (1991); Yu. Kucherenko et al., Phys. Rev. B 66, 115116 (2002). [7] Note, that within the recently proposed combination of LDA with dynamical mean-field theory (DMFT) a realistic description of Ce PE spectra becomes possible. See, e. g. M. B. Zölfl et al., Phys. Rev. Lett. 87, 276403 (2001); K. Held et al., Phys. Rev. Lett. 87, 276404 (2001). [8] O. Gunnarsson and K. Schönhammer, Phys. Rev. Lett. 50, 604 (1983); Phys. Rev. B 28, 4315 (1983); J. W. Allen et al., Adv. Phys. 35, 275 (1986); A. Kotani and S. Shin, Rev. Mod. Phys. 73, 203 (2001); R. Hayn et al., Phys. Rev. B 64, 115106 (2001). [9] A. B. Andrews et al., Phys. Rev. B 53, 3317 (1996); H. Kumigashira et al., Phys. Rev. B 55, R3355 (1997); M. Garnier et al., Phys. Rev. B 56, R11399 (1997). [10] S. Danzenbächer et al., Phys. Rev. B 72, 033104 (2005). [11] D. V. Vyalikh et al., Phys. Rev. Lett. 96, 026404 (2006). [12] S. Danzenbächer et al., Phys. Rev. Lett. 96, 106402 (2006). [13] R. J. Jung et al., Phys. Rev. Lett. 91, 157601 (2003); Yu. Kucherenko et al., Phys. Rev. B 70, 045105 (2004). [14] C. Laubschat et al., Phys. Rev. Lett. 65, 1639 (1990). [15] Yu. Kucherenko et al., Phys. Rev. B 66, 165438 (2002); S. L. Molodtsov et al., Phys. Rev. B 68, 193101 (2003). [16] J. Kessler, Polarized Electrons, 2nd ed. (Springer-Verlag, Berlin, 1985). [17] P. D. Johnson et al., Rev. Sci. Instrum. 63, 1902 (1992). [18] E. Weschke at al., Phys. Rev B 44, 8304 (1991) and references therein. [19] A. Chassé et al., Phys. Rev. B 68, 214402 (2003). [20] M. Finazzi et al., Phys. Rev. Lett. 75, 4654 (1995). [21] M. Arend et al., Phys. Rev. B 57, 2174 (1998). [22] T. Konishi et al., Phys. Rev. B 62, 14304 (2000) and references therein. FIG. 1: (Color online) LEED images obtained from (a) Fe(110) and (b) Ce/Fe(110); assumed surface crystallographic structure of the Ce/Fe(110) system (c) and simulation of the LEED-image (d). The shaded rectangle in (c) visualizes the fcc Ce(110) plane expanded by 11%. FIG. 2: (Color online) Spin-resolved PE spectra of Ce/Fe(110) system measured in on- and off- resonance at the 4d → 4f absorption threshold. Open/filled triangles denote contributions of majority/minority spin directions, respectively. Bottom part: Calculated local 4f character of the VB states (|cσ (E,Γ)|2) at the La site in the Γ point of the surface BZ for La/Fe(110) system for majority- (solid line) and minority-spin (shaded area) direction. FIG. 3: (Color online) Spin-resolved experimental (upper part) and calculated (lower part) Ce 4f emission for Ce/Fe(110). Majority- and minority-spin emissions are shown by open and solid triangles, respectively. The insets show the corresponding spin polarization P . References

0704.1255

Two-way coupling of FENE dumbbells with a turbulent shear flow

Two-way coupling of FENE dumbbells with a turbulent shear flow Thomas Peters∗ Department of Physics, Philipps-Universität Marburg, D-35032 Marburg, Germany Jörg Schumacher† Department of Mechanical Engineering, Technische Universität Ilmenau, D-98684 Ilmenau, Germany (Dated: August 3, 2021) We present numerical studies for finitely extensible nonlinear elastic (FENE) dumbbells which are dispersed in a turbulent plane shear flow at moderate Reynolds number. The polymer ensemble is described on the mesoscopic level by a set of stochastic ordinary differential equations with Brow- nian noise. The dynamics of the Newtonian solvent is determined by the Navier-Stokes equations. Momentum transfer of the dumbbells with the solvent is implemented by an additional volume forcing term in the Navier-Stokes equations, such that both components of the resulting viscoelas- tic fluid are connected by a two-way coupling. The dynamics of the dumbbells is given then by Newton’s second law of motion including small inertia effects. We investigate the dynamics of the flow for different degrees of dumbbell elasticity and inertia, as given by Weissenberg and Stokes numbers, respectively. For the parameters accessible in our study, the magnitude of the feedback of the polymers on the macroscopic properties of turbulence remains small as quantified by the global energy budget and the Reynolds stresses. A reduction of the turbulent drag by up to 20% is observed for the larger particle inertia. The angular statistics of the dumbbells shows an increasing alignment with the mean flow direction for both, increasing elasticity and inertia. This goes in line with a growing asymmetry of the probability density function of the transverse derivative of the streamwise turbulent velocity component. We find that dumbbells get stretched preferentially in regions where vortex stretching or bi-axial strain dominate the local dynamics and topology of the velocity gradient tensor. PACS numbers: 47.27.ek, 83.10.Mj, 83.80.Rs I. INTRODUCTION When a few parts per million in weight of long-chained polymers are added to a turbulent fluid its properties change drastically and a significant reduction of turbulent drag is observed. [1] Although the phenomenon is known from pipe flow experiments for almost 60 years,[2, 3] a complete understanding is still lacking. One reason for this circumstance is that the physical processes in a turbulent and dilute polymer solution cover several orders of magnitude in space and time; in other words, we are faced with a real multiscale problem. [4, 5] In case of fully developed turbulence, the integral scale L, which measures the extension of largest vortex structures in the flow, exceeds the viscous Kolmogorov scale ηK, which stands for the extension of the smallest turbulent eddies, by a factor of at least 1000. However, long-chained polymers barely exceed the viscous flow scale even in an almost stretched state. Their equilibrium extension as given by the Flory radius R0 is usually by a factor of 100 smaller than ηK.[6] In terms of time scales the situation differs slightly. The viscous Kolmogorov time τη can become smaller than the slowest relaxation time τ of the macromolecules. Although macroscopic closures can rationalize some issues of drag reduction [7], the challenging question remains of how the individual dynamics of numerous polymer chains, which is present on sub-Kolmogorov and Kolmogorov scales, adds up to a macroscopic effect at scales r <∼ L as being observed in several experiments. [8, 9, 10] The description of dilute polymer solutions relies for most studies on one of the following two models: on one side, macroscopic continuum models such as Oldroyd-B or FENE-P models [11, 12, 13, 14, 15] include the polymer dynamics as an additional additive macroscopic stress field. Only the largest scales ℓ >∼ ηK of the viscoelastic fluid are described in its full complexity. Numerical problems arise in connection with the pure hyperbolic character of the equation of motion for the polymer stress field, such as the conservation of its positivity (see e.g. Ref. [16] for a detailed discussion). In addition, the coarse graining to the macroscopic polymer stress can lead to deeper conceptional difficulties, e.g., the failure of energy stability of viscoelastic flows, which is an important building block for investigations of stability and upper bounds on the dissipation rate in Newtonian flows. [17] Further problems ∗ Present address: Institute for Theoretical Astrophysics, Ruprecht-Karls-Universität Heidelberg, D-69120 Heidelberg, Germany † Corresponding author: [email protected] http://arxiv.org/abs/0704.1255v1 arise for the macroscopic description of non-Newtonian fluids in the limits of very low and high frequencies, where they should behave as Newtonian fluids and solids, respectively. [18, 19, 20] On the other side, Brownian dynamics models [21, 22, 23, 24, 25] describe the polymer chain on a mesoscopic level as overdamped coupled oscillators arranged in bead-spring chains. The models include complex conformations of the macromolecules and screening effects due to the solvent such as hydrodynamic interaction.[26] The simplest of such mesoscopic models for a polymer chain is a dumbbell where two beads are connected by a spring. The dynamics in these models is on scales ℓ <∼ ηK. This means that the surrounding fluid is spatially smooth and either a steady [22], a start-up shear flow [23], or a white-in-time random flow. [27] In a recent work by Davoudi and Schumacher[28], numerical studies at the interface of both descriptions were conducted by combining Brownian dynamics simulations (BDS) with direct numerical simulations of a turbulent Navier-Stokes shear flow. The simplest mesoscopic model with a linear spring force - the Hookean dumbbell model - was taken there in order to study the stretching of the dumbbell as a function of the outer shear rate and the elastic properties of the springs. However, a feedback of the polymers on the shear flow was not included in their study. In the following, we want to extend these investigations into two directions. Firstly, we will model the macro- molecules more realistically as finitely extensible nonlinear elastic (FENE) dumbbells. Secondly, their feedback on the shear flow is included via a two-way coupling. The effect of the FENE dumbbells on the statistical fluctuations of the velocity and the velocity gradients will be studied. In addition, conformational properties of the dumbbells, such as their extension and angular distribution with respect to the mean flow component, will be addressed. The polymer feedback results in an additional forcing that has to be added to the right hand side of the Navier-Stokes equations for the advecting Newtonian solvent similar to the case of two-phase flows with dispersed particles [29, 30, 31] or bubbles.[32] We will keep the full dynamic equation of motion for the dumbbells, containing accelerations due to elastic, friction and stochastic forces, and cannot neglect inertia. This step is necessary in order to describe the momentum transfer of the dumbbells to the solvent as discussed in Ref.[33]. In contrast to the conventional BDS that neglect inertia effects from beginning, we will be left here with three physical parameters: the Stokes number St for the particle inertia, the Weissenberg number Wi for the elastic properties of the dumbbells, and the Reynolds number Re of the flow, respectively. The Reynolds number is defined , (1) with the characteristic (large-scale) velocity U , the characteristic length L (both are specified later in the text), and the kinematic viscosity of the Newtonian solvent ν. The Weissenberg number Wi compares the characteristic dumbbell relaxation time τ from a stretched to a coiled state with the characteristic time scale of the advecting flow, L/U , and is given by . (2) The Stokes number St relates the particle response time to changes in the surrounding velocity, τst, with the charac- teristic flow time scale. It follows to . (3) The physics of dispersed FENE dumbbells in a turbulent shear flow is thus described by three dimensionless numbers. For a fixed Reynolds numbers Re, we can basically distinguish the following four limiting cases: (i) Wi ≫ 1, St ≫ 1; (ii) Wi ≪ 1, St ≫ 1; (iii) Wi ≪ 1, St ≪ 1; (iv) Wi ≫ 1, St ≪ 1. Case (i) would stand for very heavy particles (or dumbbells) which are stretched almost to their contour length. They will behave as dispersed rods. In case (ii), the dumbbells would act as heavy spherical particles since they remain coiled in practical terms. The cases of interest for dilute polymer solutions are (iii) and (iv), respectively. Inertia effects are then very small, [34] and the Weissenberg number can vary from very small to large values implying an increasingly slower relaxation of the macromolecules from a stretched non-equilibrium to a coiled equilibrium state in comparison to the characteristic flow variation time scale. As we will discuss in the next section, the numerical treatment becomes challenging, on one hand due to the finite extensibility, on the other hand due to the small Stokes numbers we are aiming at. The Stokes time τst sets a small but finite time scale then, which can cause stiffness problems for an explicit integration algorithm. Despite these efforts, our values for the Stokes number will still exceed the realistic magnitudes for polymer chains in solution by orders of magnitude. Nevertheless, we think it is interesting and to some degree necessary to study the dumbbell dynamics under these circumstances and to provide a systematic study of how a shear flow will be affected by the presence of dispersed bead-spring chains with variable degree of inertia. This will shed some light on possible reasons for drag reduction in our model. The outline of the manuscript is as follows. In the next section the equations of motion, the two-way coupling and the numerical scheme are presented. Afterwards, we discuss the results for the macroscopic energy balance as well as for the Reynolds stresses. This is followed by studies of small-scale properties such as the statistics of the extension and orientation of the dumbbells and of their impact on the fluctuations of velocity gradients. We conclude with a discussion of our results and will give a brief outlook to extensions of the present work toward more realistic parameter settings. II. MODEL AND EQUATIONS A. The Newtonian solvent The Navier-Stokes equations that describe the dynamics of the three-dimensional incompressible Newtonian fluid are solved by a pseudo-spectral method using a second-order predictor-corrector scheme for advancement in time.[28] The equations of motion are + (u ·∇)u = −∇p+ ν∇2u+ f + fp , (4) ∇ · u = 0 , (5) where u is the (total) velocity field, p the kinematic pressure field, f the volume forcing which sustains the turbulence, and fp the feedback of the dumbbells (see section II C). The shear flow is modeled in a volume with free-slip boundary conditions in the shear direction y and periodic boundaries in the streamwise and spanwise directions x and z. The free-slip boundary conditions at y = 0, Ly are given by uy = 0 , = 0 . (6) Here, the total velocity field follows by a Reynolds (de)composition as a linear mean part with the constant shear rate S and a turbulent fluctuating part u = 〈u〉+ u′ = Syex + u′ . (7) The notation 〈·〉 stands for the ensemble average, which will be a combination of volume and time averages for most cases. The aspect ratio is Lx:Ly:Lz = 4π: 2: 2π. The characteristic length is the halfwidth of the slab, L = Ly/2. Velocities are measured in units of the laminar flow profile U(y) = − 2 cos(πy/2)ex. We will take Ux(Ly/4) as the characteristic velocity U (see also (1), (2), and (3)). The applied volume forcing sustains this laminar flow profile and follows from (4) consequently to f(y) = − 2π2/(4ν) cos(πy/2)ex. Forcing amplitude and profile will remain unchanged throughout this study. At sufficiently large Reynolds numbers this linearly stable laminar shear flow becomes turbulent when a finite perturbation is applied.[35] The volume forcing f is then a permanent source of kinetic energy injection into the shear flow which sustains turbulence in a statistically stationary state. Although the steady forcing is of cosine shape, the resulting mean turbulent flow profile will be linear except for small layers in the vicinity of both free-slip planes, where the boundary conditions have to be satisfied. Our mean profiles follow to 〈ux(y)〉 ≃ S(y − 1) for y ∈ [0, 2] with S = 0.035 − 0.04 for Re = 800. This range of S-values remained nearly unchanged for all parameter sets. In addition, 〈u′y〉 = 〈u′z〉 = 0. The shear flow can be considered therefore as being nearly homogeneous. The simulation program is run with two spectral resolutions. For Re = 400, a grid with 64 × 32 × 32 mesh points was taken. For Re = 800, we took a grid with 128 × 32 × 64 points. The spectral resolution as given by the product kmaxηK = 8πNx/(3Lx)ηK was 1.5 for the first case and 2.3 for the second. Here, ηK is the viscous Kolmogorov scale and defined as ηK = ν 3/4/〈ε′〉1/4 with the mean turbulent energy dissipation rate 〈ε′〉, where ε′(x, t) = (ν/2)(∂u′i/∂xj + ∂u j/∂xi) 2 for i, j = x, y, z. Clearly, the spectral resolutions are not very large, but they give us the opportunity to perform parametric studies in the three-dimensional space which is spanned by Re, Wi, and St. Most of our following studies will be conducted for the better resolved case of Re = 800. B. The FENE dumbbells The smallest building block for the mesoscopic description of the polymer stretching can be accomplished by considering dumbbells where two beads (that stand for several hundreds of monomers) are connected by a spring. The entropic elastic force follows the Warner force law [11] and depends on the separation vector R(t) = x2(t)−x1(t) that is spanned between both beads at positions x2(t) and x1(t), respectively. The force law is given by Fel(R) = 1−R2/L20 , (8) where L0 is the contour length of the dumbbells which cannot be exceeded. The spring constant is denoted byH . When taking into account the elastic entropic force, hydrodynamic Stokes drag, and thermal noise, the second Newtonian law for a FENE dumbbell written in relative coordinates R(t) and center-of-mass coordinates r(t) = (x1(t)+x2(t))/2 reads [27, 34] ṙ = v , (9) v̇ = −v + 1 (u1 + u2) + ξr , (10) Ṙ = V , (11) V̇ = −V +∆u− 2HR ζ (1−R2/L20) ξR , (12) where ∆u = u(x2, t) − u(x1, t) is the relative fluid velocity at the bead centers. The last terms in the velocity equations, containing ξr and ξR, stand for vectors of thermal Gaussian noise with the properties 〈ξi(t)〉 = 0 , (13) 〈ξi(t)ξj(t′〉 = δijδ(t− t′) (14) for i, j = x, y, z. The three components of each vectorial noise term are statistically independent stochastic processes. Furthermore, the vectorial noise with respect to the center-of-mass velocity is statistically independent to that for the relative velocity dynamics. The noise prevents the extension of a dumbbell to shrink below its equilibrium length , (15) with kB being the Boltzmann constant, T the temperature. Equation (15) follows from the equipartition theorem. The contour length L0 = 10R0 is used throughout this study and R0 ≃ ηK. The relaxation time of the dumbbells is given by [11] , (16) where ζ = 6πρfνa (17) is the Stokes drag coefficient of a spherical bead with radius a. The fluid mass density is ρf . Due to the current resolution contraints the dumbbells will experience both the smooth and partly rough scales of the advecting flow. Consequently, the velocity difference ∆u is kept in the equation and not approximated by the linearization ∆u ≈ (R · ∇)u as it is done in BDS where L0 ≪ ηK . For spatially smooth flows both expressions give the same results. The equations (9) through (12) introduce the other two dimensionless parameters beside the Reynolds number Re, the Weissenberg number Wi and the Stokes number St, respectively (see definitions (2) and (3)). The Stokes time τst is the response time of an inertial particle which is required to speed up to the velocity of its local surrounding. A zero Stokes time implies a behavior as a passive Lagrangian tracer. For beads, this time follows to τst = mb/ζ with ζ as given above and consequently τst = . (18) The density contrast ρp/ρf is to very good approximation unity[36], i.e. polymers are considered as neutrally buoyant. In Ref. [28], we have compared the polymer relaxation time to the microscopic stretching time scale. This is given by the inverse of the maximum Lyapunov exponent and is comparable to the microscopic time scale of the flow, the Re = 400 Re = 800 Wi = 3 Wiη = 0.8 Wiη = 0.6 Wi = 20 Wiη = 5.1 Wiη = 4.3 Wi = 100 Wiη = 25.7 Wiη = 21.5 St = 5.0 × 10−4 Stη = 1.3× 10 −4 Stη = 1.1 × 10 St = 5.0 × 10−3 Stη = 1.3× 10 −3 Stη = 1.1 × 10 St = 5.0 × 10−2 Stη = 1.3× 10 −2 Stη = 1.1 × 10 St = 5.0 × 10−1 Stη = 1.3× 10 −1 Stη = 1.1 × 10 TABLE I: The Weissenberg and Stokes numbers rescaled by the Kolmogorov time τη of the flow. Wiη = τ/τη and Stη = τst/τη. Note that τη is based on the pure Newtonian case. Only minor changes arise when polymers are added to the solvent. Kolmogorov time τη = ν/〈ε〉. Table 1 gives an overview of the values of St and Wi that have been used and of how they translate into Stη and Wiη, respectively. We see that the Stokes numbers get as low as 10 −4 when measured in viscous units, which is still orders of magnitude above the realistic estimates for dilute polymer solutions which are about three to four order of magnitude below our minimal value. In most cases, an ensemble of 6.3× 104 FENE dumbbells, i.e. 1.2× 105 beads, is advanced by a weak second-order predictor-corrector scheme simultaneously with the flow equations.[21] The finite extensibility and the small Stokes numbers require a semi-implicit time-stepping for some variables. In order to avoid a total length larger than L0, we proceed in line with Ref. [21] and solve a cubic equation for R = |R| in the corrector step. Initially, the center of mass of the dumbbells is seeded randomly in space with a uniform distribution and an initial extension of R0. All Lagrangian interpolations were done with a trilinear scheme. Details on the numerical procedure are outlined in appendix A. In order to build a bridge to macroscopic simulations we provide an estimate for the contribution of the dumbbell ensemble to the zero-shear viscosity. Following Ref. [21] it is defined as ηp = ρpνp = npkBTτ , (19) with the number density of dumbbells np. When applying (15) as well as definitions (16) and (17), and using ρf/ρp = 1 one gets 0νa (20) with the solvent viscosity ν. The bead radius a is substituted by the Stokes time τst. Recalling the definitions for the Kolmogorov length ηK = ν 3/4/〈ε′〉1/4 and for the Kolmogorov time τη = ν/〈ε′〉, one ends with the relative viscosity Stη . (21) For the present simulations, one dumbbell is seeded per grid cell and therefore np ≈ 1/η3K. Additionally, R0 ≃ ηK. Following table 1 for the runs at Re = 800, one gets ratios of s between between 0.1 for the smallest Stokes number going up to 3 for the largest one. The latter value is rather large for polymer solutions. Values below unity are usually taken, such as in DNS with the Oldroyd-B model.[14] Equation (21) is in this spirit consistent with the discussion in the introductory part. Only the lower Stokes numbers result to values of s as taken for macroscopic DNS for viscoelastic shear flows. C. Two-way coupling The back-reaction of the dumbbells on the fluid consists of contributions from the Stokes friction and the stochastic noise term. In accordance with Newton’s third law, the force contribution from each of the two beads at positions xi (i = 1, 2) follows to Fi = −F (st)i − F i = ζ(ẋi − u(xi))− 2kBTζ ξi . (22) The force density generated by all FENE dumbbells results to ρffp = i δ(x− x i ) , (23) where Np is the number of dumbbells. The volume integral of (23) gives a force since the delta function carries the dimension of an inverse volume due to δ(x − x(j)i ) d 3x = 1. Consequently, the dimensionless form of the forcing reads i − u(x i ))− δ̃(x− x(j)i ) , (24) where the bead volume follows to Vb = 4πa 3/3 = (4π/3)(9ντst/2) 3/2. The notation δ̃ is for the dimensionless delta function. We have used again ρf/ρp ≈ 1. The force density has to be evaluated at space points that are between the mesh vertices. Again the trilinear interpolation has to be used to evaluate the contributions of the point force to the eight next neighboring mesh vertices. III. LARGE-SCALE PROPERTIES A. Energy balance The first analysis step is the study of the effects of the two-way coupling on the macroscopic properties of turbulence. Given the boundary conditions for our problem, eq. (4) results in the following balance for the total kinetic energy E(t) = 1 |u|2 d3x with V = LxLyLz, = −ν〈(∂ui/∂xj)2〉V + 〈u · f〉V + 〈u · fp〉V , = −ε(t) + εin(t)− εp(t) (25) where 〈·〉V = 1V · d3x is the short notation for the volume average. In case of statistical stationarity, one gets d〈E〉t/dt = 0 and thus 〈εin〉 = 〈ε〉+ 〈εp〉 . (26) Figure 1 shows the three mean rates as a function of the Stokes number for two Weissenberg numbers Wi = 20, 100. The mean energy dissipation rate 〈ε〉 and the mean energy injection rate 〈εin〉 are of the same order of magnitude for all cases. They remain nearly unchanged with respect to Weissenberg number, which indicates that the effect of the dumbbell ensemble on the macroscopic flow properties is small. Nevertheless, one observes a slight increase of the mean energy injection rate 〈εin〉 with respect to St going in line with a decrease of 〈ε〉 (see upper and mid panel of Fig. 1). Recall that the energy injection rate will be maximal for the laminar case, i.e. for u ‖ f . The trend of the data indicates that the streamwise flow component relaminarizes slightly with growing inertia. The lower panel of the same figure shows the findings for the dissipation due to polymer stretching 〈εp〉. As an additional energy dissipation mechanism, it consumes injected energy which goes into the elastic energy budget of the dumbbell ensemble. The rate 〈εp〉 grows in magnitude with respect to both parameters, the Stokes and Weissenberg number. For Wi = 3, the dumbbells are not significantly extended and no clear trend of 〈εp〉 with St could be observed. The dissipation rate 〈εp〉 is significantly smaller in comparison to the runs with larger Wi. In order to estimate the maximum feedback of the dumbbells on the flow, we performed an “academic experiment” for our system by tethering one of the two beads of a dumbbell at a fixed position. The dumbbells get then stretched more efficiently and undergo strong conformational fluctuations. Figure 2 illustrates their dramatic effect on the total kinetic energy. We compare the freely draining case with the tethered one and observe a significant decrease of the kinetic energy. An inspection of the flow structures indicates that the turbulent fluctuations are supressed almost completely. The flow becomes basically laminar. The magnitude of the feedback for freely draining dumbbells will always remain significantly below this artifical limit with tethered dumbbells. B. Reynolds stresses Figure 3 shows the four non-vanishing components of the Reynolds stress tensor 〈u′iu′j〉/(2k) where k = 〈(u′i)2〉/2 is the turbulent kinetic energy (TKE). The moments are averages over the whole simulation volume for a sequence of about 100 statistically independent snapshots of the time evolution of the shear flow. The results can be summarized to the following trends. For the two smallest Stokes numbers, no dependence on the Weissenberg number is observed. For St = 0.05 and 0.5, the mean streamwise fluctuations are enhanced while the remaining components of the Reynolds stress tensor decrease as a function of Wi. This finding is in agreement with observations in a Kolmogorov flow by Boffetta et al. [37] Similar to the friction factor for a turbulent pipe [38], we can define a friction factor for the present flow where the applied pressure gradient term has to be substituted by an amplitude of the static volume forcing profile f that sustains the laminar cosine flow profile. Consequently, 〈ux(y = Ly)〉2 . (27) Since f(y) = − 2π2/(4ν) cos(πy/2)ex, we take F = fx(y = Ly) = 2π2/(4ν). A similar definition was suggested for a Kolmogorov flow which is also driven by a volume forcing.[37] Drag reduction by dispersed dumbbells would go in line with a decrease of the dimensionless measure cf below the Newtonian value c f . For the smallest Stokes number, the ratio goes to about unity. The slight overshoot is attributed to the strong variations of the streamwise velocity at the free-slip planes. Figure 4 indicates a reduction by 20%− 25% at St = 0.05, 0.5 and for the larger Weissenberg numbers. The series with Wi = 3 gave cf ≃ cNf . An important structural ingredient of shear flows are the asymmetric fluctuations of the three diagonal elements of the Reynolds stress tensor. The streamwise fluctuations 〈(u′x)2〉 are spatially arranged in streamwise streaks which interact with streamwise vortices in a so-called regeneration cycle of coherent structures. This cycle is sustained by the non-normal amplification mechanism.[39, 40] The impact of long-chained polymers on the extension of the streamwise streaks has been demonstrated in experiments [10] and numerical simulations.[41, 42] While streamwise fluctuations were found to increase, the fluctuations in shear and spanwise directions decreased. This is in line with our observations as discussed above. In Fig. 5, we show isolevels of the streamwise turbulent fluctuations for opposite sign at Wi = 3, 20, 100. Although not very pronounced, a slight increase in the connectivity and extension of the streamwise streaks can be observed with increasing Weissenberg number. As we can see, the statistics of macroscopic turbulent properties is affected only slightly by the dispersed FENE- dumbbells. Their impact increases with Weissenberg number as well as with Stokes number. In order to rule out that particle inertia dominates the discussed trends of our studies, we considered the case of dispersed beads in the same flow at the same Stokes numbers. This is achieved by switching off the elastic spring force, i.e. Fel = 0. The Stokes friction force remained as the only force. The quantity fp models then the feedback of the particles on the flow. We added the statistical means of injection and dissipation rates as a function of the Stokes number for this case to Fig. 1. While the mean injection and mean dissipation rates are of the same magnitude, the dissipation due to particle feedback is orders of magnitude smaller in comparison to the polymer feedback, except for the largest St. In addition, we found no clear trends for the Reynolds stress components as a function of St. IV. SMALL-SCALE PROPERTIES A. Extensional and angular statistics of dumbbells The finite extensibility of the dumbbells will affect the shape of the probability density function (PDF) of R, which is supported on scales smaller than L0 only. Figure 6 reports our findings for p(R) for different Weissenberg and Stokes numbers. For the lowest Weissenberg number, Wi = 3, the majority of the dumbbells remains at the extension of about the Kolmogorov length ηK. This picture changes for larger values of Wi. At Wi = 100, the majority of the ensemble is stretched to almost L0, which manifests in the sharp maximum at R <∼ L0. Qualitatively, the change of the shapes of the PDFs with increasing Wi agrees well with experimental findings [43] and analytical studies [27, 44] for the coil-stretch transition in random flows. The trends with the Stokes number remain small in all cases. However, the data show that growing particle inertia suppresses the stretching to very extended molecules since the response time of the molecules to the variation of the structures increases (see e.g. mid panel of Fig. 6). As we have seen in the last section, the fluctuations of the turbulent velocity field in the shear flow vary strongly from one space direction to another (see e.g. Fig. 3). The major contribution is contained in the streamwise component 〈(u′x)2〉 parallel to the direction of the mean turbulent flow. This suggests an investigation of the angular statistics of the polymers since their stretching can be expected to become anisotropic as well. The following dumbbell coordinate system will be used therefore throughout this text: Rx = R cosϕ cos θ, Ry = R sinϕ cos θ, and Rz = R sin θ, where R is the distance between both beads. The notation differs from conventional spherical coordinates, but has the advantage of giving perfect alignment with the outer mean flow direction for ϕ = θ = 0. ϕ is the azimuthal angle and θ the polar angle. While the azimuthal angle always remains in the shear plane that is spanned by the streamwise and shear directions, the polar angle θ 6= 0 indicates a dumbbell orientation out of this plane. Davoudi and Schumacher [28] discussed the statistics of both angles as a function of the Weissenberg number for passively advected Hookean dumbbells. The PDF of the polar angle was found to remain symmetric and to be less sensitive with respect to variations of Wi. Our focus will be therefore on the statistics of the azimuthal angle ϕ which can take values between −π/2 and π/2. The asymmetry between both quadrants is quantified by the following measure for the PDF p(ϕ): A(ϕ) = p(ϕ)− p(−ϕ) , (28) with ϕ ∈ [0, π/2]. The measure A(ϕ) is plotted for two Weissenberg numbers in Fig. 7. A pronounced maximum of A(ϕ) implies that the dumbbells are preferentially slightly tilted in the direction of shear, away from the mean flow direction (see an illustration in Fig. 8). We find that with increasing Weissenberg number the asymmetry of the angular distribution grows in magnitude. The same trend holds when the Stokes number grows at fixed Weissenberg number. In each case, the graph of A(ϕ) shows an increasingly sharper maximum, which is shifted towards smaller ϕ. Fluctuations of the dumbbells in the vicinity of ϕ = 0 are enhanced while the tails for very large ϕ are depleted. Growing inertia amplifies this trend. Once the dumbbells are aligned along the mean flow they remain in this orientation for longer periods of their evolution. B. Velocity gradient statistics Since the polymer dynamics takes place at the smallest scales of the turbulent flow, we study the impact of the dumbbells on the small-scale statistical properties of the flow in the following. Recent experimental and numerical studies in simple Newtonian shear flows indicate that in particular the statistics of the transverse derivative of the streamwise turbulent velocity component ∂u′x/∂y is a sensitive measure for detecting deviations from local isotropy in homogeneous or nearly homogeneous shear flows.[45, 46, 47] In a shear flow with a mean shear rate S > 0, one expects a positive value for derivative skewness and other higher odd order moments which are defined as M2n+1(∂u x/∂y) = 〈(∂u′x/∂y)2n+1〉 〈(∂u′x/∂y)2〉n+1/2 . (29) The derivative moments would be exactly zero in a perfectly isotropic flow. Their non-zero magnitudes indicate that velocity gradient fluctuations of the streamwise component along the direction of the outer shear gradient are more probable than the ones in the opposite direction. It can be expected that the asymmetry in the angular distribution, which we discussed above, will have an impact on the statistics of exactly these gradient fluctuations. Figure 9 reports our findings for the PDF of the transverse derivative, which has been normalized by its root mean square value for all cases. We observe in both figures a depletion of the left hand tail, which stands exactly for the velocity gradient fluctuations opposite to the direction of the mean shear. The results suggest that the preferential orientation fluctuations of the dumbbells at azimuthal angles ϕ > 0 go in line with a depletion of the negative tail of the PDF of the transverse derivative. As sketched in Fig. 8, negative transverse gradients would be amplified by prefential orientations with ϕ < 0 which correspond to the dumbbell colored in gray. The findings are consistent with our observations on the ϕ-statistics. They can also be rationalized (but not explained) when considering the equation for the Brownian dynamics of the FENE dumbbell [21] = R · ∇u− 2τ(1 −R2/L20) ξR . (30) In the plane shear flow geometry the component Rx along the mean flow direction is of particular interest. Since we are interested in stretched dumbbells with Rx > R0 and in Wi > 1 we neglect contributions from the spring force and the noise for a moment. With the Reynolds decomposition (7) we get Rx + ... , (31) Rx + ... (32) The important term is the first term on the r.h.s. of (31). The other three contributions will behave as noise terms. Fluctuating gradients ∂u′x/∂y along Sey lead to a more rapid growth of Rx (for an angle ϕ > 0) and a prefered alignment with the mean flow. This causes a more rapid decrease of Ry and consequently of Rx via (31). The dumbell can be kicked afterwards again to larger ϕ values and transfers momentum to the flow which corresponds exactly to a local patch of ∂u′x/∂y > 0 (see also Fig. (8)). Then Ry grows and this whole cycle starts anew. Small scale gradients with the opposite sign diminish the total shear in the surrounding of the dumbbell and cause a less efficient stretching and cycle. Clearly, this picture omits some important features such as the tumbling of the dumbbells. The depletion of gradient fluctuations goes in line with experimental observations by Liberzon et al. [48, 49] The authors found e.g. that the enstrophy production became anisotropic when polymers are added to the fluid. This quantity is directly related to transverse gradient components discussed here. C. Invariants of the velocity gradient tensor and dumbbell extension The efficient stretching of the dumbbells is connected to particular local flow topologies. They are related to the three eigenvalues λi of the velocity gradient tensor or the corresponding three velocity gradient tensor invariants, which are denoted as I1, I2, and I3. The eigenvalues of the velocity gradient tensor ∂u i/∂xj result as zeros of the following third-order characteristic polynomial[50] λ3 − I1λ2 + I2λ− I3 = 0 . (33) For an incompressible flow [53], I1 = λ1 + λ2 + λ3 = Tr = 0 , I2 = λ1λ2 + λ2λ3 + λ3λ1 = − I3 = λ1λ2λ3 = det . (34) The remaining coefficients of (33) are therefore I2 and I3, which span the I3 − I2 parameter plane. The scatter plots for turbulent flows result in a typical skewed teardrop shape. With our definitions given above the following crude classification scheme can be given. For I2 > 0, I3 > 0 vortex stretching is present corresponding to λ1 = a, λ2,3 = −a ± ib (first quadrant); for I2 > 0, I3 < 0 vortex compression is present corresponding to λ1 = −a, λ2,3 = a ± ib (second quadrant). The cases I3 < 0 are associated with bi-axial strain for I2 < 0 (third quadrant) corresponding to λ1 = a, λ2 = b, λ3 = −(a + b) and with uniaxial strain at I2 > 0 (fourth quadrant) corresponding to λ1 = a, λ2 = −b, λ3 = −(a−b). Constants a and b are larger than zero in all cases. Figure 10 relates the extension of the dumbbells to the corresponding local velocity gradients in the I3−I2 plane (and consequently to the existing local flow topology). The invariants of the velocity gradient were evaluated in the center of mass of each dumbbell. The typical teardrop shape for the turbulence data in the parameter plane is detected. Our findings can be summarized as follows. Strongly stretched dumbbells go in line with the largest excursions of the gradients in the I3 − I2 plane. The longest dumbbells are found preferentially in regions where vortex stretching or bi-axial strain dominate the local flow topology. The preferential stretching by bi-axial strain was discussed already for the passive advection of FENE dumbbells in a minimal flow unit.[25] It corresponds to the scenario that different parts of the dumbbell get pulled by counterstreaming streamwise streaks. The preferential extension close to vortex stretching means that the polymers are pulled around streamwise vortices. This point was outlined in Ref. [42] on the basis of an analysis of the energetics of viscoelastic turbulence. Here, we find both in a common description based on the analysis of the full velocity gradient tensor, i.e. the symmetric strain tensor plus the anti-symmetric vorticity tensor. We do also observe that the area of the teardrop shape shrinks with increasing Stokes number. This indicates that the small-scale velocity gradients are supressed in magnitude, which goes in line with more limited excursions across the I3 − I2 plane and a relaminarization of the turbulence. Again, this goes in line with very recent experimental observations by Liberzon et al.[49] V. SUMMARY AND DISCUSSION The presented numerical studies aimed at connecting a macroscopic description for the Newtonian turbulent shear flow to the mesoscopic description of an ensemble of FENE dumbbells which are advected in such flow. The momentum transfer of the dumbbells with the fluid is implemented by an additional volume forcing in the Navier-Stokes equations. In numerical terms, pseudospectral simulations for the solvent are coupled to a system of stochastic nonlinear ordinary equations in order to model a viscoelastic fluid. For the accessible parameters we found slight modifications of the macroscopic flow structures and mean statistical properties only. This was demonstrated for the global energy balance and the mean components of the Reynolds stress tensor. We conclude that dumbbell inertia effects are present, but remain subleading in comparison to the elastic properties. For the present viscoelastic flow a drag reduction of up to 20% is achieved. The microscopic properties of turbulence were found to be more sensitive with respect to the Weissenberg number. The statistics of the azimuthal angle ϕ is consistent with former findings for elastic Hookean dumbbells. [28] A growing number of dumbbells becomes increasingly aligned with the mean flow direction. The feedback of the FENE dumbbells on the small-scale properties of turbulence is demonstrated for two gradient measures, the PDF of the transverse derivative of the turbulent streamwise velocity component ∂u′x/∂y and the diminished scattering of the velocity gradient invariants ampiltudes in the I3 − I2 plane with increasing Wi. The asymmetry of the PDF p(∂u′x/∂y) is found to increase with increasing Wi. Furthermore, we determined that strongly stretched dumbbells can be found close to vortex stretching or biaxial strain topologies of the advecting shear flow. The present study should be considered as a first step for such class of hybrid models. One difference to the situation in a dilute polymer solution is the relatively large Stokes number that had to be taken. Our dispersed dumbbells behave in parts like deformable particles rather than polymer chains. Frequently, heavier quasi-particles are used for the study of turbulence in particle-ladden flows.[31] Extensions of our investigations will have to go into two directions. Firstly, it is desirable that larger spectral resolutions, like the ones in Ref. [28], are achieved. This will require a fully parallel implementation of the current numerical scheme. Larger computational grids and higher Reynolds numbers will give us the opportunity to decrease the ratio R0/ηK and to increase L0/R0 to more realistic values. Secondly, eq. (21) implies the efforts that have to be taken in order to approach the situation in a polymer solution. Decreasing values of R0 and St have to be compensated by np, e.g., a reduction of both – R0 and Stη – by an order of magnitude requires an increase of the concentration (or number density) by a power of 5/2. Once such operating point is reached, the time scale argument which is thought to be important for the drag reduction effect, can also be studied.[1] Finally, a recent work by Vincenzi and co-workers [51] provides an interesting ansatz for modelling the polymer dynamics. The authors studied a conformation-dependent Stokes drag coefficient that caused a significant dynamical slow-down of the coil-stretch transition in steady elongational and random flows. The test of these ideas in turbulent shear flows is still to be done. Acknowledgments This work was supported by the Deutsche Forschungsgemeinschaft (DFG) and the Deutscher Akademischer Aus- tauschdienst (DAAD) within the German-French PROCOPE program. We thank for computing ressources on the JUMP supercomputer at the John von Neumann Institute for Computing, Jülich (Germany). Further computations have been conducted at the MARC cluster (Marburg) and the MaPaCC cluster (Ilmenau). Fruitful discussions with F. de Lillo, B. Eckhardt, and D. Vincenzi are acknowledged. APPENDIX A: SEMI-IMPLICIT INTEGRATION SCHEME FOR DUMBBELLS The FENE dumbbells consist of two beads at positions x1(t) and x2(t) which are connected by a nonlinear elastic spring. The velocities of the advecting flow at both beads are denoted by u1 and u2, respectively. Note that these velocities coincide with ẋ1 and ẋ2, respectively, for St = 0 only. Since the beads are usually found between mesh vertices, the values for u1 and u2 have to be determined by trilinear interpolation from the known velocity vectors at the neighboring grid sites. The dynamical equations for the dumbbells are set up in relative and center-of-mass coordinates. The relative coordinate (or separation) vector of the dumbbell is given by R(t) = x2(t)− x1(t) . (A1) The center-of-mass coordinate vector is given by r(t) = (x1(t) + x2(t)) . (A2) The velocities which are assigned with the relative and center-of mass coordinates are denoted as V and v, respectively. The Newtonian equations for the dynamics of the FENE dumbbells in dimensionless form, which follow then from (9)-(12) with the definitions (3) and (2), are given by = ṽ , (A3) −ṽ + 1 (ũ1 + ũ2) + , (A4) = Ṽ , (A5) −Ṽ + (ũ2 − ũ1)− 1− R̃2L2/L20  . (A6) For the following, we omit the tilde symbol for the dimensionless quantities. The predictor values of the center-of-mass vector r and the distance vector R are calculated by an explicit Euler step whereas the corresponding velocities are treated by an implicit Euler step, giving r∗ = rl +∆tvl , (A7) St + ∆t St vl + (ul1 + u 2)∆t+ , (A8) R∗ = Rl +∆tV l , (A9) V ∗ = St + ∆t StV l + (ul2 − ul1)∆t− 2Wi (1− (Rl)2L2/L20) . (A10) The corrector step for the center-of-mass and distance vectors is given as rl+1 = rl + (v∗ + vl)∆t (A11) l+1 = St + ∆t Stv∗ + (u∗1 + u 2)∆t+ Stv (ul1 + u (A12) l+1 = Rl + (V l + V l+1)∆t (A13) V l+1 = St + ∆t StV ∗ + (u∗2 − u∗1)∆t+ StV l + (ul2 − ul1)∆t− 2Wi (1− (Rl+1)2L2/L20) ∆t− R 2Wi (1− (Rl)2L2/L20) ∆W l] . (A14) Note that the corrector step for the distance vector is semi-implicit in the velocity in order to avoid stiffness of the equation system at small Stokes numbers. 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Chevallard, and V. Steinberg, “Single polymer dynamics: coil-stretch transition in a random flow,” Europhys. Lett. 71 221 (2005). [44] M. Chertkov, I. Kolokolov, V. Lebedev, and K. Turitsyn, “Polymer statistics in a random flow with mean shear,” J. Fluid Mech. 531, 251 (2005). [45] Z. Warhaft, “Turbulence in nature and laboratory,” Proc. Nat. Acad. Sci. 99, 2481 (2002). [46] J. Schumacher, K. R. Sreenivasan, and P. K. Yeung, “Derivative moments in turbulent shear flows,” Phys. Fluids 15, 84 (2003). [47] L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414, 43 (2005). [48] A. Liberzon, M. Guala, B. Lüthi, W. Kinzelbach, and A. Tsinober, “Turbulence in dilute polymer solutions,” Phys. Fluids 17, 031707 (2005). [49] A. Liberzon, M. Guala, W. Kinzelbach, and A. Tsinober, “On the kinetic energy production and dissipation in dilute polymer solutions,” Phys. Fluids 18, 125101 (2006). [50] P. A. Davidson, Turbulence, Oxford University Press, Oxford, 2004. [51] A. Celani, A. Puliafito, and D. Vincenzi, “Dynamical slowdown of polymers in laminar and random flows,” Phys. Rev. Lett. 97, 118301 (2006). [52] T. Needham, Visual Complex Analysis, Oxford University Press, Oxford, 1997. [53] We will use the definitions as given in Ref. [50] (p. 264) with I1 = P , I2 = Q, and I3 = R PSfrag replacements Wi = 20 Wi = 20 Wi = 100 Wi = 100 Fel = 0 Fel = 0 FIG. 1: Mean dissipation and injection rates as a function of the Stokes and Weissenberg numbers. Upper picture: mean energy injection rate due to shear flow forcing 〈εin〉. Mid panel: mean energy dissipation rate 〈ε〉. Lower panel: mean dissipation rate which arises from the coupling to the dumbbell ensemble 〈εp〉. The Reynolds number is Re = 800. The case with Fel = 0 is for the case without spring force and stands for a shear flow with dispersed inertial particles. PSfrag replacements Wi = 20, St = 0.05 tethered 0 5 10 15 20 FIG. 2: Comparison of the kinetic energy for two cases at Wi = 20 and St = 0.05: for the tethered case one bead of each dumbbell is fixed at a grid site while the second bead can fluctuate. The freely draining case is the usual situation which allows the free motion of the dumbbells through the turbulent flow volume. The Reynolds number is Re = 400. PSfrag replacements St = 0.0005 St = 0.005 St = 0.05 St = 0.5 1.0 · 10−4 2.0 · 10−4 3.0 · 10−4 4.0 · 10−4 5.0 · 10−4 2.5 · 10−1 2.0 · 10−1 1.5 · 10−1 1.0 · 10−1 0.5 · 10−1 9.0 · 10−2 8.0 · 10−2 7.0 · 10−2 6.0 · 10−2 5.0 · 10−2 4.0 · 10−2 7.0 · 10−1 8.0 · 10−1 9.0 · 10−1 9.0 · 10−1 8.5 · 10−1 8.0 · 10−1 7.5 · 10−1 7.0 · 10−1 FIG. 3: Reynolds stresses 〈u′iu j〉 normalized by the turbulent kinetic energy k = 〈(u 2〉/2 as a function of the Weissenberg and Stokes numbers. From top to bottom: streamwise fluctuations, fluctuations in shear direction, spanwise fluctuations, and shear stress. The Reynolds number is Re = 800. 10−2 10−1 1.2 Wi = 20 Wi = 100 FIG. 4: Ratio of friction factors as a function of St for the largest Wi. The friction factor for the fluid with the dispersed dumbbells is cf (cf. Eq. (27)). The quantity c f is the friction factor of the Newtonian fluid. FIG. 5: Isosurface plot of the fluctuations of the streamwise turbulent velocity component u′x. The snapshots are for Re = 800 and St = 0.0005. The isolevels are for ±0.04 in each case. Wi = 3 St = 0.0005 St = 0.005 St = 0.05 St = 0.5 0.0 0.2 0.4 0.6 0.8 1.0 Wi = 20 St = 0.0005 St = 0.005 St = 0.05 St = 0.5 0.0 0.2 0.4 0.6 0.8 1.0 Wi = 100 St = 0.0005 St = 0.005 St = 0.05 St = 0.5 0.0 0.2 0.4 0.6 0.8 1.0 FIG. 6: Probability density function (PDF) of the extension R normalized by the contour length L0. Three different Weissenberg numbers are shown. The Stokes numbers of the data are indicated in the legend. Data are for Re = 800. PSfrag replacements Wi = 3 Wi = 100 −3π/8 St = 0.0005 St = 0.0005 St = 0.005 St = 0.005 St = 0.05 St = 0.05 St = 0.5 St = 0.5 FIG. 7: Asymmetry of the probability density function (PDF) of the azimuthal angle ϕ. It is defined as A(ϕ) = p(ϕ)− p(−ϕ). The upper panel shows the data for Wi = 3 and four different Stokes numbers. The lower panel shows the data for Wi = 100 and four different Stokes numbers. The analysis is for Re = 800. FIG. 8: Sketch of the orientation of a dumbbell in the turbulent shear flow. The mean turbulent flow profile is indicated. The dark-colored dumbbell stands for the preferentially oriented one while the gray-colored orientation is less probable. This orientation asymmetry leads to the asymmetry in the angular distribution as given in Fig. 7. 19PSfrag replacements x/(∂yu x)rms x/(∂yu x)rms Newtonian Newtonian Wi = 3 Wi = 20 Wi = 20 Wi = 100 Wi = 100 St = 0.0005 St = 0.5 FIG. 9: Probability density function (PDF) of the transverse velocity gradient of the streamwise turbulent fluctuations, ∂u′x/∂y. The Newtonian case is compared with the two larger values of the Weissenberg number at St = 5× 10−4 and 0.5, respectively. The data are for Re = 800. FIG. 10: Relation between the extension of the dumbbells and the local velocity gradient at the center of mass of the dumbbells. The local flow topology that is related to the velocity gradient is quantified by the second and third invariants I2 and I3 (see eqns. (34) for the definition). Quadrant I stands for vortex stretching, II for vortex compression, III for bi-axial strain, and IV for uniaxial strain, respectively. The gray color coding of the bins for 0 < R/L0 < 0.25, 0.25 ≤ R/L0 < 0.5, 0.5 ≤ R/L0 < 0.75, 0.75 ≤ R/L0 ≤ 1 is indicated by the legend for each figure. Data are for Re = 800. Introduction Model and equations The Newtonian solvent The FENE dumbbells Two-way coupling Large-scale properties Energy balance Reynolds stresses Small-scale properties Extensional and angular statistics of dumbbells Velocity gradient statistics Invariants of the velocity gradient tensor and dumbbell extension Summary and discussion Acknowledgments Semi-implicit integration scheme for dumbbells References

0704.1256

A Novel String Derived Z' With Stable Proton, Light-Neutrinos and R-parity violation

arXiv:0704.1256v2 [hep-ph] 10 Oct 2007 LTH–742 arXiv:????.???? A Novel String Derived Z ′ With Stable Proton, Light–Neutrinos and R–parity violation Claudio Corianò†1, Alon E. Faraggi♦2 and Marco Guzzi†3 †Dipartimento di Fisica, Universita’ di Lecce, I.N.F.N. Sezione di Lecce Via Arnesano, 73100 Lecce, Italy ♦Department of Mathematical Sciences University of Liverpool, Liverpool, L69 7ZL, United Kingdom Abstract The Standard Model indicates the realization of grand unified structures in nature, and can only be viewed as an effective theory below a higher energy cutoff. While the renormalizable Standard Model forbids proton decay medi- ating operators due to accidental global symmetries, many extensions of the Standard Model introduce such dimension four, five and six operators. Fur- thermore, quantum gravity effects are expected to induce proton instability, indicating that the higher energy cutoff scale must be above 1016 GeV. Quasi– realistic heterotic string models provide the arena to explore how perturbative quantum gravity affects the particle physics phenomenology. An appealing ex- planation for the proton longevity is provided by the existence of an Abelian gauge symmetry that suppresses the proton decay mediating operators. Addi- tionally, such a low–scale U(1) symmetry should: allow the suppression of the left–handed neutrino masses by a seesaw mechanism; allow fermion Yukawa couplings to the electroweak Higgs doublets; be anomaly free; be family uni- versal. These requirements render the existence of such U(1) symmetries in quasi–realistic heterotic string models highly non–trivial. We demonstrate the existence of a U(1) symmetry that satisfies all of the above requirements in a class of left–right symmetric heterotic string models in the free fermionic for- mulation. The existence of the extra Z ′ in the energy range accessible to future experiments is motivated by the requirement of adequate suppression of proton decay mediation. We further show that while the extra U(1) forbids dimension four baryon number violating operators it allows dimension four lepton number violating operators and R–parity violation. 1E-mail address: [email protected] 2E-mail address: [email protected] 3E-mail address: [email protected] http://arxiv.org/abs/0704.1256v2 1 Introduction The Standard Model of particle physics successfully accounts for all observations in the energy range accessible to contemporary experiments. Despite this enormous success the Standard Model can only be viewed as an effective low energy field theory below a higher energy cutoff. In the least, the existence of a Landau pole in the hypercharge sector, albeit at an enormously high scale, unequivocally demonstrates the formal inconsistency of the Standard Model. In this regard, the renormalizability of the Standard Model is an approximate feature and effects of nonrenormalizable operators, suppressed by powers of the high scale cutoff, must be considered. The high precision analysis of the Standard Model parameters, achieved at LEP and other particle physics experiments, indicates that the Standard Model remains an approximate renormalizable quantum field theory up to a very large energy scale. Possibly the grand unification scale, or the Planck scale. The logarithmic evolution of the Standard Model parameters is in agreement with the available data, and is compatible with the notion of unification at a high energy scale in the gauge and heavy matter sectors of the Standard Model. Preservation of the logarithmic evolution in the scalar sector necessitates the introduction of a new symmetry between bosons and fermions, dubbed supersymmetry. Perhaps the most important observation indicative that the Standard Model cutoff scale is a very high scale is the longevity of the proton. Renormalizability insures that baryon and lepton violating operators are absent in the perturbative Standard Model. Hence, in the renormalizable Standard Model baryon and lepton numbers are accidental global symmetries. However, at the cutoff scale dimension six operators are induced and the proton is in general expected to decay. The observed proton lifetime implies that the cutoff scale is of order 1016GeV. The problem is exacerbated in supersymmetric extensions of the Standard Model that allow dimension four and five baryon and lepton violating operators [1]. Indeed, one would expect proton decay mediating operators to arise in most extensions of the Standard Model. In the Minimal Supersymmetric Standard Model one imposes a global symmetry, R–parity, which forbids the dimension four baryon and lepton number violating operators. The difficulty with dimension five operators can only be circumvented if one further assumes that the relevant Yukawa couplings are suppressed. However, as global symmetries are not expected to survive quantum gravity effects [2], the proton lifetime problem becomes especially acute in the context of theories that unify the Standard Model with gravity. This question has been examined extensively in the context of quasi–realistic heterotic string models. In this context, the most appealing suggestion is that the suppression of the proton decay mediating operators is a result of a gauged U(1) symmetry, under which the undesired nonrenormalizable dimension four and five operators are not invariant. If the U(1) symmetry remains unbroken down to sufficiently low scales the problematic operators will be suppressed by at least the VEV that breaks the additional U(1) symmetry over the cutoff scale. The free fermionic heterotic string models are among the most realistic string models constructed to date [3, 4, 5, 6, 7, 8]. The issue of proton stability was sporad- ically explored in these models [9, 10, 11, 12, 13], as well as explorations of possible U(1) symmetries that can ensure proton longevity [9, 11, 12, 13]. However, non of the current proposals is satisfactory. The U(1) symmetry of ref. [9] is the U(1) combination of B − L and T3R which is embedded in SO(10) and is orthogonal to the electroweak hypercharge. However, this U(1) symmetry in general needs to be broken to allow for the suppression of the left–handed neutrino masses by a seesaw mechanism. Similarly, the U(1) symmetries studied in ref. [11, 12, 13], that arise in the string models from combinations of the U(1) symmetries that are external to SO(10) are flavour dependent U(1) symmetries that in general must be broken near the string scale to allow for generation of fermion masses. In ref. [12] it was concluded that non of the symmetries suggested in ref. [11] can remain unbroken down to low energies and provide for the suppression of the proton decay mediat- ing operators. Furthermore, a family non–universal U(1) symmetry is restricted by constraints on flavour changing neutral currents, and cannot exist in energy range accessible to forthcoming experiments. The proton longevity, together with the Standard Model multiplet structure, therefore provide the most important clues for the origin of the Standard Model particle spectrum. These favour the embedding of Standard Model in a Grand Uni- fied Theory, possibly broken to the Standard Model at the string level. The GUT embedding of the Standard Model, and its supersymmetric extension, leads to pro- ton decay mediating operators. The most robust and economical way to suppress the dangerous operators is by the existence of an additional Abelian gauge symmetry which is broken above the electroweak scale and does not interfere with the other phenomenological constraints. Such a U(1) symmetry should fulfill the following requirements: • Forbid dimension four, five and six proton decay mediating operators. • Allow suppression of left–handed neutrino masses by a seesaw mechanism. • Allow the fermion Yukawa couplings to electroweak Higgs doublets. • Be family universal. • Be anomaly free. This list of requirements render the existence of such a U(1) symmetry in string models highly nontrivial. For example, in models with an underlying SO(10) GUT embedding the U(1)B−L symmetry is gauged. It forbids the dimension four baryon and lepton number violating operators, but not the dimension five operator. Further- more, suppression of left–handed neutrino masses by a seesaw mechanism in general necessitates that the symmetry is broken near the GUT scale. Hence, it cannot remain unbroken down to low energies, and in general fast proton decay from dimen- sion four operators is expected to ensue. Similarly, the U(1)A symmetry external to SO(10) in E6 → SO(10) × U(1)A is anomalous in many of the quasi–realistic string models constructed to date [14] and is broken by a generalised Green–Schwarz mechanism. The additional U(1)s investigated in refs. [11, 12, 13] are either flavour non–universal or constrain the fermion Yukawa mass terms and must therefore be similarly broken near the Planck scale. Thus, of all the extra U(1)’s investigated to date non seems to remain viable down to low energies, and to provide the coveted proton protection symmetry. In this paper we therefore explore further the possibility that quasi–realistic string models give rise to Abelian gauged symmetries that can play the role of the proton lifetime guard. We demonstrate the existence of a U(1) symmetry satisfying all of the above requirements in the class of left–right symmetric string–derived models of ref. [7]. The key to obtaining the U(1) symmetry satisfying the above requirements is the SO(10) symmetry breaking pattern particular to the left–right symmetric models [7]. The key distinction is that in these models the U(1)A, which is external to the unbroken SO(10) subgroup, is anomaly free, and may remain unbroken down to low energies. It is does not restrict the charged fermion mass terms, and it allows for the suppression of the left–handed neutrino masses by a seesaw mechanism. Its existence at low energies is motivated by the longevity of the proton lifetime. Furthermore, as we discuss below, while it forbids the supersymmetric dimension four and five baryon number violating operators, it allows the dimension four lepton number violating operator. Hence, while proton decay from dimension four operators does not ensue, lepton number and R–parity violation do arise. This observation has far reaching implications in terms of the phenomenology and collider signatures of the models. 2 The structure of the free fermionic models In this section we describe the structure of the quasi–realistic free fermionic models and the properties of the proton protecting U(1) symmetry. The free fermionic for- mulation the 4-dimensional heterotic string, in the light-cone gauge, is described by 20 left–moving and 44 right–moving two dimensional real fermions [16]. The models are constructed by specifying the phases picked up by the world–sheet fermions when transported around the torus non-contractible loops. Each model corresponds to a particular choice of fermion phases consistent with modular invariance that can be generated by a set of basis vectors vi, i = 1, . . . , n, vi = {αi(f1), αi(f2), αi(f3)) . . .} . The basis vectors span a space Ξ which consists of 2N sectors that give rise to the string spectrum. The spectrum is truncated by a Generalised GSO (GGSO) projec- tions [16]. The U(1) charges, Q(f), with respect to the unbroken Cartan generators of the four dimensional gauge group, which are in one to one correspondence with the U(1) currents f ∗f for each complex fermion f , are given by: Q(f) = α(f) + F (f), (1) where α(f) is the boundary condition of the world–sheet fermion f in the sector α. F (f) is the fermion number operator counting each mode of f once (and if f is complex, f ∗ minus once). For periodic fermions, α(f) = 1, the vacuum is a spinor in order to represent the Clifford algebra of the corresponding zero modes. For each periodic complex fermion f there are two degenerate vacua |+〉, |−〉 , annihilated by the zero modes f0 and f0 ∗ and with fermion numbers F (f) = 0,−1, respectively. The two dimensional world–sheet fermions are divided in the following way: the eight left–moving real fermions ψ1,2 and χ1,···,6 correspond to the eight Ramond– Neveu–Schwarz fermions of the ten dimensional heterotic string in the light–cone gauge; the twenty–four real–fermions {yi, ωi|ȳi, ω̄i}, i = 1, . . . , 6 correspond to the fermionized internal coordinates of a compactified manifold in a bosonic formulation; the complex right–moving fermions φ̄1,···,8 generate the rank eight hidden gauge group; ψ̄1,···,5 generate the SO(10) gauge group; η̄1,2,3 generate the three remaining U(1) generators in the Cartan sub-algebra of the observable rank eight gauge group. A combination of these U(1) currents will play the role of the proton lifetime guard. The free fermionic models are defined in terms of the basis vectors and one– loop GGSO projection coefficients. The quasi–realistic free fermionic heterotic–string model are typically constructed in two stages. The first stage consists of the NAHE– set, {1, S, b1, b2, b3} [17, 18]. The gauge group at this stage is SO(10)×SO(6) 3×E8, and the vacuum contains forty–eight multiplets in the 16 chiral representation of SO(10). The second stage consists of adding three or four basis vectors to the NAHE– set, typically denoted by {α, β, γ}. The additional basis vectors reduce the number of generations to three, with one arising from each of the basis vectors b1, b2 and b3. Additional non–chiral generations may arise from the basis vectors that extend the NAHE–set. This distribution of the chiral generations is particular to the class of quasi–realistic free fermionic models that has been explored to date, and other possibilities may exist [15]. Additionally, the basis vectors that extend the NAHE– set break the four dimensional gauge group. The SO(10) symmetry is broken to one of the subgroups: SU(5)× U(1) [3]; SO(6)× SO(4) [5]; SU(3)× SU(2)× U(1)2 [6]; SU(3)×SU(2)2×U(1) [7]; or SU(4)×SU(2)×U(1) [8]. The three generations from the sectors b1, b2 and b3 are decomposed under the final SO(10) subgroup. The flavour SO(6)3 groups are broken to products of U(1)n with 3 ≤ n ≤ 9. The U(1)1,2,3 factors arise from the three right–moving complex fermions η̄1,2,3. Additional U(1) currents may arise from complexifications of right–moving fermions from the set {ȳ, ω̄}1,···,6. The U(1) symmetry that will serve as the proton lifetime guard is a combination of the three U(1) symmetries generated by the world–sheet complex fermions η̄1,2,3. The states from each of the sectors b1, b2 and b3 are charged with respect to one of these U(1) symmetries, i.e. with respect to U(1)1, U(1)2 and U(1)3, respectively. Hence the U(1) combination U(1)ζ = U1 + U2 + U3 (2) is family universal. In the string derived models of ref. [3, 4, 5, 6] U(1)1,2,3 are anomalous. Therefore, also U(1)ζ is anomalous and must be broken near the string scale. In the string derived left–right symmetric models of ref [7] U(1)1,2,3 are anomaly free, and hence also the combination U(1)ζ is anomaly free. It is this property of these models which allows this U(1) combination to remain unbroken. Subsequent to constructing the basis vectors and extracting the massless spectrum the analysis of the free fermionic models proceeds by calculating the superpotential. The cubic and higher-order terms in the superpotential are obtained by evaluating the correlators AN ∼ 〈V 3 · · ·VN〉, (3) where V i ) are the fermionic (scalar) components of the vertex operators, using the rules given in [19]. Generically, correlators of the form (3) are of order O(gN−2), and hence of progressively higher orders in the weak-coupling limit. Typically, one of the U(1) factors in the free-fermion models is anomalous, and generates a Fayet– Ilioupolos term which breaks supersymmetry at the Planck scale [20]. The anomalous U(1) is broken, and supersymmetry is restored, by a non–trivial VEV for some scalar field that is charged under the anomalous U(1). Since this field is in general also charged with respect to the other anomaly-free U(1) factors, some non-trivial set of other fields must also get non–vanishing VEVs V, in order to ensure that the vacuum is supersymmetric. Some of these fields will appear in the nonrenormalizable terms (3), leading to effective operators of lower dimension. Their coefficients contain fac- tors of order V/M∼ 1/10. Typically the solution of the D– and F–flatness constraints break most or all of the horizontal U(1) symmetries. 3 The proton lifeguard In this section we discuss the characteristics of U(1)ζ in the left–right symmetric string derived models [7], versus those of U(1)A in the string derived models of refs. [3, 4, 5, 6]. We note that both U(1)ζ as well as U(1)A are obtained from the same combination of complex right–moving world–sheet currents η̄1,2,3, i.e. both are given by a combination of U1, U2, and U3. The distinction between the two cases, as we describe in detail below, is due to the charges of the Standard Model states, arising from the sectors b1, b2 and b3, under this combination. The key feature of U(1)ζ in the models of ref. [7] is that it is anomaly free. To study the characteristics of the proton protecting U(1) symmetry it is instructive to examine in combinatorial notation the vacuum structure of the chiral generations from the sectors b1,2,3. The vacuum of the sectors bj contains twelve periodic fermions. Each periodic fermion gives rise to a two dimensional degenerate vacuum |+〉 and |−〉 with fermion numbers 0 and −1, respectively. The GSO operator, is a generalised parity operator, which selects states with definite parity. After applying the GSO projections, we can write the degenerate vacuum of the sector b1 in combinatorial form )] {( where 4 = {y3y4, y5y6, ȳ3ȳ4, ȳ5ȳ6}, 2 = {ψµ, χ12}, 5 = {ψ̄1,···,5} and 1 = {η̄1}. The combinatorial factor counts the number of |−〉 in the degenerate vacuum of a given state. The first term in square brackets counts the degeneracy of the multiplets, being eight in this case. The two terms in the curly brackets correspond to the two CPT conjugated components of a Weyl spinor. The first term among those corresponds to the 16 spinorial representation of SO(10), and fixes the space–time chirality properties of the representation, whereas the second corresponds to the CPT conjugated anti–spinorial 16 representation. Similar vacuum structure is obtained for b2 and b3. The periodic boundary conditions of the world–sheet fermions η̄ entails that the fermions from each sector bj are charged with respect to one of the U(1)j symmetries. The charges, however, depend on the SO(10) symmetry breaking pattern, induced by the basis vectors that extend the NAHE–set, and may, or may not, differ in sign between different components of a given generation. In the models of ref. [3, 6, 5] the charges of a given bj generation under U(1)j is of the same sign, whereas in the models of ref. [7] they differ. In general, the distinction is by the breaking of SO(10) to either SU(5) × U(1) or SO(6) × SO(4). In the former case they will always have the same sign, whereas in the later they may differ. This distinction fixes the charges of the Standard Model states under the U(1) symmetry which safeguards the proton from decaying, while not obstructing the remaining constraints listed above. In the free fermionic standard–like models the SO(10) symmetry is broken to4 SU(3)× SU(2)× U(1)C × U(1)L. The weak hypercharge is given by U(1)Y = U(1)C + U(1)L, (5) and the orthogonal U(1)Z′ combination is given by U(1)Z′ = U(1)C − U(1)L. (6) The three twisted sectors b1, b2 and b3 produce three generations in the sixteen representation of SO(10) decomposed under the final SO(10) subgroup. In terms of 4U(1)C = 3/2U(1)B−L ; U(1)L = 2U(1)T3 the SU(3)C × U(1)C × SU(2)L × U(1)L decomposition they take the values E ≡ [(1, 3/2); (1, 1)]; U ≡ [(3̄,−1/2); (1,−1)]; Q ≡ [(3, 1/2); (2, 0)]; N ≡ [(1, 3/2); (1,−1)]; D ≡ [(3̄,−1/2); (1, 1)]; L ≡ [(1,−3/2); (2, 0)]. (7) In terms of the SO(6)×SO(4) Pati–Salam decomposition [21] the Standard Model fermion fields are embedded in the FL ≡ (4, 2, 1) = Q + L ; FR ≡ (4̄, 1, 2) = U +D + E +N , (8) representations of SU(4) × SU(2)L × SU(2)R. In terms of the left–right symmetric decomposition of ref. [7] the embedding is in the following representations: QL = (3, 2, 1, ) , (9) QR = (3̄, 1, 2,− ) = U +D , (10) LL = (1, 2, 1,− ) , (11) LR = (1, 1, 2, ) = E +N , (12) of SU(3) × SU(2)L × SU(2)R × U(1)C . The Higgs fields in the later case are in a bi–doublet representation h = (1, 2, 2, 0) = hu+ h hu0 h . (13) Using the combinatorial notation introduced in eq. (4) the decomposition of the 16 representation of SO(10) in the Pati–Salam string models is )] [( )] [( } (14) The crucial point is that the Pati–Salam breaking pattern allows the first and second terms in curly brackets to come with opposite charges under U(1)j . This results from the operation of the GSO projection operator, which differentiates between the two terms. Thus, in models that descend from SO(10) via the SU(5) × U(1) breaking pattern the charges of a generation from a sector bj j = 1, 2, 3, under the corresponding symmetry U(1)j are either +1/2, or −1/2, for all the states from that sector. In contrast, in the left–right symmetric string models the corresponding charges, up to a sign are, Qj(QL;LL) = +1/2 ;Qj(QR;LR) = −1/2, (15) i.e. the charges of the SU(2)L doublets have the opposite sign from those of the SU(2)R doublets. This is in fact the reason that in the left–right symmetric string models [7] it was found that, in contrast to the case of the FSU5 [3], Pati–Salam [5] and standard–like [6], string models, the U(1)j symmetries are not part of the anomalous U(1) symmetry [7]. It is therefore noted that the U(1)ζ = U1 + U2 + U3 (16) combination is a family–universal, anomaly free5, U(1) symmetry, and allows the quark and lepton fermion mass terms QLQRh and LLLRh . (17) The two combinations of U(1)1, U(1)2 and U(1)3, that are orthogonal to U(1)ζ , are family non–universal and may be broken at, or slightly below, the string scale. The left–right symmetric heterotic string models of ref. [7] provide explicit quasi– realistic string models, that realize the charge assignment of eq. (15). Furthermore, the dimension four and five baryon number violating operators that arise from QLQLQLLL → QQQL (18) QRQRQRLR → {UDDN,UUDE} (19) are forbidden, while the lepton number violating operators that arise from QLQRLLLR → QDLN (20) LLLLLRLR → LLEN (21) are allowed. The crucial observation is the opposite charge assignment of the left and right– handed fields under U(1)ζ . This is available in models that descend from the Pati– Salam symmetry breaking pattern of the underlying SO(10) GUT symmetry. In this case the left– and right–moving fields carry opposite sign under the GSO projection operator, induced by the basis vector that breaks SO(10) → SO(6) × SO(4). An additional symmetry breaking stage of the Pati–Salam models [5], or left–right sym- metric models [7], can be obtained at the string level or in the effective low–energy 5We note that there may exist string models in the classes of [3, 5, 6] in which U(1)ζ is anomaly free. This may be the case in the so called self–dual vacua of ref. [15]. Such quasi–realistic string models with an anomaly free U(1)ζ have not been constructed to date. field theory by the Higgs fields in the representations {QH , Q̄H} = {(4̄, 1, 2), (4, 1, 2)} or {LH , L̄H} = {(1̄, 1, 2, ), (1, 1, 2,−3 )}. The breaking can be achieved at the string level, while preserving the desired charge assignment, as long as a basis vector of the form 2γ of refs. [6], or b6 of ref. [5], are not introduced. The boundary condi- tion assignments in these basis vectors entails that the N = 4 vacuum that we start with factorizes the gauge degrees of freedom into E8 ×E8 or SO(16)× SO(16). The consequence of this is that all the states from the twisted matter sectors bj carry the same charge under U(1)j . Thus, this result is circumvented by not including the vectors 2γ of [6], or b6 of [5] in the construction. In effect, such models are descending from a different N = 4 underlying vacuum [7, 8]. Being SO(16) × E7 × E7 in the models of ref. [7], which explicitly realize the desired breaking pattern in a class of quasi–realistic string models. We assume below that the SU(2)R symmetry is bro- ken directly at the string level in which case the remnant U(1)Z′ given in eq. (6) has to be broken by the Higgs fields {NH , N̄H} = (1, 1, 0, 5/2), (1, 1, 0,−5/2) under SU(3)× SU(2)× U(1)Y × U(1)Z′. 4 An effective string inspired Z ′ model Inspired by the U(1) charge assignment in the left–right symmetric string derived models [7], we present an effective field theory model incorporating these features. At this stage our aim is to build an effective model that can be used in correspondence with experimental data, rather than a complete effective field theory model below the string scale, which is of further interest and will be discussed in future publications. The charges of the fields in the low energy effective field theory of the string inspired model are given by Field U(1)Y U(1)Z′ U(1)ζ U(1)ζ′ Li −1 U i −2 Ei 1 1 N i 0 5 φi 0 0 0 0 φ0 0 0 0 0 −1 0 −1 HD −1 1 0 1 N̄H 0 − ζH 0 0 1 1 ζ̄H 0 0 −1 −1 with i = 1, 2, 3. The U(1)ζ′ symmetry is the combination of U(1)Z′ and U(1)ζ left unbroken by the vevs of NH and N̄H . The fields ζH and ζ̄H are needed to break the residual U(1)ζ′ symmetry. States with the required quantum numbers in (22) exist in the string models [7]. The fields φi are employed in an extended seesaw mechanism. Using the superpotential terms LiNjH U , NiN̄Hφj , φiφjφk . (23) The neutrino seesaw mass matrix takes the form ( νi Nk φm ) 0 (kM )ij 0 )ij 0 Mχ 0 Mχ O(Mφ) , (24) with Mχ ∼ 〈N̄H〉 and Mφ ∼ 〈φ0〉. The mass eigenstates are mainly νi, Nk and φm with a small mixing and with the eigenvalues mνj ∼Mφ kM ju mNj , mφ ∼Mχ . A detailed fit to the neutrino data was discussed in ref [22]. We emphasize, however, that our aim here is merely to demonstrate that the extra U(1)ζ′ , introduced below, is not in conflict with the requirement of light neutrino masses. Alternatively, the VEV of 〈N̄H〉 induces heavy Majorana mass terms for the right–handed neutrinos from nonrenormalizable terms NiNjN̄HN̄H . (25) The effective Majorana mass scale of the right–handed neutrinos is then Mχ ∼ 〈N̄H〉 2/M , which for 〈N̄H〉 ∼ 10 16GeV gives Mχ ∼ 10 14GeV. The VEV of 〈NH〉 may induce unsuppressed dimension four baryon and lepton number violating interactions η1QDL+ η2UDD (26) from the nonrenormalizable terms given in eqs. (19) and (20). Therefore, if the VEV of NH is of the order of the GUT, or intermediate, scale, as is required in the seesaw mass matrix in eq. (24), then unsuppressed proton decay will ensue. However, this VEV leaves the unbroken combination of U(1)Z′ and U(1)ζ given by U(1)ζ′ = U(1)Z′ − U(1)ζ . (27) The induced dimension four lepton number violating operator that arises from eq. (20) is invariant under U(1)ζ′ , whereas the induced dimension four baryon number violating operator that arises from eq. (19) is not. Hence, to generate an unsup- pressed dimension four baryon number violating operator we must break also U(1)ζ′. Therefore, if U(1)ζ′ remains unbroken down to low energies, it suppresses proton de- cay from dimension four operators. Similarly, the dimension five baryon and lepton number violating operators given in eqs. (18) and (19) are not invariant under U(1)ζ′ and hence suppressed if U(1)ζ′ remains unbroken down to low energies. 5 Estimate of the U(1)ζ ′ mass scale The dimension four and five proton decay mediating operators are forbidden by the U(1)Z′ and U(1)ζ gauge symmetries. These symmetries are broken by some fields and we can estimate the required symmetry breaking scale in order to ensure suffi- cient suppression. In turn this will indicate the possible mass scale of the additional Zζ′ vector boson, and whether it may exist in the range accessible to forthcoming experiments. The dimension four operators that give rise to rapid proton decay, η1UDD + η2QLD, are induced from the non–renormalizable terms of the form η1(UDDN)Φ + η2(QLDN)Φ ′ (28) where, Φ and Φ′ are combinations of fields that fix gauge invariance and the string selection rules. The field NH can be the Standard Model singlet in the 16 represen- tation of SO(10), or it can be a product of two fields, which effectively reproduces the SO(10) charges of NH [12]. We take the VEV of NH , which breaks the B − L symmetry, to be of the order of the GUT scale, i.e. 〈NH〉 ∼ 10 16GeV. This is the case as the VEV of N̄H induces the seesaw mechanism, which suppresses the left–handed neutrino masses. The VEVs of Φ and Φ′ then fixes the magnitude of the effective proton decay mediating operators, with η′1 ∼ ; η′2 ∼ . (29) We take M to be the heterotic string unification scale, M ∼ 1018GeV. Similarly, the dimension five proton decay mediating operator QQQL can effectively be induced from the nonrenormalizable terms λ1QQQL(Φ ′′) (30) The VEV of φ′′ then fixes the magnitude of the effective dimension five operator to λ′1 ∼ λ1 〈φ′′〉 The experimental limits impose that the product (η′1η 2) ≤ 10 −24 and (λ′1/M) ≤ 10 Hence, for M ∼ Mstring ∼ 10 18GeV we must have λ′1 ≤ 10 −7, to guarantee that the proton lifetime is within the experimental bounds. The induced dimension four lepton number violating operator is invariant under U(1)ζ′. Hence, we can take n ′ = 0. The dimension five baryon number violating operator is not invariant under U(1)ζ′ . Hence we must have at least n′′ = 1. We assume that the dimension four baryon number violating operator in eq. (26) is induced at the quintic order. The corresponding nonrenormalizable term in eq. (28) contain one additional field that breaks the proton protecting U(1)ζ′ at intermediate energy scale Λζ′. Hence, we have n = 1 in eq. (29), and (η′1η Taking 〈N〉 ∼ 1016GeV and M ∼ 1018GeV, we obtain the estimate Λζ′ ≤ 10 −2GeV, which is clearly too low. Taking 〈N〉 ∼ 1013GeV yields Λζ′ ≤ 10 4GeV. We also have that in this case λ′1/M < 10 −14. Hence, the baryon and lepton number violat- ing dimension five operator is adequately suppressed. On the other hand, we have η′2 ∼ 10 −5. This may be too small to produce sizable effects in forthcoming col- lider experiments, but may have interesting consequences for neutralino dark matter searches. 6 Conclusions The Standard Model gauge and matter spectrum clearly indicates the realization of grand unification structures in nature. Most appealing in this respect is the struc- ture of unification in the context of embedding the Standard Model chiral spectrum into spinorial representations of SO(10). In this case each Standard Model gener- ation together with the right–handed neutrino fits into a single SO(10) spinorial representation. While this can be a mirage, it is the strongest hint from the avail- able experimental data, accumulated over the past century. On the other hand, grand unified theories, and many other extensions of the renormalizable Standard Model, predict processes that lead to proton instability and decay. Proton longevity is therefore another key ingredient in trying to understand the fundamental origin of the Standard Model matter spectrum and interactions. A model that provides a robust explanation for these two key observations, while not interfering with other experimental and theoretical constraints, may indeed stand a good chance to pass further experimental scrutiny. String theory provides a viable framework for perturbative quantum gravity, while at the same time giving rise to the gauge and matter structures that describe the in- teractions of the Standard Model. In this respect string theory is unique and enables the development of a phenomenological approach to the unification of the gauge and gravitational interactions. Heterotic–string theory has the further distinction that by giving rise to spinorial representations in the massless spectrum it also enables the embedding of the Standard Model chiral spectrum in SO(10) spinorial representa- tions. The free fermionic models provide examples of quasi–realistic three generation heterotic–string models, in which the chiral spectrum arises from SO(10) spinorial representations. These models therefore admit the SO(10) embedding of the Stan- dard Model matter states. They satisfy the two pivotal criteria suggested by the Standard Model data. These models are related to Z2×Z2 orbifolds at special points in the moduli space. Other classes of quasi–realistic perturbative heterotic–string models have also been studied on unrelated compactifications and using different techniques [23]. Perhaps the most appealing explanation for the stability of the proton is the existence of additional gauge symmetries that forbid the proton decay mediating operators. However, such gauge symmetries should not interfere, or obstruct, the other phenomenological requirements that must be imposed on any extension of the Standard Model. Therefore, they should allow for generation of fermion masses and suppression of neutrino masses. They should be anomaly free. Gauge symmetries that may be observed in forthcoming collider experiments should also be family universal. In this paper we examined the question of such an additional U(1) gauge symme- try in the free fermionic models. While in most cases the additional gauge symmetries that arise in the string models do not satisfy the needed requirements, we demon- strated the existence of a U(1) symmetry in the class of models of ref. [7] that indeed does pass all the criteria. The existence of this U(1) symmetry at low energies is therefore motivated by the fact that it protects the proton from decaying, and it may indeed exist in the range accessible to forthcoming experiments. It is noted that although we investigated the additional U(1) in the context of the free fermionic string models, the properties of the U(1) symmetry, and the charges of the Standard Model state under it, rely solely on the weight charges of the string states under the rank 16 gauge symmetry of the ten dimensional theory. A U(1) symmetry with the properties that we extracted here may therefore arise in other classes of string compactifications. We emphasize that the characteristics of the extra U(1) that we extracted from a particular class of free fermionic models, do not depend on the specific string compactification. It ought to be further noted that compactifications that yielded the U(1) and the peculiar Standard Model charges under it, are not decedent from the E8 ×E8 heterotic string in 10 dimensions. This is because a U(1) symmetry which descends from the E8 × E8 (or SO(16) × SO(16)) will necessarily have an embedding in E6 and as we demonstrated here the Standard Model U(1) charges derived in this paper do not possess an E6 embedding, and do not descend from E8. The properties of this U(1) symmetry therefore differ from those that have been predominantly explored in the literature, which are inspired from compactifi- cations of the E8 × E8 heterotic string. The investigation of the phenomenological characteristics of this additional U(1) is therefore of further interest and we shall return to it in future publications. 7 Acknowledgments AEF would to thank the Oxford theory department for hospitality during the comple- tion of this work. 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0704.1257

Complexity of Janet basis of a D-module

Complexity of Janet basis of a D-module Alexander Chistov Steklov Institute of Mathematics, Fontanka 27, St. Petersburg 191023, Russia [email protected] Dima Grigoriev CNRS, IRMAR, Université de Rennes Beaulieu, 35042, Rennes, France [email protected] http://perso.univ-rennes1.fr/dmitry.grigoryev Abstract We prove a double-exponential upper bound on the degree and on the complexity of constructing a Janet basis of a D-module. This generalizes a well known bound on the complexity of a Gröbner basis of a module over the algebra of polynomials. We would like to emphasize that the obtained bound can not be immediately deduced from the commutative case. Introduction Let A be the Weyl algebra F [X1, . . . , Xn, , . . . , ∂ ] (or the algebra of differential operators F (X1, . . . , Xn)[ , . . . , ∂ ]). Denote for brevity Di = , 1 ≤ i ≤ n. Any A–module is called D–module. It is well known that an A–module which is a submodule of a free finitely generated A-module has a Janet basis. Historically, it was first introduced in [9]. In more recent times of developing computer algebra Janet bases were studied in [5], [13], [10]. Janet bases generalize Gröbner bases which were widely elaborated in the algebra of polynomials (see e. g.[3]). For Gröbner bases a double-exponential complexity bound was obtained in [12], [6] relying on [1] and which was made more precise (with a self–contained proof) in [4]. Surprisingly, no complexity bound on Janet bases was established so far; in the present paper we fill this gap and prove a double-exponential complexity bound. On the other hand, a double-exponential complexity lower bound on Gröbner bases [12], [14] provides by the same token a bound on Janet bases. There is a folklore opinion that the problem of constructing a Janet basis is easily reduced to the commutative case by considering the associated graded module, and, on the other hand, in the commutative case [6], [12], [4] the double– exponential upper bound is well known. But it turns out to be a fallacy! From a known system of generators of a D-module one can not obtain immediately any system of generators (even not necessarily a Gröbner basis) of the associ- ated graded module. The main problem here is to construct such a system of generators of the graded module. It may have the elements of degrees (dl)2 http://arxiv.org/abs/0704.1257v1 http://perso.univ-rennes1.fr/dmitry.grigoryev see the notation below. Then, indeed, to the last system of generators of big degrees one can apply the result known in the commutative case and get the bound ((dl)2 = (dl)2 . So new ideas specific to non–commutative case are needed. We are interested in the estimations for Janet bases of A-submodules of Al. The Janet basis depends on the choice of the linear order on the monomials (we define them also for l > 1). In this paper we consider the most general linear orders on the monomials from Al. They satisfy conditions (a) and (b) from Section 1 and are called admissible. We prove the following result. THEOREM 1 For any admissible linear order on the monomials from Al any A-submodule I of Al generated by elements of degrees at most d (with respect to the filtration in the corresponding algebra, see Section 1 and Section 9) has a Janet basis with the degrees and the number of its elements less than (dl)2 We prove in detail this theorem for the case of the Weyl algebra A. The proof for the case of the algebra of differential operators is similar. It is sketched in Section 9. ¿From Theorem 1 we get that the Hilbert function H(I,m), see Section 1, of the A-submodule from this theorem is stable for m ≥ (dl)2 and the absolute values of all coefficients of the Hilbert polynomial of I are bounded from above by (dl)2 , cf. e.g., [12]. This fact follows directly from (10), Lemma 12 from Appendix 1, Lemma 2 and Theorem 2. We mention that in [7] the similar bound was shown on the leading coefficient of the Hilbert polynomial. Now we outline the plan for the proof of Theorem 1. The main tool in the proof is a homogenized Weyl algebra hA (or respectively, a homogenized alge- bra of differential operators hB). It is introduced in Section 3 (respectively, Section 9). The algebra hA (respectively hB) is generated over the ground field F by X0, . . . , Xn, D1, . . . , Dn (respectively over the field F (X1, . . . , Xn) by X0, D1, . . . , Dn). Here X0 is a new homogenizing variable. In the algebra (respectively hB) relations (12) Section 3 (respectively (50) Section 9) hold for these generators in hA. We define the homogenization hI of the module I. It is a hA–submodule of hAl. The main problem is to estimate the degrees of a system of generators of hI. These estimations are central in the paper. They are deduced from Theorem 2 Section 7. This theorem is devoted to the problem of solving systems of linear equations over the ring hA; we discuss it below in more detail. The system of generators of hI gives a system of generators of the graded gr(A)–module gr(I) corresponding to I. But gr(A) is a polynomial ring. Hence using Lemma 12 Appendix 1 we get a double–exponential bound (dl)2 the stabilization of the Hilbert function of gr(I) and the absolute values of the coefficients of the Hilbert polynomial of gr(I). Therefore, the similar bound holds for the stabilization of the Hilbert functions of I and the coefficients of the Hilbert polynomial of I, see Section 2. But the Hilbert functions of the modules I and hI coincide, see Section 3. Hence the last bound holds also for the stabilization of the Hilbert functions of hI and the coefficients of the Hilbert polynomial of hI. In Section 5 we introduce the linear order on the monomials from hAl induced by the initial linear order on the monomials from Al (the homogenizing variable X0 is the least possible in this ordering). Further, we define the Janet basis of hI with respect to the induced linear order on the monomials. Such a basis can be obtained by the homogenization of the elements of a Janet basis of I with respect to the initial linear order, see Lemma 3. Let Hdt(hI) be the monomial module (i.e., the module which has a system of generators consisting of monomials) generated by the greatest monomials of all the elements of the module hI, see Section 4. Let cI, see Section 4, be the module over the polynomial ring cA = F [X0, . . . , Xn, D1, . . . , Dn] generated by all the monomials from Hdt(hI) (they are considered now as elements of cA). Then the Hilbert functions of the modules hI and cI coincide. Thus, we have the same as above double–exponential estimation for the stabilization of the Hilbert functions of cI and the coefficients of the Hilbert polynomial of cI. Now using Lemma 13 we get the estimation (dl)2 on the monomial system of generators of cI, hence also of Hdt(hI). This gives the bound for the degrees of the elements of the Janet bases of hI and hence also for the required Janet basis of I, and proves Theorem 1. The problem of solving systems of linear equations over the homogenized algebra is central in this paper, see Theorem 2. It is studied in Sections 5– 7. A similar problem over the Weyl algebra (without a homogenization) was considered in [7]. The principal idea is to try to extend the well known method due to G.Hermann [8] which was elaborated for the algebra of polynomials, to the homogenized Weyl algebra. There are two principal difficulties on this way. The first one is that in the method of G.Hermann the use of determinants is essential which one has to avoid dealing with non-commutative algebras. The second is that one needs a kind of the Noether normalization theorem in the situation under consideration. So it is necessary to choose the leading elements in the analog of the G.Hermann method with the least ordX0 , where X0 is a homogenizing variable, see Section 3. The obtained bound on the degree of a Janet basis implies a similar bound on the complexity of its constructing. Indeed, by Corollary 1 (it is formulated for the case of Weyl algebra but the analogous corollary holds for the case of algebra of differential operators) one can compute the linear space of all the elements z ∈ I of degrees bounded from above by (dl)2 . Further, by Theorem 1 the module Hdt(I), see Section 1, is generated by all the elements Hdt(z) with z ∈ I of degrees bounded from above by (dl)2 . Hence one can compute a system of generators of Hdt(I) and a Janet basis of I solving linear systems over F of size bounded from above (dl)2 (just by the enumeration of all monomials of degrees at most (dl)2 which are possible generators of Hdt(I)). If one needs to construct the reduced Janet basis it is sufficient to apply additionally Remark 1 Section 4. For the sake of self–containedness in Appendix 1, see Lemma 12, we give a short proof of the double–exponential estimation for stabilization of the Hilbert function of a graded module over a homogeneous polynomial ring. A conversion of Lemma 12 also holds, see Appendix 1 Lemma 13. It is essential for us. The proof of Lemma 13 uses the classic description of the Hilbert function of a homogeneous ideal in F [X0, . . . , Xn] via Macaulay constants bn+2, . . . , b1 and the constant b0 introduced in [4]. In Appendix 2 we give an independent and instructive proof of Proposition 1 which is similar to Lemma 13. In some sence Proposition 1 is even more strong than Lemma 13 since to apply it one does not need a bound for the stabilization of the Hilbert function. Of course, the reference to Proposition 1 can be used in place of Lemma 13 in our paper. 1 Definition of the Janet basis Let A = F [X1, . . . , Xn, D1, . . . , Dn], n ≥ 1, be a Weyl algebra over a field F of zero–characteristic. So A is defined by the following relations XvXw = XwXv, DvDw = DwDv, DvXv−XvDv = 1, XvDw = DwXv, v 6= w. By (1) any element f ∈ A can be uniquely represented in the form i1,...,in,j1,...,jn≥0 fi1,...,in,j1,...,jnX 1 . . .X 1 . . . D n , (2) where all fi1,...,in,j1,...,jn ∈ F and only a finite number of fi1,...,in,j1,...,jn are nonzero. Denote for brevity Z+ = {z ∈ Z : z ≥ 0} to the set of all nonnegative integers and i = (i1, . . . , in), j = (j1, . . . , jn), fi,j = fi1,...,in,j1,...,jn X i = X i11 . . . X n , D j = D 1 . . . D n , f = fi,jX iDj , |i| = i1 + . . .+ in, i+ j = (i1 + j1, . . . , in + jn). So i, j ∈ Zn+ are multiindices. By definition the degree of f deg f = degX1,...,Xn,D1,...,Dn f = max{|i|+ |j| : fi,j 6= 0}. LetM be a left A-module given by its generatorsm1, . . . ,ml, l ≥ 0, and relations 1≤w≤l av,wmw, 1 ≤ v ≤ k. (4) where k ≥ 0 and all av,w ∈ A. We assume that deg av,w ≤ d for all v, w. By (4) we have the exact sequence → M → 0 (5) of left A-modules. Denote I = ι(Ak) ⊂ Al. If l = 1 then I is a left ideal of A and M = A/I. In the general case I is generated by the elements (av,1, . . . , av,l) ∈ A l, 1 ≤ v ≤ k. For an integer m ≥ 0 put Am = {a : deg a ≤ m}, Mm = π(A m), Im = I ∩ A m. (6) So now A, M , I are filtered modules with filtrations Am, Mm, Im, m ≥ 0, respectively and the sequence of homomorphisms of vector spaces 0 → Im → A m → Mm → 0 induced by (5) is exact for every m ≥ 0. The Hilbert function H(M,m) of the module M is defined by the equality H(M,m) = dimF Mm, m ≥ 0. Each element ofAl can be uniquely represented as an F -linear combination of elements ev,i,j = (0, . . . , 0, X iDj , 0, . . . , 0), herewith i, j ∈ Zn+ are multiindices, see (3), and the nonzero monomial X iDj is at the position v, 1 ≤ v ≤ l. So every element f ∈ Al can be represented in the form v,i,j fv,i,jev,i,j , fv,i,j ∈ F. (7) The elements ev,i,j will be called monomials. Consider a linear order < on the set of all the monomials ev,i,j or which is the same on the set of triples (v, i, j), 1 ≤ v ≤ l, i, j ∈ Zn+. If f 6= 0 put o(f) = max{(v, i, j) : fv,i,j 6= 0}, (8) see (7). Set o(0) = −∞ < o(f) for every 0 6= f ∈ A. Let us define the leading monomial of the element 0 6= f ∈ Al by the formula Hdt(f) = fv,i,jev,i,j , where o(f) = (v, i, j). Put Hdt(0) = 0. Hence o(f−Hdt(f)) < o(f) if f 6= 0. For f1, f2 ∈ A l if o(f1) < o(f2) we shall write f1 < f2. We shall require additionally (a) for all multiindices i, j, i′, j′ for all 1 ≤ v ≤ l if i1 ≤ i 1, . . . , in ≤ i n and j1 ≤ j 1, . . . , jn ≤ j n then (v, i, j) ≤ (v, i ′, j′). (b) for all multiindices i, j, i′, j′, i′′, j′′ for all 1 ≤ v, v′ ≤ l if (v, i, j) < (v′, i′, j′) then (v, i+ i′′, j + j′′) < (v′, i′ + i′′, j′ + j′′). Conditions (a) and (b) imply that for all f1, f2 ∈ A l for every nonzero a ∈ A if f1 < f2 then af1 < af2, i.e., the considered linear order is compatible with the products. Any linear order on monomials ev,i,j satisfying (a) and (b) will be called admissible. Hdt(I) = AHdt(f). So Hdt(I) is an ideal of A. By definition the family f1, . . . , fm of elements of I is a Janet basis of the module I if and only if 1) Hdt(I) = AHdt(f1)+ . . .+AHdt(fm), i.e., the submodule of A l generated by Hdt(f1), . . . ,Hdt(fm) coincides with Hdt(I). Further, the Janet basis f1, . . . , fm of I is reduced if and only if the following conditions hold. 2) f1, . . . , fm does not contain a smaller Janet basis of I, 3) Hdt(f1) > . . . > Hdt(fm). 4) the coefficient from F of every monomial Hdt(fv), 1 ≤ v ≤ l, is 1. 5) Let fα = v,i,j fα,v,i,jev,i,j be representation (2) for fα, 1 ≤ α ≤ m. Then for all 1 ≤ α < β ≤ m for all 1 ≤ v ≤ l and multiindices i, j the monomial fα,v,i,jev,i,j 6∈ Hdt(Afβ \ {0}). Since the ringA is Noetherian for considered I there exists a Janet basis. Further the reduced Janet basis of I is uniquely defined. 2 The graded module corresponding to a D–mo- Put Av = Iv = Mv = 0 for v < 0 and gr(A) = ⊕m≥0Am/Am−1, gr(I) = ⊕m≥0Im/Im−1, gr(M) = ⊕m≥0Mm/Mm−1. The structure of the algebra on A induces the structure of a graded algeb- ra on gr(A). So we have gr(A) = F [X1, . . . , Xn, D1, . . . , Dn] is an algebra of polynomials with respect to the variables X1, . . . , Xn, D1, . . . , Dn. Further, gr(I) and gr(M) are graded gr(A)-modules. From (6) we get the exact sequences 0 → Im/Im−1 → (Am/Am−1) l → Mm/Mm−1 → 0, m ≥ 0. (9) The Hilbert function of the module gr(M) is defined as follows H(gr(M),m) = dimF Mm/Mm−1, m ≥ 0. Obviously H(M,m) = 0≤v≤m H(gr(M), v), H(gr(M),m) = H(M,m)−H(M,m− 1). for every m ≥ 0. Denote for an arbitrary a ∈ M by gr(a) ∈ gr(M) the image of a in gr(M). LEMMA 1 Assume that b1, . . . , bs is a system of generators of I. Let νi = deg bi, 1 ≤ i ≤ s. Suppose that for every m ≥ 0 1≤v≤µ cvbv : cv ∈ A, deg cv ≤ m− νv, 1 ≤ i ≤ s . (11) Then gr(b1), . . . , gr(bs) is a system of generators of the gr(A)-module gr(I). PROOF This is straightforward. So it is sufficient to construct a system of generators b1, . . . , bs of I satisfying (11). 3 Homogenization of the Weyl algebra Let X0 be a new variable. Consider the algebra hA = F [X0, X1, . . . , Xn, D1, . . . , Dn] given by the relations XvXw = XwXv, DvDw = DwDv, for all v, w, DvXv −XvDv = X 0 , 1 ≤ v ≤ n, XvDw = DwXv for all v 6= w. The algebra hA is Noetherian similarly to the Weyl algebra A. By (12) an element f ∈ hA can be uniquely represented in the form i0,i1,...,in,j1,...,jn≥0 fi0,...,in,j1,...,jnX 0 . . . X 1 . . .D n , (13) where all fi0,...,in,j1,...,jn ∈ F and only a finite number of fi0,...,in,j1,...,jn are nonzero. Let i, j be multiindices, see (3). Denote for brevity i = (i1, . . . , in), j = (j1, . . . , jn), fi0,i,j = fi0,...,in,j1,...,jn i0,i,j fi0,i,jX iDj . By definition the degrees of f deg f = degX0,...,Xn,D1,...,Dn f = max{i0 + |i|+ |j| : fi0,i,j 6= 0}, degD1,...,Dn f = max{|j| : fi0,i,j 6= 0}, degDα f = max{jα : fi0,i,j 6= 0}, 1 ≤ α ≤ n degXα f = max{iα : fi0,i,j 6= 0}, 1 ≤ α ≤ n Set ord 0 = ordX0 0 = +∞. If 0 6= f ∈ hA then put ord f = ordX0 f = µ if and only if f ∈ X hA) \X hA), µ ≥ 0. (15) For every z = (z1, . . . , zl) ∈ hAl put ord z = min 1≤i≤l {ord zi}, deg z = max 1≤i≤l {deg zi}. Similarly one defines ord b and deg b for an arbitrary (k × l)–matrix b with coefficients from hA. More precisely, one consider here b as a vector with kl entries. The element f ∈ hA is homogeneous if and only if fi0,i,j 6= 0 implies i0+ |i|+ |j| = deg f , i.e., if and only if f is a sum of monomials of the same degree deg f . The homogeneous degree of a nonzero homogeneous element f is its degree. The homogeneous degree of 0 is not defined (0 belongs to all the homogeneous components of hA, see below). The m-th homogeneous component of hA is the F -linear space (hA)m = z ∈ hA : z is homogeneous & deg z = m or z = 0 for every integer m. Now hA is a graded ring with respect to the homogeneous degree. By definition the ring hA is a homogenization of the Weyl algebra A. We shall consider the category of finitely generated graded modules G over the ring hA. Such a module G = ⊕m≥m0Gm is a direct sum of its homogeneous components Gm, where m,m0. are integers. Every Gm is a finite dimensional F -linear space and (hA)pGm ⊂ Gp+m for all integers p,m. If G and G ′ are two finitely generated graded hA-modules then ϕ : G → G′ is a morphism (of degree 0) of the graded modules if and only if ϕ is a morphism of hA-modules and ϕ(Gm) ⊂ G m for every integer m. The element z ∈ hA (respectively z ∈ A) is called to be the term if and only if z = λz1 · . . . · zν for some 0 6= λ ∈ F , integer ν ≥ 0 and zw ∈ {X0, . . . , Xn, D1, . . . , Dn} (respectively zw ∈ {X1, . . . , Xn, D1, . . . , Dn}), 1 ≤ w ≤ ν. Let z = zj ∈ A be an arbitrary element of the Weyl algebraA represented as a sum of terms zj and deg z = maxj deg zj. One can take here, for example, representation (3) for z. Then we define the homogenization hz ∈ hA by the formula deg z−deg zj By (1), (12) the right part of the last equality does not depend on the chosen representation of z as a sum of terms. Hence hz is defined correctly. If z ∈ hA then az ∈ A is obtained by substituting X0 = 1 in z. Hence for every z ∈ A we have ahz = z, and for every z ∈ hA the element z = hazX 0 , where µ = ord z. For an element z = (z1, . . . , zl) ∈ A l put deg z = max1≤i≤l{deg zi} and deg z−deg z1 0 , . . . , deg z−deg zl ∈ hAl. Similarly one defines deg a and the homogenization ha = (av,w)1≤v≤k, 1≤w≤l for an arbitrary k×l–matrix a with coefficients from A. More precisely, one consider here a as a vector with kl entries. Hence if b = (bv,w)1≤v≤k, 1≤w≤l = ha then bv,w = hav,wX deg a−deg av,w 0 for all v, w. The m-th homogeneous component of hAl is (hAl)m = hz : z ∈ Al & deg z = m or z = 0 For an F -linear subspace X ⊂ Al put hX to be the least linear subspace of hAl containing the set {hz : z ∈ X}. If X is a (finitely generated) A-submodule of Al then hX is a (finitely generated) graded submodule of hAl. The graduation on hX is induced by the one of hAl. For an element z = (z1, . . . , zl) ∈ hAl put az = (az1, . . . , azl) ∈ A l. For a subset X ⊂ hAl put aX = {az : z ∈ X} ⊂ Al. If X is a F -linear space then aX is also a F -linear space. If X is a finitely generated graded submodule of hAl then aX is finitely generated submodule of Al. Now hI is a graded submodule of hAl. Further, ahI = I. Let (hI)m be the m-th homogeneous component of hI. Then h(Im) = ⊕0≤j≤m( hI)j , m ≥ 0, (16) a((hI)m) = Im, m ≥ 0. (17) and (17) induces the isomorphism ι : (hI)m → Im. Set hM = hAl/hI. Hence hM is a graded hA-module and we have the exact sequence 0 → hI → hAl → hM → 0. (18) The m-th homogeneous component (hM)m of (hM)m = ( hAl)m/( hI)m ≃ A m/Im. (19) by the isomorphism ι. We have the exact sequences 0 → (hI)m → ( hAl)m → ( hM)m → 0, m ≥ 0. (20) By definition the Hilbert function of the module hM is H(hM,m) = dimF ( hM)m, m ≥ 0. By (19) we have H(M,m) = H(hM,m) for every m ≥ 0, i.e., the Hilbert functions of M and hM coincide. LEMMA 2 Let b1, . . . , bs be a system of homogeneous generators of the module hI. Then gr(ab1), . . . , gr( abs) ∈ gr(A) is a system of generators of gr(A)-module gr(I). PROOF By (17) a((hI)m) = Im. Now the required assertion follows from Lemma 1. The lemma is proved. 4 The Janet bases of a module and of its ho- mogenization Each element of hAl can be uniquely represented as an F -linear combination of elements ev,i0,i,j = (0, . . . , 0, X iDj , 0, . . . , 0), herewith 0 ≤ i0 ∈ Z, i, j ∈ Z are multiindices, see (3), and the nonzero monomial X i00 X iDj is at the position v, 1 ≤ v ≤ l. So every element f ∈ hAl can be represented in the form v,i0,i,j fv,i0,i,jev,i0,i,j , fv,i0,i,j ∈ F. (21) and only a finite number of fv,i0,i,j are nonzero. The elements ev,i0,i,j will be called monomials. Let us replace everywhere in Section 1 after the definition of the Hilbert function the ring A, the monomials ev,i,j , the multiindices i, i ′, i′′, triples (v, i, j), (v, i′, j′), the module I and so on by the ring hA, monomials ev,i0,i,j , the pairs (i0, i), (i ′), (i′′0 , i ′′) (they are used without parentheses), quadruples (v, i0, i, j), (v, i ′, j′), the homogenization hI and so on respectively. Thus, we get the definitions of o(f), Hdt(f) for f ∈ hAl, new conditions (a) and (b) which define admissible linear order on the monomials of hAl, new conditions 1)–5), the definitions of the Janet basis and reduced Janet basis of hI. For example, the new conditions (a) and (b) are (a) for all indices i0, i 0, all multiindices i, j, i ′, j′ for all 1 ≤ v ≤ l if i0 ≤ i i1 ≤ i 1, . . . , in ≤ i n and j1 ≤ j 1, . . . , jn ≤ j n then (v, i0, i, j) ≤ (v, i ′, j′). (b) for all indices i0, i 0 , all multiindices i, j, i ′, j′, i′′, j′′ for all 1 ≤ v, v′ ≤ l if (v, i0, i, j) < (v ′, i′0, i ′, j′) then (v, i0+ i 0 , i+ i ′′, j+ j′′) < (v′, i′0+ i 0 , i i′′, j′ + j′′). The Janet basis of hI is homogeneous if and only if it consists of homogeneous elements from hAl. Let< be an admissible linear order on the monomials fromAl, or which is the same, on the triples (v, i, j), see Section 1. So < satisfies conditions (a) and (b). Let us define the linear order on the monomials ev,i0,i,j or, which is the same, on the quadruples (v, i0, i, j). This linear order is induced by < on the triples (v, i, j) and will be denoted again by <. Namely, for two quadruples (v, i0, i, j) and (v′, i′0, i ′, j′) put (v, i0, i, j) < (v ′, i′0, i ′, j′) if and only if (v, i, j) < (v′, i′, j′), or (v, i, j) = (v′, i′, j′) but i0 < i 0. Notice that this induced linear order satisfies conditions (a) and (b) (in the new sense). REMARK 1 If f1, . . . , fm is a Janet basis of I (respectively homogeneous Janet basis of hI) satisfying 1)–4) then there are the unique cα,β ∈ A (respec- tively cα,β ∈ hA), 1 ≤ α < β ≤ m, such that α<β≤m cα,βfβ , 1 ≤ α ≤ m, is a reduced Janet basis of I (respectively reduced homogeneous Janet basis of hI), cf. [3]. LEMMA 3 Let f1, . . . , fm be a (reduced) Janet basis of I with respect to the linear order <. Then hf1, . . . , hfm is a (reduced) homogeneous Janet basis of the module hI with respect to the induced linear order <. Conversely, let g1, . . . , gm be a (reduced) homogeneous Janet basis of the module hI with respect to the induced linear order <. Then ag1, . . . , agm is a (reduced) Janet basis of I with respect to the linear order <. PROOF This follows immediately from the definitions. Let f ∈ hAl and the module hI be as above. Then there is the unique element g ∈ hAl such that v,i0,i,j gv,i0,i,jev,i0,i,j , gv,i0,i,j ∈ F, f −g ∈ hI and if gv,i0,i,j 6= 0 then ev,i0,i,j 6∈ Hdt( hI). The element g is called the normal form of f with respect to the module hI. We shall denote g = nf(hI, f). Obviously nf(hI, (hAl)m) ⊂ ( hAl)m is a linear subspace. Let cA = F [X0, . . . , Xn, D1, . . . , Dn] be the polynomial ring in the variables X0, . . . , Xn, D1, . . . , Dn. Each monomial ev,i0,i,j can be considered also as an element of cAl. Denote by cI ⊂ cAl the graded submodule of cAl generated by all the monomials ev,i0,i,j such that there is 0 6= f ∈ hI with o(f) = (v, i0, i, j). The Hilbert functions H(cI,m) = dimF {(z1, . . . , zl) ∈ cI : ∀ i ( deg zi = m or zi = 0 )}, H(cAl/cI,m) = m+ 2n −H(cI,m). Let us replace in the definition of the normal form above hA, hI by cA, cI respec- tively. Thus, for f ∈ cAl we get the definition of the normal form nf(cI, f) ∈ cAl, cf. [4]. Obviously, nf(cI, (cAl)m) ⊂ ( cAl)m is a linear subspace. Since the ideals and Hdt(hI) are generated by the same monomials we have dimnf(cI, (cAl)m) = dimnf(hI, (hAl)m). Hence the Hilbert functions H(hAl/hI,m) = H(cAl/cI,m), H(hI,m) = H(cI,m), m ≥ 0, coincide. Therefore, see Section 3, H(I,m) = H(cI,m), m ≥ 0 (22) 5 Bound on the kernel of a matrix over the ho- mogenized Weyl algebra LEMMA 4 Let k = l− 1 and l ≥ 1 be integers. Let b = (bi,j)1≤i≤k, 1≤j≤l be a matrix where bi,j ∈ hA are homogeneous elements for all i, j. Let deg bi,j < d, d ≥ 1, for all i, j. Assume that there are integers dj ≥ 0, 1 ≤ i ≤ k, and d i ≥ 0, 1 ≤ j ≤ l, such that deg bi,j = di − d j (23) for all nonzero bi,j, and additionally min1≤j≤l{d j} = 0 (hence di < d, d j < d for all i, j), d ≥ 1. Then there are homogeneous elements z1, . . . , zl ∈ hA such that (z1, . . . , zl) 6= (0, . . . , 0), 1≤j≤l bi,jzj = 0, 1 ≤ i ≤ l − 1, (24) all nonzero bi,jzj have the same degree depending only on i and deg zj ≤ (2n+ 3)ld, 1 ≤ j ≤ l. (25) Besides that, if all bi,j do not depend on Xn (i.e., they can be represented as sums of monomials which do not contain Xn) then one can choose also z1, . . . , zl satisfying additionally the same property. Finally, dividing by an appropriate power of X0 one can assume without loss of generality that min{ord zi : 1 ≤ i ≤ l} = 0. PROOF We shall assume without loss of generality that l ≥ 2. At first suppose that that deg bi,j = deg b for all nonzero bi,j . Consider the linear mapping (hA)lm−deg b −→ ( hA)l−1m , ( z1, . . . , zl ) 7→ 1≤j≤l bi,jzj 1≤i≤l−1 m− deg b+ 2n > (l − 1) m+ 2n then the kernel of (26) is nonzero. But (27) holds if deg b m+ 2n− deg b deg b m+ 2n− 1− deg b . . . deg b m− deg b l − 1 Further, (28) is true if (1+ deg b/(m− deg b))2n < l/(l− 1). The last inequality follows from m ≥ (2n + 1) deg b/ log(l/(l − 1)). Hence also from m ≥ (2n + 1)l deg b. Notice that (2n + 2)ld ≥ 1 + (2n + 1)l deg b. Thus, the existence of z1, . . . , zl is proved, and even more all nonzero bi,jzj have the same degree which does not depend on i, j. Notice that in the considered case we prove a more strong inequality deg zj ≤ (2n+ 2)ld for all 1 ≤ j ≤ l. Suppose that a1, . . . , al do not depend on Xn. We represent zi = zi,jX 1 ≤ i ≤ l, where all zi,j do not on Xn. Let α = maxi{degXn zi}. Obviously in this case one can replace (z1, . . . , zl) by (z1,α, . . . , zl,α). Let us return to general case of arbitrary deg bi,j . We shall reduce it to the considered one. Namely, multiplying the i-th equation of system (24) to maxi{di}−di 0 we shall suppose without loss of generality that all di are equal. Let us substitute zjX 0 for zj in (24). Now the degrees of all the nonzero coefficients of the obtained system coincide. Thus, we get the required reduction and estimation (25). The lemma is proved. REMARK 2 Lemma 4 remains true if one replaces in its statement condition (24) by ∑ 1≤j≤l zjbi,j = 0, 1 ≤ i ≤ l − 1, (29) The proof is similar. REMARK 3 Let the elements bi,j be from Lemma 4. Notice that there are integers δ′i ≥ 0, 1 ≤ i ≤ k, and δj ≥ 0, 1 ≤ j ≤ l, such that deg bi,j = δj − δ for all nonzero bi,j, and min1≤i≤k{δ i} = 0. Namely, δ i = −di +max1≤i≤k{di}, δj = −d j +max1≤i≤k{di}. 6 Transforming a matrix with coefficients from A to the trapezoidal form Let b be the matrix from Lemma 4 but now k, l are arbitrary. Hence (23) holds. Let b = (b1, . . . , bl) where b1, . . . , bl ∈ hAk be the columns of the matrix b (notice that in Lemma 1 and Lemma 2 bi are rows of size l; so now we change the notation). By definition b1, . . . , bl are linearly independent over hA from the right (or just linearly independent if it will not lead to an ambiguity) if and only if for all z1, . . . , zl ∈ hA the equality b1z1+. . .+blzl = 0 implies z1 = . . . = zl = 0. By (23) in this definition one can consider only homogeneous z1, . . . , zl. For an arbitrary family b1, . . . , bl from Lemma 4 (with arbitrary k, l) one can choose a maximal linearly independent from the right subfamily bi1 , . . . , bir of b1, . . . , bl. It turns out that r does not depend on the choice of a subfamily. More precisely, we have the following lemma. LEMMA 5 Let cj = 1≤i≤l bizi,j, 1 ≤ j ≤ r1, where zi,j ∈ hA are homo- geneous elements. Suppose that there are integers d′′j , 1 ≤ j ≤ r1, such that for all i, j the degree deg zi,j = d i − d j . Assume that cj, 1 ≤ j ≤ r1, are lin- early independent over hA from the right. Then r1 ≤ r, and if r1 < r there are cr1+1, . . . , cr ∈ {bi1 , . . . , bir} such that cj, 1 ≤ j ≤ r, are linearly independent over hA from the right. PROOF The proof is similar to the case of vector spaces over a field and we leave it to the reader. We denote r = rankr{b1, . . . , bl} and call it the rank from the right of b1, . . . , bl. In the similar way one can define rank from the left of b1, . . . , bl. Denote it by rankl{b1, . . . , bl}. It is not difficult to construct examples when rankr{b1, . . . , bl} 6= rankl{b1, . . . , bl}. The aim of this section is to prove the following result. LEMMA 6 Let b be the matrix with homogeneous coefficient from hA satisfying (23), see above. Suppose that deg bi,j < d for all i, j. Assume that k ≥ l ≥ 1. Let l1 = rankr{b1, . . . , bl} and b1, . . . , bl1 be linearly independent. Hence 0 ≤ l1 ≤ l. Then there is a matrix (zj,r)1≤j,r≤l1 with homogeneous entries zj,r ∈ hA and a square permutation matrix σ of size k satisfying the following properties. (i) All the nonzero elements bi,jzj,r for 1 ≤ j ≤ l have the same degree depending only on i, r and deg zj,r ≤ (2n+ 3)ld. (30) (ii) Set the matrix e = (ei,j)1≤i≤k, 1≤j≤l1 = σbz. Then the matrix where e′ = diag(e′1,1, . . . , e l1,l1 ) is a diagonal matrix with l1 columns and each e′j,j, 1 ≤ j ≤ l1, is nonzero. (iii) ord ei,j ≥ ord e j,j for all 1 ≤ i ≤ k, 1 ≤ j ≤ l1. Besides that, if all ai,j (and hence all bi,j) do not depend on Xn (i.e., they can be represented as sums of monomials which do not contain Xn) then one can choose also zj,r satisfying additionally the same property. Finally, dividing by an appropriate power of X0 one can assume without loss of generality that min{ord zj,r : 1 ≤ j ≤ l1} = 0 for every 1 ≤ r ≤ l1. PROOF At first we shall show how to construct z and e such that (ii) and (iii) hold. We shall use a kind of Gauss elimination and Lemma 4. Namely, we transform the matrix e. At the beginning we put e = (e1, . . . , el1) = (b1, . . . , bl1). We shall perform some hA-linear transformations of columns and permutations of rows of the matrix e and replace each time e by the obtained matrix. These transformation do not change the rank from the right of the family of columns of e. At the end we get a matrix e satisfying the required properties (ii), (iii). We have rankr(e) = l1. If l1 = 0, i.e, e is an empty matrix, then this is the end of the construction: z′ is an empty matrix. Suppose that l1 > 0. Let us choose indices 1 ≤ i0 ≤ k, 1 ≤ j0 ≤ l1 such that ord ei0,j0 = min1≤j≤l1{ord ej}. Permuting rows and columns of e we shall assume without loss of generality that (i0, j0) = (1, 1). By Lemma 4 we get elements wi,1, wi,i ∈ hA of degrees at most (2n + 3)2d such that e1,1w1,i = e1,iwi,i, 1 ≤ i ≤ l1, and ordwi,i = 0 for every 1 ≤ i ≤ l1. Set w′ = (−w1,2, . . . ,−w1,l1), and w ′′ = diag(w2,2, . . . , wl1,l1) to be the diagonal matrix. Put 1, w′ 0, w′′ to be the square matrix with l1 rows. We replace e by ew. Now e1,1, 0 E2,1, E2,2 where E2,2 has l1 − 1 columns and 1≤j≤l1 {ord bj} = ord e1,1 = min 1≤j≤l1 {ord ej} (31) (for the new matrix e). Let us apply recursively the described construction to the matrix E2,2 in place of e. So using only linear transformations of columns with indices 2, . . . , l1 and permutation of rows with indices 2, . . . , k we transform e to the form σeτ = e1,1, 0 E′2,1, E E′′2,1 E  , τ = 0, τ ′ where σ is a permutation matrix and τ ′ is a square matrix with l1 − 1 rows (it transforms E2,2), the matrix E 2,2 = diag(e2,2, . . . , el1,l1) is a diagonal matrix with l1 − 1 ≥ 0 columns, and all the elements e2,2, . . . , el1,l1 ∈ hA are nonzero. We shall assume without loss of generality that σ = 1 is the identity matrix. We replace e by eτ . Conditions (ii) and (iii) hold for the obtained e and, more than that, by (iii) applied recursively for (E2,2, E 2,2, E 2,2) (in place of (e, e ′, e′′)), and (31) the same equalities are satisfied for the new obtained matrix e. Let E′2,1 = (e2,1, . . . , el1,1) t where t denotes transposition. By Lemma 4 there are nonzero elements v1,1, . . . , vl1,1 ∈ hA of degrees at most (2n+ 3)(max{deg ei,i : 1 ≤ i ≤ l1}+ 1)l1 (32) such that ei,1v1,1 = ei,ivi,1 and min{ord v1,1, ord v1,i} = 0 for all 1 ≤ i ≤ l1 − 1. Let v′ = (−v2,1, . . . ,−vl1,1) t and v′′ be the identity matrix of size l1 − 1. Put v1,1, 0 v′, v′′ Let us replace e by ev. Put z = wτv, where the matrix z has l1 columns. Recall that without loss of generality σ = 1 is the identity permutation. We have e = (b1, . . . , bl1)z. These Gauss elimination transformations of e do not change the rank from the right of the family of columns of e. It can be easily proved using the recursion on l, cf. Lemma 8 below. Now the matrix e satisfies required conditions (ii), (iii) and σ = 1. Let us change the notation. Denote the obtained matrix z by z′. Let z′ = (z′1, . . . , z ) where z′j is the j-th column of z ′. Our aim now is to prove the existence of the matrix z satisfying (i)–(iii). By Lemma 4 for every 1 ≤ r ≤ l1 there are homogeneous elements zj,r ∈ hA, 1 ≤ j ≤ l, such that (z1,r, . . . , zl,r) 6= (0, . . . , 0), ∑ 1≤j≤l1 bi,jzj,r = 0 for every 1 ≤ i ≤ l1, i 6= r, (33) and estimations for degrees (30) hold. Put the matrix z = (zj,r)1≤j,r≤l1 . Let z = (z1, . . . , zl1) where zj is the j-th column of z. Hence zj = (z1,r, . . . , zl,r) LEMMA 7 For every 1 ≤ r ≤ l1 we have 1≤j≤l1 br,jzj,r 6= 0. (34) Further, for every 1 ≤ r ≤ l1 there are nonzero homogeneous elements g r, gr ∈ hA such that z′rg r = zrgr. PROOF Consider the matrix (z′, zr) with l1 rows and l1 + 1 columns. By Lemma 4 there are homogeneous elements h1, . . . , hl1+1 ∈ hA (they depend on r) such that (h1, . . . , hl1+1) 6= (0, . . . , 0) and the following property holds. Denote h = (h1, . . . , hl1+1) t, h′ = (h1, . . . , hl1) t. Then z′h′ + zrhl1+1 = 0 (35) (we don’t need at present any estimation on degrees from Lemma 4; only the existence of h). Denote by b′′ the submatrix consisting of the first l1 rows of the matrix (b1, . . . , bl1). Multiplying (35) to b ′′ from the left we get b′′z′h′ + b′′zrhl1+1 = 0. (36) But b′′z′ is a diagonal matrix with nonzero elements on the diagonal, see (ii) (for z′ in place of z). Hence by (33) and (36) hj = 0 for every j 6= r. Now h 6= (0, . . . , 0)t implies hr 6= 0 and hl1+1 6= 0. Therefore, (34) holds. Put g′r = hr and gr = hl1+1. We have z r = zrgr by (36). The lemma is proved. Let us return to the proof of Lemma 6. Now (i)–(iii) are satisfied by Lemma 7. The last assertions of Lemma 6 are proved similarly to the ones of Lemma 4. Lemma 6 is proved. 7 An algorithm for solving linear systems with coefficients from hA. Let u = (u1, . . . , ul) t ∈ hAl. Let all nonzero uj be homogeneous elements of the degree −d′j+ρ for an integer ρ. Suppose that −d j+ρ < d ′ for an integer d′ > 1. Let b = (bi,j)1≤i≤k, 1≤j≤l be the matrix with k rows and l columns from the statement of Lemma 6 (but now k and l are arbitrary). So deg bi,j = di−d j < d for all i, j. Let Z = (Z1, . . . , Zk) be unknowns. Consider the linear system 1≤i≤k Zibi,j = uj , 1 ≤ j ≤ l, (37) or, which is the same, Zb = u. Denote ordu = min 1≤i≤k {ordui}. (38) The similar notations will be used for other vectors and matrices. In this section we shall describe an algorithm for solving linear systems over hA and prove the following theorem. THEOREM 2 Suppose that system (37) has a solution over hA. One can represent the set of all solutions of (37) over hA in the form J + z∗, where J ⊂ hAl is a hA-submodule of all the solutions of the homogeneous system corresponding to (37) (i.e., system (37) with all uj = 0) and z ∗ is a particular solution of (37). Moreover, the following assertions hold. (A) One can choose z∗ such that ord z∗ ≥ ordu− ν, where ν ≥ 0 is an integer bounded from above by (dl)2 (and depends only on d and l). The degree deg z∗ is bounded from above by d′(dl)2 (B) There exists a system of generators of J of degrees bounded from above by (dl)2 . The number of elements of this system of generators is bounded from above by k(dl)2 Besides that, if all bi,j and uj do not depend on Xn (i.e., they can be represented as sums of monomials which do not contain Xn) then z ∗ and all the generators of the module J also satisfy this property. PROOF Let l1 = rankr(b1, . . . , bl). Permuting equations of (37) we shall assume without loss of generality that (b1, . . . , bl1) are linearly independent from the right over hA. Let σ, z, e, e′, e′′ be the matrices from Lemma 6. Similarly to the proof of Lemma 6 we shall assume without loss of generality that σ = 1. Denote by b′ the submatrix of b consisting of the first l1 columns of b, i.e., b′ = (b1, . . . , bl1). By Lemma 4 there are nonzero elements q1,1, . . . , ql1,l1 of degrees at most (32) such that e1,1q1,1 = ei,iqi,i and min{ord q1,1, ord qi,i} = 0 for all 2 ≤ i ≤ l1. Set q = diag(q1,1, . . . , ql1,l1) to be the diagonal matrix. Let ν0 = ord e1,1q1,1. Then by Lemma 6 (iii) ord(b ′zq) ≥ ν0. Let X 0 δ = b Then δ is a matrix with coefficients from hA and where δ′ = diag(δ1,1, . . . , δl1,l1) is a diagonal matrix with homogeneous coef- ficients from hA and all the elements on the diagonal are nonzero and equal, i.e., δj,j = δ1,1 for every 1 ≤ j ≤ l1. Besides that, ord δ1,1 = 0. Fur- ther, δ′′ = (δi,j)l1+1≤i≤k, 1≤j≤l1 . We have ord(uzq) ≥ ν0, since, otherwise, system (37) does not have a solution. Obviously ordu ≤ ord(uzq). Denote u′ = (u′0, . . . , u t = X−ν00 uzq ∈ hAl. Hence ordu′ ≥ ord(u) − ν0. Consider the linear system Zδ = u′. (39) LEMMA 8 Suppose that system (37) has a solution over hA. Then linear system (39) is equivalent to (37), i.e., the sets of solutions of systems (39) and (37) over hA coincide. PROOF The system Zb′z = uz is equivalent to (37) by Lemma 5. System (39) is equivalent to Zb′z = uz since the ring hA does not have zero–divisors. The lemma is proved. REMARK 4 Since rankr(b1, . . . , bl) = l1 and by Lemma 6 for every l1 + 1 ≤ j ≤ l there are homogeneous zj,j, z1,j , . . . , zl1,j ∈ hA such that zj,j 6= 0 and bjzj,j + 1≤r≤l1 brzr,j = 0 and all deg zj,j, deg zr,j are bounded from above by (2n+3)(l1+1)d. Put u j = ujzj,j + 1≤r≤l1 urzr,j, l1+1 ≤ j ≤ l. Then system (37) has a solution if and only if system (39) has a solution and u′j = 0 for all l1 + 1 ≤ j ≤ l. This follows from Lemma 8 and Lemma 5. But in what follows for our aims it is sufficient to use only Lemma 8. REMARK 5 Assume that degXn bi,j ≤ 0 for all i, j, i.e., the elements of the matrix b do not depend on Xn. Then by Lemma 4 and the described construction all the elements of the matrices b, z, q, δ, δ′, δ′′ also do not depend on Xn. By Lemma 4 and Remark 2 for every l1 + 1 ≤ j ≤ k there are homogeneous elements gj,j , gj,i ∈ hA, 1 ≤ i ≤ l1, such that gj,jδj,i = gj,iδ1,1, 1 ≤ i ≤ l1, all the degrees deg gj,j , deg gj,i, 1 ≤ i ≤ l1, are bounded from above by (2n+ 3)(l1 + 1)(max{deg δj,i : 1 ≤ i ≤ k}+ 1) and min1≤i≤l1{ord gj,j , ord gj,i} = 0. Hence ord gj,j = 0 for every l1+1 ≤ j ≤ k since ord δ1,1 = 0. Denote h = δ1,1gl1+1,l1+1gl1+2,l1+2 . . . gk,k. So h ∈ hA is a nonzero homoge- neous element and ordh = 0. Set ε = deg h. We need an analog of the Noether normalization theorem from commutative algebra, cf. also Lemma 3.1 [7]. LEMMA 9 There is a linear automorphism of the algebra hA α : hA → hA, α(Xi) = 1≤j≤n (α1,i,jXj + α2,i,jDj), α(Di) = 1≤j≤n (α3,i,jXj + α4,i,jDj), α(X0) = X0, 1 ≤ i ≤ n, such that all αs,i,j ∈ F , degDn α(h) = ε. If degXn h = 0 then one can choose additionally α(Xn) = Xn, all α1,n,j = 0 for 1 ≤ j ≤ n − 1 and α3,n,j = 0 for 1 ≤ j ≤ n. PROOF Recall that ordh = 0. Hence at first it is not difficult to construct a linear automorphism β such that β(X0) = X0, β(Xi) = β1,iXi + β2,iDi, β(Di) = β3,iXi + β4,iDi, 1 ≤ i ≤ n, (40) and β(h) contains a monomial ai1,...,inD 1 , . . . , D n with ai1,...,in 6= 0 and i1 + . . .+in = ε, i.e., ε = degD1,...,Dn β(h). After that one can find an automorphism γ such that γ(X0) = X0, γ(Xi) = 1≤j≤n γ1,i,jXj , γ(Di) = 1≤j≤n γ4,i,jDj , 1 ≤ i ≤ n, (41) and (γ ◦ β)(h) contains a monomial aDεn with a coefficient 0 6= a ∈ F . Put α = γ ◦ β. We leave to prove the last assertion to the reader. The lemma is proved. We apply the automorphism α. In what follows to simplify the notation we shall suppose without loss of generality that α = 1. So h contains a monomial aDεn with a coefficient 0 6= a ∈ F , where ε = deg h. It follows from here that degDn δ1,1 = deg δ1,1, degDn gj,j = deg gj,j , l1 + 1 ≤ j ≤ k. (42) Let z = (z1, . . . , zk) ∈ hAk be a solution of (39). Then (42) implies that one can uniquely represent zj = z jgj,j + 0≤s<deg gj,j zj,sD n, l1 + 1 ≤ j ≤ k, (43) where z′j , zj,s ∈ hA, the degrees degDn zj,s ≤ 0 for all l1 + 1 ≤ j ≤ k, 0 ≤ s < degD1 gj,j . Again by (42) one can uniquely represent u′i = u i δ1,1 + 0≤s<deg δ1,1 u′i,sD n, 1 ≤ i ≤ l, where u′′i , u i,s ∈ hA, the degrees degDn u i,s ≤ 0 for all 1 ≤ i ≤ l, 0 ≤ s < degD1 gj,j . Finally, by (42) for all l1 +1 ≤ j ≤ k, 1 ≤ i ≤ l1, 0 ≤ r < degD1 gj,j, one can uniquely represent Drnδj,i = δj,r,iδ1,1 + 0≤r<deg δ1,1 δj,r,i,sD where δj,r,i, δj,r,i,s ∈ hA, the degrees degDn δj,r,i,s ≤ 0 for all considered j, r, i, s. I = { (j, r) : l1 + 1 ≤ j ≤ k&0 ≤ r < deg gj,j } , J = { (i, s) : 1 ≤ i ≤ l1 &1 ≤ s < deg δ1,1 } . Therefore, zi = − l1+1≤j≤k z′jgj,i − (j,r)∈I zj,rδj,r,i + u i , 1 ≤ i ≤ l1, (44) (j,r)∈I zj,rδj,r,i,s = u i,s, (i, s) ∈ J . (45) Let us introduce new unknowns Zj,r, (j, r) ∈ I. By (43)–(45) system (37) is reduced to the linear system (j,r)∈I Zj,rδj,r,i,s = u i,s, (i, s) ∈ J . (46) More precisely, any solution of system (37) is given by (43), (44) where z′j ∈ are arbitrary and zj,r is a solution of system (45) over hA (we underline that here this solution zj,r may depend on Dn although one can restrict oneself by solutions zj,r which do not depend on Dn). Note that all δj,r,i,s and u are homogeneous elements of hA and there are integers dj,r, (j, r) ∈ I, d (i, s) ∈ J , ρ̃ such that deg δj,r,i,s = dj,r − d i,s and deg u i,s = −d i,s + ρ̃ for all (j, r) ∈ I, (i, s) ∈ J . This follows immediately from the described construction. Now all the coefficients of system (46) do not depend on Dn. As we have proved if the coefficients of (37) do not depend on Xn then the coefficients of (46) also do not depend on Xn, and hence in the last case they do not depend on Xn, Dn. If the coefficients of (46) depend on Xn we perform an automorphism Xn 7→ Dn Dn 7→ −Xn, Xi 7→ Xi, Di 7→ Di, 1 ≤ i ≤ n − 1. Now the coefficients of system (46) do not depend on Xn (but depend on Dn). After that we apply our construction recursively to system (46). The final step of the recursion is n = 0 (although in the statement of theorem n ≥ 1, see Section 1; we are interested only in Weyl algebras). In this case I = J = ∅. Hence using (44) for n = 0 we get the required z∗ and J for n = 0. Thus, by the recursive assumption we get a particular solution Zj,r = z (j, r) ∈ I, of system (46), an integer ν1 (in place of ν from assertion (A)) such (j,r)∈I {ord z∗j,r} ≥ min (i,s)∈J {ordu′i,s} − ν1, (47) and a system of generators ( zα,j,r )(j,r)∈I , 1 ≤ α ≤ β, (48) of the module J ′ of solutions of the homogeneous system corresponding to (46). Notice that if the coefficients of (37) do not depend on Xn then J ′ is a module over the homogenization F [X0, X1, . . . , Xn−1, D1, . . . , Dn−1] of the Weyl alge- bra of X1, . . . , Xn−1, D1, . . . , Dn−1. But obviously in the last case (48) gives also a system of generators of the hA-module J ′′ = hAJ ′ of solutions of the homogeneous system corresponding to (46). Put z∗i = − (j,r)∈I z∗j,rδj,r,i + u i , 1 ≤ i ≤ l1, z∗j = 0≤s<deg gj,j z∗j,sD n, l1 + 1 ≤ j ≤ k, z∗ = (z∗1 , . . . , z Then z∗ is a particular solution of (37). Put zα,i = − (j,r)∈I zα,j,rδj,r,i, 1 ≤ i ≤ l1, 1 ≤ α ≤ β, zα,j = 0≤s<deg gj,j zα,j,sD n, l1 + 1 ≤ j ≤ k, 1 ≤ α ≤ β, zβ−l1+j,i = 0, l1 + 1 ≤ i, j ≤ k, i 6= j, zβ−l1+j,j = gj,j , l1 + 1 ≤ j ≤ k, zβ−l1+j,i = −gj,i, 1 ≤ i ≤ l1, l1 + 1 ≤ j ≤ k. Then J = 1≤α≤β+k−l1 hA(zα,1, . . . , zα,k). Hence (zα,1, . . . , zα,k), 1 ≤ α ≤ β+k− l1, is a system of generators of the module J . By (47) and the definitions of u′, u′′i and u i,s we have ord z ∗ ≥ ordu− ν0 − ν1. Put ν = ν0 + ν1. LEMMA 10 All the degrees deg δj,i, deg gj,i, deg δj,r,i, deg δj,r,i,s and ν, see above, are bounded from above by (nld)O(1), the degrees deg u′i are bounded from above d′ + (nld)O(1), the degrees deg u′′i , deg u i,s are bounded from above by d′(nld)O(1). Further, all ordu′′i , ordu i,s are bounded from below by ordu − ν. Finally, in system (46) the number of equations #J is bounded from above by (nld)O(1) and the number of unknowns #I is bounded from above by k(nld)O(1). PROOF This follows immediately from the described construction. Let us return to the proof of Theorem 2. Applying Lemma 10 and recursively assertions (A) and (B) for the formulas giving z∗ and J we get (A) and (B) from the theorem. The last assertion (related to the case when all bi,j and uj do not depend on Dn) has been already proved. The theorem is proved. 8 Proof of Theorem 1 for Weyl algebra Let a be the matrix from Section 1. We shall suppose without loss of generality that the vectors (ai,1, . . . , ai,l), 1 ≤ i ≤ k, are linearly independent over the field F . We have deg ai,j < d. This implies k ≤ l Put the matrix b = ha. Let us define the graded submodules of hI hA(b1,1, . . . , b1,l) + . . .+ hA(bk,1, . . . , bk,l), Jν = J0 : (X 0 ) = {z ∈ hAl : zXν0 ∈ J0}, ν ≥ 1. We have the exact sequence of graded hA-modules hAk → J0 → 0. Further, Jν ⊂ Jν+1 ⊂ hI for every ν ≥ 0 and hI = ν≥0 Jν . Since hA is Noetherian there is N ≥ 0 such that hI = JN . So to construct a system of generators of hI it is sufficient to compute the least N such that hI = JN and to find a system of generators of JN . LEMMA 11 hI = JN for some N bounded from above by (dl) 2O(n) . There is a system of generators b1, . . . , bs of the module JN such that s and all the degrees deg bv, 1 ≤ v ≤ s, are bounded from above by (dl) 2O(n) . PROOF Let us show that the module JN+1 ⊂ JN for N ≥ ν. Let u ∈ JN+1. Consider system (37). By assertion (A) of Theorem 2 there is a particular solution z∗ of (37) such that ord z∗ ≥ 1. Hence u ∈ X0JN ⊂ JN . The required assertion is proved. Hence hI = Jν . Let us replace in (37) (u1, . . . , ul) by (U1X 0 , . . . , UlX 0 ), where U1, . . . , Ul are new unknowns. Then applying (B) from Theorem 2 to this new homogeneous linear system with respect to all unknowns U1, . . . , Ul, Z1, . . . , Zk we get the required estimations for the number of generators of Jν and the degrees of these generators. The lemma is proved. COROLLARY 1 Let (ai,1, . . . , ai,l), 1 ≤ i ≤ l, be from the beginning of the section and the integer N be from Lemma 3. Then for every integer m ≥ 0 the F–linear space Am+N (a1,1, . . . , a1,l) + . . .+Am+N (ak,1, . . . , ak,l) ⊃ Im. (49) PROOF By Lemma 3 we have (J0)m+N ⊃ X 0 (JN )m = X hI)m. Taking the affine parts we get (49). The corollary is proved. Now everything is ready for the proof of Theorem 1. By Lemma 11 and Lemma 1 there is a system of generators of the module gr(I) with degrees bounded from above by (dl)2 . By Lemma 12 from Appendix 1 the Hilbert function H(gr(I),m) is stable for m ≥ (dl)2 . By (10) Section 2 the Hilbert function H(I,m) is stable for all m ≥ (dl)2 Consider the linear order < on the monomials from hAl which is induced by the linear order < on the monomials from Al, see Section 4. Then the monomial submodule cI ⊂ cAl is defined, see Section 4, where cA = F [X0, . . . , Xn, D1, . . . , Dn] is the polynomial ring. By (22) Section 4 the Hilbert function H( cI,m) is stable for all m ≥ (dl)2 . Hence all the coefficients of the Hilbert polynomial of cI are bounded from above (dl)2 . Therefore, according to (31) the module cI has a system of generators with degrees (dl)2 . This means, see Section 4, that the module Hdt(hI) has a system of generators with degrees (dl)2 Therefore, the degrees of all the elements of the Janet basis of hI with respect to the induced linear order < are bounded from above by (dl)2 . Hence by Lemma 3 Section 4 the same is true for the Janet basis of the module I with respect to the linear order < on the monomials from Al. Theorem 1 is proved for Weyl algebra. 9 The case of algebra of differential operators Denote by B = F (X1, . . . , Xn)[D1, . . . , Dn] the algebra of differential operators. Recall that A ⊂ B and hence relations (1) are satisfied. Further, each element f ∈ B can be uniquely represented in the form j1,...,jn≥0 fj1,...,jnD 1 . . . D where all fj1,...,jn = fj ∈ F (X1, . . . , Xn) and F (X1, . . . , Xn) is a field of rational functions over F . Let us replace everywhere in Section 1 and Section 2 A, X iDj , deg f = degX1,...,Xn,D1,...,Dn f , dimF M , ev,i,j , fv,i,j ∈ F , (v, i, j), (i, j), (i′, j′), (i′′, j′′) by B, Dj , deg f = degD1,...,Dn f , dimF (X1,...,Xn) M , ev,j , fv,j ∈ F (X1, . . . , Xn), (v, j), j, j ′, j′′ respectively. Thus, we get the definition of the Janet basis and all other objects from Section 1 for the case of the algebra of differential operators. We define the homogenization hB of B similarly to hA, see Section 3. Namely, hB = F (X1, . . . , Xn)[X0, D1, . . . , Dn] given by the relations XiXj = XjXi, DiDj = DjDi, for all i, j, DiXi −XiDi = X0, 1 ≤ i ≤ n, XiDj = DjXi for all i 6= j. Further, the considerations are similar to the case of the Weyl algebra A with minor changes. We leave them to the reader. For example, Theorem 2 for the case of the algebra of differential operators is the same. One need only to replace everywhere in its statement A, hA and Xn by B, hB and Dn respectively. Thus, one can prove Theorem 1 for the case when A is an algebra of differential operators (but now it is B). Theorem 1 is proved completely. One can consider more general algebra of differential operators. Let F be a field with n derivatives D1, . . . , Dn. Then Kn = F [D1, . . . , Dn] is the algebra of differential operators and similarly one can define its homogenization hKn by means of adding the variable X0 satisfying the relations DiDj = DjDi, X0Di = DiX0, Dif − fDi = fDiX0 for all i, j and any element f ∈ F where fDi ∈ F denotes the result of the application of Di to f . Following the proof of Theorem 1 one can deduce the following statement. REMARK 6 A similar bound to Theorem 1 holds for Kn. Appendix 1: Degrees of generators of a graded module over a polynomial ring and its Hilbert function. We give a short proof of the following result, cf. [1], [12], [6], [4]. LEMMA 12 Let I ⊂ Al be a graded submodule over the graded polynomial ring A = F [X0, . . . , Xn], and I is given by a system of generators f1, . . . , fm of degrees less than d. Then the Hilbert function H(Al/I,m) = dimF (A l/I)m is stable for m ≥ (dl)2 O(n+1) . Further, all the coefficients of the Hilbert polynomial of Al/I are bounded from above by (dl)2 O(n+1) PROOF Denote M = Al/I. Let L ∈ F [X0, . . . , Xn] be a linear form in gen- eral position. Denote byK the kernel of the morphismM → M of multiplication to L. We have K = {z ∈ Al : Lz = 1≤i≤m fizi,& zi ∈ A}. Hence solving a linear system over A, we get that K has a system of generators g1, . . . , gµ with degrees bounded from above by (dl)2 O(n+1) . Let P be an arbitrary associ- ated prime ideal of the module M such that P 6= (X0, . . . , Xn). Since L is in general position we have L 6∈ P. Hence P is not an associated prime ideal of K. Therefore, KN = 0 for all sufficiently big N . So X i gj ∈ I for sufficiently big N and all i, j. Hence gj = 1≤i≤m yj,ifi where yj,i ∈ F (Xi)[X0, . . . , Xn]. Solving a linear system over the ring F (Xi)[X0, . . . , Xn] we get an estimation for denominators from F [Xi] of all yj,i. Since all gj and fi are homogeneous we can suppose without loss of generality that all the denominators are XNi . Thus, we get an upper bound for N . Namely, N is bounded from above by (dl)2 O(n+1) Therefore, the sequence 0 → Mm → Mm+1 → (M/LM)m+1 → 0 (51) is exact for m ≥ (dl)2 O(n+1) . But M/LM = Al/(I + LAl) is a module over a polynomial ring of F [X0, . . . , Xn]/(L) ≃ F [X0, . . . , Xn−1]. Hence by the inductive assumption the Hilbert function H(Al/(I + LAl),m) is stable for m ≥ (dl)2 . Therefore, (51) implies that the Hilbert function H(Al/I,m) is stable for m ≥ (dl)2 O(n+1) Obviously for m < (dl)2 O(n+1) the values H(Al/I,m) are bounded from above by (dl)2 O(n+1) . Hence by the Newton interpolation all the coefficients of the Hilbert polynomial of Al/I are bounded from above by (dl)2 O(n+1) . The lemma is proved. We need also a conversion of Lemma 12. LEMMA 13 Let I ⊂ Al be a graded submodule over the graded polynomial ring A = F [X0, . . . , Xn]. Assume that the Hilbert function H(A l/I,m) = dimF (A l/I)m is stable for m ≥ D and all absolute values of the coefficients of the Hilbert polynomial of the module Al/I are bounded from above by D for some integer D > 1. Then I has a system of generators f1, . . . , fm with degrees O(n+1) PROOF Let us choose f1, . . . , fm to be the reduced Gröbner basis of I with respect to an admissible linear order < on the monomials from Al, cf. the definitions from Section 1 and Section 4. The degree of a monomial from Al is defined similarly to Section 1 and Section 4. We shall suppose additionally that the considered linear order is degree compatible, i.e., for any two monomials z1, z2 if deg z1 < deg z2 then z1 < z2. For every z ∈ A the greatest monomial Hdt(z) is defined. Further the monomial ideal Hdt(I) is generated by all Hdt(z), z ∈ I. Now Hdt(f1), . . . ,Hdt(fm) is a minimal system of generators of Hdt(I) and deg fi = degHdt(fi) for every 1 ≤ i ≤ m. The values of Hilbert functions H(Al/Hdt(I),m) = H(Al/I,m) coincide for all m ≥ 0. Thus, replacing I by Hdt(I) we shall assume in what follows in the proof that I is a monomial module. For every 1 ≤ i ≤ l denote by Ai ⊂ A l the i-th direct summand of Al. Put Ii = I ∩ Ai, 1 ≤ i ≤ l. Then I ≃ ⊕1≤i≤lIi since I is a monomial module. Further, for every 1 ≤ α ≤ m there is 1 ≤ i ≤ l such that fα ∈ Ii. Let us identify Ai = A. Then Ii ⊂ A is a homogeneous monomial ideal. The case Ii = A is not excluded for some i. For the Hilbert functions we have H(Al/I,m) = 1≤i≤l H(A/Ii,m), m ≥ 0. (52) If (A/Ii)D = 0 for some i then (A/Ii)m = 0 for every m ≥ D. In this case the ideal Ii is generated by 0≤m≤D(Ii)m. Hence in (52) for the values m ≥ D one can omit this index i in the sum from the right part. Therefore, in this case the proof is reduced to a smaller l. So we shall assume without loss of generality that (A/Ii)D 6= 0, 1 ≤ i ≤ l. Further, we use the exact description of the Hilbert function of a homoge- neous ideal, see [4] Section 7. Namely there are the unique integers bi,0 ≥ bi,1 ≥ . . . ≥ bi,n+2 = 0 such that H(A/Ii,m) = m+ n+ 1 1≤j≤n+1 m− bi,j + j − 1 for all sufficiently big m and bi,0 = min{d : d ≥ bi,1 & ∀m > d (53) holds }. (54) This description (without constants bi,0) is originated from the classical paper [11]. The integers bi,0, . . . , bi,n+2 are called the Macaulay constants of the ideal Ii. Besides that, h(i,m) = H(A/Ii,m)− m+ n+ 1 + 1 + 1≤j≤n+1 m− bi,j + j − 1 for every m ≥ bi,1, see [4] Section 7. By Lemma 7.2 [4] for all 1 ≤ α ≤ m if fα ∈ Ii then deg fα ≤ bi,0. Hence it is sufficient to prove that all bi,0, 1 ≤ i ≤ l, are bounded from above by D2 O(n+1) By (52) and (53) the coefficient at mn−j , 0 ≤ j ≤ n, of the Hilbert polyno- mial of Al/I is (n+ 1− j)! 1≤i≤l bi,n+1−j + 0≤v≤j−1 1≤i≤l (n+ 1− v)! µj,v(bi,n+1−v), (56) where 0 6= µj is an integer and µj,v ∈ Z[Z], 0 ≤ v ≤ j − 1, is a polynomial with integer coefficients with deg µj,v = j − v + 1. Moreover, |µj | and absolute values of all the coefficients of all the polynomials µj,v are bounded from above by, say, 2O(n 2). Denote bj = 1≤i≤l bi,j , 0 ≤ j ≤ n + 2. By the condition of the lemma all the coefficients of the Hilbert polynomial of Al/I are bounded from above by D. Hence from (56) one can recursively estimate bn+1, bn, . . . , b1. Namely, bn+1−j = (2 O(j+1) , 0 ≤ j ≤ n. Hence b1 = (lD) 2O(n+1) . Notice that bi,1 ≤ max1≤i≤l bi,1 ≤ b1 for every 1 ≤ i ≤ m. Now let m ≥ max1≤i≤l bi,1. By (55) if h(i,m) 6= 0 for some 1 ≤ i ≤ l then m < D, i.e., m is less than the bound D for the stabilization of the Hilbert function of Al/I. Thus, bi,0 ≤ max{bi,1, D} by (54). Hence bi,0 is bounded from above by (lD)2 O(n+1) We have (A/Ii)D 6= 0 for every 1 ≤ i ≤ l. This implies H(A l/I,D) ≥ l. Denote by cj the j-th coefficient of the Hilbert polynomial of the module A Now |cj |D j ≥ l/(n + 1) for at least one j. Hence Dn+1(n + 1) ≥ l by the condition of the lemma. This implies that l2 O(n+1) is bounded from above by O(n+1) . Therefore, bi,0 is bounded from above by D 2O(n+1) . The lemma is proved. Appendix 2: Bound on the Gröbner basis of a monomial module via the coefficients of its Hilbert polynomial Denote by Cl = Z + ∪ · · · ∪ Z + the disjoint union of l copies of the semigrid + = {(i1, . . . , in) : ij ≥ 0, 1 ≤ j ≤ n}. A subset of Cl which intersects each disjoint copy of Zn+ by a semigroup closed with respect to addition of elements from Zn+ is called an ideal of Cl. Any ideal I in Cl has a unique finite Gröbner basis V = VI , denote T = Cl \ I. Clearly, I corresponds to a monomial submodule in the free module (F [X1, . . . , Xn]) l. The degree of an element u = (k; i1, . . . , in) ∈ Cl, 1 ≤ k ≤ l is defined as |u| = i1 + · · · + in. The degree of a subset in Cl is defined as the maximum of the degrees of its elements. The Hilbert function HT (z) equals to the number of vectors u ∈ T such that |u| ≤ z. Then HT (z) = 0≤s≤m csz s, z ≥ z0 for suitable z0, integers c0, . . . , cm where the degree m ≤ n. Denote c = max0≤s≤m |cs|s! + 1. PROPOSITION 1 (cf. [6], [12], [4]). The degree of V does not exceed (cn)2 PROOF An s-cone we call a subset of a k-th copy of Zn+ in Cl for a certain 1 ≤ k ≤ l of the form P = {Xj1 = i1, . . . , Xjn−s = in−s} (57) for suitable 1 ≤ j1, . . . , jn−s ≤ n. The degree of (57) we define as |P | = i1 + · · · + in−s (note that this definition is different from the one in [4]). By a predessesor of (57) we mean each s-cone in the same k-th copy of Zn+ of the {Xj1 = i1, . . . , Xjp−1 = ip−1, Xjp = ip − 1, Xjp+1 = ip+1, . . . , Xjn−s = in−s} for some 1 ≤ p ≤ n− s, provided that ip ≥ 1. Fix an arbitrary linear order on s-cones compatible with the relation of predessesors. By inverse recursion on s we fill gradually T (as a union) by s-cones. For the base we start with s = m. Assume that a current union T0 ⊂ T of m-cones is already constructed (at the very beginning we put T0 = ∅) and an m-cone of the form (57) with s = m is the least one (with respect to the fixed linear order on m-cones) which is contained in T not being a subset of T0. Observe that each predessesor of this m-cone was added to T0 at earlier steps of its construction. Since the total number of m-cones added to T0 does not exceed cmm! < c we deduce that the degree of every such m-cone is less than cmm! (taking into account that the very first m-cone added to T0 has the degree 0). For the recursive step assume that the current T0 is a union of all possible m-cones, (m− 1)-cones,...,(s+1)-cones and perhaps, some s-cones. This can be expressed as deg(HT −HT0) ≤ s. Again as in the base take the least s-cone of the form (57) which is contained in T not being a subset of T0. Observe that each predessesor of the type (58) of this s-cone is contained in an appropriate r-cone Q, r ≥ s, such that Q was added to T0 at earlier steps of its constructing and Q ⊂ {Xjp = ip − 1}. Hence |Q| ≥ ip − 1. (59) The described construction terminates when T0 = T . Denote by ts the number of s-cones added to T0 and by ks the maximum of their degrees. We have seen already that tm, km < c. Now by inverse induction on s we prove that ts, ks ≤ (cn) 2O(m−s) . To this end we introduce a relevant semilattice on cones. Let C = {Cα,β}α,β, 0 ≤ β ≤ γα be a family of cones of the form (57) where dimCα,β = α. By an α-piece we call an α-cone being the intersection of a few cones from C. All the pieces constitute a semilattice L with respect to the intersection and with maximal elements from C. We treat L also as a partially ordered set with respect to the inclusion relation. Clearly, the depth of L is less than n. Our nearest purpose is to bound from above the size of L. For the sake of simplifying the bound we assume (and this will suffice for our goal in the sequel) that γα ≤ (cn) 2O(m−α) for s ≤ α ≤ m and γα = 0 when α < s, although one could write a bound in general in the same way. Besides that we assume that the constant in O(. . .) is sufficiently big. In what follows all the constants in O(. . .) coincide. LEMMA 14 Under the assumption on the numbers γα ≤ (cn) 2O(m−α) , s ≤ α ≤ m of maximal elements of all dimensions from C, the number of α-pieces in L does not exceed (cn)2 O(m−α)+1 for s ≤ α ≤ m or (cn)2 O(m−s)(s−α+1)+1 when α < s. PROOF For each α-piece choose its arbitrary irredundant representation as the intersection of the cones from C. Let δ be the minimal dimension among these cones. Then this intersection contains at most δ−α+1 cones. Therefore, the number of possible α-pieces does not exceed max{α,s}≤δ≤m (cn)2 O(m−δ)(δ−α+1), that proves the lemma. Now we come back to estimating ts, ks by inverse induction on s. Let in the described above construction the current T0 is the union of all added m- cones, (m − 1)-cones,...,s-cones. Denote this family of cones by C and consider the corresponding semilattice L (see above). Our next purpose is to represent T0 as a Z-linear combination of the pieces from L by means of a kind of the inclusion-exclusion formula. We assign the coefficients of this combination by recursion in L. As a base we assign 1 to each maximal piece, so to the elements of C. As a recursive step, if for a certain piece P ∈ L the coefficients are already assigned to all the pieces greater than P , we assign to P the coefficient ǫP in such a way that the sum of the assigned coefficients to P and to all the greater pieces equals to 1. Therefore, we get where the sum is understood in the sense of multisets. Hence HT0(z) = z − |P |+ dimP for large enough z. We recall that deg(HT −HT0) ≤ s− 1. Now we majorate the coefficients |ǫP | by induction in the semilattice L. The inductive hypothesis on tα ≤ (cn) 2O(m−α) , s ≤ α ≤ m and Lemma 14 imply that dimP=λ |ǫP | ≤ (cn) 2O(m−λ) , s− 1 ≤ λ ≤ m. by inverse induction on λ following the assigning ǫP . In fact, one could majorate in a similar way also dimP=λ |ǫP | when λ < s− 1, but we don’t need it. The inductive hypothesis on kα ≤ (cn) 2O(m−α) , s ≤ α ≤ m and (60) entail that the coefficient of HT0(z) at the power z α does not exceed (cn)2 O(m−α) , s− 1 ≤ α ≤ m (actually, due to the inequality deg(HT −HT0) ≤ s− 1 the coefficients at the powers zα for s ≤ α ≤ m are less than c). In particular, the coefficient at the power zs−1 does not exceed (cn)2 O(m−s+1) . Denote HT −HT0 = ηz s−1+ · · ·. By constructing T0 we add to it ts−1 = η(s− 1)! of (s− 1)-cones, which justifies the inductive step for ts−1 ≤ (cn) 2O(m−s+1) . To conduct the inductive step for ks−1 ≤ (cn) 2O(m−s+1) we observe that for each (s − 1)-cone P added to T0 either every its predessesor is contained in a cone of dimension at least s, or some its predessesor is an (s − 1)-cone as well. In the former case |P | ≤ (maxs≤α≤m kα + 1)(n − s + 1) (due to (59)), while in the latter case |P | is greater by 1 than the degree of this predessesor, hence ks−1 ≤ (maxs≤α≤m kα + 1)(n − s + 1) + ts−1. Finally, exploit the inductive hypothesis for km, . . . , ks, and the just obtained inequality on ts−1. To complete the proof of the proposition it suffices to notice that for any vector from the basis V treated as an 0-cone, each its predessesor of the type (58) for s = 0 is contained in an appropriate r-cone, whence the degree of V does not exceed (max0≤α≤m kα + 1)n again due to (59) (cf. above). Acknowledgement. The authors are grateful to the Max-Planck Institut für Mathematik, Bonn for its hospitality during the stay where the paper was written. References [1] D.A. Bayer, The division algorithm and the Hilbert scheme, Ph.D. Thesis, Harvard, 1982. [2] A. Chistov, D. Grigoriev, Complexity of quantifier elimination in the theory of algebraically closed fields, Lect. Notes Comput. Sci., 176 (1984), 17-31. [3] D. Cox, J. Little, D. O’Shea, Using Algebraic Geometry, Springer, 1998. [4] T. Dubé, The structure of polynomial ideals and Gröbner bases, SIAM J. Comput., 19 (1990), 750–775. [5] A. Galligo, Some algorithmical questions on ideals of differential operators, Lect.Notes Comput.Sci., 204 (1985), 413–421. [6] M. Giusti, Some effective problems in polynomial ideal theory, Lect. Notes Comput. Sci, 174 (1984), 159–171. [7] D. Grigoriev, Weak Bézout inequality for D-modules, J.Complexity, 21, (2005), 532–542. [8] G. Hermann Die Frage der endlich vielen Schritte in der Theorie der Poli- nomideale, Math. Ann. 95, (1926), 736-788. [9] M. Janet, Les modules de formes algébriques et la théorie générale des systèmes différentiels, Annals Sci. Ecole Normale Supér., 41 (1924), 27–65. [10] H. Li, Noncommutative Gröbner bases and filtered-graded transfer, Lect. Notes Math., 1795, 2002. [11] F.S. Macaulay Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531–555. [12] M. Möller, T. Mora, Upper and lower bounds for the degree of Groebner bases, Lect. Notes Comput. Sci., 174 (1984), 172–183. [13] F. Schwarz, Janet bases for symmetry groups, Groebner bases and applica- tions, in London Math. Society, Lecture Note Ser. 251, 221-234, Cambridge University Press, Cambridge, 1998. [14] C. Yap, A new lower bound construction for commutative Thue systems, with applications, J. Symb. Comput., 12 (1991), 1–27. Definition of the Janet basis The graded module corresponding to a D–module Homogenization of the Weyl algebra The Janet bases of a module and of its homogenization Bound on the kernel of a matrix over the homogenized Weyl algebra Transforming a matrix with coefficients from h-A to the trapezoidal form An algorithm for solving linear systems with coefficients from h-A. Proof of Theorem ?? for Weyl algebra The case of algebra of differential operators

0704.1258

Evidence for a merger of binary white dwarfs: the case of GD 362

Draft version October 31, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 EVIDENCE FOR A MERGER OF BINARY WHITE DWARFS: THE CASE OF GD 362 E. Garćıa–Berro , P. Lorén–Aguilar and A.G. Pedemonte Departament de F́ısica Aplicada, Universitat Politècnica de Catalunya, Av. del Canal Oĺımpic s/n, E-08860 Castelldefels (Barcelona), Spain J. Isern Institut de Ciències de l’Espai (CSIC), Facultat de Ciències, Campus UAB, Torre C5-parell, E-08193 Bellaterra (Barcelona), Spain P. Bergeron, P. Dufour and P. Brassard Département de Physique, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, Québec, Canada H3C 3J7 Draft version October 31, 2018 ABSTRACT GD 362 is a massive white dwarf with a spectrum suggesting a H–rich atmosphere which also shows very high abundances of Ca, Mg, Fe and other metals. However, for pure H–atmospheres the diffusion timescales are so short that very extreme assumptions have to be made to account for the observed abundances of metals. The most favored hypothesis is that the metals are accreted from either a dusty disk or from an asteroid belt. Here we propose that the envelope of GD 362 is dominated by He, which at these effective temperatures is almost completely invisible in the spectrum. This assumption strongly alleviates the problem, since the diffusion timescales are much larger for He– dominated atmospheres. We also propose that the He–dominated atmosphere of GD 362 is likely to be the result of the merger of a binary white dwarf, a very rare event in our Galaxy, since the expected galactic rate is ∼ 10−2 yr−1. Subject headings: stars: white dwarfs — stars: chemically peculiar — stars: individual (GD 362) 1. INTRODUCTION GD 362 has been interpreted as a massive, rather cool (Teff ≈ 9740±50 K), white dwarf with a heavy accretion disk surrounding it (Kilic et al. 2005; Becklin et al. 2005; Gianinas, Dufour, & Bergeron 2004). The dusty disk around GD 362 produces an excess of infrared ra- diation which amounts to ∼ 3% of the total stellar lu- minosity. The chemical composition of GD 362 is also rather singular, showing a hydrogen rich atmosphere with very high abundances of Ca, Mg, Fe and other met- als (Gianinas, Dufour, & Bergeron 2004). Thus, it is classified as a massive DAZ (hydrogen–rich) white dwarf. The origin of such particularly high photospheric abun- dances — log(NCa/NH) = −5.2, log(NMg/NH) = −4.8 and log(NFe/NH) = −4.5 — and of the surrounding dusty disk around it still remains a mystery. Since the diffusion timescales for metals in H–rich white dwarfs are of only a few years (Koester & Wilken 2006) very extreme assumptions have to be made in order to ex- plain these abundances. At present the most widely ac- cepted scenario is disruption and accretion of a plane- tary body, although for this scenario to be feasible the planetary system should survive during the advanced stages of stellar evolution, which by no means is guaran- teed. Thus, the formation of an asteroid would require the previous existence of a disk around this white dwarf (Livio, Pringle, & Saffer 1992; Livio, Pringle, & Wood 2005). Particularly, a recent analysis (Villaver & Livio Electronic address: [email protected], [email protected], [email protected] Electronic address: [email protected] Electronic address: [email protected], [email protected], [email protected] 1 Institut d’Estudis Espacials de Catalunya, Ed. Nexus-201, c/ Gran Capità 2–4, E-08034 Barcelona, Spain 2007) has shown that planets around white dwarfs with masses MWD > 0.7M⊙ are generally expected to be found at orbital radii r > 15 AU because they cannot survive the planetary nebula phase and that if planets are to be found at smaller orbital radii around mas- sive white dwarfs, they had to form as the result of the merger of two white dwarfs. It is also interesting to note that there have been previous suggestions about white dwarfs that are merger products — see for instance Liebert, Bergeron & Holberg (2005) — but these claims do not have yet any observational support. 2. THE SCENARIO Another possibility is that some massive white dwarfs are the result of the merger of a double white dwarf close binary system. This scenario has been stud- ied in several papers. However, in most of these pa- pers either the resulting nucleosynthesis was not ad- dressed (Segretain, Chabrier, & Mochkovitch 1997), or the spatial resolution was poor (Benz et al. 1990), or the calculations were performed using crude approxi- mations (Mochkovitch & Livio 1990). Very recently, and using a Smoothed Particle Hydrodynamics code, a series of simulations with the adequate spatial reso- lution were performed and the nucleosynthesis of the merger was studied (Guerrero, Isern, & Garćıa–Berro 2004; Lorén–Aguilar et al. 2005). The main results of such simulations are that the less massive white dwarf of the binary system is totally disrupted in a few or- bital periods. A fraction of the secondary is directly accreted onto the primary whereas the remnants of the secondary form a heavy, rotationally–supported accre- tion disk around the primary and little mass is ejected http://arxiv.org/abs/0704.1258v1 mailto:[email protected], [email protected], [email protected] mailto:[email protected] mailto:[email protected], [email protected], [email protected] 2 Garćıa–Berro et al. TABLE 1 Main results of the SPH simulations. Run 0.4+0.8 0.4+1.0 0.4+1.2 0.6+0.6 0.6+0.8 MWD/M⊙ 0.99 1.16 1.30 0.90 1.09 Mdisk/M⊙ 0.21 0.24 0.30 0.30 0.29 Mej/M⊙ 10 −3 10−3 10−3 10−2 10−3 He 0.94 0.93 0.99 0 0 C 3× 10−2 2× 10−2 5× 10−3 0.4 0.4 O 1× 10−2 3× 10−3 3× 10−3 0.6 0.6 Ca 4× 10−5 2× 10−4 9× 10−6 0 0 Mg 3× 10−5 3× 10−5 6× 10−6 0 0 S 8× 10−5 2× 10−4 5× 10−7 0 0 Si 1× 10−4 2× 10−4 3× 10−5 0 0 Fe 9× 10−3 7× 10−3 5× 10−4 0 0 from the system. The resulting temperatures are rather high (∼ 9× 108 K) during the most violent phases of the merger, allowing for extensive nuclear processing. The enhancement of the abundances of the most rele- vant nuclear isotopes occurs when one of the coalescing white dwarfs is made of pure He. Table 1 shows the average chemical composition of the resulting disk and the main characteristics of some selected simulations. It should be noted, however, that the distribution of the different elements in the disk is rather inhomogeneous. Obviously those parts of the disk in which the mate- rial of the secondary has been shocked have undergone major nuclear processing. Hence, these regions are C– and O–depleted and Si– and Fe–enhanced. In fact, the innermost regions (R < 0.1R⊙) of the merged object, which have approximately the shape of an ellipsoid, are C– and O–rich. It is expected that this region would be eventually accreted during the the first moments of the cooling phase of the central object, leading to a more massive white dwarf. We also find that the abundance of intermediate–mass and iron–group elements is consid- erably larger than that of C and O in the remnants of the accretion stream (Guerrero, Isern, & Garćıa–Berro 2004) which are at larger distances, thus favoring smaller accretion rates in order to explain the Ca abundance. In any case, if the photospheric abundances of GD 362 are to be explained with this scenario the accretion of He– rich material is required. Since He is also accreted onto the surface of GD 362, the photospheric layers may contain significant amounts of He which, at the effective temperature of GD 362 would be almost spectroscopically invisible. Thus, GD 362 would still be classified as a DA white dwarf provided that some H is present in its atmosphere. Con- sequently, the H/He ratio can be regarded as a free parameter. However, the presence of He in a cool hydrogen-rich atmosphere affects the surface gravity de- termined from spectroscopy, and thus the mass deter- mination (Bergeron, Wesemael, & Fontaine 1991). In Fig. 1 we show three almost identical synthetic spec- tra representative of GD 362 with various assumed He abundances. If He/H=10 is adopted then log g = 8.25 is obtained (MWD ∼ 0.8M⊙) whereas if we adopt He/H=1 then the surface gravity turns out to be log g = 8.72. This corresponds to a mass of the primary of MWD ∼ 1.0M⊙, which can be obtained from the coalescence of a 0.4 + 0.8M⊙ binary system. Additionally, in this case the largest abundances of the relevant elements are ob- Fig. 1.— Spectrum of GD 362 for three different helium abun- dances. The black line shows the spectrum of GD 362 when a pure hydrogen atmosphere is assumed, leading to a surface grav- ity of log g = 9.12. For increasing amounts of He — namely N(He)/N(H)=1, red curve, and N(He)/N(H)=10, blue curve — the corresponding surface gravities are smaller. The inset shows an expanded view of the predicted He line at 5876 Å for N(He)/N(H)=10. High quality spectroscopic observations should be able to confirm its presence, which has been recently reported (Jura et al. 2007). See the electronic edition of the Journal for a color version of this figure. tained. Thus, we choose the 0.4 + 0.8M⊙ simulation as our reference model, although reasonable results can be obtained adopting other masses. In passing, we note that nevertheless the He abundance is rather uncertain since equally good fits to the observed spectrum of GD 362 can be obtained with very different He abundances. Thus, the mass of GD 362 is also rather uncertain. More im- portantly, if the mechanism producing the unusual pho- tospheric abundance pattern of GD 362 were to be ac- cretion from the inner regions of the disk — which are C– and O–rich — atomic lines of CI, and C2 molecular bands should be rather apparent in the spectrum. But the strength of these spectral features depends very much on the adopted He abundance, because the atmospheric pressure and the opacity also depend very much on the H/He ratio, which is rather uncertain. In order to know whether the chemical abundances of GD 362 can be reproduced by direct accretion from the keplerian disk we proceed as follows. Given the sur- face gravity and the effective temperature of our model we compute the luminosity, the radius and the cooling time of the white dwarf according to a set of cooling sequences (Salaris et al. 2000). We obtain respectively log(LWD/L⊙) ≃ −3.283, log(RWD/R⊙) ≃ −2.096, and tcool ≃ 2.2 Gyr. Hence, in this scenario GD 362 has had enough time from the moment in which the merger occurred to cool down, to accrete most of the C– and O–rich region, settle down the accretion disk, and to form dust. Additionally, the central white dwarf has had time enough to accrete (at a rate much smaller than the Bondi–Hoyle accretion rate) the small amount of hydro- gen from the ISM to show spectroscopic hydrogen fea- tures. We further assume that the accretion luminosity: Lacc = GṀMWD Evidence for a merger of binary white dwarfs: the case of GD 362 3 Fig. 2.— Spectral energy distribution of GD 362. The figure shows the spectral energy distribution of GD 362. The dotted line shows the spectrum of a white dwarf with and effective temperature of 9740 K and log g = 8.72, which corresponds to a mass of about one solar mass, the dashed line shows the spectrum of a passive flat, opaque dust disk and the solid line depicts the composite spectrum. The observational data were obtained from Becklin et al. (2005). is, in the worst of the cases, smaller than the luminosity of the white dwarf. This provides us with an (extreme) upper limit to the accretion rate, which turns out to be 1.3 × 10−13M⊙ yr −1. Next, we assume that the abun- dance of Ca is the result of the equilibrium between the accreted material and gravitational diffusion: ṀXdisk = MenvXobs τdiff where Xdisk is the abundance in the accretion disk, Xobs is the photospheric abundance, Menv is the mass of the envelope of GD 362 and τdiff is the diffusion timescale. The diffusion timescale of Ca for H–rich at- mospheres is of the order of a few years. However, the accreted material is He–rich, so the diffusion timescale is probably more typical of a He–rich envelope, which is much larger (Paquette et al. 1986), of the order of τdiff ∼ 1.5 × 10 4 yr. Unfortunately, diffusion timescales for mixed H/He envelopes do not exist. However, the diffusion characteristic times scale as τdiff ∝ ρT −1/2g−2 (Alcock & Illarionov 1980). We have computed detailed atmosphere models for pure H, He/H=1 and He/H=10 and scaled the diffusion timescale using the values of the density and the temperature at the base of the convective zones and the appropriate chemical composi- tion. For our fiducial composition (He/H=10) we obtain τdiff ∼ 8.5 × 10 3 yr. From this we obtain the mass of the region where diffusion occurs, which turns out to be Menv ∼ 7.2× 10 −9M⊙, which is much smaller than that obtained by accretion from the interstellar medium at the Bondi–Hoyle accretion rate (∼ 1.5 × 10−6M⊙). Hence, the photospheric abundances of GD 362 can be success- fully explained by direct accretion from the surrounding disk. Now we assess whether the flux from the accretion disk can be fitted by the results of our SPH simulations. In Fig. 3.— Evolution of the rotational velocity for several field strengths, the observational upper limit is shown as a horizontal dashed line. order to compute the flux radiated away from the system two contributions must be taken into account. The first one is the expected photospheric flux from the star, for which we use the spectral energy distribution (BWD) of a white dwarf of mass 1M⊙, at Teff ≈ 9740 K: FWD = π BWD(Teff), (3) Given the luminosity of our model and the apparent magnitude of GD 362 we obtain a distance of DWD = 33 pc. The second contribution to the total flux comes from the emission of the disk, which for a passive flat, opaque dust disk is (Chiang & Goldreich 1997; Jura 2003): Fdisk ≃ 12π 1/3 cos i 2kBTs ∫ xout ex − 1 dx (4) where i is the inclination of the disk (which we adopt to be face–on), xin = hν/kBTin and Tin = 1200 K is the condensation temperature of silicate dust. The outer ra- dius is taken from the results of our SPH simulations and turns out to be Rout ≈ 1R⊙. The result is displayed in Fig. 2. The dots are the observational data for GD 362. The proposed scenario has apparently two weak points. The first one is that infrared observations indicate the presence of SiO. This requires that O should be more abundant than C in order to form it. However our simu- lations show that the ratio of C to O is a function of the distance to the primary and, in some regions of the disk the ratio is smaller than 1, allowing for the formation of SiO in the accretion disk. Furthermore, after 2.2 Gyr of evolution the resulting disk has had time to form planets or asteroids with the subsequent chemical differentiation. The second apparent drawback of the model is that the central white dwarf rotates very fast. However, an 4 Garćıa–Berro et al. unobservable magnetic field can brake down the central star to acceptable velocities. Using the observed spec- trum of GD 362 it is possible to set an upper limit to the rotation speed of v sin i . 500 km s−1. We assume that the central white dwarf has a weak magnetic field, B. The magnetic torques that lead to spin–down are caused by the interaction between the white dwarf and the surrounding disk. The evolution of the angular ve- locity due to the coupling of the white dwarf magneto- sphere and the disk is given by (Livio & Pringle 1992; Armitage & Clark 1996; Benacquista et al. 2003): 2µ2Ω3 sin2 α+ (RcRm)3/2 ṀR2mΩ where µ = BR3WD, Rm is the magnetospheric radius of the star, I is the moment of inertia, α is the angle be- tween the rotation and magnetic axes (which we adopt to be 30◦) and is the corotation radius. The first term in this expression corresponds to the magnetic dipole radiation emission, the second to the disk–field coupling and the last one to the angular momentum transferred from the disk to the white dwarf. The magnetic linkage between the star and the disk leads to a spin–down torque on the star if the magnetospheric radius is large enough relative to the corotation radius: ≥ 2−2/3 (7) We adopt Rm = Rc. Solving numerically the previous differential equation with the appropriate parameters for our case, the evolution of the rotation velocity is shown in figure 3. As can be seen a weak magnetic field of about 50 kG is able to brake down the white dwarf to velocities below the observational limit. This magnetic field is much smaller than the upper limit of about 0.7 MG obtained from the spectrum of GD 362. Hence, our scenario also accounts for the low rotational velocity of GD 362, without adopting extreme assumptions. 3. CONCLUSIONS We have shown that the anomalous photospheric chem- ical composition of the DAZ white dwarf GD 362 and of the infrared excess of surrounding disk can be quite nat- urally explained assuming that this white dwarf is the result of the coalescence of a binary white dwarf system. This scenario provides a natural explanation of both the observed photospheric abundances of GD 362 and of its infrared excess without the need to invoke extreme as- sumptions, like the accretion of a planet or an asteroid, since direct accretion from the disk surrounding disk pro- vides a self–consistent way of polluting the envelope of the white dwarf with the required amounts of Ca, Mg, Si and Fe. Moreover, this last scenario can be also well accomodated within the framework of our scenario given that the formation of planets and other minor bodies is strongly enhanced in metal–rich disks. Hence, GD 362 could be the relic of a very rare event in our Galaxy: the coalescence of a double white dwarf binary system. This work has been partially supported by the MEC grants AYA05–08013–C03–01 and 02, by the European Union FEDER funds, by the AGAUR and by the Barcelona Supercomputing Center (National Supercom- puter Center). This work was also supported in part by the NSERC (Canada). P. Bergeron is a Cottrell Scholar of the Research Corporation. 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0704.1259

Intersection local time for two independent fractional Brownian motions

Intersection Local Time for two Independent Fractional Brownian Motions David Nualart Department of Mathematics, University of Kansas 405 Snow Hall, Lawrence, 66045 KS, USA [email protected], http://www.math.ku.edu/˜nualart/ Salvador Ortiz-Latorre Facultat de Matemàtiques, Universitat de Barcelona Gran Via 585 08007 Barcelona, Spain [email protected] Abstract Let BH and eBH be two independent, d-dimensional fractional Brow- nian motions with Hurst parameter H ∈ (0, 1) . Assume d ≥ 2. We prove that the intersection local time of BH and eBH s )dsdt exists in L2 if and only if Hd < 2. Keywords: Fractional Brownian motion. Intersection local time. Mathematics Subject Classification MSC2000: 60G15, 60F25, 60G18, 60J55. 1 Introduction We consider two independent fractional Brownian motions onRd, d ≥ 2, with the same Hurst parameter H ∈ (0, 1) . This means that we have two d-dimensional independent centered Gaussian processesBH = BHt , t ≥ 0 and B̃H = {B̃Ht , t ≥ 0} with covariance structure given by s ] = E[B̃ s ] = δijRH (s, t) , where i, j = 1, ..., d, s, t ≥ 0 and RH (s, t) ≡ t2H + s2H − |t− s| http://arxiv.org/abs/0704.1259v1 http://www.math.ku.edu/~nualart/ The object of study in this paper will be the intersection local time of BH and B̃H , which is formally defined as I(BH , B̃H) ≡ t − B̃ s )dsdt, where δ0 (x) is the Dirac delta function. It is a measure of the amount of time that the trajectories of the two processes, BH and B̃H , intersect on the time interval [0, T ] . As we pointed out before, this definition is only formal. In order to give a rigorous meaning to I(BH , B̃H) we approximate the Dirac function by the heat kernel pε (x) = (2πε) exp(− |x| /2ε), in Rd. Then, we can consider the following family of random variables indexed by ε > 0 H , B̃H) ≡ t − B̃ s )dsdt, that we will call the approximated intersection local time of BH and B̃H . We are interested in the L2 (Ω) convergence of Iε(B H , B̃H) as ε tends to zero. For H = 1/2, the processes BH and B̃H are classical Brownian motions. The intersection local time of independent Brownian motions has been studied by several authors (see Wolpert [9] and Geman, Horowitz and Rosen [2]). The approach of these papers rely on the fact that the intersection local time of independent Brownian motions can be seen as the local time at zero of some Gaussian vector field. This approach easily allows to consider the intersection of k independent Wiener processes, k ≥ 2. The applications of the intersection local time theory for Brownian motions range from the construction of relativis- tic quantum fields, see Wolpert [10], to the construction of the self-intersection local time for the Brownian motion, see LeGall [4]. Further research has been done in order to study such problems for other types of stochastic processes, mainly Lévy processes with a particular structure (strongly symmetric), see Marcus and Rosen [6]. In the general case, that is H 6= 1/2, only the self-intersection local time has been studied. Rosen studied in [11] the planar case and a recent paper by Hu and Nualart [3] gives a complete picture for the multidimensional case. On the other hand, Nualart et al. [8] used a weighted version of the 3-dimensional self- intersection local time for the study of probabilistic models for vortex filaments based on the fractional Brownian motion . In recent years the fBm has become an object of intense study. A stochastic calculus with respect to this process has been developed by many authors, see Nualart [7] for an extensive account on this subject. Because of its interesting properties, such as short/long range dependence and selfsimilarity, the fBm it’s being widely used in a variety of areas such finance, hydrology and telecommunications engineering, see [8]. Therefore, it seems interesting to study the intersection local time for this kind of processes. The aim of this paper is to prove the existence of the intersection local time of BH and B̃H , for an H 6= 1/2 and d ≥ 2. We have obtained the following result. Theorem 1 (i) If Hd < 2, then the family of random variables Iε(B H , B̃H) converges in L2 (Ω). We will denote this limit by I(BH , B̃H). (ii) If Hd ≥ 2, then E[Iε(B H , B̃H)] = +∞ Var[Iε(B H , B̃H)] = +∞. If {B t , t ≥ 0} is a planar Brownian motion, then B1/2s −B diverges almost sure, when ε tends to zero. Varadhan, in [12], proved that the renormalized self-intersection local time defined as limε→0(Iε − E[Iε]), exists in L2 (Ω). Condition (ii) implies that Varadhan renormalization does not converge in this case. For Hd ≥ 2, according to the previous theorem, Iε(B H , B̃H) doesn’t con- verge in L2 (Ω) and therefore I(BH , B̃H), the intersection local time of BH and B̃H , doesn’t exist. The proof of Theorem 1.1 rest on Lemma 4, which deals with the integral of a negative power of the determinant of some covariance matrix. The paper is organized as follows. In Section 2 we prove Theorem 1.1. In order to clarify the exposition, some technical lemmas needed in the proof are stated and proved in the Appendix. 2 Intersection Local Time of BH and B̃H , Case Hd < 2 Let BH and B̃H two independent fractional Brownian motions on Rd with the same Hurst parameter H ∈ (0, 1) . Using the following classical equality pε (x) = ei〈ξ,x〉e−ε from Fourier analysis, and the definition of Iε(B H , B̃H), we obtain H , B̃H) = ei〈ξ,B t − eB s 〉e−ε 2 dξdsdt. (1) Therefore, E[Iε(B H , B̃H)] = E[ei〈ξ,B t − eB s 〉]e−ε 2 dξdsdt e−(ε+s 2H+t2H) 2 dξdsdt (ε+ s2H + t2H)−d/2dsdt, (2) where we have used that 〈ξ, BHt − B̃ s 〉 ∼ N(0, |ξ| s2H + t2H ), so E[ei〈ξ,B t − eB s 〉] = e−(s 2H+t2H ) and the fact that e−(ε+s 2H+t2H ) 2 dξ = ε+ s2H + t2H According to the representation (1) for Iε(B H , B̃H), we have that E[I2ε (B H , B̃H)] = [0,T ]4 E[ei(〈ξ,B t − eB s 〉+〈η,B v − eB u 〉)] × e−ε |ξ|2+|η|2 2 dξdηdsdtdudv. (3) Let introduce some notation that we will use throughout this paper, λ = λ (s, t) = s2H + t2H , ρ = ρ (u, v) = u2H + v2H , µ = µ (s, t, u, v) = s2H + t2H + u2H + v2H − |t− v| − |s− u| Notice that λ is the variance of B t − B s , ρ is the variance of B v − B and µ is the covariance between B s and B u , where B H,1 and BH,2 are independent one-dimensional fractional Brownian motions with Hurst parameter H. Using that 〈ξ, BHt − B̃ s 〉 + 〈η,B v − B̃ u 〉 ∼ N(0, λ |ξ| + ρ |η| + 2µ〈ξ, η〉) and (3) we can write for all ε > 0 E[I2ε (B H , B̃H)] [0,T ]4 {(λ+ε)|ξ|2+(ρ+ε)|η|2+2µ〈ξ,η〉}dξdηdsdtdudv [0,T ]4 ((λ + ε)(ρ+ ε)− µ2)−d/2dsdtdudv. (4) The last equality follows from the well known fact that 〈x,Ax〉dx = (detA) A = Idd ⊗ λ+ ε µ µ ρ+ ε where Idd is the d-dimensional identity matrix and ⊗ denotes the Kronecker product of matrices. We also have that detA = det Idd ⊗ λ+ ε µ µ ρ+ ε = (det (Idd)) λ+ ε µ µ ρ+ ε = ((λ + ε)(ρ+ ε)− µ2)d. Proof of Theorem 1. Suppose first that Hd < 2. A slight extension of (4) yields E[Iε(B H , B̃H)Iη(B H , B̃H)] = [0,T ]4 ((λ + ε)(ρ+ η)− µ2)−d/2dsdtdudv. Consequently, a necessary and sufficient condition for the convergence in L2 (Ω) of Iε(B H , B̃H) is that [0,T ]4 (λρ− µ2)−d/2dsdtdudv < +∞. Then the result follows from Lemma 4. Now suppose that Hd ≥ 2, then from (2) and using monotone convergence theorem E[Iε(B H , B̃H)] = s2H + t2H )−d/2 dsdt, and this integral is divergent by Lemma 3. According to the expression (2) for E[Iε(B H , B̃H)] and the expression (4) for E[I2ε (B H , B̃H)] we obtain Var[Iε(B H , B̃H)] = lim E[Iε(B H , B̃H)2]− E[Iε(B H , B̃H)] [0,T ]4 (λρ− µ2)−d/2 − (λρ) dsdtdudv. Dε := {(s, t, u, v) ∈ R + | s 2 + t2 + u2 + v2 ≤ ε2}. (5) We can find ε > 0 such that Dε ⊂ [0, T ] . Making a change to spherical coor- dinates, as the integrand is always positive, we have [0,T ]4 (λρ− µ2)−d/2 − (λρ) dsdtdudv (λρ− µ2)−d/2 − (λρ) dsdtdudv = r3−2Hddr Ψ(θ) dθ, where the integral in r is convergent if and only if Hd < 2, and the angular in- tegral is different from zero thanks to the positivity of the integrand. Therefore, if Hd ≥ 2, then Var[Iε(B H , B̃H)] = +∞. 3 Appendix For clarity of exposition, we state and prove some technical lemmas in this appendix. Lemma 2 Let α > 0, and let γ (α, x) ≡ e−yyα−1dy (6) be the lower incomplete gamma function. Then for all ε < α and x > 0, γ (α, x) ≤ K (α)xε, where K (α) ≡ 1 ∨ Γ (α) and Γ (α) = γ (α,+∞). Proof. If x ≥ 1, γ (α, x) ≤ Γ (α) xε, for all ε > 0. On the other hand, if x < 1, γ (α, x) ≤ yα−1dy = if ε < α. Lemma 3 The following integral s2H + t2H )−d/2 dsdt, is finite if and only if Hd < 2. Proof. It easily follows from a polar change of coordinates. Lemma 4 Let [0,T ]4 (λρ− µ2)−d/2dsdtdudv, then AT < +∞ if and only if Hd < 2. Proof. The necessary condition follows from a spherical change of coordinates. We can find ε > 0 such that Dε ⊂ [0, T ] , where Dε is given in (5) . As the integrand in AT is always positive we have (λρ− µ2)−d/2dsdtdudv = r3−2Hddr φ (θ) dθ, where the integral in r is convergent if and only if Hd < 2, and the angular in- tegral is different from zero thanks to the positivity of the integrand. Therefore, if Hd ≥ 2, then AT = +∞. Suppose now that Hd < 2. By symmetry we have that AT = 4 λρ− µ2 )−d/2 dsdtdudv, where T ≡ {(s, t, u, v) : 0 < v < t, 0 < t ≤ T, 0 < u < s, 0 < s ≤ T )}. Notice that λρ− µ2 = detVar (Z) , where Z ≡ (B t − B̃ s , B v − B̃ u ). Due to the independence of B H and B̃H , we have that Var (Z) = Var(B t , B v ) + Var(B̃ s , B̃ λρ− µ2 ≥ det(Var(B t , B v )) + det(Var(B̃ s , B̃ u )), because the matrices Var(B t , B v ) and Var(B̃ s , B̃ u ) are strictly positive definite (see A8, (viii) in [5]). Then AT ≤ 4 (ϕ (t, v) + ϕ (s, u))−d/2dsdtdudv, where ϕ (t, v) ≡ det(Var(B t , B v )) = t 2Hv2H − t2H + v2H − |t− v| Using Fubini’s Theorem and λ−α = Γ (α) e−λzzα−1dz, for all λ, α > 0, we obtain (ϕ (t, v) + ϕ (s, u))−d/2dsdtdudv e−(ϕ(t,v)+ϕ(s,u))zz −1dzdsdtdudv −1A2 (z) dz, (7) where A (z) ≡ e−ϕ(t,v)zdvdt. As A (z) < +∞, for all z ∈ [0, 1], the integral (7) is convergent in a neighborhood of zero. Hence, we have to study the convergence of −1A2 (z) dz. Due to the homogeneity of order 4H of ϕ (t, v) , if we make the change of coor- dinates t = z− 4H x, v = z− 4H y, we obtain −1A2 (z) dz = −1− 1 ∫ Tz 14H e−ϕ(x,y)dydx Now, using that {(x, y) : 0 < x < Tz 4H , 0 < y < x} ⊂ {(x, y) : x2 + y2 ≤ 2T 2z 2H } ≡ S, and making a polar change of coordinates we have ∫ Tz 14H e−ϕ(x,y)dydx ≤ e−ϕ(x,y)dydx = ∫ π/4 ∫ √2Tz 14H 4Hϕ(θ)drdθ, where ϕ (θ) ≡ ϕ (cos θ, sin θ) . After the new change of variable x = r4Hϕ (θ) , the last integral is equal to ∫ π/4 ϕ (θ) ∫ 22HT 4Hzϕ(θ) −1dxdθ ∫ π/4 ϕ (θ) (2H)−1, 22HT 4Hzϕ (θ) where γ (α, x) is given by (6) . Applying Lemma 2, −1A2 (z)dz −1− 1 (∫ π/4 ϕ (θ) (2H)−1, 22HT 4Hzϕ (θ) 24HεT 8Hε )∫ +∞ −1− 1 +2εdz (∫ π/4 ϕ (θ) 2H dθ The integral in z is convergent provided ε < 2−Hd . It’s an exercise of compu- tation of limits to prove that ϕ (θ) ∼ θ2H as θ ↓ 0 and ϕ (θ) ∼ (π/4 − θ)2H as θ ↑ π/4, the main tool is to substitute the trigonometric functions by their first order approximations at the respective points. As a consequence, the integral ∫ π/4 ϕ (θ) 2H dθ is always convergent. References [1] Doukhan P., Oppenheim G., Taqqu M.S. (2003). Theory and Applications of Long Range Dependence. Birkhäuser, Boston. [2] Geman, D., Horowitz, J., Rosen, J. (1984). A Local Time Analysis of Inter- sections of Brownian Paths in the Plane. Annals of Probability. 12 86-107. [3] Hu, Y., Nualart, D. (2005). Renormalized Self-Intersection Local Time for Fractional Brownian Motion. Annals of Probability. 33 948-983. [4] LeGall, J.F. (1985). Sur le Temps Local d’Intersection du Mouvement Brownien Plan et la Méthode de Renormalisation de Varadhan.Séminaire de Probabilités XIX. Lecture Notes in Math. 1123, 314-331. Springer, Berlin. [5] Muirhead, R.J. (1982) Aspects of Multivariate Statistical Theory. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York. [6] Marcus, M.B., Rosen, J. (1999).Additive Functionals of Several Levy Pro- cesses and Intersection Local Times. Annals of Probability. 27 1643-1678. [7] Nualart, D. (2003) Stochastic Integration with Respect to Fractional Brow- nian Motion and Applications. Contemporary Mathematics. 336 3-39. [8] Nualart, D., Rovira, C. and Tindel S. (2003) Probabilistic Models for Vor- tex Filaments Based on Fractional Brownian Motion. Annals of Probability. 31 1862-1899. [9] Wolpert, R. (1978). Wiener Path Intersections and Local Time. Journal of Functional Analysis. 30 329-340. [10] Wolpert, R. (1978). Local Time and a Particle Picture for Euclidean Field Theory. Journal of Functional Analysis. 30 341-357. [11] Rosen, J. (1987). The intersection local time of fractional brownian motion in the plane. Journal of Multivariate Analysis. 23 37-46. [12] Varadhan, S.R.S. (1969) Appendix to ”Euclidian quantum field theory”. By K. Symanzik, in : ”Local Quantum Theory”. Jost, R. (ed). Academic Press, New York. Introduction Intersection Local Time of BH and B"0365BH, Case Hd<2 Appendix

0704.1260

Lapse of transmission phase and electron molecules in quantum dots

Lapse of transmission phase and electron molecules in quantum dots S.A. Gurvitz∗ Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel (Dated: October 25, 2018) The puzzling behavior of the transition phase through a quantum dot can be understood in a natural way via formation of the electron molecule in the quantum dot. In this case, the resonance tunneling takes place through the quasistationary (doorway) state, which emerges when the number of electrons occupying the dot reaches a certain “critical” value, Ncr. Our estimation of this quantity agrees with the experimental data. The dependence of Ncr on the dot’s size is predicted as well. PACS numbers: 73.23.Hk, 73.43.Jn, 73.50.Bk, 73.63.Kv One of the challenging problems in mesoscopic physics is the puzzling behavior of the transmission phase through a quantum dot, embedded in an Aharonov-Bohm ring. It was found in series of experiments performed by the Weizmann group1,2,3 that all transmission am- plitudes through different resonant levels of a quantum dot are in phase. This necessarily implies an unexpected lapse in the evolution of the transmission phase between different resonant levels. In addition, it was found in re- cent measurements3 that this phenomenon takes place when the number of electrons inside the dot reaches a certain “critical” value (Ncr & 15) 3. In spite of many publications addressed to these experiments no fully sat- isfactory understanding has been found yet4,5. In this Rapid Communication we demonstrate that the observed phase-lapse behavior of the transmission am- plitude can be naturally explained by implying the for- mation of electron (Wigner) molecules inside quantum dots, proposed in recent publications6,7,8,9. Moreover, this framework allows us to estimate Ncr and then to determine how it is varied with a size of the dot. In order to explain our model in a proper way, we first elab- orate the physical nature of the transmission phase in the case of noninteracting and interacting electrons. In particular, we concentrate on the role of the Pauli prin- ciple that prevents different conductance resonances to carry essentially the same internal wave function. This point represents a formidable obstacle for resolving the puzzling behavior of the transmission phase for different models of the quantum dot. We demonstrate, however, that this difficulty can be overcame in the context of the Wigner-molecule when an unstable state is developed in the middle of the dot. Let us consider the resonant tunneling through a quan- tum dot, represented by a potential UD(x), Fig. 1. The bottom of this potential can be moved by the plunger electrode, so that one observes the current sweeping trough different resonant states (Eλ) of the dot. We would treat this problem in the framework of a tunnel Hamiltonian approach. This approach is more trans- parent for evaluation of the transmission phase than the standard scattering theory, in particular, when the Pauli principle and the electron-electron interaction are taken into account. We introduce therefore the following tun- neling Hamiltonian: H = HL +HR +HD +HT , where xxl xr U (x)D Lµ Ωr (λ)Ωl U (x)D FIG. 1: (Color online) Resonant tunneling trough a quantum dot. µL(R) are Fermi energies in the left (right) reservoir. The dotted lines show the potential ŪD(x), needed for evaluation of the bound state wave functions in the Bardeen formula. HL(R) = El(r)a al(r) , HD = kdk +HC , kal + l ↔ r +H.c. (1) Here, a l,r(al,r) is the creation (annihilation) operator of an electron in the reservoirs and d k(dk) is the same op- erator for an electron inside the dot. For simplicity, we consider electrons as spin-less fermions. The term HC denotes the Coulomb interaction between electrons in the dot and Ω r ] is the coupling between the states El(Er) and Ek of the reservoir and the dot, respectively. In the absence of magnetic field, all couplings Ω are real. All parameters of the tunneling Hamiltonian (1) are re- lated to the initial microscopic description of the system in the configuration space. For instance, the coupling is given by the Bardeen formula10 = − ~ x∈Σl(r) φk(x) ∇n χl(r)(x)dσ , (2) where φk(x) and χl(r)(x) are the electron wave functions inside the dot and the reservoir, respectively, and Σl(r) is a surface inside the left (right) barrier that separates the dot from the corresponding reservoir. It is impor- tant to point out that φk(x) in Eq. (2) is a bound state wave function for the “inner” potential. The latter co- incides with the original potential inside the surface Σ http://arxiv.org/abs/0704.1260v3 and a constant outside this region. On the other hand, χl(r)(x) is a non-resonant scattering wave function in the “outer” potential, which coincides with the original po- tential outside the surface Σ and a constant inside this region11. In one-dimensional case (Fig. 1), the separation surface Σ becomes the separation point, x̄, inside the barrier, Fig. 1. Then Eq. (2) can be rewritten as12 = −(κk/m)φk(x̄l(r))χl(r)(x̄l(r)) , (3) where κk = 2m[UD(x̄l,r)− Ek] and φk(x) is the bound state wave function in the potential ŪD(x) (Fig. 1). The separation points x̄l,r are to be taken inside the left (right) barrier as indicated in Fig. 1 and far away from the classical turning points13. We start with non-interacting electrons, HC = 0 in Eq. (1). Then the electron transport through the level Eλ can be described by the time-dependent Schrödinger equation i~∂t|Ψ(t)〉 = H |Ψ(t)〉 for a single electron. Tak- ing the stationary limit we obtain the Landauer formula for the total current, with the transmission amplitude given by the Bright-Wigner formula tλ(E) = N E − Eλ + i(Γ(λ)L + Γ R )/2 , (4) where N = −2π(̺L̺R)1/2 and Γ(λ)L,R = 2π(Ω 2̺L,R are the partial widths, and ̺L(R) is the density of states in the left (right) reservoir. We assumed that Ω l,r ≡ Ω are weakly dependent on El,r. The corresponding evolution of the resonance trans- mission phase for different states |λ〉 is determined by sign of the product of Ω R . Since the reservoir states χl,r are not affected by the plunger voltage, one finds from Eq. (3) that the evolution of the sign [Ω R ] is given by the sign of the product φλ(x̄l)φλ(x̄r). The lat- ter is positive or negative, depending on the number of nodes of φλ(x). Hence, it is clear that the non-interacting electron model cannot explain the same sign for all reso- nances, observed in Ref.2 (see also Refs.4,14). Consider N interacting electrons trapped inside the dot. Despite the electron-electron interaction, the cou- pling amplitudes ΩL,R can still be evaluated by us- ing the same multi-dimensional overlapping formula (2), as in the non-interacting case. Indeed, the many-body tunneling can be considered as one-body tunneling, but in the many-dimensional space. In this case, the wave-function χl(r)(x) is replaced by χl(r)(xN+1)Φ N (x1, . . . , xN ), where χl(r) is the wave function of tunneling electron in the left (right) reser- voir and Φ N is the ground state wave function of N electrons inside the dot. The wave-function φk(x) cor- responds to Φ N+1(x1, . . . , xN+1), which is the lowest en- ergy state (ground state) of N + 1 electrons in the inner potential of the dot (ŪD in Fig. 1). Taking n along a coordinate of the tunneling electron, xN+1, we can integrate over x1, . . . , xN in Eq. (2) thus reducing this equation to Eq. (3) with φn being replaced by the overlap function ϕN (xN+1) = 〈xN+1,Φ(0)N |Φ N+1〉 . (5) Therefore, the sign of Ω R is determined by the sign of ϕN (x̄l)ϕN (x̄r). By applying the mean-field approximation, we can write |Φ(0)N 〉 and |Φ N+1〉 as a product of one-electron states (orbitals) in the effective single-particle potential, ŪD + UC , where ŪD is the inner part of quantum-dot potential (Fig. 1) and UC(x) is the mean-field describ- ing the electron-electron interaction. As a result, the overlap function ϕN (x) is a bound state wave function in the potential ŪD(x) + UC(x), corresponding to one of the orbitals. Since the lowest energy state is always nodeless15, one might assume that ϕN (x) is also a node- less one, so that the sign of ϕN (x̄l)ϕN (x̄r) would be the same sign for all resonances. This, however, is not correct because of the Pauli principle. Indeed, due to the anti- symmetrization, any two orbitals in the product of the wave functions representing |Φ(0)N+1〉 cannot be the same. Since the lowest state is already occupied, the wave func- tion ϕN (xN+1) must correspond to a higher non-occupied orbital, and therefore it cannot be nodeless. Hence, the Pauli principle would create serious problems in any at- tempt to explain the same sign for all resonances2 in a framework of the mean-field description of the electron- electron interaction. Note that this problem cannot be resolved even by as- suming large coupling with reservoirs, so that the reso- nances are overlap. Indeed, the problem is related only to the inner component of the resonant state, Eqs. (2) and (3). The latter is eventually brought by the plunger below the Fermi level, µR, blocking an appearance of the resonance above the Fermi level with a similar inner component. The same situation holds in a more general case, when the interaction term UC varies with each new electron trapped inside the dot, UC → U (N)C (Koopman‘s theo- rem is violated). One finds that due to the central sym- metry of the self-consistent potential such a variation of UC with N would not affect the number of nodes in the overlap function ϕN (xN+1). As a result, the sign of the transmission amplitude would fluctuate between ±1 for different resonances. We illustrate this point by evaluating the overlap func- tion ϕN (xN+1), Eq. (5), for N = 0 and N = 1. In the first case, ϕ0(x1) coincides with the wave function of the lowest energy state, Φ 1 (x1) ≡ φ̃0(x1), in the in- ner potential ŪD, Fig. 1. This wave function is nodeless. The second overlap function is ϕ1(x2) = 〈x2,Φ(0)1 |Φ where Φ 2 (x1, x2) = [φ0(x1)φ1(x2) − φ0(x2)φ1(x1)]/ is the lowest energy state of two electrons in the potential ŪD + U C . Here φ0,1 represent the two first orbitals in this potential. One easily finds that ϕ1(x2) = 1 (x1)Φ 2 (x1, x2)dx1 = c0φ1(x2) , (6) where c0 = φ̃0(x1)φ0(x1)dx1/ 2. (The second term is zero, since φ̃0 and φ1 are orthogonal due to the opposite parities). Therefore, ϕ1 contains one node, so that the corresponding transition amplitude changes its sign. The same behavior of the overlap function would per- sist for any N . For instance, one easily obtains for N = 2 that ϕ2(x3) ∝ c0φ3(x3)−c13φ1(x3), where the coefficients c0 = 〈φ̃1|φ1〉, c13 = 〈φ̃0|φ2〉 and φ̃, φ are the orbitals in the potentials, ŪD(x)+U C (x) and ŪD(x)+U C (x), re- spectively. Since c13 ≪ c0, the overlap function ϕ2 would contain an additional node in a comparison to ϕ1. Thus, by assuming the N dependence of the mean-field effec- tive potential, we are still not able to explain the puzzling behavior of the transmission phase. The above consideration was based on symmetry argu- ments applied to electrons moving in a spherically sym- metric mean-field central potential. In fact, the central mean-field picture for two-dimensional quantum dots was challenged in recent publications6,7,8,9. It was suggested that due to the strong inter-electron repulsion inside the dot, spontaneous symmetry breaking takes place lead- ing to the formation of electron molecules. As a re- sult, the electrons appear on the ring (rings) around the dot’s center. This idea was substantiated by unrestricted Hartree-Fock calculations or by using other computa- tional techniques6,7,8,9. In principle, if the symmetry is broken, the overlap function ϕN (xN+1), Eq. (5), could be very different from the corresponding orbital φN (xN+1) in the spherical sym- metric potential. Therefore, it is desirable to investigate the evolution of the transmission phase in this case. Con- sider again the overlap function ϕ1(x2) = 〈x2,Φ(0)1 |Φ but now without the mean-field approximation, as in Eq. (6). In fact, by taking the parabolic confining poten- tial, the two-electron wave function |Φ(0)2 〉 can be exactly calculated16, since relative and center-of-mass coordi- nates of two electrons are decoupled in the total Hamilto- nian. As a result, Φ 2 (x1, x2) = φcm(x1+x2)φr(x2−x1), where φr(−x) = −φr(x) due to the Pauli principle. Such a wave function peaks for x1 = −x2 and therefore it would bear the features of a two-electron molecule9. One finds from Eq. (5), ϕ1(x2) = φ̃0(x1)φcm(x1 + x2)φr(x2 − x1)dx1 . (7) Taking into account that the values of x1 which mainly contribute to the integral (7) are localized inside the dot and that the wave function φr(x) is the odd one, we find that the overlap function changes its sign when the ar- gument varies from x̄l to x̄r, Fig. 1. Hence, ϕ1 displays one node, as in the spherically symmetric mean-field po- tential, Eq. (6). One can continue with the same arguments for the three and more electron molecules, where the electrons are placed on the ring. The corresponding ground state wave functions |Φ(0)N 〉 would represent a fully anti-symmetrized product of the original (site) nodeless orbitals7,8. Yet, the overlap function Eq. (5) cannot be nodeless. As a result, the sign of ϕN (x̄l)ϕN (x̄r) would fluctuate with N . One can demonstrate it rather easily for N = 3, 4. Although it would be hard to extend such a demonstration for large N , there is no reason to expect that the sign of ϕN (x̄l)ϕN (x̄r) ceases to fluctuate when N increases. It seems from the above arguments that the rota- tional symmetry breaking (the electron-molecule forma- tion) cannot explain the evolution of the transmission phase observed in the experiments2,3. Nevertheless, there is an additional feature of the electron molecule, which has not been yet utilized. That is due to the electrons located on the ring (rings) would develop an additional (inner) electrostatic trap inside the dot when their num- ber (N) is large enough. As an example, we display in Fig. 2a such a potential, VC(x) = 2/ǫ|x− xj |, pro- duced by 14 electrons equally distributed on the ring, where ǫ = 13.6 is the dielectric constant of the medium. The radius of the ring (R = 50 nm) is taken close to the dot’s size in Ref.3. The radial profile of this potential along the angle θ = π/N , where the potential height on the ring is minimal, is shown in Fig. 2b for two values of N . It appears that the trap is not well developed for N = 6, but it is already pronounced for N = 14. A minimum number of electrons in the dot sufficient to develop the trap with one bound state inside it can be estimated from the condition that the barrier height, hN in the Fig. 2b, reaches the ground state energy ε0. We estimate the latter as π2~2/m∗R2, where m∗ is the effective electron mass (m∗/m0 = 0.067). For instance, one finds ε0 = 4.5 meV for R = 50 nm. Then the condi- tion hN = ǫ0 corresponds to N ≃ 10, which is a minimal (“critical”) number of electrons, Ncr, enabled to hold a resonance state. This value is an approximate agreement with that found in3. In fact, a more elaborate, semi- classical estimations of Ncr approximately produce the same number [hN ≃ 10 meV for N = 14, Fig. 2b]17. The state |ε0〉 in the inner part of the trap, V̄C , Fig. 2b, is not stable due to the symmetry breaking, leading to formation of the electron molecule. Nevertheless, this state is important in formation of the (N + 1)-electron molecule by adding an additional electron to the N - electron system. Indeed, one expects that the overlap function (5) for the electron states on the ring is sup- pressed in comparison to the same overlap for the cen- tral mean-field potential. The reason is that all electrons are shifted from their positions whenever an additional electron is placed on the ring. This is in contrast to the mean-field description, where the N -electron core is not modified. On the other hand, if the electron is placed in the center of the ring, it distorts the remaining N elec- trons in a minimal way. We expect therefore that the 10 20 30 40 50 60 [meV]V V ( ) Nθ=π/ρ,C VCε0 hN FIG. 2: (Color online) (a) Electrostatic trap generated by 14 electrons placed on the ring of the radius 50 nm. (b) The radial profile of the Coulomb potential along the potential valley. ε0 is the ground state energy in the potential V̄C , representing the inner part of the trap VC . corresponding overlap function is large, as in the case of the central mean-field potential. Hence, such an unstable state |ε0〉 in the middle of the dot would play a role of a “doorway” state in formation of the (N + 1)-electron molecule. It follows from the same arguments that the electron transport would proceed through such an unstable state when the quantum dot coupled with the reservoirs. Since this doorway state is of the lowest energy in the inner trap, V̄C (Fig. 2), it is nodeless. The crucial point here is that this state is eventually not occupied, when it is brought by the plunger below the Fermi levels of the reservoirs. Indeed, it is not turned to a stable state be- low the Fermi levels due to the symmetry breaking, but it always decays to the ring states. Therefore, this state is never blocked by the Pauli principle to carry the res- onant transport through it, when it is above the Fermi level µR, Fig. 1. As a result, all transmission amplitudes for any N > Ncr would be in phase. In fact, by taking into account the electron spin, one finds that two electrons with the same spatial (nodeless) wave functions are allowed to occupy the lowest energy states. Therefore, even if the state |ε0〉 in the center of the dot becomes a stable one for some values of N , the resonant transport would proceed through an unstable state of the two electrons (with opposite spin) inside the dot. The corresponding overlap function would be again nodeless. 40 50 60 70 80 90 R@nmD FIG. 3: Dependence of the “critical” number of electrons on the dot’s radius. Note that although the doorway-state energy is the lowest one for the inner trap, V̄C , it exceeds the energy of the ring states. Therefore, the ring states would ap- pear inside the bias voltage before the doorway state. We can assume, however, that the ring states are not well separated in the energy from the doorway state, which dominates the resonant current. It was also taken into account that in the presence of the Coulomb interaction, the shift of the resonance energy due to tunneling is dif- ferent for different levels18. In particular, the broad reso- nance is shifted down more than the narrow one18. As a result the doorway state could have a lower energy than the ring states. One of the consequences of our model is an existence of the critical number of electrons in the dot, which is nec- essary for formation of the resonant state inside the dot (Ncr). This number would vary with the dot’s size. Such a dependence of Ncr on the radius of the dot (R), ob- tained from our estimation, hN = ε0, is shown in Fig. 3. One finds from this figure that this dependence is rather weak. The critical number slightly decreases with an in- crease of the dot’s size. In summary, we demonstrated that the unusual be- havior of the resonant phase, observed in interference ex- periments, can be considered as a strong evidence for formation of electron molecules in quantum dots. This structure would produce an electrostatic trap, contain- ing an unstable (doorway) state localized in the center of the dot, whenever the number of electrons occupying the dot is large enough, N > Ncr. Then such an unstable state would carry the electron transport through the dot irrespective of the value of N . This would appear as if the different transmission amplitudes are in phase. Our prediction for the dependence of Ncr on the dot’s radius can be experimentally verified. ∗ Electronic address: [email protected] 1 A. Yacoby, M. Heiblum, D. Mahalu, and H. Shtrikman, Phys. Rev. Lett. 74, 4047 (1995). 2 R. Schuster, E. Buks, M. Heiblum, D. Mahalu, V. Uman- sky and H. Shtrikman, Nature 385, 417 (1997). 3 M. Avinun-Kalish, M. Heiblum, O. Zarchin, D. Mahalu and V. Umansky, Nature, 436, 529 (2005). 4 G. Hackenbroich, Phys. Rep. 343, 463 (2001). 5 C. Karrasch, T. Hecht, A. Weichselbaum, Y. Oreg, J. von- Delft and V. Meden, Phys. Rev. Lett. 98, 186802 (2007). 6 S. Reimann and M. Manninen, Rev. Mod. Phys. 74, 1283 (2002). 7 C. Yannouleas and U. Landman, Phys. Rev. B 68, 035325 (2003). 8 A. Ghosal, A.D. Güçlü, C.J. Umrigar, D. Ullmo and H.U. Baranger, Nature Phys. 2, 336(2006). 9 C. Yannouleas and U. Landman, Rep. Prog. Phys. 70, 2067 (2007). 10 J. Bardeen, Phys. Rev. Lett. 6, 57 (1961). 11 S.A. Gurvitz, in Multiple facets of quantization and su- persymmetry, Michael Marinov Memorial Volume, p. 91 (World Scientific, 2002). 12 S.A. Gurvitz, Phys. Rev. A 38, 1747 (1988). 13 S.A. Gurvitz, P.B. Semmes, W. Nazarewicz, and T. Vertse, T, Phys. Rev. A69, 042705 (2004). 14 A. Silva, Y. Oreg, and Y. Gefen, Phys. Rev. B66, 195316 (2002). 15 M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. IV (Academic Press, New York, 1977). 16 M. Wagner et al., Phys. Rev. B 45, 1951 (1992). 17 In fact, a two-ring structure is expected for N = 14 electrons6,9. Although this creates a more complicated trap, our simple estimations of Ncr, based on the effective one-dimentional potential, Fig. 2(b), remain the same. For such estimations it is sufficient to consider all electrons on one ring of the average radius of two rings. Note that Ncr is weakly dependent on the ring’s radius, Fig. 3. 18 P.G. Silvestrov and Y. Imry, Phys. Rev. Lett. 85, 2565 (2000); ibid, New J. Phys. 9, 125 (2007). mailto:[email protected]

0704.1261

Quantum-corrected black hole thermodynamics to all orders in the Planck length

Quantum-corrected black hole thermodynamics to all orders in the Planck length Khireddine Nouicer∗ Laboratory of Theoretical Physics and Department of Physics, Faculty of Sciences, University of Jijel Bp 98, Ouled Aissa, 18000 Jijel, Algeria. Abstract We investigate the effects to all orders in the Planck length from a generalized un- certainty principle (GUP) on black holes thermodynamics. We calculate the corrected Hawking temperature, entropy, and examine in details the Hawking evaporation process. As a result, the evaporation process is accelerated and the evaporation end-point is a zero entropy, zero heat capacity and finite non zero temperature black hole remnant (BHR). In particular we obtain a drastic reduction of the decay time, in comparison with the result obtained in the Hawking semi classical picture and with the GUP to leading order in the Planck length. PACS: 04.60.-m, 05.70.-a Key Words: Quantum Gravity, Generalized Uncertainty Principle, Thermodynamics of Black Holes 1 Introduction Recently a great interest has been devoted to the study of the effects of generalized uncertainty principles (GUPs) and modified dispersion relations (MDRs) on black holes thermodynamics. The concepts of GUPs and MDRs originates from several studies in string theory approach to quantum gravity [1- 4], loop quantum gravity [5], noncommutative space-time algebra [6 - 8] and black holes gedanken experiments [9 - 10]. All these approaches indicate that the standard Heisenberg uncertainty principle must be generalized to incorporate additional uncertainties when quantum gravitational effects are taken into account. Actually it is believed that any ∗Email: [email protected] / [email protected] http://arxiv.org/abs/0704.1261v1 promising candidate for a quantum theory of gravity must include the GUPs and/or MDRs as central ingredients. The main consequence of the GUP is the appearance of a minimal length scale of the order of the Planck length which cannot be probed, providing a natural UV cut-off, and thus corrections to black holes thermodynamic parameters are expected at the Planck scale. The consequences of GUPs and/or MDRs on black holes thermodynamics have been considered intensively in the recent literature on the subject [11 - 16]. Notably, it has been shown that GUP prevents black holes from complete evaporation, exactly like the standard Heisenberg principle prevents the hydrogen atom from total collapse [17]. Then at the final stage of the Hawking radiation process of a black hole, a inert black hole remnant (BHR) continue to exist with zero entropy, zero heat capacity and a finite non zero temperature. The inert character of the BHR, besides gravitational interactions, makes this object a serious candidate to explain the nature of dark matter [18, 19]. On the other hand, a particular attention has been also devoted to the computation of the entropy of a black hole and the sub-leading logarithmic correction [20 - 34]. All the above studies have been performed with a GUP to leading order in the Planck length. However, recent generalization of the GUP induces quantitative corrections to the entropy and then influences the evaporation phase of the black hole [35]. Besides this growing interest in quantum gravity phenomenology, a intense activity is actually devoted to possible production of black holes at particle colliders [36, 37] and in ultrahigh energy cosmic ray (UHECR) airshowers [38, 39]. The next generation of particle colliders are planned to reach a c-m energy of the order of few TeV , a scale at which the complete evaporation of BH is expected to end, leaving up in a scenario with GUP a inert BHR. Then, it is phenomenologically relevant, to obtain the corrections to BH thermodynamic parameters in the framework of a GUP beyond the leading order in the Planck length. In this paper we discuss the effects, brought by a generalization of the GUP to all orders in the Planck length, on thermodynamic parameters of the Schwarzschild black hole . Hereafter, we refer to this version of GUP as GUP∗. The organization of this work is as follows. In section 2, we introduce a deformed position and momentum operators algebra leading to GUP∗ and examine its various implications. In section 3, the Hawking temperature and entropy are computed and the departures from the standard case shown. In section 4, we calculate the deviation from the standard Stefan-Boltzmann law of the black body radiation spectrum and investigate the Hawking evaporation process of black holes by a calculation of the evaporation rate, the decay time and the heat capacity. Finally we compare our results with the ones obtained in the context of the GUP to leading order in the Planck length commonly used in the literature. Our conclusions are summarized in the last section. 2 Generalized uncertainty principle Loop quantum gravity and string theory approach to quantum gravity predict slight deviations in the laws describing photons propagation in vacuum. It is expected that these effects, leading to a modified dispersion relation (MDR), could be amplified by cosmological distances and then become observables [40]. On the other hand, quantum gravity phenomenology has been tackled within effective models based on MDRs and/or GUPs and containing the minimal length as a natural UV cut-off. Recently the relation between these approaches has been clarified and established [41]. The idea of a minimal length can be modelled in terms of a quantized space-time and goes back to the early days of quantum field theory [42] (see also [40− 43] ). An other approach is to consider deformations to the standard Heisenberg algebra [7, 8], which lead to generalized uncertainty principles. In this section we follow the latter approach and exploit results recently obtained. Indeed, it has been shown in the context of canonical noncommutative field theory in the coherent states representation [47] and field theory on non-anticommutative superspace [48, 49], that the Feynman propagator display an exponential UV cut-off of the form exp (−ηp2), where the parameter η is related to the minimal length. This framework has been further applied, in series of papers [50], to the black hole evaporation process. At the quantum mechanical level, the essence of the UV finiteness of the Feynman propagator can be also captured by a non linear relation, p = f(k), between the momentum and the wave vector of the particle [41]. This relation must be invertible and has to fulfill the following requirements: 1. For smaller energies than the cut-off the usual dispersion relation is recovered. 2. For large energies, the wave vector asymptotically reaches the cut-off. In this case, the usual momentum measure dnp is deformed and becomes dnp . In the following, we will restrict ourselves to the isotropic case and work with one space-like dimension. Following [47, 49] and setting η = we have = ~exp α2L2P l , (1) where α is a dimensionless constant of order one. From Eq.(1) we obtain the dispersion relation k (p) = 2αLP l αLP l , (2) from which we have the following minimum Compton wavelength λ0 = 4 παLP l. (3) Let us show that these results can be obtained from the following representation of the position and momentum operators X = i~ exp α2L2P l ∂p P = p. (4) The corrections to the standard Heisenberg algebra become effective in the so-called quantum regime where the momentum and length scales are of the order of the Planck mass MP l and of the Planck length LP l respectively. The hermiticity condition of the position operator implies the following modified completeness relation p2|p〉〈p| = 1 (5) and modified scalar product 〈p| p′〉 = e δ (p− p′) . (6) From Eq.(5) we observe that we have reproduced the Gaussian damping factor in the Feyn- man propagator [47, 49]. The algebra defined by Eq. (4) leads to the following generalized commutator and generalized uncertainty principle (GUP∗) [X,P ] = i~ exp α2L2P l , (δX) (δP ) ≥ α2L2P l . (7) In order to investigate the quantum implications of this deformed algebra, we consider the saturate GUP∗ and solve for (δP ). Using the property 〈P 2n〉 ≥ 〈P 2〉n and (δP )2 = 〈P 2〉−〈P 〉2 the saturate GUP∗ is then given by (δX) (δP ) = α2L2P l (δP ) + 〈P 〉2 . (8) Taking the square of this expression we obtain W (u) eW (u) = u, (9) where we have set W (u) = −2α (δP ) and u = − α 2(δX) 〈P 〉2 The equation given by Eq.(9) is exactly the definition of the Lambert function [51]. The LambertW function is a multivalued functions. Its different branches are labelled by the integer k = 0,±1,±2, · · · . When u is a real number Eq.(9) have two real solutions for 0 ≥ u ≥ −1 denoted by W0(u) and W−1(u), or it can have only one real solution for u ≥ 0, namely W0(u) . For -∞ < u < −1 , Eq.(9) have no real solutions. Using Eq.(9) the uncertainty in momentum is then given by (δP ) = 〈P 〉2 2 (δX) 2L2P le 2α2L2 〈P 〉2 2 (δX)  . (10) Then from the argument of the Lambert function in Eq.(10) we have the following condition α2L2P le 2α2L2 〈P 〉2 2 (δX) , (11) which leads to a minimal uncertainty in position given by αLP le 〈P 〉2 . (12) The absolutely smallest uncertainty in position or minimal length is obtained for physical states for which we have 〈P 〉 = 0 and (δP ) = ~/ 2αLP l , and is given by (δX)0 = αLP l (13) In terms of the minimal length the momentum uncertainty becomes (δP ) = 2 (δX) (δX)0 . (14) Here we observe that 1 (δX)0 < 1 is a small parameter, by virtue of the GUP∗, and perturbative expansions to all orders in the Planck length can be safely performed. Indeed a series expansion of Eq.(14) gives the corrections to the standard Heisenberg principle 2 (δX) (δX)0 (δX)0 (δX)0 + . . . . (15) This expression of (δP ) containing only odd powers of (δX) is consistent with a recent analysis in which string theory and loop quantum gravity, considered as the most serious candidates for a theory of quantum gravity, put severe constraints on the possible forms of GUPs and MDRs [20]. Before ending this section and for later use let us recall the form of the GUP to leading order in the Planck length widely used in the literature on quantum gravity phenomenology. This GUP is given by (δX) (δP ) ≥ α2L2P l (δP ) . (16) A simple calculation leads to the following minimal length (δX)0 = αLP l, (17) which is of order of the Planck length. However, as nicely noted in [41], this form of GUP do not fulfill the second requirement listed above. In the following sections we use the form of the GUP given by Eq.(14) and investigate the thermodynamics of the Schwarzschild black hole. We use units ~ = c = kB = 1 which imply LP l = M P l = T P l = 3 Black hole thermodynamics The metric of a four-dimensional Schwarzschild black hole is given by ds2 = 1− 2MG dt2 − 1− 2M dr2 − r2dΩ2, (18) where M represents the mass of the black hole. The Schwarzschild horizon radius, located at rh, is defined by rh = 2MG. (19) Near-horizon geometry considerations suggests to set δX ≃ rh, and then Eq.(19) leads to minimum horizon radius and minimum mass given by rh = (δX)0 = αLP l, M0 = MP l. (20) Therefore, black holes with mass smaller than M0 do not exist. In the standard Hawking picture, temperature and entropy of the Schwarzschild black hole of mass M are [52, 53] , S = 4πGM2. (21) Let us then examine the corrections to the above expressions due to the GUP∗. Following the heuristic argument of Bekenstein we have . (22) Using Eq.(14), the GUP∗-corrected Hawking temperature is 8πML2P l . (23) On substituting Eq.(20) into Eq.(23) we obtain the following black hole maximum temperature TmaxH = . (24) The corrections to the standard Hawking temperature are obtained by expanding Eq.(23) in terms of 1 (M0/M). Indeed we obtain 8πML2P l + . . . . (25) The variation of the Hawking temperature, Eq.(23), with the mass of the black hole is shown in figure 1. It is interesting to inverse Eq.(23) and write the mass of the black hole as a function of the temperature 8πTHL TmaxH . (26) This relation shows that for temperatures larger than TmaxH , the black hole mass increases with temperature. In our framework, such a behavior is forbidden by the cut-off brought by GUP∗. However, in the noncommutative approach to radiating black hole, this behavior is allowed because of a lack of a generalized uncertainty principle [50]. 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 1: The temperature versus BH mass. From left to right: the Hawking result (black solid line), GUP (doted line) and GUP∗ results (solid line) for α=0.75 (red), α=1 (green), α=1.25 (blue) respectively. We turn now to the calculation of the micro canonical entropy of a large black hole. In the stan- dard situation the entropy is proportional to the black hole horizon-area. Following heuristic considerations due to Bekenstein, the minimum increase of the area of a black hole absorbing a classical particle of energy E and size R is given by (∆A)0 ≃ 4L2P l (ln 2)ER. At the quantum mechanical level the size and the energy of the particle are constrained to verify R ∼ 2δX and E ∼ δP . Then we have (∆A)0 ≃ 8L2P l (ln 2) δXδP. Extending this approach to the case with GUP∗ and using near horizon geometry considerations, we obtain (∆A)0 ≈ 4L P l ln 2 exp , (27) where A = 4π (δX) and A0 = 4π (δX) 0 are respectively the horizon area and minimum horizon area of the black hole. With the aid of the Bekenstein calibration factor for the minimum increase of entropy (∆S)0 = ln 2 we have (∆S)0 (∆A)0 4L2P l . (28) Before integrating over A we note that the existence of a minimum horizon area enforces us to set the lower limit of integration as A0. Then the entropy, up to a irrelevant constant, is 4L2P l dA.. (29) The relation e W (x) x/W (x) allows us to write Eq.(29) as 4eL2P l ∫ − 1 2 [W (y)] 2 dy, (30) where PV means the Cauchy principal value of the integral. Setting y = −1 and performing the integration we obtain the GUP∗-corrected black hole entropy 8eL2P l ))− 1 e− Ei , (31) where Ei (x) is the exponential function. Expanding Eq.(31) in the parameter 1 (A0/A) we have 4L2P l 8L2P le 25πα2 192e2 343πα2 2304e3 + . . .+ C where the constant is given by 8L2P le γ − 1− 2 ln (2e)− 2 e− Ei ≃ −4.60 L2P l and γ is the Euler constant. The dependence on the Planck length is contained in A0 ∼ L2P l. We observe that we have reproduced, in our framework with GUP∗, the log-area correction with a negative sign. Other approaches like string theory, loop quantum gravity and effectif models with GUPs and/or MDRs, lead to the same sub-leading logarithmic correction. Setting ρ = −πα2 and β = 3π in Eqs. (25) and (32) we obtain M2P l ρ2 + β/4 , (34) 4L2P l + ρ ln L2P l βL2P l . (35) These expressions are exactly the temperature and entropy obtained in loop quantum gravity and string theory approach quantum gravity. 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 2: The entropy versus BH mass. From left to right: the Hawking result (black solid line), GUP (doted line) and GUP∗ results (solid line) for α=0.75 (red), α = 1 (green) and α=1.25 (blue) respectively. From figures 1 and 2 it follows that the GUP∗-corrected temperature and entropy are respec- tively higher and smaller than the semi classical results. 4 Black holes evaporation As a warming to study the Hawking radiation process of the Schwarzschild black hole, we examine the effects of GUP∗ on the black body radiation spectrum. With the aid of the squeezed momentum measure given by Eq.(5), which suppress the contribution of unwanted high momenta, the energy density of a black body at temperature T is defined by d3pe−α T − 1 . (36) Using the variable y = βp (β = 1/T ) and expanding the exponential, equation (36) becomes Eγ = 8πT 4 (−1)n (αT/TP l) y2n+3 ey − 1 . (37) Now with the help of the following definition of the Riemann zeta function ey − 1 = Γ (s) ζ (s) , (38) we obtain Eγ = 8πT 4 (−1)n (αT/TP l) Γ (2n+ 4) ζ (2n+ 4) . (39) This energy density is defined only for values of temperatures below some characteristic scale. In fact Eq.(39) is an alternating series which converge if and only if (αT/TP l) Γ (2n + 4) ζ (2n+ 4) = 0. (40) From this relation it follows that T < α−1TP l, (41) as expected from the Gaussian damping factor in Eq.(36). However, we note that we have a stronger condition on T . Indeed in our framework, the maximum temperature of the black hole is given by Eq.(24) and it is approximately 0.1TP l for α of order one. Then the condition on the BH temperature is rewritten as T/TP l < 0.1. For our purpose, the latter constraint allows us to cut the series in Eq.(39) at n = 1. Using ζ (4) = π and ζ (6) = π and Eq.(24) we finally obtain, from Eq.(39) , the following expression Eγ (T ) = 1− 15 TmaxH . (42) The first term is the standard Stefan-Boltzmann law while the second term is the correction brought by GUP∗. We are now ready to study the Hawking evaporation process. The intensity emitted by a black hole of mass M is defined by I = AEγ (TH) , (43) where A is the BH horizon area. Invoking energy conservation, the evaporation rate of the black hole is = −AEγ (TH) . (44) Using Eq. (23) for the corrected Hawking temperature we obtain = − γ1 M2L4P l 1− 8γ2 with γ1 = , γ2 = 16128 . The deviations from the standard expression are obtained by applying a series expansion in 1 (M0/M) = − γ1 M2L4P l 1− 2γ2 1− 72γ2 25eγ1 + . . . The variation of the evaporation rate with the black hole mass is shown in Figure 3. We clearly observe that the divergence for M → 0 in the standard description of the black hole evaporation process is now completely regularized by the GUP∗. This regularization is also reflected by the constraint (41), which suppress the evaporation process beyond the Planck temperature. This phenomenon is similar to the prevention, by the standard uncertainty principle, of the hydrogen atom from total collapse. In our picture, the regularization can be considered as a dynamical effect and not as a consequence of any quantum symmetry in the theory. 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 3: The evaporation rate versus BH mass. From left to right: the Hawking result (black solid line) , GUP (doted line) and GUP∗ results (solid line) for α = 0.90 (red) , α = 1. (green) and α = 1.25 (blue). On the other hand, we observe that the evaporation phase ends when the BH mass becomes equal to M0 with a minimum rate given by (γ1 − 8γ2)M4P l. (47) Thus the evaporation process of a black hole with initial mass M > M0 continue until the horizon radius becomes (δX)0 , leaving a massive relic referred to, in the literature, as a black hole remnant (BHR). To find the nature of the BHR we calculate the heat capacity defined by . (48) Using the expression of temperature given by (23) we easily obtain C = −8πM2L2P l . (49) This expression vanishes when 1 + W (M0/M) = 0, whose solution is M = M0. We conclude that the heat capacity of the black hole vanishes at the end point of the evaporation process characterized by a BHR with mass M0.Besides the gravitational interaction with the surrounding, the vanishing of the heat capacity reveals the inert character of the BHRs and thus make them as potential candidates to explain the origin of dark matter [18, 19]. Finally we note that, as it is the case with the form of the GUP to leading order in the Planck length, the BHRs are also a consequence of GUP∗ [17, 55]. We have drawn the variation of the heat capacity with BH mass in figure 4. In it we see that, the heat capacity vanishes for M0 ≃ 0.50, 0.75 in the case with GUP and M0 ≃ 0.58, 0.87 in the case with GUP∗. 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 4: The heat capacity versus BH mass. From left to right: the standard result (black solid line), GUP (doted line) and GUP∗ results (solid line) for α = 1 (red) and α = 1.5 (blue) respectively. Taylor expanding Eq.(49) we have C = −8πM2L2P l − . . . . (50) The standard expression of the heat capacity C = −8πM2 is reproduced in the limit of black holes with mass larger than the minimum mass M0. The correction terms to the heat capacity due to GUP∗ are all positive indicating that the evaporation process is accelerated and leading to a corrected decay time smaller than the decay time in the standard case. Let us consider a black hole starting the evaporation process with a mass M and ending the process with the minimum mass M0. Using (45) and the variable y = −1e (M0/M) , the decay time is given by t = (−1)7/2 ∫ − 1 (M0/M) W−5/2 (y) e− W (y)dy + (−1)5/2 2γ21e 2L2P l ∫ − 1 (M0/M) W−3/2 (y) e− W (y)dy. (51) Performing the integration we obtain 4 (1− ǫ) W (y) −W (y) W 2 (y) + C, (52) where the constant C is the value for y = −1/e and ǫ = 3γ2 ∼ 10−6 for α of order one. Ignoring ǫ and performing a series expansion in y we have M3L4P l + . . . Then to first order in 1 (M0/M) the relative correction to the decay time is , (54) where t0 = is the decay time without GUP∗. From Eq.(54) , it follows that black holes with GUP∗ are hotter and decay faster than in the standard case. Let us now turn to a comparison of the corrected BH thermodynamics with GUP∗ with the corrections brought by the GUP to leading order in the Planck length. Since our comparison is quantitative we use the Planck units. Repeating the same calculations as above with the GUP given by Eq.(16), the temperature, the entropy and the heat capacity of the black hole are respectively given by TGUP = , (55) SGUP = 2πM CGUP = πα 1− α2 1− α2 . (57) The minimum black hole mass and maximum temperature allowed by GUP areM0 = (δX)0 /2 = and Tmax = 1/2πα. In figures 1, 2, 3 and 4 we have plotted, besides the results obtained with GUP∗, the variation of temperature, entropy, evaporation rate and heat capacity with GUP as functions of the black hole mass for different values of the parameter α. Figure 2 shows, that in the scenario with GUP∗, the BH entropy decreases compared to the entropy in the standard case and the scenario with GUP. This reveals the deeper quantum nature of the black hole in the scenario with GUP∗. Thus quantum effects become manifest at an earlier stage of the evaporation phase than was predicted by the semi classical Hawking analysis [54] and the GUP analysis [55]. The calculation of the evaporation rate in the framework with GUP requires a careful analysis. In all the calculations done until now, the validity of the Stefan-Boltzmann law is assumed, ignoring the UV cut-off implemented by GUP. However, it was pointed in [16] that the effect of the GUP should be also reflected in a modification of the de Broglie wave length relation 1 + α2p2 . (58) This relation must be translated into a modification of the momentum measure such that the contributions of high momenta are suppressed. As shown in [7], the GUP to leading order in the Planck length leads to a squeezing of the momentum measure by a factor 1 (1+α2L2Plp2) . Then following the same calculation leading to Eq.(39) , the energy density of a black body with GUP (1 + α2L2P lp T − 1 . (59) Performing the integral and using the same argument as before, we obtain the expression given by Eq. (42) . We note, that in a recent calculation of the Stefan-Boltzmann law with GUP [24], the sign of the correction term is positive, in contradiction with the role of the UV cut-off implemented by the GUP. The correct evaporation rate with GUP is then given by 128π2M6 1024π2M8 63α10 . (60) In figure 3 we observe that the evaporation process with GUP is retarded compared to the process with GUP∗ and that the process ends at a mass M0 = α/2 with a minimum rate given min,GUP = − 32π 1260M20 , (61) which is greater (in absolute value) than the one obtained with GUP∗. In table 1 we show the GUP and GUP∗-corrected thermodynamics of two black holes with initial mass equal to 2MP l and 5MP l for α = 1. The first row gives the semi classical Hawking results. The second row gives the GUP-corrected results and the third row the GUP∗−corrected ones. It is interesting to note that, in the scenario with GUP∗, the final stage of the evaporation phase is a remnant with a mass larger than the one obtained with GUP and that the decay time is drastically reduced. In a scenario with extra dimensions, these results may have important consequences on possible black holes production at particle colliders and in ultrahigh energy cosmic ray (UHECR) air-showers. Finally, let us notice that the corrections to the black hole thermodynamics become indistin- guishable in the two version of GUP in the limit of large mass and small values of α. However, for growing values of the parameter α, corresponding to strong gravitational effects, the predictions of the two GUPs concerning the entropy become different even for massive black holes. Table 1. GUP and GUP∗-corrected thermodynamics for two BHs with mass M = 2 and M = 5 (in Planck units). The deviations from the Hawking results are also given. M = 2 α Minimum mass Initial temp Final temp Decay time Entropy 0 - 0.019 ∞ 129.69 50.27 1.0 0.5 0.020 (+3%) 0.16 111.92 (−14%) 44.66 (−11%) 1.0 0.58 (+16%) 0.020 (+3%) 0.11 (−31%) 3.33 (−97%) 43.73 (−13%) M = 5 α Minimum mass Initial temp Final temp Decay time Entropy 0 - 0.008 ∞ 2026.42 314.16 1.0 0.5 0.008 0.16 1976.60 (−2.5%) 307.10 (−2%) 1.0 0.58 (+16%) 0.008 0.11 (−31%) 22.17 (−99%) 306.18 (−2.2%) 5 Conclusion In this paper we have studied how black holes thermodynamic parameters are affected by a GUP to all orders in the Planck length. We have obtained exact analytic expressions for the Hawking temperature and entropy. Particularly we found that a black hole with a mass smaller than a minimum mass do not exist. The existence of a energy scale which is one order below the Planck scale allowed us to calculate, to leading order, the deviations from the standard Stefan- Boltzmann law. Then we investigated the Hawking radiation process of the Schwarzschild black hole and shown that at the end of the evaporation phase a inert massive relic continue to exist as a black hole remnant (BHR) with zero entropy, zero heat capacity and non zero finite temperature. For completeness, we have also compared our results with the semi classical results and the predictions of the GUP to leading order in the Planck length. In particular, we have shown that the entropy in our framework is smaller than the entropy in the standard case and with GUP. We have also made the correct calculation of the evaporation rate with GUP. Finally, we have shown that black holes with the form of GUP used in this paper are hotter, shorter-lived and tend to evaporate less than black holes in the semi classical and the GUP to leading order pictures. 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0704.1262

Symmetry Breaking Study with Deformed Ensembles

Symmetry Breaking Study with Deformed Ensembles ∗ J. X. de Carvalho1,2, M. S. Hussein†1,2, M. P. Pato2 and A. J. Sargeant2 1Max-Planck-Institut für Physik komplexer Systeme Nöthnitzer Straβe 38, D-01187 Dresden, Germany 2Instituto de F́ısica, Universidade de São Paulo C.P. 66318, 05315-970 São Paulo, S.P., Brazil A random matrix model to describe the coupling of m-fold symmetry is con- structed. The particular threefold case is used to analyze data on eigenfrequencies of elastomechanical vibration of an anisotropic quartz block. It is suggested that such experimental/theoretical study may supply powerful means to discern intrinsic symmetries of physical systems. The standard ensembles of Random Matrix Theory (RMT) [1] have had wide appli- cation in the description of the statistical properties of eigenvalues and eigenfunctions of complex many-body systems. Other ensembles have also been introduced [2], in order to cover situations that depart from universality classes of RMT. One such class of en- sembles is the so-called Deformed Gaussian Orthogonal Ensemble (DGOE) [3, 4, 5, 6] that proved to be particularly useful when one wants to study the breaking of a discrete symmetry in a many-body system such as the atomic nucleus. In fact, the use of spectral statistics as a probe of symmetries in physical systems has been a subject of intensive experimental and theoretical investigation following the pioneering work of Bohigas, Giannoni and Schmit [7] which showed that the quantal behaviour of classically chaotic systems exhibits the predictions supplied by the RMT. Examples of symmetry breaking in physical systems that have been studied include nuclei [8, 9], atoms [10, 11] and mesoscopic devices such as quantum dots [12]. In the case of nuclei, the Mitchell group at the Triangle Universities Nuclear Labo- ratory [8, 9], studied the effect of isospin symmetry breaking, in odd-odd nuclei such as 26Al. They detected the breakdown of this important symmetry by the applications of two statistics: the short-range, nearest neighbor level spacing distribution (NND) and the long range Dyson’s ∆-statistics [8, 9]. These results were well described by a DGOE in which a pair of diagonal blocks is coupled. The strength of the coupling needed to account for the symmetry breaking can be traced to the average matrix element of the Coulomb interaction responsible for this discrete symmetry breaking [4, 13]. The justi- fication for the use of block matrices to describe the statistics of a superposition of R spectra with different values of the conserved quatum number can be traced to Refs. [1, 14]. In the case of non-interacting spectra, i.e. if the quantum number is exactly conserved, the answer is a superposition of the R spectra. Since the level repulsion is present in each one of the R spectra, their superposition does not show this feature. Thus, we can say that for each spectra of states of a given value of the quantum number, one attaches a random matrix (GOE). For R spectra each of which has a given value of the conserved quantum number, one would have an R × R block diagonal matrix. Each ∗Supported in part by the CNPq and FAPESP (Brazil). †Martin Gutzwiller Fellow, 2007/2008. http://arxiv.org/abs/0704.1262v2 block matrix will have a dimension dictated by the number state of that spectra. If the quantum number is not conserved then the R × R block matrix acquires non-diagonal matrices that measure the degree of the breaking of the associated symmetry. This idea was employed by Guhr and Weidenmüller [13] and Hussein and Pato [3] to discuss isopin violation in the nucleus 26Al. In reference [3], the random block matrix model was called the Deformed Gaussian Orthogonal Ensemble (DGOE). In order to study transitions amongst universal classes of ensembles such as order- chaos (Poisson→GOE), symmetry violation transitions (2GOE→1GOE), experiments on physical systems are more complicated due to the difficulty of tuning the interaction (except, e.g. in highly excited atoms where the application of a magnetic field allows the study of GOE-GUE transitions). To simulate the microscopic physical systems, one relies on analog computers such as microwave cavities, pioneered by A. Richter and collaborators [15] and acoustic resonators of Ellegaard and collaborators[16, 17, 18]. It is worth mentioning at this point that the first to draw attention to the applicability of RMT to accoustic waves in physical system was Weaver [19]. In the experiment of Ellegaard et al. [17] what was measured were eigenfrequencies of the elastomechanical vibrations of an anisotropic crystal block with a D3 point-group symmetry. The rectangular crystal block employed by Ellegard was so prepared as to have only a two-fold flip symmetry retained. Then, to all effects, the quartz specimen resembles a system of two three-dimensional Sinai billiards. The statistical treatment of the eigenfrequencies of such a block would follow that of the superposition of two uncoupled GOE’s. Then, by removing octants of progressively larger radius from a corner of the crystal block this remnant two-fold symmetry was gradually broken. The spectral statistics show a transition towards fully a chaotic system as the octant radius increases. What was then seen was that the measured NND is compatible with a two block DGOE description but the ∆-statistics was discrepant. This discrepancy was attributed to pseudo integrable behavior and this explanation was later implemented with the result that the long-range behavior was fitted at the cost, however, of loosing the previous agreement shown by the NND[23]. Here we reanalyse this experiment following the simpler idea of extending the DGOE matrix model [5] to consider the coupling of three instead of two GOE’s [6]. We show that within this extension both, the short- and the long-range statistics, are reasonably fitted suggesting that the assumption of the reduction of the complex symmetries of anisotropic quartz block may not be correct. Our findings have the potential of supplying very precise means of testing details of symmetry breaking in pysical systems. To define the ensembles of random matrices we are going to work with, we recall the construction based on the Maximum Entropy Principle [3], that leads to a random Hamiltonian which can be cast into the form H = H0 + λH1, (1) where the block diagonal H0 is a matrix made of m uncoupled GOE blocks and λ (0 ≤ λ ≤ 1) is the parameter that controls the coupling among the blocks represented by the H1 off-diagonal blocks. For λ = 1, the H1 part completes the H0 part and H = H These two matrices H0 and H1 are better expressed introducing the following m pro- jection operators | j >< j |, (2) where Ii defines the domain of variation of the row and column indexes associated with ith diagonal block of size Mi. Since we are specifically interested in the transition from a set of m uncoupled GOE’s to a single GOE, we use the above projectors to generalize our previous model [3, 4] by writing GOEPi (3) GOEQi (4) where Qi = 1− Pi. It is easily verified that H = HGOE for λ = 1. The joint probability distribution of matrix elements can be put in the form [3, 20] P (H,α, β) = Z−1N exp −αtrH2 − βtrH2 with the parameter λ being given in terms of α and β by λ = (1 + )−1/2. (6) Statistical measures of the completely uncoupled m blocks have been derived. They show that level repulsion disappears which can be understood since eigenvalues from different blocks behave independently. In fact, as m increases the Poisson statistics are gradually approached. In the interpolating situation of partial coupling, some approxi- mate analytical results have been derived. In Ref. [20], for instance, it has been found that the density ρ(E) for arbitrary λ and m is given by ρ(E) = ρi(E) (7) where ρi(E) = a2i − E2, | E |≤ a 0, | E |> a is Wigner’s semi-circle law with a = N/α and a2i = a 1− Mi . (9) Eq. (5) can be used to calculate exactly analytically the NND for 2 × 2 and 3 × 3 matrices [6]. For the 2× 2 case the DGOE, Eq. (5), gives P2×2(s, β) = αs exp ) exp , (10) where I0 is the modified Bessel function, whose asymptotic form is I0(x) → . (11) Thus,there is no level repulsion for β → ∞, P2×2(s,∞) = 2πα exp , which repre- sents the 2x2 Poisson distribution where the usual exponential is replaced by a Gaussian. The prefactor is just 1 if 2α is taken to be π. In the opposite limit, β → 0, I0(x) ≈ 1−x2/4 and one obtains the Wigner distribution, P2×2(s, β → 0) ≈ s (12) Note that the parameter λ of eq (6) is 0 if β is ∞ and 1 if β is 0. For higher dimensions Eq. (5) can only be used for numerical simulations. This is what we are now reporting, using 2 and 3 bolck matrices of sizes 105 x 105 and 70 x 70 each, respectively. The size of the whole matrix is 210 x 210. Further, we take an ensemble of 1000 elements and fix α to be 1. We apply our model to analyse the eigenfrequency data of the elastomechanical vibrations of an anisotropic quartz block used in [17]. In this reference in order to break the flip symmetry of the crystal block gradually they removed an octant of a sphere of varying size at one of the corners. The rectangular quartz block has the dimensions 14 × 25 × 40mm3. The radii of the spheres containing the octants are r = 0.0, 0.5, 0.8, 1.1, 1.4 and 1.7mm representing figures (a)− (f). Figs. 1x and 2x of Ref. [17] correspond to an octant of a huge sphere of radius r = 10.0mm, whose center is inside the crystal and close to one of the corners. They found 1424, 1414, 1424, 1414, 1424 and 1419 frequency eigenmodes in cases (a)− (f), respectively. These eigenfrequecies were measured in the frequency range between 600 and 900 kHz. Thus the average spacing between the modes is about 214Hz. The histograms and circles in the two figures of Ref. [17] represent the short-range nearest-neighbor distributions P (s) (Fig. 1) and the long range ∆3(L) statistics (Fig. 2). In our DGOE simulation the unfolding of the calculated spectra is performed with the DGOE density given by Eq. (7) above. In figures 1 and 2, we show the results of our simlulations as compared to the data of Ellegaard et al. [17] for the spacing distribution and in figures 3 and 4 the long range correlation exemplified by the spectral rigidity ∆3(L). We simulate the gradual breaking of the 2- or 3-fold symmetry by changing the value of the parameter λ above. We see clearly that in so far as the ∆3(L) is concerned a 3-GOE description works much better than a 2-GOE one. It is clear, however that both descriptions fall below the data, specially at large L. We shall analyse this discrepancy in the following using the missing level effect[21]. It is often the case that there are some missing levels in the statistical sample analysed. Such a situation was addressed recently by Bohigas and Pato [21] who have shown that if g fraction of the levels or eigenfrequencies is missing, the ∆3(L) becomes (L) = g + (1− g)2∆3 . (13) The presence of the linear term, even if small, could explain the large L behavior of the measured ∆3(L). We call this effect the Missing Level (ML) effect. Another possible deviation of ∆3 could arise from the presence of pseudo-integrable effect (PI) [22, 23]. This also modifies ∆3 by adding a Poisson term just like Eq. (13). In the following we show that there is no need for the PI effects to explain the large-L data on the ∆3 if the ML effect is taken into account. We take a study case Figs. 3b and 4b which correspond to r = 0.5mm and where 1414 frequency eigenvalues were found. We consider this a potential ML case and take for ∆3, the expression given in Eq. (13) and apply to our simulations. We find perfect fit to the data, if g is taken to be 0.1, namely only 90% of the eigenfrequencies were in fact taken into account in the statistical analysis. There is, threfore, room to account much better for all cases (Fig. 2a, 2c, . . . ) by appropiately choosing the correponding value of g. We have also verified that if a 2GOE description is used, namely, m = 2 , then an account of the large-L behaviour of ∆3 can also be obtained if a much larger number of levels were missing in the sample. In our particular case of Fig. 2b, we obtained g = 0.3. This is 3 times larger than the ML needed in the 3GOE description. We consider the large value of g needed in the 2GOE description, much too large to conform to the reported data in Ref [17]. Figure 5 summarizes our the above. It is therefore clear that the 3GOE description of the spectral rigidity of the eifen- frequency spectra of [17] for the crystal block does work very well if a small fraction of the levels is taken to be missing, without resort to pseudointegrable trajectories or levels that do not feel the symmetry breaking [23]. On the other hand, the 2GOE description, which does as good as the 3GOE one in fitting the measured P (s), fails dramatically in accounting for the spectral rigidity, even if as much as 30 per cent of the levels are taken as missing. In conclusion, a random matrix model to describe the coupling of m-fold symmetry is constructed. The particular threefold case is used to analyse data on eigenfrequencies of elastomechanical vibration of a anisotropic quartz block. By properly taking into account the ML effect we have shown that the quartz block could very well be described by 3 uncoupled GOE’s , which are gradually coupled by the breaking of the three-fold sym- metry (through the gradual removal of octants of increasing sizes), till a 1GOE situation is attained. This, therefore, indicates that the unperturbed quartz block may possess another symmetry, besides the flip one. A preliminary version of the formal aspect of this work has previously appeared in [24]. [1] M.L. Mehta, Random Matrices 2nd Edition (Academic Press, Boston, 1991); T.A. Brody et al., Rev. Mod. Phys. 53, 385 (1981); T. Guhr, A. Müller-Groeling and A. Weidenmüller, Phys. Rep. 299, 189 (1998). [2] F.J. Dyson, J. Math. Phys. 3, 1191 (1962). [3] M. S. Hussein, and M. P. Pato, Phys. Rev. Lett. 70, 1089 (1993). [4] M. S. Hussein, and M. P. 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Rev. Lett. 98, 074103 (2007). [16] C. Ellegaard, T. Guhr, K. Lindemann, H. Q. Lorensen, J. Nygard and M. Oxborrow, Phys. Rev. Lett. 75, 1546 (1995). [17] C. Ellegaard, T. Guhr, K. Lindemann, J. Nygard and M. Oxborrow, Phys. Rev. Lett. 77, 4918 (1996). [18] P. Bertelsen, C. Ellegaard, T. Guhr, M. Oxborrow and K. Schaadt, Phys. Rev.Lett. 83, 2171 (1999). [19] R. L. Weaver, J. Acoustic. Soc. Am. 85, 1005 (1989). [20] A. C. Bertuola, J. X. de Carvalho, M. S. Hussein, M. P. Pato, and A. J. Sargeant, Phys. Rev. E 71, 036117 (2005). [21] O. Bohigas and M. P. Pato, Phys. Lett. B, 595, 171 (2004). [22] D. Biswas and S. R. Jain, Phys. Rev. A 42, 3170 (1990). [23] A. Abd El-Hady, A. Y. Abul-Magd, and M. H. Simbel, J. Phys. A 35, 2361 (2002). [24] M. S. Hussein, J. X. de Carvalho, M. P. Pato and A. J. Sargeant, Few-Body Systems, 38, 209 (2006). P(s) d 0 1 2 3 4 FIG. 1: Nearest Neighbour Distributions. Histograms show data (a)-(x) from Ref. [17]. Thick histograms show the three coupled GOE fits to the data carried out using the DGOE numerical simulations using Eq. (5). Also shown as the full thin line the three uncoulped GOE P (s). In graph (x) the dotted line is the Poisson distribution, the dashed line is the two uncoupled GOE P (s). The very thin line is Wigner distribution which is hidden behind histograms. The values of λ that adjust the data are 0.0032, 0.0071, 0.0158, 0.0250, 0.0333, 0.9950, 1.000 for cases (a)-(x). See text for details. 0 1 2 3 4 FIG. 2: Nearest Neighbour Distributions. Histograms show data (a)-(x) from Ref. [17]. Thick histograms show the two coupled GOE fits to the data carried out using the DGOE numerical simulations using Eq. (5). Also shown as the full thin line the two uncoulped GOE P (s). In graph (x) the dotted line is the Poisson distribution, the dashed line is three uncoupled GOE P (s). The very thin line is Wigner distribution which is hidden behind histograms. The values of λ that adjust the data are 0.000, 0.0258, 0.0200, 0.0400, 0.0705, 0.0600, 1.000 for cases (a)-(x). See text for details. ∆3(L) 0 10 20 30 40 50 FIG. 3: Spectral Rigidities. The thick lines are the DGOE simulation for the three coupled GOE’s.The same values of λ as in Fig. 1 were used. The thin lines correspond the three uncoupled GOE’s case. The data points are from Ref. [17]. See text for details. 0.8 a 0 10 20 30 40 50 FIG. 4: Spectral Rigidities. The thick lines are the DGOE simulation for the two coupled GOE’s. The same values of λ as in Fig. 2 were used. The thin lines correspond the two uncoupled GOE’s case. The data points are from Ref. [17]. See text for details. 0 10 20 30 40 50 ∆3(L) 2GOE, g=0.30 3GOE, g=0.10 FIG. 5: The ML effect. The data points correspond to case (b) of Ref. [17], r = 0.5mm. The full line corresponds to our three coupled GOE’s fit with λ = 0.0071, figure 3b and g = 0.10. The dashed line corresponds to our two coupled GOE’s fit with λ = 0.0258, figure 4b and g = 0.30. See text for details. References

0704.1263

The Measurement Calculus

The Measurement Calculus Vincent Danos Université Paris 7 & CNRS [email protected] Elham Kashefi IQC - University of Waterloo Christ Church - Oxford [email protected] Prakash Panangaden McGill University [email protected] Abstract Measurement-based quantum computation has emerged from the physics community as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model which is based on unitary operations. Among measurement-based quantum computation methods, the recently introduced one-way quantum computer [RB01] stands out as fundamental. We develop a rigorous mathematical model underlying the one-way quantum computer and present a concrete syntax and operational semantics for programs, which we call patterns, and an algebra of these patterns derived from a denotational semantics. More importantly, we present a calculus for reasoning locally and compositionally about these patterns. We present a rewrite theory and prove a general standardization theorem which allows all patterns to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements and expressiveness theorems. Furthermore we formalize several other measurement-based models e.g.Teleportation, Phase and Pauli models and present compositional embeddings of them into and from the one-way model. This allows us to transfer all the theory we develop for the one-way model to these mod- els. This shows that the framework we have developed has a general impact on measurement- based computation and is not just particular to the one-way quantum computer. 1 Introduction The emergence of quantum computation has changed our perspective on many fundamental aspects of computing: the nature of information and how it flows, new algorithmic design strategies and complexity classes and the very structure of computational models [NC00]. New challenges have been raised in the physical implementation of quantum computers. This paper is a contribution to a nascent discipline: quantum programming languages. This is more than a search for convenient notation, it is an investigation into the structure, scope and limits of quantum computation. The main issues are questions about how quantum processes are defined, how quantum algorithms compose, how quantum resources are used and how classical and quantum information interact. Quantum computation emerged in the early 1980s with Feynman’s observations about the dif- ficulty of simulating quantum systems on a classical computer. This hinted at the possibility of turning around the issue and exploiting the power of quantum systems to perform computational tasks more efficiently than was classically possible. In the mid 1980s Deutsch [Deu87] and later http://arxiv.org/abs/0704.1263v1 Deutsch and Jozsa [DJ92] showed how to use superposition – the ability to produce linear combi- nations of quantum states – to obtain computational speedup. This led to interest in algorithm design and the complexity aspects of quantum computation by computer scientists. The most dramatic results were Shor’s celebrated polytime factorization algorithm [Sho94] and Grover’s sub- linear search algorithm [Gro98]. Remarkably one of the problematic aspects of quantum theory, the presence of non-local correlation – an example of which is called “entanglement” – turned out to be crucial for these algorithmic developments. If efficient factorization is indeed possible in practice, then much of cryptography becomes insecure as it is based on the difficulty of factorization. However, entanglement makes it possible to design unconditionally secure key distribution [BB84, Eke91]. Furthermore, entanglement led to the remarkable – but simple – protocol for transferring quantum states using only classical communication [BBC+93]; this is the famous so-called “teleportation” protocol. There continues to be tremendous activity in quantum cryptography, algorithmic design, complexity and information theory. Parallel to all this work there has been intense interest from the physics community to explore possible implementations, see, for example, [NC00] for a textbook account of some of these ideas. On the other hand, only recently has there been significant interest in quantum programming languages; i.e. the development of formal syntax and semantics and the use of standard machinery for reasoning about quantum information processing. The first quantum programming languages were variations on imperative probabilistic languages and emphasized logic and program develop- ment based on weakest preconditions [SZ00, Ö01]. The first definitive treatment of a quantum programming language was the flowchart language of Selinger [Sel04b]. It was based on combining classical control, as traditionally seen in flowcharts, with quantum data. It also gave a denotational semantics based on completely positive linear maps. The notion of quantum weakest preconditions was developed in [DP06]. Later people proposed languages based on quantum control [AG05]. The search for a sensible notion of higher-type computation [SV05, vT04] continues, but is problem- atic [Sel04c]. A related recent development is the work of Abramsky and Coecke [AC04, Coe04] where they develop a categorical axiomatization of quantum mechanics. This can be used to verify the correct- ness of quantum communication protocols. It is very interesting from a foundational point of view and allows one to explore exactly what mathematical ingredients are required to carry out certain quantum protocols. This has also led to work on a categorical quantum logic [AD04]. The study of quantum communication protocols has led to formalizations based on process algebras [GN05, JL04] and to proposals to use model checking for verifying quantum protocols. A survey and a complete list of references on this subject up to 2005 is available [Gay05]. These ideas have proven to be of great utility in the world of classical computation. The use of logics, type systems, operational semantics, denotational semantics and semantic-based inference mechanisms have led to notable advances such as: the use of model checking for verification, reasoning compositionally about security protocols, refinement-based programming methodology and flow analysis. The present paper applies this paradigm to a very recent development: measurement-based quantum computation. None of the cited research on quantum programming languages is aimed at measurement-based computation. On the other hand, the work in the physics literature does not clearly separate the conceptual layers of the subject from implementation issues. A formal treatment is necessary to analyze the foundations of measurement-based computation. So far the main framework to explore quantum computation has been the circuit model [Deu89], based on unitary evolution. This is very useful for algorithmic development and complexity analysis [BV97]. There are other models such as quantum Turing machines [Deu85] and quantum cellular automata [Wat95, vD96, DS96, SW04]. Although they are all proved to be equivalent from the point of view of expressive power, there is no agreement on what is the canonical model for exposing the key aspects of quantum computation. Recently physicists have introduced novel ideas based on the use of measurement and entangle- ment to perform computation [GC99, RB01, RBB03, Nie03]. This is very different from the circuit model where measurement is done only at the end to extract classical output. In measurement-based computation the main operation to manipulate information and control computation is measure- ment. This is surprising because measurement creates indeterminacy, yet it is used to express deterministic computation defined by a unitary evolution. The idea of computing based on measurements emerged from the teleportation protocol [BBC+93]. The goal of this protocol is for an agent to transmit an unknown qubit to a remote agent without actually sending the qubit. This protocol works by having the two parties share a maximally en- tangled state called a Bell pair. The parties perform local operations – measurements and unitaries – and communicate only classical bits. Remarkably, from this classical information the second party can reconstruct the unknown quantum state. In fact one can actually use this to com- pute via teleportation by choosing an appropriate measurement [GC99]. This is the key idea of measurement-based computation. It turns out that the above method of computing is actually universal. This was first shown by Gottesman and Chuang [GC99] who used two-qubit measurements and given Bell pairs. Later Nielsen [Nie03] showed that one could do this with only 4-qubit measurements with no prior Bell pairs, however this works only probabilistically. Leung [Leu04] improved this to two qubits, but her method also works only probabilistically. Later Perdrix and Jorrand [Per03, PJ04] gave the minimal set measurements to perform universal quantum computing – but still in the probabilistic setting – and introduced the state-transfer and measurement-based quantum Turing machine. Finally the one-way computer was invented by Raussendorf and Briegel [RB01, RB02] which used only single-qubit measurements with a particular multi-party entangled state, the cluster state. More precisely, a computation consists of a phase in which a collection of qubits are set up in a standard entangled state. Then measurements are applied to individual qubits and the outcomes of the measurements may be used to determine further measurements. Finally – again depending on measurement outcomes – local unitary operators, called corrections, are applied to some qubits; this allows the elimination of the indeterminacy introduced by measurements. The phrase “one-way” is used to emphasize that the computation is driven by irreversible measurements. There are at least two reasons to take measurement-based models seriously: one conceptual and one pragmatic. The main pragmatic reason is that the one-way model is believed by physicists to lend itself to easier implementations [Nie04, CAJ05, BR05, TPKV04, TPKV06, WkJRR+05, KPA06, BES05, CCWD06, BBFM06]. Physicists have investigated various properties of the cluster state and have accrued evidence that the physical implementation is scalable and robust against decoherence [Sch03, HEB04, DAB03, dNDM04b, dNDM04a, MP04, GHW05, HDB05, DHN06]. Conceptually the measurement-based model highlights the role of entanglement and separates the quantum and classical aspects of computation; thus it clarifies, in particular, the interplay between classical control and the quantum evolution process. Our approach to understanding the structural features of measurement-based computation is to develop a formal calculus. One can think of this as an “assembly language” for measurement-based computation. Ours is the first programming framework specifically based on the one-way model. We first develop a notation for such classically correlated sequences of entanglements, measurements, and local corrections. Computations are organized in patterns1, and we give a careful treatment of the composition and tensor product (parallel composition) of patterns. We show next that such pattern combinations reflect the corresponding combinations of unitary operators. An easy proof of universality follows. So far, this is primarily a clarification of what was already known from the series of papers introducing and investigating the properties of the one-way model [RB01, RB02, RBB03]. However, we work here with an extended notion of pattern, where inputs and outputs may overlap in any way one wants them to, and this results in more efficient – in the sense of using fewer qubits – implementations of unitaries. Specifically, our universal set consists of patterns using only 2 qubits. From it we obtain a 3 qubit realization of the Rz rotations and a 14 qubit realization for the controlled-U family: a significant reduction over the hitherto known implementations. The main point of this paper is to introduce a calculus of local equations over patterns that exploits some special algebraic properties of the entanglement, measurement and correction op- erators. More precisely, we use the fact that that 1-qubit XY measurements are closed under conjugation by Pauli operators and the entanglement command belongs to the normalizer of the Pauli group; these terms are explained in the appendix. We show that this calculus is sound in that it preserves the interpretation of patterns. Most importantly, we derive from it a simple algorithm by which any general pattern can be put into a standard form where entanglement is done first, then measurements, then corrections. We call this standardization. The consequences of the existence of such a procedure are far-reaching. Since entangling comes first, one can prepare the entire entangled state needed during the computation right at the start: one never has to do “on the fly” entanglements. Furthermore, the rewriting of a pattern to stan- dard form reveals parallelism in the pattern computation. In a general pattern, one is forced to compute sequentially and to strictly obey the command sequence, whereas, after standardization, the dependency structure is relaxed, resulting in lower computational depth complexity. Last, the existence of a standard form for any pattern also has interesting corollaries beyond implementation and complexity matters, as it follows from it that patterns using no dependencies, or using only the restricted class of Pauli measurements, can only realize a unitary belonging to the Clifford group, and hence can be efficiently simulated by a classical computer [Got97]. As we have noted before, there are other methods for measurement-based quantum comput- ing: the teleportation technique based on two-qubit measurements and the state-transfer approach based on single qubit measurements and incomplete two-qubit measurements. We will analyze the teleportation model and its relation to the one-way model. We will show how our calculus can be smoothly extended to cover this case as well as new models that we introduce in this paper. We get several benefits from our treatment. We get a workable syntax for handling the dependencies of operators on previous measurement outcomes just by mimicking the one obtained in the one-way model. This has never been done before for the teleportation model. Furthermore, we can use this embedding to obtain a standardization procedure for the models. Finally these extended calculi can be compositionally embedded back in the original one-way model. This clarifies the relation between different measurement-based models and shows that the one-way model of Raussendorf 1We use the word “pattern” rather than “program”, because this corresponds to the commonly used terminology in the physics literature. and Briegel is the canonical one. This paper develops the one-way model ab initio but certain concepts that the reader may be unfamiliar with: qubits, unitaries, measurements, Pauli operators and the Clifford group are in an appendix. These are also readily accessible through the very thorough book of Nielsen and Chuang [NC00]. In the next section we define the basic model, followed by its operational and denotational semantics, for completeness a simple proof of universality is given in section 4, this has appeared earlier in the physics literature [DKP05], in section 5 we develop the rewrite theory and prove the fundamental standardization theorem. In section 6 we develop several examples that illustrate the use of our calculus in designing efficient patterns. In section 7 we prove some theorems about the expressive power of the calculus in the absence of adaptive measurements. In section 8 we discuss other measurement-based models and their compositional embedding to and from the one-way model. In section 9 we discuss further directions and some more related work. In the appendix we review basic notions of quantum mechanics and quantum computation. 2 Measurement Patterns We first develop a notation for 1-qubit measurement based computations. The basic commands one can use in a pattern are: • 1-qubit auxiliary preparation Ni • 2-qubit entanglement operators Eij • 1-qubit measurements Mαi • and 1-qubit Pauli operators corrections Xi and Zi The indices i, j represent the qubits on which each of these operations apply, and α is a parameter in [0, 2π]. Expressions involving angles are always evaluated modulo 2π. These types of command will be referred to as N , E, M and C. Sequences of such commands, together with two distinguished – possibly overlapping – sets of qubits corresponding to inputs and outputs, will be called measurement patterns, or simply patterns. These patterns can be combined by composition and tensor product. Importantly, corrections and measurements are allowed to depend on previous measurement outcomes. We shall prove later that patterns without these classical dependencies can only realize unitaries that are in the Clifford group. Thus, dependencies are crucial if one wants to define a universal computing model; that is to say, a model where all unitaries over ⊗nC2 can be realized. It is also crucial to develop a notation that will handle these dependencies. This is what we do 2.1 Commands Preparation Ni prepares qubit i in state |+〉i. The entanglement commands are defined as Eij := ∧Zij (controlled-Z), while the correction commands are the Pauli operators Xi and Zi. Measurement Mαi is defined by orthogonal projections on |+α〉 := 1√ (|0〉+ eiα|1〉) |−α〉 := 1√ (|0〉 − eiα|1〉) followed by a trace-out operator. The parameter α ∈ [0, 2π] is called the angle of the mea- surement. For α = 0, α = π , one obtains the X and Y Pauli measurements. Operationally, measurements will be understood as destructive measurements, consuming their qubit. The out- come of a measurement done at qubit i will be denoted by si ∈ Z2. Since one only deals here with patterns where qubits are measured at most once (see condition (D1) below), this is unambiguous. We take the specific convention that si = 0 if under the corresponding measurement the state collapses to |+α〉, and si = 1 if to |−α〉. Outcomes can be summed together resulting in expressions of the form s = i∈I si which we call signals, and where the summation is understood as being done in Z2. We define the domain of a signal as the set of qubits on which it depends. As we have said before, both corrections and measurements may depend on signals. Depen- dent corrections will be written Xsi and Z i and dependent measurements will be written t[Mαi ] where s, t ∈ Z2 and α ∈ [0, 2π]. The meaning of dependencies for corrections is straightforward: X0i = Z i = I, no correction is applied, while X i = Xi and Z i = Zi. In the case of dependent measurements, the measurement angle will depend on s, t and α as follows: t[Mαi ] s := M (−1)sα+tπ i (1) so that, depending on the parities of s and t, one may have to modify the α to one of −α, α + π and −α+ π. These modifications correspond to conjugations of measurements under X and Z: i Xi = M i (2) i Zi = M i (3) accordingly, we will refer to them as the X and Z-actions. Note that these two actions commute, since −α+ π = −α− π up to 2π, and hence the order in which one applies them does not matter. As we will see later, relations (2) and (3) are key to the propagation of dependent corrections, and to obtaining patterns in the standard entanglement, measurement and correction form. Since the measurements considered here are destructive, the above equations actually simplify to Mαi Xi = M i (4) Mαi Zi = M i (5) Another point worth noticing is that the domain of the signals of a dependent command, be it a measurement or a correction, represents the set of measurements which one has to do before one can determine the actual value of the command. We have completed our catalog of basic commands, including dependent ones, and we turn now to the definition of measurement patterns. For convenient reference, the language syntax is summarized in Figure 1. 2.2 Patterns Definition 1 Patterns consists of three finite sets V , I, O, together with two injective maps ι : I → V and o : O → V and a finite sequence of commands An . . . A1, read from right to left, applying to qubits in V in that order, i.e. A1 first and An last, such that: (D0) no command depends on an outcome not yet measured; S := 0, 1, si, S + S Signals A := Ni Preparations Eij Entanglements t[Mαi ] s Measurements Xsi , Z i Corrections Figure 1: 1-qubit based measurement language syntax (D1) no command acts on a qubit already measured; (D2) no command acts on a qubit not yet prepared, unless it is an input qubit; (D3) a qubit i is measured if and only if i is not an output. The set V is called the pattern computation space, and we write HV for the associated quantum state space ⊗i∈V C2. To ease notation, we will omit the maps ι and o, and write simply I, O instead of ι(I) and o(O). Note, however, that these maps are useful to define classical manipulations of the quantum states, such as permutations of the qubits. The sets I, O are called respectively the pattern inputs and outputs, and we write HI , and HO for the associated quantum state spaces. The sequence An . . . A1 is called the pattern command sequence, while the triple (V, I,O) is called the pattern type. To run a pattern, one prepares the input qubits in some input state ψ ∈ HI , while the non-input qubits are all set to the |+〉 state, then the commands are executed in sequence, and finally the result of the pattern computation is read back from outputs as some φ ∈ HO. Clearly, for this procedure to succeed, we had to impose the (D0), (D1), (D2) and (D3) conditions. Indeed if (D0) fails, then at some point of the computation, one will want to execute a command which depends on outcomes that are not known yet. Likewise, if (D1) fails, one will try to apply a command on a qubit that has been consumed by a measurement (recall that we use destructive measurements). Similarly, if (D2) fails, one will try to apply a command on a non-existent qubit. Condition (D3) is there to make sure that the final state belongs to the output space HO, i.e., that all non-output qubits, and only non-output qubits, will have been consumed by a measurement when the computation ends. We write (D) for the conjunction of our definiteness conditions (D0), (D1), (D2) and (D3). Whether a given pattern satisfies (D) or not is statically verifiable on the pattern command se- quence. We could have imposed a simple type system to enforce these constraints but, in the interests of notational simplicity, we chose not to do so. Here is a concrete example: H := ({1, 2}, {1}, {2},Xs12 M01E12N2) with computation space {1, 2}, inputs {1}, and outputs {2}. To run H, one first prepares the first qubit in some input state ψ, and the second qubit in state |+〉, then these are entangled to obtain ∧Z12(ψ1 ⊗ |+〉2). Once this is done, the first qubit is measured in the |+〉, |−〉 basis. Finally an X correction is applied on the output qubit, if the measurement outcome was s1 = 1. We will do this calculation in detail later, and prove that this pattern implements the Hadamard operator H. In general, a given pattern may use auxiliary qubits that are neither input nor output qubits. Usually one tries to use as few such qubits as possible, since these contribute to the space complexity of the computation. A last thing to note is that one does not require inputs and outputs to be disjoint subsets of V . This, seemingly innocuous, additional flexibility is actually quite useful to give parsimonious implementations of unitaries [DKP05]. While the restriction to disjoint inputs and outputs is unnecessary, it has been discussed whether imposing it results in patterns that are easier to realize physically. Recent work [HEB04, BR05, CAJ05] however, seems to indicate it is not the case. 2.3 Pattern combination We are interested in how one can combine patterns in order to obtain bigger ones. The first way to combine patterns is by composing them. Two patterns P1 and P2 may be composed if V1 ∩ V2 = O1 = I2. Provided that P1 has as many outputs as P2 has inputs, by renaming the pattern qubits, one can always make them composable. Definition 2 The composite pattern P2P1 is defined as: — V := V1 ∪ V2, I = I1, O = O2, — commands are concatenated. The other way of combining patterns is to tensor them. Two patterns P1 and P2 may be tensored if V1 ∩ V2 = ∅. Again one can always meet this condition by renaming qubits in a way that these sets are made disjoint. Definition 3 The tensor pattern P1 ⊗ P2 is defined as: — V = V1 ∪ V2, I = I1 ∪ I2, and O = O1 ∪O2, — commands are concatenated. In contrast to the composition case, all the unions involved here are disjoint. Therefore commands from distinct patterns freely commute, since they apply to disjoint qubits, and when we say that commands have to be concatenated, this is only for definiteness. It is routine to verify that the definiteness conditions (D) are preserved under composition and tensor product. Before turning to this matter, we need a clean definition of what it means for a pattern to implement or to realize a unitary operator, together with a proof that the way one can combine patterns is reflected in their interpretations. This is key to our proof of universality. 3 The semantics of patterns In this section we give a formal operational semantics for the pattern language as a probabilistic labeled transition system. We define deterministic patterns and thereafter concentrate on them. We show that deterministic patterns compose. We give a denotational semantics of deterministic patterns; from the construction it will be clear that these two semantics are equivalent. Besides quantum states, which are non-zero vectors in some Hilbert space HV , one needs a classical state recording the outcomes of the successive measurements one does in a pattern. If we let V stand for the finite set of qubits that are still active (i.e. not yet measured) and W stands for the set of qubits that have been measured (i.e. they are now just classical bits recording the measurement outcomes), it is natural to define the computation state space as: S := ΣV,WHV × ZW2 . In other words the computation states form a V,W -indexed family of pairs2 q, Γ, where q is a quantum state from HV and Γ is a map from some W to the outcome space Z2. We call this classical component Γ an outcome map, and denote by ∅ the empty outcome map in Z∅2 . We will treat these states as pairs unless it becomes important to show how V and W are altered during a computation, as happens during a measurement. 3.1 Operational semantics We need some preliminary notation. For any signal s and classical state Γ ∈ ZW2 , such that the domain of s is included in W , we take sΓ to be the value of s given by the outcome map Γ. That is to say, if s = I si, then sΓ := I Γ(i) where the sum is taken in Z2. Also if Γ ∈ ZW2 , and x ∈ Z2, we define: Γ[x/i](i) = x, Γ[x/i](j) = Γ(j) for j 6= i which is a map in Z W∪{i} We may now view each of our commands as acting on the state space S, we have suppressed V and W in the first 4 commands: Ni−→ q ⊗ |+〉i,Γ Eij−→ ∧Zijq,Γ i−→ XsΓi q,Γ i−→ ZsΓi q,Γ V ∪ {i},W, q,Γ −→ V,W ∪ {i}, 〈+αΓ |iq,Γ[0/i] V ∪ {i},W, q,Γ −→ V,W ∪ {i}, 〈−αΓ |iq,Γ[1/i] where αΓ = (−1)sΓα + tΓπ following equation (1). Note how the measurement moves an index from V to W ; a qubit once measured cannot be neasured again. Suppose q ∈ HV , for the above relations to be defined, one needs the indices i, j on which the various command apply to be in V . One also needs Γ to contain the domains of s and t, so that sΓ and tΓ are well-defined. This will always be the case during the run of a pattern because of condition (D). All commands except measurements are deterministic and only modify the quantum part of the state. The measurement actions on S are not deterministic, so that these are actually binary relations on S, and modify both the quantum and classical parts of the state. The usual convention has it that when one does a measurement the resulting state is renormalized and the probabilities are associated with the transition. We do not adhere to this convention here, instead we leave the states unnormalized. The reason for this choice of convention is that this way, the probability of reaching a given state can be read off its norm, and the overall treatment is simpler. As we will show later, all the patterns implementing unitary operators will have the same probability for all the branches and hence we will not need to carry these probabilities explicitly. 2These are actually quadruples of the form (V,W, q,Γ), unless necessary we will suppress the V and the W . We introduce an additional command called signal shifting : i−→ q,Γ[Γ(i) + sΓ/i] It consists in shifting the measurement outcome at i by the amount sΓ. Note that the Z-action leaves measurements globally invariant, in the sense that |+α+π〉, |−α+π〉 = |−α〉, |+α〉. Thus changing α to α+ π amounts to swapping the outcomes of the measurements, and one has: t[Mαi ] s = Sti 0[Mαi ] s (6) and signal shifting allows to dispose of the Z action of a measurement, resulting sometimes in convenient optimizations of standard forms. 3.2 Denotational semantics Let P be a pattern with computation space V , inputs I, outputs O and command sequence An . . . A1. To execute a pattern, one starts with some input state q in HI , together with the empty outcome map ∅. The input state q is then tensored with as many |+〉s as there are non- inputs in V (the N commands), so as to obtain a state in the full space HV . Then E, M and C commands in P are applied in sequence from right to left. We can summarize the situation as follows: // HO HI × Z∅2 prep // HV × Z∅2 A1...An // HO × ZVrO2 If m is the number of measurements, which is also the number of non outputs, then the run may follow 2m different branches. Each branch is associated with a unique binary string s of length m, representing the classical outcomes of the measurements along that branch, and a unique branch map As representing the linear transformation from HI to HO along that branch. This map is obtained from the operational semantics via the sequence (qi,Γi) with 1 ≤ i ≤ n+ 1, such that: q1,Γ1 = q ⊗ |+ . . .+〉,∅ qn+1 = q ′ 6= 0 and for all i ≤ n : qi,Γi Ai−→ qi+1,Γi+1. Definition 4 A pattern P realizes a map on density matrices ρ given by ρ 7→ s(ρ)As. We write [[P]] for the map realized by P. Proposition 5 Each pattern realizes a completely positive trace preserving map. Proof. Later on we will show that every pattern can be put in a semantically equivalent form where all the preparations and entanglements appear first, followed by a sequence of measurements and finally local Pauli corrections. Hence branch maps decompose as As = CsΠsU , where Cs is a unitary map over HO collecting all corrections on outputs, Πs is a projection from HV to HO rep- resenting the particular measurements performed along the branch, and U is a unitary embedding from HI to HV collecting the branch preparations, and entanglements. Note that U is the same on all branches. Therefore, sAs = U †Π†sC sCsΠsU U †Π†sΠsU = U †( = U †U = I where we have used the fact that Cs is unitary, Πs is a projection and U is independent of the branches and is also unitary. Therefore the map T (ρ) := As(ρ)A s is a trace-preserving completely-positive map (cptp-map), explicitly given as a Kraus decomposition. ✷ Hence the denotational semantics of a pattern is a cptp-map. In our denotational semantics we view the pattern as defining a map from the input qubits to the output qubits. We do not explicitly represent the result of measuring the final qubits; these may be of interest in some cases. Techniques for dealing with classical output explicitly are given by Selinger [Sel04b] and Unruh [Unr05]. Definition 6 A pattern is said to be deterministic if it realizes a cptp-map that sends pure states to pure states. A pattern is said to be strongly deterministic when branch maps are equal. This is equivalent to saying that for a deterministic pattern branch maps are proportional, that is to say, for all q ∈ HI and all s1, s2 ∈ Zn2 , As1(q) and As2(q) differ only up to a scalar. For a strongly deterministic pattern we have for all s1, s2 ∈ Zn2 , As1 = As2 . Proposition 7 If a pattern is strongly deterministic, then it realizes a unitary embedding. Proof. Define T to be the map realized by the pattern. We have T = sAs. Since the pattern in strongly deterministic all the branch maps are the same. Define A to be 2n/2As, then A must be a unitary embedding, because A†A = I. ✷ 3.3 Short examples For the rest of paper we assume that all the non-input qubits are prepared in the state |+〉 and hence for simplicity we omit the preparation commands NIc . First we give a quick example of a deterministic pattern that has branches with different proba- bilities. Its type is V = {1, 2}, I = O = {1}, and its command sequence is Mα2 . Therefore, starting with input q, one gets two branches: q ⊗ |+〉,∅ (1 + e−iα)q,∅[0/2] (1− e−iα)q,∅[1/2] Thus this pattern is indeed deterministic, and implements the identity up to a global phase, and yet the two branches have respective probabilities (1 + cosα)/2 and (1 − cosα)/2, which are not equal in general and hence this pattern is not strongly deterministic. There is an interesting variation on this first example. The pattern of interest, call it T , has the same type as above with command sequence X 2E12. Again, T is deterministic, but not strongly deterministic: the branches have different probabilities, as in the preceding example. Now, however, these probabilities may depend on the input. The associated transformation is a cptp-map, T (ρ) := AρA† +BρB† with: , B := One has A†A+B†B = I, so T is indeed a completely positive and trace-preserving linear map and T (|ψ〉〈ψ|) = 〈ψ,ψ〉|0〉〈0| and clearly for no unitary U does one have T (ρ) := UρU †. For our final example, we return to the pattern H, already defined above. Consider the pattern with the same qubit space {1, 2}, and the same inputs and outputs I = {1}, O = {2}, as H, but with a shorter command sequence namely M01E12. Starting with input q = (a|0〉 + b|1〉)|+〉, one has two computation branches, branching at M01 : (a|0〉+ b|1〉)|+〉,∅ E12−→ 1√ (a|00〉 + a|01〉+ b|10〉 − b|11〉),∅ ((a+ b)|0〉 + (a− b)|1〉),∅[0/1] ((a− b)|0〉 + (a+ b)|1〉),∅[1/1] and since ‖a+ b‖2 + ‖a− b‖2 = 2(‖a‖2 + ‖b‖2), both transitions happen with equal probabilities 1 Both branches end up with non proportional outputs, so the pattern is not deterministic. However, if one applies the local correction X2 on either of the branches’ ends, both outputs will be made to coincide. If we choose to let the correction apply to the second branch, we obtain the pattern H, already defined. We have just proved H = UH, that is to say H realizes the Hadamard operator. 3.4 Compositionality of the Denotational Semantics With our definitions in place, we will show that the denotational semantics is compositional. Theorem 1 For two patterns P1 and P2 we have [[P1P2]] = [[P2]][[P1]] and [[P1⊗P2]] = [[P2]]⊗ [[P1]]. Proof. Recall that two patterns P1, P2 may be combined by composition provided P1 has as many outputs as P2 has inputs. Suppose this is the case, and suppose further that P1 and P2 respectively realize some cptp-maps T1 and T2. We need to show that the composite pattern P2P1 realizes T2T1. Indeed, the two diagrams representing branches in P1 and P2: // HO1 HI2 // HO2 HI1 × Z p1// HV1 × Z // HO1 × Z V1rO1 HI2 × Z p2// HV2 × Z // HO2 × Z V2rO2 can be pasted together, since O1 = I2, and HO1 = HI2 . But then, it is enough to notice 1) that preparation steps p2 in P2 commute with all actions in P1 since they apply on disjoint sets of qubits, and 2) that no action taken in P2 depends on the measurements outcomes in P1. It follows that the pasted diagram describes the same branches as does the one associated to the composite P2P1. A similar argument applies to the case of a tensor combination, and one has that P2 ⊗ P1 realizes T2 ⊗ T1. ✷ If one wanted to give a categorical treatment3 one can define a category where the objects are finite sets representing the input and output qubits and the morphisms are the patterns. This is clearly a monoidal category with our tensor operation as the monoidal structure. One can show that the denotational semantics gives a monoidal functor into the category of superoperators or into any suitably enriched strongly compact closed category [AC04] or dagger category [Sel05a]. It would be very interesting to explore exactly what additional categorical structures are required to interpret the measurement calculus presented below. Duncan Ross[Dun05] has skectched a polycategorical presentation of our measurement calculus. 4 Universality Define the two following patterns on V = {1, 2}: J (α) := Xs12 M 1 E12 (7) ∧Z := E12 (8) with I = {1}, O = {2} in the first pattern, and I = O = {1, 2} in the second. Note that the second pattern does have overlapping inputs and outputs. Proposition 8 The patterns J (α) and ∧Z are universal. Proof. First, we claim J (α) and ∧Z respectively realize J(α) and ∧Z, with: J(α) := 1√ 1 eiα 1 −eiα We have already seen in our example that J (0) = H implements H = J(0), thus we already know this in the particular case where α = 0. The general case follows by the same kind of computation.4 The case of ∧Z is obvious. Second, we know that these unitaries form a universal set for ⊗nC2 [DKP05]. Therefore, from the preceding section, we infer that combining the corresponding patterns will generate patterns realizing any unitary in ⊗nC2. ✷ These patterns are indeed among the simplest possible. As a consequence, in the section devoted to examples, we will find that our implementations often have lower space complexity than the traditional implementations. Remarkably, in our set of generators, one finds a single measurement and a single dependency, which occurs in the correction phase of J (α). Clearly one needs at least one measurement, since patterns without measurements can only implement unitaries in the Clifford group. It is also true that dependencies are needed for universality, but we have to wait for the development of the measurement calculus in the next section to give a proof of this fact. 3The rest of the paragraph can be omitted without loss of continuity. 4Equivalently, this follows from J(α) = HP (α), with P (α) = 0 eiα 1 E12 = X 1P (α)1E12 = HP (α)1. 5 The measurement calculus We turn to the next important matter of the paper, namely standardization. The idea is quite simple. It is enough to provide local pattern-rewrite rules pushing Es to the beginning of the pattern and Cs to the end. The crucial point is to justify using the equations as rewrite rules. 5.1 The equations The expressions appearing as commands are all linear operators on Hilbert space. At first glance, the appropriate equality between commands is equality as operators. For the deterministic com- mands, the equality that we consider is indeed equality as operators. This equality implies equality in the denotational semantics. However, for measurement commands one needs a stricter definition for equality in order to be able to apply them as rewriting rules. Essentially we have to take into the account the effect of different branches that might result from the measurement process. The precise definition is below. Definition 9 Consider two patterns P and P ′ we define P = P ′ if and only if for any branch s, we have APs = A s , where A s and A s are the branch map As defined in Section 3.2. The first set of equations gives the means to propagate local Pauli corrections through the entangling operator Eij. i = X jEij (9) j = X iEij (10) i = Z iEij (11) j = Z jEij (12) These equations are easy to verify and are natural since Eij belongs to the Clifford group, and therefore maps under conjugation the Pauli group to itself. Note that, despite the symmetry of the Eij operator qua operator, we have to consider all the cases, since the rewrite system defined below does not allow one to rewrite Eij to Eji. If we did allow this the reqrite process could loop forever. A second set of equations allows one to push corrections through measurements acting on the same qubit. Again there are two cases: t[Mαi ] sXri = t[Mαi ] s+r (13) t[Mαi ] sZri = t+r[Mαi ] s (14) These equations follow easily from equations (4) and (5). They express the fact that the measure- ments Mαi are closed under conjugation by the Pauli group, very much like equations (9),(10),(11) and (12) express the fact that the Pauli group is closed under conjugation by the entanglements Eij . Define the following convenient abbreviations: [Mαi ] s := 0[Mαi ] s, t[Mαi ] := t[Mαi ] 0, Mαi := 0[Mαi ] Mxi :=M i , M i :=M Particular cases of the equations above are: Mxi X i = M i = [M s = s[M i ] = M The first equation, follows from the fact that −0 = 0, so the X action on Mxi is trivial; the second equation, is because −π is equal π + π modulo 2π, and therefore the X and Z actions coincide on i . So we obtain the following: t[Mxi ] s = t[Mxi ] (15) s = s+t[M i ] (16) which we will use later to prove that patterns with measurements of the form Mx and My may only realize unitaries in the Clifford group. 5.2 The rewrite rules We now define a set of rewrite rules, obtained by orienting the equations above5: i ⇒ Xsi ZsjEij EX j ⇒ XsjZsiEij EX i ⇒ ZsiEij EZ j ⇒ ZsjEij EZ t[Mαi ] sXri ⇒ t[Mαi ]s+r MX t[Mαi ] sZri ⇒ r+t[Mαi ]s MZ to which we need to add the free commutation rules, obtained when commands operate on disjoint sets of qubits: EijA~k ⇒ A~kEij where A is not an entanglement i ⇒ XsiA~k where A is not a correction i ⇒ ZsiA~k where A is not a correction where ~k represent the qubits acted upon by command A, and are supposed to be distinct from i and j. Clearly these rules could be reversed since they hold as equations but we are orienting them this way in order to obtain termination. Condition (D) is easily seen to be preserved under rewriting. Under rewriting, the computation space, inputs and outputs remain the same, and so do the entanglement commands. Measurements might be modified, but there is still the same number of them, and they still act on the same qubits. The only induced modifications concern local corrections and dependencies. If there was no dependency at the start, none will be created in the rewriting process. In order to obtain rewrite rules, it was essential that the entangling command (∧Z) belongs to the normalizer of the Pauli group. The point is that the Pauli operators are the correction operators and they can be dependent, thus we can commute the entangling commands to the beginning without inheriting any dependency. Therefore the entanglement resource can indeed be prepared at the outset of the computation. 5Recall that patterns are executed from right to left. 5.3 Standardization Write P ⇒ P ′, respectively P ⇒⋆ P ′, if both patterns have the same type, and one obtains the command sequence of P ′ from the command sequence of P by applying one, respectively any number, of the rewrite rules of the previous section. We say that P is standard if for no P ′, P ⇒ P ′ and the procedure of writing a pattern to standard form is called standardization6. One of the most important results about the rewrite system is that it has the desirable properties of determinacy (confluence) and termination (standardization). In other words, we will show that for all P, there exists a unique standard P ′, such that P ⇒⋆ P ′. It is, of course, crucial that the standardization process leaves the semantics of patterns invariant. This is the subject of the next simple, but important, proposition, Proposition 10 Whenever P ⇒⋆ P ′, [[P]] = [[P ′]]. Proof. It is enough to prove it when P ⇒ P ′. The first group of rewrites has been proved to be sound in the preceding subsections, while the free commutation rules are obviously sound. ✷ We now begin the main proof of this section. First, we prove termination. Theorem 2 (Termination) All rewriting sequences beginning with a pattern P terminate after finitely many steps. For our rewrite system, this implies that for all P there exist finitely many P ′ such that P ⇒⋆ P ′ where the P ′ are standard. Proof. Suppose P has command sequence An . . . A1; so the number of commands is n. Let e ≤ n be the number of E commands in P. As we have noted earlier, this number is invariant under ⇒. Moreover E commands in P can be ordered by increasing depth, read from right to left, and this order, written <E, is also invariant, since EE commutations are forbidden explicitly in the free commutation rules. Define the following depth function d on E and C commands in P: d(Ai) = i if Ai = Ejk n− i if Ai = Cj Define further the following sequence of length e, dE(P)(i) is the depth of the E-command of rank i according to <E. By construction this sequence is strictly increasing. Finally, we define the measure m(P) := (dE(P), dC (P)) with: dC(P) = C∈P d(C) We claim the measure we just defined decreases lexicographically under rewriting, in other words P ⇒ P ′ implies m(P) > m(P ′), where < is the lexicographic ordering on Ne+1. To clarify these definitions, consider the following example. Suppose P’s command sequence is of the form EXZE, then e = 2, dE(P) = (1, 4), and m(P) = (1, 4, 3). For the command sequence EEX we get that e = 2, dE(P) = (2, 3) and m(P) = (2, 3, 2). Now, if one considers the rewrite EEX ⇒ EXZE, the measure of the left hand side is (2, 3, 2), while the measure of the right hand side, as said, is (1, 4, 3), and indeed (2, 3, 2) > (1, 4, 3). Intuitively the reason is clear: the Cs are being pushed to the left, thus decreasing the depths of Es, and concomitantly, the value of dE . 6We use the word “standardization” instead of the more usual “normalization” in order not to cause terminological confusion with the physicists’ notion of normalization. Let us now consider all cases starting with an EC rewrite. Suppose the E command under rewrite has depth d and rank i in the order <E. Then all Es of smaller rank have same depth in the right hand side, while E has now depth d − 1 and still rank i. So the right hand side has a strictly smaller measure. Note that when C = X, because of the creation of a Z (see the example above), the last element of m(P) may increase, and for the same reason all elements of index j > i in dE(P) may increase. This is why we are working with a lexicographical ordering. Suppose now one does an MC rewrite, then dC(P) strictly decreases, since one correction is absorbed, while all E commands have equal or smaller depths. Again the measure strictly decreases. Next, suppose one does an EA rewrite, and the E command under rewrite has depth d and rank i. Then it has depth d− 1 in the right hand side, and all other E commands have invariant depths, since we forbade the case when A is itself an E. It follows that the measure strictly decreases. Finally, upon an AC rewrite, all E commands have invariant depth, except possibly one which has smaller depth in the case A = E, and dC(P) decreases strictly because we forbade the case where A = C. Again the claim follows. So all rewrites decrease our ordinal measure, and therefore all sequences of rewrites are finite, and since the system is finitely branching (there are no more than n possible single step rewrites on a given sequence of length n), we get the statement of the theorem. The final statement of the theorem follows from the fact that we have finitely many rules so the system is finitely branching. In any finitely branching rewrite system with the property that every rewrite sequence terminates, it is clearly true that there can be only finitely many standard forms. The next theorem establishes the important determinacy property and furthermore shows that the standard patterns have a certain canonical form which we call the NEMC form. The precise definition is: Definition 11 A pattern has a NEMC form if its commands occur in the order of Ns first, then Es , then Ms, and finally Cs. We will usually just say “EMC” form since we can assume that all the auxiliary qubits are prepared in the |+〉 state we usually just elide these N commands. Theorem 3 (Confluence) For all P, there exists a unique standard P ′, such that P ⇒⋆ P ′, and P ′ is in EMC form. Proof. Since the rewriting system is terminating, confluence follows from local confluence 7 by Newman’s lemma, see, for example, [Bar84]. The uniqueness of the standard is form an immediate consequence. We look for critical pairs, that is occurrences of three successive commands where two rules can be applied simultaneously. One finds that there are only five types of critical pairs, of these the three involve the N command, these are of the form: NMC, NEC and NEM ; and the remaining two are: EijMkCk with i, j and k all distinct, EijMkCl with k and l distinct. In all cases local confluence is easily verified. Suppose now P ′ does not satisfy the EMC form conditions. Then, either there is a pattern EA with A not of type E, or there is a pattern AC with A not of type C. In the former case, E and 7This means that whenever two rewrite rules can be applied to a term t yielding t1 and t2, one can rewrite both t1 and t2 to a common third term t3, possibly in many steps. A must operate on overlapping qubits, else one may apply a free commutation rule, and A may not be a C since in this case one may apply an EC rewrite. The only remaining case is when A is of type M , overlapping E’s qubits, but this is what condition (D1) forbids, and since (D1) is preserved under rewriting, this contradicts the assumption. The latter case is even simpler. ✷ We have shown that under rewriting any pattern can be put in EMC form, which is what we wanted. We actually proved more, namely that the standard form obtained is unique. However, one has to be a bit careful about the significance of this additional piece of information. Note first that uniqueness is obtained because we dropped the CC and EE free commutations, thus having a rigid notion of command sequence. One cannot put them back as rewrite rules, since they obviously ruin termination and uniqueness of standard forms. A reasonable thing to do, would be to take this set of equations as generating an equivalence relation on command sequences, call it ≡, and hope to strengthen the results obtained so far, by proving that all reachable standard forms are equivalent. But this is too naive a strategy, since E12X1X2 ≡ E12X2X1, and: 2 ⇒⋆ Xs1Zs2Xt2Zt1E12 ≡ Xs1Zt1Zs2Xt2E12 obtaining an expression which is not symmetric in 1 and 2. To conclude, one has to extend ≡ to include the additional equivalence Xs1Z 1 ≡ Zt1Xs1 , which fortunately is sound since these two operators are equal up to a global phase. Thus, these are all equivalent in our semantics of patterns. We summarize this discussion as follows. Definition 12 We define an equivalence relation ≡ on patterns by taking all the rewrite rules as equations and adding the equation Xs1Z 1 ≡ Zt1Xs1 and generating the smallest equivalence relation. With this definition we can state the following proposition. Proposition 13 All patterns that are equivalent by ≡ are equal in the denotational semantics. This≡ relation preserves both the type (the (V, I,O) triple) and the underlying entanglement graph. So clearly semantic equality does not entail equality up to ≡. In fact, by composing teleportation patterns one obtains infinitely many patterns for the identity which are all different up to ≡. One may wonder whether two patterns with same semantics, type and underlying entanglement graph are necessarily equal up to ≡. This is not true either. One has J(α)J(0)J(β) = J(α + β) = J(β)J(0)J(α) (where J(α) is defined in Section 4), and this readily gives a counter-example. We can now formally describe a simple standardization algorithm. Algorithm 1 Input: A pattern P on |V | = N qubits with command sequence AM · · ·A1. Output: An equivalent pattern P ′ in NEMC form. 1. Commute all the preparation commands (new qubits) to the right side. 2. Commute all the correction commands to the left side using the EC and MC rewriting rules. 3. Commute all the entanglement commands to the right side after the preparation commands. Note that since each qubit can be entangled with at most N − 1 other qubits, and can be measured or corrected only once, we have O(N2) entanglement commands and O(N) measurement commands. According to the definiteness condition, no command acts on a qubit not yet prepared, hence the first step of the above algorithm is based on trivial commuting rules; the same is true for the last step as no entanglement command can act on a qubit that has been measured. Both steps can be done in O(N2). The real complexity of the algorithm comes from the second step and the EX commuting rule. In the worst case scenario, commuting an X correction to the left might create O(N2) other Z corrections, each of which has to be commuted to the left themselves. Thus one can have at most O(N3) new corrections, each of which has to be commuted past O(N2) measurement or entanglement commands. Therefore the second step, and hence the algorithm, has a worst case complexity of O(N5). We conclude this subsection by emphasizing the importance of the EMC form. Since the entanglement can always be done first, we can always derive the entanglement resource needed for the whole computation right at the beginning. After that only local operations will be performed. This will separate the analysis of entanglement resource requirements from the classical control. Furthermore, this makes it possible to extract the maximal parallelism for the execution of the pattern since the necessary dependecies are explicitly expressed, see the example in section 6 for further discussion. Finally, the EMC form provides us with tools to prove general theorems about patterns, such as the fact that they always compute cptp-maps and the expressiveness theorems of section 7. 5.4 Signal shifting One can extend the calculus to include the signal shifting command Sti . This allows one to dispose of dependencies induced by the Z-action, and obtain sometimes standard patterns with smaller computational depth complexity, as we will see in the next section which is devoted to examples. t[Mαi ] s ⇒ Sti [Mαi ]s i ⇒ StiX s[t+si/si] i ⇒ StiZ s[t+si/si] t[Mαj ] sSri ⇒ Sri t[r+si/si][Mαj ]s[r+si/si] Ssi S j ⇒ StjS s[t+sj/sj ] where s[t/si] denotes the substitution of si with t in s, s, t being signals. Note that when we write a t explicitly on the upper left of an M , we mean that t 6= 0. The first additional rewrite rule was already introduced as equation (6), while the other ones merely propagate the signal shift. Clearly one can dispose of Sti when it hits the end of the pattern command sequence. We will refer to this new set of rules as ⇒S. Note that we always apply first the standardization rules and then signal shifting, hence we do not need any commutation rule for E and S commands. It is important to note that both theorem 2 and 3 still hold for this extended rewriting system. In order to prove termination one can start with the EMC form and then adapt the proof of Theorem 2 by defining a depth function for a signal shift similar to the depth of a correction command. As with the correction, signal shifts can also be commuted to the left hand side of a command sequence. Now our measure can be modified to account for the new signal shifting terms and shown to be decreasing under each step of signal shifting. Confluence can be also proved from local confluence using again Newman’s Lemma [Bar84]. One typical critical pair is t[Mαj ]S i where i appears in the domain of signal t and hence the signal shifting command Ssi will have an effect on the measurement. Now there are two possible ways to rewrite this pair, first, commute the signal shifting command and then replace the left signal of the measurement with its own signal shifting command: t[Mαj ] S i ⇒ Ssi t+s[Mαj ] ⇒ Ssi S The other way is to first replace the left signal of the measurement and then commute the signal shifting command: t[Mαj ] S i ⇒ StjMαj Ssi ⇒ Stj Ssi Mαj Now one more step of rewriting on the last equation will give us the same result for both choices. Stj S j ⇒ Ssi S All other critical terms can be dealt with similarly. 6 Examples In this section we develop some examples illustrating pattern composition, pattern standardization, and signal shifting. We compare our implementations with the implementations given in the refer- ence paper [RBB03]. To combine patterns one needs to rename their qubits as we already noted. We use the following concrete notation: if P is a pattern over {1, . . . , n}, and f is an injection, we write P(f(1), . . . , f(n)) for the same pattern with qubits renamed according to f . We also write P2 ◦ P1 for pattern composition, in order to make it more readable. Finally we define the computational depth complexity to be the number of measurement rounds plus one final correction round. More details on depth complexity, specially on the preparation depth, i.e. depth of the entanglement commands, can be found in [BK06]. Teleportation. Consider the composite pattern J (β)(2, 3)◦J (α)(1, 2) with computation space {1, 2, 3}, inputs {1}, and outputs {3}. We run our standardization procedure so as to obtain an equivalent standard pattern: J (β)(2, 3) ◦ J (α)(1, 2) = Xs23 M 2 E23X 1 E12 ⇒EX Xs23 M 1 E23E12 ⇒MX Xs23 Z s1M−α1 E23E12 Let us call the pattern just obtained J (α, β). If we take as a special case α = β = 0, we get: 1E23E12 and since we know that J (0) implements H and H2 = I, we conclude that this pattern implements the identity, or in other words it teleports qubit 1 to qubit 3. As it happens, this pattern obtained by self-composition, is the same as the one given in the reference paper [RBB03, p.14]. x-rotation. Here is the reference implementation of an x-rotation [RBB03, p.17], Rx(α): Xs23 Z s1Mx1E23E12 with type {1, 2, 3}, {1}, and {3}. There is a natural question which one might call the recognition problem, namely how does one know this is implementing Rx(α) ? Of course there is the brute force answer to that, which we applied to compute our simpler patterns, and which consists in computing down all the four possible branches generated by the measurements at qubits 1 and 2. Another possibility is to use the stabilizer formalism as explained in the reference paper [RBB03]. Yet another possibility is to use pattern composition, as we did before, and this is what we are going to do. We know that Rx(α) = J(α)H up to a global phase, hence the composite pattern J (α)(2, 3) ◦ H(1, 2) implements Rx(α). Now we may standardize it: J (α)(2, 3) ◦ H(1, 2) = Xs23 M 2 E23X ⇒EX Xs23 Z 1E23E12 ⇒MX Xs23 Z s1Mx1E23E12 obtaining exactly the implementation above. Since our calculus preserves the semantics, we deduce that the implementation is correct. z-rotation. Now, we have a method here for synthesizing further implementations. Let us replay it with another rotation Rz(α). Again we know that Rz(α) = HRx(α)H, and we already know how to implement both components H and Rx(α). So we start with the pattern H(4, 5) ◦ Rx(α)(2, 3, 4) ◦ H(1, 2) and standardize it: H(4, 5) ◦ Rx(α)(2, 3, 4) ◦ H(1, 2) = H(4, 5)Xs34 Z 1+s2Mx2E34 E23X 1E12 ⇒EX H(4, 5)Xs34 Z 1+s2Mx2X 2 E34Z 1E123 ⇒EZ H(4, 5)Xs34 Z 1+s2Z 1E1234 ⇒MX H(4, 5)Xs34 Z 1+s2Zs13 M 1E1234 ⇒MZ 4 E45X s1 [Mα3 ] 1+s2Mx2M 1E1234 ⇒EX Xs45 Z s1 [Mα3 ] 1+s2Mx2M 1E12345 ⇒MX s1 [Mα3 ] 1+s2Mx2M 1E12345 ⇒MZ Xs45 Z s2 [Mx4 ] s3s1 [Mα3 ] 1+s2Mx2M 1E12345 To aid reading E23E12 is shortened to E123, E12E23E34 to E1234, and t[Mαi ] 1+s is used as shorthand for t[M−αi ] Here for the first time, we see MZ rewritings, inducing the Z-action on measurements. The resulting standardized pattern can therefore be rewritten further using the extended calculus: Xs45 Z s2 [Mx4 ] s3s1 [Mα3 ] 1+s2Mx2M 1E12345 ⇒S s2+s4 s1+s3 1+s2Mx2M 1E12345 obtaining the pattern given in the reference paper [RBB03, p.5]. However, just as in the case of the Rx rotation, we also have Rz(α) = HJ(α) up to a global phase, hence the pattern H(2, 3)J (α)(1, 2) also implements Rz(α), and we may standardize it: H(2, 3) ◦ J (α)(1, 2) = Xs23 Mx2 E23X 1 E12 ⇒EX Xs23 Z 1 E123 ⇒MX Xs23 Z 1 E123 obtaining a 3 qubit standard pattern for the z-rotation, which is simpler than the preceding one, because it is based on the J (α) generators. Since the z-rotation Rz(α) is the same as the phase operator: P (α) = 0 eiα up to a global phase, we also obtain with the same pattern an implementation of the phase oper- ator. In particular, if α = π , using the extended calculus, we get the following pattern for P (π Xs23 Z 1E123. General rotation. The realization of a general rotation based on the Euler decomposition of rotations asRx(γ)Rz(β)Rx(α), would results in a 7 qubit pattern. We get a 5 qubit implementation based on the J(α) decompo- sition [DKP05]: R(α, β, γ) = J(0)J(−α)J(−β)J(−γ) (The parameter angles are inverted to make the computation below more readable.) The extended standardization procedure yields: J (0)(4, 5)J (−α)(3, 4)J (−β)(2, 3)J (−γ)(1, 2) = Xs45 M 4E45X 3 E34X 2 E23X 1E12 ⇒EX 4E45X 3 E34X 1E123 ⇒MX 4E45X 3 E34X 1E123 ⇒EXZ Xs45 M 4E45X 1E1234 ⇒MXZ 4 E45X s1 [Mα3 ] s2 [M 1E1234 ⇒EXZ Xs45 M s1 [Mα3 ] s2 [M 1E12345 ⇒MXZ s2 [M04 ] s1 [Mα3 ] s2 [M 1E12345 ⇒S s2+s4 s1+s3 s2 [M 1E12345 CNOT (∧X). This is our first example with two inputs and two outputs. We use here the trivial pattern I with computation space {1}, inputs {1}, outputs {1}, and empty command sequence, which implements the identity over H1. One has ∧X = (I ⊗ H)∧Z(I ⊗ H), so we get a pattern using 4 qubits over {1, 2, 3, 4}, with inputs {1, 2}, and outputs {1, 4}, where one notices that inputs and outputs intersect on the control qubit {1}: (I(1)⊗ 〈(3, 4))∧Z(1, 3)(I(1) ⊗ 〈(2, 3)) = Xs34 Mx3E34E13X By standardizing: 3E34 E13X 2E23 ⇒EX 3 E34X 2E13E23 ⇒EX Xs34 Z 2E13E23E34 ⇒MX Xs34 Z 2E13E23E34 Note that, in this case, we are not using the E1234 abbreviation, because the underlying struc- ture of entanglement is not a chain. This pattern was already described in Aliferis and Leung’s paper [AL04]. In their original presentation the authors actually use an explicit identity pattern (us- ing the teleportation pattern J (0, 0) presented above), but we know from the careful presentation of composition that this is not necessary. We present now a family of patterns preparing the GHZ entangled states |0 . . . 0〉 + |1 . . . 1〉. One GHZ(n) = (Hn ∧Zn−1n . . . H2 ∧Z12)|+. . .+〉 and by combining the patterns for ∧Z and H, we obtain a pattern with computation space {1, 2, 2′, . . . , n, n′}, no inputs, outputs {1, 2′, . . . , n′}, and the following command sequence: MxnEnn′E(n−1)′n . . . X Mx2E22′E12 With this form, the only way to run the pattern is to execute all commands in sequence. The situation changes completely, when we bring the pattern to extended standard form: MxnEnn′E(n−1)′n . . . X Mx3E33′ E2′3X Mx2E22′E12 ⇒ MxnEnn′E(n−1)′n . . . X Mx3 Z 2E33′E2′3E22′E12 ⇒ MxnEnn′E(n−1)′n . . . X s2 [Mx3 ]M 2E33′E2′3E22′E12 ⇒⋆ Xsnn′ . . . X sn−1 [Mxn ] . . . s2 [Mx3 ]M 2Enn′E(n−1)′n . . . E33′E2′3E22′E12 ⇒S Xs2+s3+···+sn . . . Xs2+s3 Mxn . . .M 2Enn′E(n−1)′n . . . E33′E2′3E22′E12 All measurements are now independent of each other, it is therefore possible after the entanglement phase, to do all of them in one round, and in a subsequent round to do all local corrections. In other words, the obtained pattern has constant computational depth complexity 2. Controlled-U . This final example presents another instance where standardization obtains a low computational depth complexity, the proof of this fact can be found in [BK06]. For any 1-qubit unitary U , one has the following decomposition of ∧U in terms of the generators J(α) [DKP05]: ∧U12 = J01Jα 2 ∧Z12J −π−δ−β 2 ∧Z12J −β+δ−π with α′ = α + β+γ+δ . By translating each J operator to its corresponding pattern, we get the following wild pattern for ∧U : BEBCX A EABX j EjkX i Eij h EhiX g EghX fEfgEAfX e Eef Xsde M d EdeX π+δ+β c EcdX bEbcEAbX β−δ+π a Eab In order to run the wild form of the pattern one needs to follow the pattern commands in sequence. It is easy to verify that, because of the dependent corrections, one needs at least 12 rounds to complete the execution of the pattern. The situation changes completely after extended standard- ization: si+sg+se+sc+sa sj+sh+sf+sd+sb sA+se+sc sh+sf+sd+sb [M ]sg+se+sc+sa [M sf+sd+sb M0f [M sd+sb [M sc+sa [M π+δ+β sbM0bM β−δ+π EBCEABEjkEijEhiEghEfgEAfEefEdeEcdEbcEabEAb Now the order between measurements is relaxed, as one sees in Figure 2, which describes the depen- dency structure of the standard pattern above. Specifically, all measurements can be completed in 7 rounds. This is just one example of how standardization lowers computational depth complexity, and reveals inherent parallelism in a pattern. Figure 2: The dependency graph for the standard ∧U pattern. 7 The no dependency theorems From standardization we can also infer results related to dependencies. We start with a simple observation which is a direct consequence of standardization. Lemma 14 Let P be a pattern implementing some cptp-maps T , and suppose P’s command se- quence has measurements only of the Mx and My kind, then U has a standard implementation, having only independent measurements, all being of the Mx and My kind (therefore of computa- tional depth complexity at most 2). Proof. Write P ′ for the standard pattern associated to P. By equations (15) and (16), the X- actions can be eliminated from P ′, and then Z-actions can be eliminated by using the extended calculus. The final pattern still implements T , has no longer any dependent measurements, and has therefore computational depth complexity at most 2. ✷ Theorem 4 Let U be a unitary operator, then U is in the Clifford group iff there exists a pattern P implementing U , having measurements only of the Mx and My kind. Proof. The “only if” direction is easy, since we have seen in the example section, standard patterns for ∧X, H and P (π ) which had only independent Mx and My measurements. Hence any Clifford operator can be implemented by a combination of these patterns. By the lemma above, we know we can actually choose these patterns to be standard. For the “if” direction, we prove that U belongs to the normalizer of the Pauli group, and hence by definition to the Clifford group. In order to do so we use the standard form of P written as P ′ = CP ′MP ′EP ′ which still implements U , and has only Mx and My measurements. Recall that, because of equations (15) and (16), these measurements are independent. Let i be an input qubit, and consider the pattern P ′′ = P ′Ci, where Ci is either Xi or Zi. Clearly P ′′ implements UCi. First, one has: CP ′MP ′EP ′Ci ⇒⋆EC CP ′MP ′C ′EP ′ for some non-dependent sequence of corrections C ′, which, up to free commutations can be written uniquely as C ′OC ′′, where C ′O applies on output qubits, and therefore commutes to MP ′ , and C applies on non-output qubits (which are therefore all measured in MP ′). So, by commuting C both through MP ′ and CP ′ (up to a global phase), one gets: CP ′MP ′C ′EP ′ ⇒⋆ C ′OCP ′MP ′C ′′EP ′ Using equations (15), (16), and the extended calculus to eliminate the remaining Z-actions, one gets: MP ′C ′′ ⇒⋆MC,S SMP ′ for some product S = {j∈J} S j of constant shifts 8, applying to some subset J of the non-output qubits. So: C ′OCP ′MP ′C ′′EP ′ ⇒⋆MC,S C ′OCP ′SMP ′EP ′ ⇒⋆ C ′OC ′′OCP ′MP ′EP ′ where C ′′O is a further constant correction obtained by signal shifting CP ′ with S. This proves that P ′′ also implements C ′OC ′′OU , and therefore UCi = C ′OC ′′OU which completes the proof, since C ′OC ′′O is a non dependent correction. ✷ The “only if” part of this theorem already appears in previous work [RBB03, p.18]. The “if” part can be construed as an internalization of the argument implicit in the proof of Gottesman-Knill theorem [NC00, p.464]. We can further prove that dependencies are crucial for the universality of the model. Observe first that if a pattern has no measurements, and hence no dependencies, then it follows from (D2) that V = O, i.e., all qubits are outputs. Therefore computation steps involve only X, Z and 8Here we have used the trivial equations Za+1i = ZiZ i and X i = XiX ∧Z, and it is not surprising that they compute a unitary which is in the Clifford group. The general argument essentially consists in showing that when there are measurements, but still no dependencies, then the measurements are playing no part in the result. Theorem 5 Let P be a pattern implementing some unitary U , and suppose P’s command sequence doesn’t have any dependencies, then U is in the Clifford group. Proof. Write P ′ for the standard pattern associated to P. Since rewriting is sound, P ′ still implements U , and since rewriting never creates any dependency, it still has no dependencies. In particular, the corrections one finds at the end of P ′, call them C, bear no dependencies. Erasing them off P ′, results in a pattern P ′′ which is still standard, still deterministic, and implementing U ′ := C†U . Now how does the pattern P ′′ run on some input φ ? First φ⊗|+. . .+〉 goes by the entanglement phase to some ψ ∈ HV , and is then subjected to a sequence of independent 1-qubit measurements. Pick a basis B spanning the Hilbert space generated by the non-output qubits HVrO and associated to this sequence of measurements. Since HV = HO ⊗HVrO and HVrO = ⊕φb∈B[φb], where [φb] is the linear subspace generated by φb, by distributivity, ψ uniquely decomposes as: φb∈B xb ⊗ φb where φb ranges over B, and xb ∈ HO. Now since P ′′ is deterministic, there exists an x, and scalars λb such that xb = λbx. Therefore ψ can be written x ⊗ ψ′, for some ψ′. It follows in particular that the output of the computation will still be x (up to a scalar), no matter what the actual measurements are. One can therefore choose them to be all of the Mx kind, and by the preceding theorem U ′ is in the Clifford group, and so is U = CU ′, since C is a Pauli operator. ✷ From this section, we conclude in particular that any universal set of patterns has to include dependencies (by the preceding theorem), and also needs to use measurements Mα where α 6= 0 modulo π (by the theorem before). This is indeed the case for the universal set J (α) and ∧Z. 8 Other Models There are several other approaches to measurement-based computation as we have mentioned in the introduction. However, it is only for the one-way model that the importance of having all the entanglement in front has been emphasized. For example, Gottesman and Chuang describe computing with teleportation in the setting of the circuit model and hence the computation is very sequential [GC99]. What we will do is to give a general treatment of a variety of measurement- based models – including some that appear here for the first time – in the setting of our calculus. More precisely we would like to know other potential definitions for commands N , E, M and C that lead to a model that still satisfies the properties of: (i) being closed under composition; (ii) universality and (iii) standardization. Moreover we are interested in obtaining a compositional embedding of these models into a single one-qubit measurement-based model. The teleportation model can indeed be embedded into the one-way model. There is, however, a new model, the Pauli model – formally defined here for the first time – which is motivated by considerations of fault tolerance [RAB04, DK05b, DKOS06]. The Pauli model can be embedded into a slight generalization of the one-way model called the phase model; also given here for the first time. The one-way model will trivially embed in the phase model so by composition all the measurement-based models will embed in the phase model. We could have done everything ab initio in terms of the phase model but this would have made much of the presentation unnecessarily complicated at the outset. We recall the remark from the introduction that these embeddings have three advantages: first, we get a workable syntax for handling the dependencies of operators on previous measurement outcomes, second, one can use these embeddings to transfer the measurement calculus previously developed for the one-way model to obtain a calculus for the new model including, of course, a standardization procedure that we get automatically; lastly, one can embed the patterns from the phase model into the new models and vice versa. In essence, these compositional embeddings will allow us to exhibit the phase model as being a core calculus for measurement-based computation. However different models are interesting from the point of view of implementation issues like fault- tolerance and ease of preparation of entanglement resources. Our embeddings allow one to move easily between these models and to concentrate on the one-way model for designing algorithms and proving general theorems. This section has been structured into several subsections, one for each model and its embedding. 8.1 Phase Model In the one-way model the auxiliary qubits are initialized to be in the |+〉 state. We extend the one-way model to allow the auxiliary qubits to be in a more general state. We define the extended preparation command Nαi to be the preparation of the auxiliary qubit i in the state |+α〉. We also add a new correction command Zαi , called a phase correction to guarantee that we can obtain determinate patterns. The dependent phase correction is written as Z i with Z i = I and 0 eiα . Under conjugation, the phase correction, defines a new action over measurement: †Mαi Z and since the measurement is destructive, it simplifies to Mαi Z i = M i . This action does not commute with Pauli actions and hence one cannot write a compact notation for dependent measurement, as we did before, and the computation of angle dependencies is a bit more compli- cated. Thereafter, a measurement preceded by a sequence of corrections on the same qubit will be called a dependent measurement. Note that, by the absorption equations, this indeed can be seen as a measurement, where the angle depends on the outcomes of some other measurements made beforehand. To complete the extended calculus it remains to define the new rewrite rules: i ⇒ Z i Eij EP Mαi X i ⇒ M (−1)sα Mαi Z i ⇒ M Mαi Z i ⇒ M The above rules together with the rewriting rules of the one-way model described in Section 5, lead to a standardization procedure for the model. It is trivial that the one-way model is a fragment of this generalized model and hence universality immediately follows. It is also easy to check that the model is closed under composition and all the semantical properties of the one-way model can be extended to this general model as well. The choice of extended preparations and its concomitant phase correction is actually quite delicate. One wishes to keep the standardizability of the calculus which constrains what can be added but one also wishes to have determinate patterns which forces us to put in appropriate corrections. The phase model is only a slight extension of the original one-way model, but it allows a discussion of the next model which is of great physical interest. 8.1.1 Pauli model An interesting fragment of the phase model is defined by restricting the angles of measurements to {0, π , π,−π } i.e. Pauli measurements and the angles of preparation to 0 and π . Also the correction commands are restricted to Pauli corrections X, Z and Phase correction Z 8 . One readily sees that the subset of angles is closed under the actions of the corrections and hence the Pauli model is closed under composition. Proposition 15 The Pauli model is approximately universal. Proof. We know that the set consisting of J(0) (which is H), J(π ), and ∧Z is approximately universal. Hence, to prove the approximate universality of Pauli model, it is enough to exhibit a pattern in the Pauli model for each of these three unitaries. We saw before that J(0) and ∧Z are computed by the following 2-qubit patterns: J (0) := Xs12 M01E12 ∧Z := E12 where both belong also to the Pauli model. The pattern for Jπ in the one-way model is expressed as follows: ) := Xs12 M 1 E12 = Xs12 M 1E12Z The above forms do not fit in the Pauli model, since the first one uses a measurement with an angle and the second uses Z 4 . However by teleporting the input qubit and then applying the Z 4 and finally running the standardization procedure we obtain the following pattern in the Pauli model for J(π 1E12Z 3E34 Z 1E12E23 = Xs34 M 3E34Z 1E12E23 Z 3E34Z 1E12E23 Z s3+s2 −(−1)s1s2 π2 1E12E23E34N Approximate universality for the Pauli model is now immediate. ✷ Note that we cannot really expect universality (as we had for the phase model) because the angles are restricted to a discrete set. On the other hand it is precisely this restriction that makes the Pauli model interesting from the point of view of implementation. The other particular interest behind this model, apart from its simple structure, is based on the existence of a novel fault tolerant technique for computing within this framework [BK05, RAB04, DKOS06]. 8.2 Teleportation Another class of measurement-based models – older, in fact, than the one-way model – uses 2- qubit measurements. These are collectively referred to as teleportation models [Leu04]. Several papers that are concerned with the relation and possible unification of these models [CLN05, AL04, JP05] have already appeared. One aspect of these models that stands in the way of a complete understanding of this relation, is that, whereas in the one-way model one has a clearly identified class of measurements, there is less agreement concerning which measurements are allowed in teleportation models. We propose here to take as our class of 2-qubit measurements a family obtained as the conjugate under the operator ∧Z of tensors of 1-qubit measurements. We show that the resulting teleportation model is universal. Moreover, almost by construction, it embeds into the one-way model, and thus exposes completely the relation between the two models. Before embarking on the specifics of our family of 2-qubit measurements, we remark that the situation commented above is more general: Lemma 16 Let A be an orthonormal basis in ⊗nC2, with associated n-qubit measurement MA, and Ai with i = 1, . . . , n be orthonormal bases in C2, with associated 1-qubit measurements MAii . Then there exists a unique (up to a permutation) n-qubit unitary operator U such that: MA1···n = U1···n(⊗iM 1···n Proof. Take U to map ⊗iAi to A. ✷ This simple lemma says that general n-qubit measurements can always be seen as conjugated 1-qubit measurements, provided one uses the appropriate unitary to do so. As an example consider the orthogonal graph basis G = ∧Z12{|±〉 ⊗ |±〉} then the two-qubit graph basis measurements are defined as MG12 = ∧Z12(M01 ⊗M02 )∧Z12. It is now natural to extend our definition of M 12 to obtain the family of 2-qubit measurements of interest: 12 := ∧Z12(M 2 )∧Z12 (17) corresponding to projections on the basis Gα,β := ∧Z12(P1(α)⊗P2(β))({|±〉 ⊗ |±〉}. This family of two-qubit measurements together with the preparation, entanglement and corrections commands of the one-way model define the teleportation model. Before we carry on, a clarification about our choice of measurements in the teleportation model is necessary. The usual teleportation protocol uses Bell basis measurement defined with B = ∧X12{|±〉 ⊗ |0/1〉} MB12 = ∧X12(Mz1 ⊗Mz2 )∧X12 where Mz is the computational-basis measurement. Note how similar these equations are to the equations defining the graph basis measurements. This is a clear indication that everything that follows can be transferred to the case where X replaces Z, and B replaces G. However, since the methodology we adopt is to embed the 2-qubit measurement based model in the one-way model, and the latter is based on ∧Z and G, we will work with the graph-basis measurements. Furthermore, since ∧Z is symmetric, whereas ∧X (a.k.a. as C-NOT) is not, the algebra is usually nicer to work with. Now we prove that the family of measurements in Equation 17 leads to a universal model, which embeds nicely into the one-way model, but first we need to describe the important notion of dependent measurements. These will arise as a consequence of standardization; they were not considered in the existing teleportation models. In what follows we drop the subscripts on the ∧Z unless they are really necessary. We write (s(i), s(j)) ∈ Z2 × Z2 to represent outcome of a 2-qubit measurement, with the specific convention that (0, 0), (0, 1), (1, 0), and (1, 1), correspond respectively to the cases where the state collapses to ∧Z|+α〉|+α〉, ∧Z|+α〉|−α〉, ∧Z|−α〉|+α〉, and ∧Z|−α〉|−α〉. We will use two types of dependencies for measurements associated with X-action and Z-action: (s,t) = M (−1)sα,(−1)tβ (u,v)[M ij ] = M α+uπ,β+vπ where s, t, u and v are in Z2. The two actions commute, so the equations above define unambigu- ously the full dependent measurement (u,v)[M (s,t). Here are some useful abbreviations: (0,0)[Mα,β ](s,t) := [Mα,β ](s,t) (u,v)[Mα,β ](0,0) := (u,v)[Mα,β ] (0,0)[Mα,β ](0,0) :=Mα,β Mα,x := Mα,0 Mα,y :=Mα, As in the 1-qubit measurement case we obtain the following rewriting rules for the teleportation model: i ⇒ Xsi ZsjEij EX i ⇒ ZsiEij EZ (u,v)[M (s,t)Xri ⇒ (u,v+r)[M (s+r,t) MX (u,v)[M (s,t)Xrj ⇒ (u+r,v)[M (s,t+r) MX (u,v)[M (s,t)Zri ⇒ (u+r,v)[M (s,t) MZ (u,v)[M (s,t)Zrj ⇒ (u,v+r)[M (s,t) MZ to which we add also the trivial commutation rewriting which are possible between commands that don’t overlap (meaning, acting on disjoint sets of qubits). 8.2.1 Embedding We describe how to translate 2-qubit EMC patterns to 1-qubit patterns and vice versa. The following equation plays the central role in the translation: ij = Eij(M j )Eij (18) Note that this immediately gives the denotational semantics of two-qubit measurements as cptp- maps. Furthermore, all other commands in the teleportation model are the same as in the one-way model, so we have right away a denotational semantics for the entire teleportation model in terms of cptp-maps. We write P for the collection of patterns in the one-way model and T for the collection of patterns in the teleportation model. Theorem 6 There exist functions [·]f : P → T and [·]b : T → P such that 1. ∀P ∈ P : [[P]] = [[[P]f ]]; 2. ∀T ∈ T : [[T ]] = [[[T ]b]]; 3. [·]f ◦ [·]b and [·]b ◦ [·]f are both identity maps. Proof. We first define the forward map [·]f in stages as follows for any patterns P = (V, I,O,An . . . A1): 1. For any i ∈ V rO (i.e. measured qubits) we add an auxiliary qubit id called a dummy qubit to the space V . 2. For any i ∈ V rO we replace any occurrence of Mαi with Mαi Mxid . 3. We then replace each of the newly created occurrences of Mαi M Eiid . Now we show that the first condition stated in the theorem holds; we do this stage wise. The first two stages are clear because we are just adding qubits that have no effect on the pattern because they are not entangled with any pre-existing qubit, and no other command depends on a measurement applied to one of the dummy qubits. Furthermore, we add qubits in the state |+〉 and measure them in the |±〉 basis. The invariance of the semantics under stage 3 is an immediate consequence of Equation 18 and the fact that all the measurements are destructive, and hence an entanglement command on qubits appearing after a measurement of any of those qubits can just be removed. The map [·]b is defined similarly except that there is no need to add dummy qubits. One only needs to replace any two-qubit measurement M ij with M j Eij . Again, this clearly pre- serves the semantics of patterns because of Equation 18 and the above remark about destructive measurements. Thus condition 2 of the theorem holds. The fact that the two maps are mutual inverses follows easily. As all the steps in the translations are local we can reason locally. Looking at the forward mapping followed by the backward mapping we get the following sequence of transformations Mαi ⇒stage 1,2 Mαi Mxid ⇒Equation 18 Mα,xiid Eiid ⇒Equation 18 Mαi MxidEiidEiid ⇒ Mαi Mxid ⇒ Mαi This shows that we have the third condition of the theorem. ✷ Note that the translations are compositional since the denotational semantics is and also it fol- lows immediately that the teleportation model is universal and admits a standardization procedure. Example. Consider the teleportation pattern in the teleportation model given by the command sequence: Xs13 Z 12 E23, we perform the above steps: 12 E23 ⇒Equation 18 Xs13 Z 2E12E23 and hence obtain the teleportation pattern with 1-qubit measurements. Example. We saw before, the following EMC 1-qubit pattern for Rz(α) which can be embedded to an EMC 2-qubit pattern using the above steps: s1M−α1 E12E23 ⇒stage 1,2 s1Mx2dM E12E23 ⇒Equation 18 and standardization Xs23 Z ](s1,0)M E11dE22dE12E23 Note that we have explicit algorithmic translations between the models and not just illustrative examples. This is the main advantage of our approach in unifying these two models compared to the extant work [CLN05, AL04, JP05]. 9 Conclusion We have presented a calculus for the one-way quantum computer. We have developed a syntax of patterns and, much more important, an algebra of pattern composition. We have seen that pattern composition allows for a structured proof of universality, which also results in parsimonious implementations. We develop an operational and denotational semantics for this model; in this simple first-order setting their equivalence is clear. We have developed a rewrite system for patterns which preserves the semantics. We have shown further that our calculus defines a polynomial-time standardization algorithm transforming any pattern to a standard form where entanglement is done first, then measurements, then local corrections. We have inferred from this procedure that the denotational semantics of any pattern is a cptp-map and also proved that patterns with no dependencies, or using only Pauli measurements, may only implement unitaries in the Clifford group. In addition we introduced some variations of the one-way and teleportation models and pre- sented compositional back-and-forth embeddings of these models into the one-way model. This allows one to carry forward all the theory we have developed: semantics, rewrite rules, standard- ization, no-dependency theorems and universality. In fact the result of making the connection between the one-way model and the teleportation model is to introduce ideas: dependent mea- surements, standard forms for patterns and a standardization procedure which had never been considered before for the teleportation model. This shows the generality of our formalism: we expect that any yet to be discovered measurement-based computation frameworks can be treated in the same way. Perhaps the most important aspect of standardization is the fact that now we can make patterns maximally parallel and distributed because all the entanglement operators, i.e. non-local operators, can be performed at the beginning of the computation. Then from the dependency structure that can be obtained from the standard form of a pattern the measurements can be organized to be as parallel as possible. This is the essence of the difference between measurement-based computation and the quantum circuit model or the quantum Turing machine. We feel that our measurement calculus has shown the power of the formalisms developed by the programming languages community to analyze quantum computations. The ideas that we use: rewriting theory, (primitive) type theory and above all, the importance of reasoning compositionally, locally and modularly, are standard for the analysis of traditional programming languages. However, for quantum computation these ideas are in their infancy. It is not merely a question of adapting syntax to the quantum setting; there are fundamental new ideas that need to be confronted. What we have done here is to develop such a theory in a new, physically-motivated setting. There were prior discussions about putting patterns in a standard form [RB02] but these worked only with strongly deterministic patterns, furthermore one needs to know which unitary is being implemented. In our case the rewrite rules are entirely local and work equally well with all patterns. An interesting question related to the measurement calculus is whether one can give sufficient conditions – depending only on the entanglement structure of a pattern – that guarantee deter- minacy. In a related paper the first two authors have solved this problem [DK05a]. In effect given an entanglement structure with distinguished inputs and outputs one can enumerate all the unitaries that can be implemented with it. This gives a precise handle on the entanglement re- sources needed in the design of specific algorithms and protocols directly in the measurement-based model [dBDK06]. Finally, there is also a compelling reading of dependencies as classical communications, while local corrections can be thought of as local quantum operations in a multipartite scenario. From this point of view, standardization pushes non-local operations to the beginning of a distributed computation, and it seems the measurement calculus could prove useful in the analysis of dis- tributed quantum protocols. To push this idea further, one needs first to articulate a definition of a distributed version of the measurement calculus; this was done in a recent paper [DDKP05]. The distributed version of the calculus was then used to analyze a variety of quantum protocols and to examine the notion of knowledge flow in them [DP05]. Acknowledgments We thank the anonymous referees for their helpful comments. 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A Background on Quantum Mechanics and Quantum Computa- We give a brief summary of quantum mechanics and quantum computing. We develop some of the algebra, define some notations, and prove a couple of equations which we have used in the paper. Although the paper is self-contained, the reader will find the expository book of Nielsen and Chuang [NC00] useful for quantum computation or the excellent book by Peres [Per95] for general background on quantum mechanics. http://arxiv.org/abs/quant-ph/0405174 http://arxiv.org/abs/quant-ph/0412156 A.1 Linear Algebra for Quantum Mechanics We assume that the reader is familiar with the basic notion of a vector space. In quantum mechanics we always consider vector spaces over the complex numbers. For quantum computation the vector spaces are always finite dimensional. The vector spaces that arise in quantum mechanics are Hilbert spaces and are thus usually written H; that is they have an inner product usually written 〈u, v〉 where u and v are vectors. The inner product is a map from H×H to the complex numbers C. The inner product is linear in the second argument but anti -linear in the first argument. In general, there is a topological completeness condition on Hilbert spaces but, in the finite dimensional case this is automatic and we will ignore it. Following Dirac, it is customary to call elements of H kets and write them in the form |u〉 or whatever symbol is appropriate inside the half-bracket. The dual vectors are called bras and are written 〈v|; the pairing thus can naturally be identified – conceptually and notationally – with the inner product. Linear operators come naturally with vector spaces; a linear operator is a linear map from a vector space to itself. Linear operators on finite dimensional spaces are often represented as matrices. The most important notion for an operator on a Hilbert space is that of an adjoint. Definition 17 If A : H → H′ is a linear operator then the adjoint, written A†, is a linear operator from H′ to H such that ∀u ∈ H′, v ∈ H〈u,Av〉 = 〈A†u, v〉. In terms of matrices this just amounts to transposing the matrix and complex conjugation each of the matrix entries; sometimes this is called the hermitian conjugate. An inner product preserving linear map is called a unitary embedding. When H = H′ we can also define the following operators. A hermitian operator A is one such that A = A† and a unitary operator U is one such that U−1 = U †. A projection P is a linear operator such that P 2 = P and P = P †. A projection operator can be identified with a subspace, namely its range. The eigenvalues of a hermitian operator are always real. Suppose U is a unitary, and P a projection, then UPU † is also a projection. It is common to use the Dirac notation to write projection operators as follows: given a vector |a〉 of unit norm, the projection onto the subspace spanned by |a〉 is written |a〉〈a|. To see why this makes sense, suppose that |b〉 is another vector then its component along |a〉 is the inner product 〈a, b〉. Now if we just juxtapose the expressions |a〉〈a| and |b〉 we get |a〉〈a, b〉, viewing the 〈a, b〉 as a number and moving it to the front we get 〈a, b〉|a〉 as the result, which is the right answer for the projection of |b〉 onto |a〉. Thus one can apply the projection operator just by juxtaposing it with the vector. This kind of suggestive manipulation is part of the appeal of the Dirac notation. One important fact – the spectral theorem for hermitian operators – states that if M is a hermitian operator, λi are its eigenvalues and Pi are projection operators onto the corresponding eigenspaces then one can write λiPi. If we have |i〉 as the normalized eigenvectors for the eigenvalues λi then we can write this in Dirac notation as: λi|i〉〈i|. Finally we need to combine Hilbert spaces. Definition 18 Given two Hilbert spaces H with basis vectors {ai|1 ≤ i ≤ n} and H′ with basis {bj |1 ≤ j ≤ m} we define the tensor product, written H⊗H′, as the vector space of dimension n ·m with basis ai ⊗ bj . There are more elegant, basis-independent ways of describing the tensor product but this definition will serve our needs. We almost never write the symbol ⊗ between the vectors. In the Dirac notation this is always omitted and one writes, for example, |uv〉 instead of |u〉 ⊗ |v〉. The important point is that there are vectors that cannot be written as the tensor product of vectors. For example, we can write a1 ⊗ b1 + a2 ⊗ b2 where the ai and the bi are basis vectors of two 2-dimensional Hilbert spaces. This means that given a general element of H ⊗ H′ one cannot produce elements of H and H′; this is very different from the cartesian product of sets. This is the mathematical manifestation of entanglement. A very important function on square matrices is the trace. The usual trace – i.e. the sum of the diagonal entries – is basis independent and is actually equal to the sum of the eigenvalues, counted with multiplicity. The trace of A is written tr(A) and satisfies the cyclicity property tr(AB) = tr(BA); applying this repeatedly one gets tr(A1 . . . An) = tr(Aσ(1) . . . Aσ(n)) where σ is a cyclic permutation. The explicit formula for the trace of A : V → V is tr(A) = i〈i|A|i〉 where |i〉 is a basis for V . One often needs to compute a partial trace. Consider a linear map L : V ⊗ W → V ⊗W . Suppose that |vi〉 is a basis for V and |wi〉 is a basis for W then |viwj〉 is a basis for V ⊗W . Now we can define the partial trace over V as trV (A) : W → W = 〈vi|A|vi〉. This corresponds to removing the V dependency; often we use the phrase “tracing out the V component.” A.2 Quantum Mechanics We state the basic facts of quantum mechanics and will not discuss the experimental basis for this framework. The key aspects of quantum mechanics are: • the states of a quantum system form a Hilbert space, • when two quantum systems are combined, the state space of the composite system is obtained as the tensor product of the state spaces of the individual systems, and • the evolution of a quantum system is given by a unitary operator, and • the effect of a measurement is indeterminate. The first says that one can form superpositions of the states. This is one of the most striking features of quantum mechanics. Thus states are not completely distinct as they are in classical systems. The inner product measures the extent to which states are distinct. The fact that systems are combined by tensor product says that there are states that of composite systems that cannot be decomposed into individual pieces. This is the phenomenon of entanglement or non-locality. Measurement is what gives quantum mechanics its indeterminate character. The usual case, called projective measurements, is when the quantity being measured is described by a hermitian operator M . The possible outcomes are the eigenvalues of M . If M is an observable (hermitian operator) with eigenvalues λi and eigenvectors |φi〉 and we have a generic state |ψ〉 = i ci|φi〉 then the probabilities and expectation values of the measurement outcomes are given by: • Prob(λi||ψ〉) = |ci|2 • E[M ||ψ〉] = i |ci|2λi = i cic̄i〈φi,Mφi〉 = 〈ψ,Mψ〉. It is important to note that the effect of the measurement is that the projection operator Pi is applied when the result λi is observed. The operator M does not describe the effect of the measurement. The formulas above reveal that no aspect of a measurement is altered if the vector describing a quantum state is multiplied by a complex number of absolute value 1. Thus we can multiply a state by eiθ without changing the state. This is called changing the phase. While the phase is not important phase differences are very important. Multiplying a vector by a complex number is a change of phase as well as a change in its length. Usually we normalize the state so that we can read the results of measurements as probabilities. Given a vector the subspace spanned by it - always one dimensional – is called a ray. Thus a state is really a ray rather than a vector. However, it is customary to blur this distinction. A.3 Some qubit algebra Quantum computation is carried out with qubits the quantum analogues of bits. Just as a bit has two possible values, a qubit is a two dimensional complex Hilbert space, in other words it is (isomorphic to) the two dimensional complex vector space C2. One works with a preferred basis, physically this corresponds to two distinguishable states, like “spin up” and “spin down”. One writes |0〉, and |1〉 for its canonical basis, so that any vector ψ can be written as α|0〉 + β|1〉 with α, β in C. Furthermore, C2 can be turned into a Hilbert space with the following inner product: 〈α|0〉 + β|1〉, α′|0〉+ β′|1〉〉 := α⋆α′ + β⋆β′ where α⋆ is the complex conjugate of α. One then obtains the norm of a vector as: ‖ψ‖ := 〈ψ,ψ〉 2 = (α⋆α+ β⋆β) Given V a finite set, one writes HV for the Hilbert space ⊗u∈VC2; the notation means an n-fold tensor product of the C2 where n is the size of V . A vector in HV is said to be decomposable if it can be written ⊗u∈V ψu for some ψu ∈ C2. Such decomposable vectors will be written ǫ in the sequel. Decomposable vectors can be represented by a map from V to C2, and we will use both notations depending on which is more convenient. As we have noted before there are some vectors that are not decomposable. As in the case of C2, there is a canonical basis for HV , sometimes also called the computational basis, containing decomposable vectors ǫ such that for all v ∈ V , ǫ(v) = |0〉 or ǫ(v) = |1〉. The inner product on HV , according to the general definition given above, is defined on decom- posable vectors as: 〈ǫ, ǫ′〉 := v∈V 〈ǫ(v), ǫ′(v)〉 Note that all vectors in the computational basis are orthogonal and of norm 1. The vectors of norm 1 are usually called unit vectors; we always assume that states are described by unit vectors as noted before. Here are some common states that arise in quantum computation: |0〉 = | ↑〉 = , |1〉 = | ↓〉 = , |+〉 = , |−〉 = It is easy to see that a linear operator is unitary if it preserves the inner product and hence the norm. Thus unitaries can be viewed as maps from quantum states to quantum states. Some particularly useful unitaries are the Pauli operators given by the following matrices in the canonical basis of C2: , Y = , Z = We note that all these operators are involutive, self-adjoint, and therefore unitaries. All these matrices have determinant = −1. We will not discuss the geometric significance of these operators here; their real importance in quantum mechanics comes from the fact that they can be used to describe rotations, thus they are usually called the “Pauli spin matrices” by physicists. Some basic algebra of these matrices are given below. First they all square to the identity. X2 = Y 2 = Z2 = I. The Pauli operators do not commute. We use the notation [A,B] for AB − BA, the commutator of A and B. The commutator measures the extent to which two operators fail to commute: it is customary to present the algebra of operators using it. One also uses the symbol {A,B} to stand for AB + BA: it is called the anti-commutator. For the Pauli operators we have the following commutators and anti-commutators : XY = iZ Y X = −iZ [X,Y ] = 2iZ {X,Y } = 0 ZX = iY XZ = −iY [Z,X] = 2iY {Z,X} = 0 Y Z = iX ZY = −iX [Y,Z] = 2iX {Y,Z} = 0 Definition 19 Define the Pauli group, Pn, as the group consisting of tensor products of I, X, Y, and Z on n qubits, with an overall phase of ±1 or ±i. Given a group G the operation x 7→ g−1xg is called conjugation by g. These conjugations give the effect of switching operators around. If G is a group and H is a subgroup of G then the normalizer of H is another subgroup of G, say K, with the property that for all h ∈ H, k ∈ K we have k−1hk in H. The effect of conjugating measurements and other corrections by Pauli operators is a key part of the rewrite rules described in the main text. They can be verified using the algebra given here. A very important related group is called the Clifford group. Definition 20 The Clifford group, Cn, is the group of unitary operators that leave the Pauli group invariant under conjugation, i.e. it is the normalizer of the Pauli group viewed as a subgroup of the unitary group. The Clifford group on n qubits can be generated by the Hadamard transform, the controlled-X (CNOT ) or controlled-Z (∧Z), and the single-qubit phase rotation: H = 1√ , CNOT = 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 , ∧Z = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 −1 , P = The importance of the Clifford group for quantum computation is that a computation consisting of only Clifford operations on the computational basis followed by final Pauli measurements can be efficiently simulated by a classical computer, this is the Gottesman-Knill theorem [Got97, NC00]. A.4 Density Matrices In order to capture partial information about quantum systems one uses density matrices. Before we describe density matrices we review some linear algebra in the bra-ket notation. Given a ket |ψ〉 the notation |ψ〉〈ψ| denotes the projection operator onto the one dimensional subspace spanned by |ψ〉. To verify this note that (|ψ〉〈ψ|)(|ψ〉 = |ψ〉‖ψ‖ = |ψ〉 (|ψ〉〈ψ|)(|φ〉) = |ψ〉〈ψ|φ〉 = 〈ψ|φ〉|ψ〉. If |ψi〉 is an orthonormal basis for H the identity matrix is written i |ψi〉〈ψi|. If Q is a linear operator with eigenvalues qi and eigenvectors |qi〉, which form an orthonormal basis for H, we can represent Q as i qi|qi〉〈qi|. To see this, let |ψ〉 = i ci|qi〉 then Q|ψ〉 = ciQ|qi〉 = ciqi|qi〉 now using our representation for Q we calculate Q|ψ〉 = qi|qi〉〈qi|(|ψ〉) = cjqi|qi〉〈qi|qj〉 = ciqi|qi〉. This is a version of the spectral theorem that we mentioned in the first subsection of this appendix. A state (i.e. a ray in H) is called a pure state. If a and b are distinct eigenvalues of some observable A with corresponding eigenvectors |a〉 and |b〉 it is perfectly possible to prepare a state of the form 1√ (|a〉 + |b〉). A measurement of A on such a state will yield either a or b each with probability 1 . However, it is also possible that a mixture is prepared. That is to say instead of a quantum superposition a classical stochastic mixture is prepared. In order to describe these we will use density matrices. For a system in a pure state |ψ〉, the density matrix is just the projection operator |ψ〉〈ψ|. If we have an observable Q with eigenvalues qi – assumed nondegenerate for simplicity – then we can expand |ψ〉 in terms of the eigenvectors by |ψ〉 = ci|qi〉. Now the probability of observing qi when measuring Q in the state |ψ〉 is |〈qi|ψ〉|2. Recalling that the identity is given by I = j |qj〉〈qj | we get that Prob(qi, |ψ〉) = 〈qi|ψ〉〈ψ|qj〉〈qj|qi〉 which after rearranging and using the definition of trace of an operator yields Tr((|qi〉〈qi|)(|ψ〉〈ψ|)). If as is typical we write ρψ for the density matrix and Pi for the projection operator onto the subspace spanned by the eigenvector |qi〉 we get Prob(qi, |ψ〉) = Tr(Piρ). It is an easy calculation to show that the expectation value for Q in the state |ψ〉 is Tr(Qρ). What if the state is not known completely? Suppose that we only know that a system is one of several possible states |ψ1〉, . . . , |ψk〉 with probabilities p1, . . . , pk respectively. We define the density matrix for such a state to be pi|ψi〉〈ψi|. The same formulas for the probability of observing a value qi , i.e. Tr(Piρ) and for the expectation value of Q, i.e. Tr(Qρ) apply. One can check directly that a density matrix has the following two properties. Proposition 21 An operator ρ on H is a density matrix if and only if • ρ has trace 1 and • ρ is a positive operator, which means that it has only positive eigenvalues or, equivalently, that for any x ∈ H we have 〈x, ρx〉 ≥ 0. Furthermore, if ρ is a density operator, Tr(ρ2) ≤ 1 with equality if and only if ρ is a pure state (i.e. a projection operator). Suppose that we have a density matrix ρ describing a pure state of an n+m dimensional system. Now suppose that an observer can only see the first n dimensions. The density matrix ξ describing what he can see is contained by taking the partial trace over the m dimensions that the observer cannot see. Doing this gives, in general, a nonpure state. Similarly a complementary observer who sees only the m dimensions would construct her density matrix σ by taking the appropriate partial trace. Taking these traces loses information; in fact, one cannot reconstruct ρ even from both ξ and σ. Certainly the tensor product of ξ and σ does not give back ρ. This is due to the loss of the cross-correlation information that was encoded in ρ but is not represented in either ξ or σ. The axioms of quantum mechanics are easily stated in the language of density matrices. For example, if evolution from time t1 to time t2 is described by the unitary transformation U and ρ is the density matrix for time t1, then the evolved density matrix ρ ′ for time t2 is given by the formula ρ′ = UρU †. Similarly, one can describe measurements represented by projective operators in terms of density matrices [NC00, Pre98]. Thus if a projector P acts on a state |ψ〉 then the result is P |ψ〉; the resulting transofrmation of density matrices is |ψ〉〈7→ |P |ψ〉〈P |. For a general density matrix ρ we have ρ 7→ PρP , note that since P is self-adjoint we do not have to write P †. A.5 Operations on Density matrices What are the legitimate “physical” transformations on density matrices? Density matrices are positive operators and they have trace either equal to 1 if we insist on normalizing them or bounded by 1. These properties muct be preserved by any transformations on them. We need first to define what it means for a vector to be positive. Any vector space V can be equipped with a notion of positivity. Definition 22 A subset C of V is called a cone if • x ∈ C implies that for any positive α, αx ∈ C, • x, y ∈ C implies that x+ y ∈ C and • x and −x both in C means that x = 0. We can define x ≥ 0 to mean x ∈ C and x ≥ y to mean x− y ∈ C. Definition 23 An ordered vector space is just a vector space equipped with a cone. It is easy to check the following explicitly. Proposition 24 The collection of positive operators in the vector space of linear operators forms a cone. Now we can say what it means for a map to be a positive map. Definition 25 Abstractly, L : (V,≤V ) → (W,≤W ) is a positive map if ∀v ∈ V. v ≥V 0 ⇒ L(v) ≥W 0. It is important to not confuse “positive maps” and “positive operators.” If we are transforming states (density matrices) then the legitimate transformations obviously take density matrices to density matrices. They have to be positive maps considered as maps between the appropriate ordered vector spaces. The appropriate ordered vector spaces are the vector spaces of linear operators on H the Hilbert space of pure states. Unfortunately the tensor product of two positive maps is not positive in general. We really want this! If one can perform transformation T1 on density matrix ρ1 and transformation T2 on density matrix ρ2 then it should be possible to regard ρ1 ⊗ ρ2 as a composite system and carry out T1 ⊗ T2 on this system. We certainly want this if, say, T2 is the identity. But even when T2 is the identity this may fail; the usual example is the transposition map, see, for example [NC00]. The remedy is to require the appropriate condition by fiat. Definition 26 A completely positive map K is a positive map such that for every identity map In : C n → Cn the tensor product K ⊗ In is positive. It is not hard to show that the tensor of completely positive maps is always a completely positive map. This condition satisfies one of the requirements. We can insist that they preserve the bound on the trace to satisfy the other requirement as well. However we would like an explicit way of recognizing this. The important result in this regard is the Kraus representation theorem [Cho75]. Theorem 7 (Kraus) The general form for a completely positive map E : B(H1) → B(H2) is E(ρ) = where the Am : H1 → H2. Here B(H) is the Banach space of bounded linear operators on H. If, in addition, we require that the trace of E(ρ) ≤ 1 then the Am will satisfy A†mAm ≤ I. The following term is common in the quantum computation literature. Definition 27 A superoperator T is a linear map from BV to BU that is completely positive and trace preserving. Introduction Measurement Patterns Commands Patterns Pattern combination The semantics of patterns Operational semantics Denotational semantics Short examples Compositionality of the Denotational Semantics Universality The measurement calculus The equations The rewrite rules Standardization Signal shifting Examples The no dependency theorems Other Models Phase Model Pauli model Teleportation Embedding Conclusion Background on Quantum Mechanics and Quantum Computation Linear Algebra for Quantum Mechanics Quantum Mechanics Some qubit algebra Density Matrices Operations on Density matrices

0704.1264

Aharonov-Bohm oscillations in the presence of strong spin-orbit interactions

Aharonov-Bohm oscillations in the presence of strong spin-orbit interactions Boris Grbić∗, Renaud Leturcq∗, Thomas Ihn∗, Klaus Ensslin∗, Dirk Reuter +, and Andreas D. Wieck+ Solid State Physics Laboratory, ETH Zurich, 8093 Zurich, Switzerland Angewandte Festkörperphysik, Ruhr-Universität Bochum, 44780 Bochum, Germany We have measured highly visible Aharonov-Bohm (AB) oscillations in a ring structure defined by local anodic oxidation on a p-type GaAs heterostructure with strong spin-orbit interactions. Clear beating patterns observed in the raw data can be interpreted in terms of a spin geometric phase. Besides h/e oscillations, we resolve the contributions from the second harmonic of AB oscillations and also find a beating in these h/2e oscillations. A resistance minimum at B = 0 T, present in all gate configurations, is the signature of destructive interference of the spins propagating along time-reversed paths. Interference phenomena with particles have challenged physicists since the foundation of quantum mechanics. A charged particle traversing a ring-like mesoscopic struc- ture in the presence of an external magnetic flux Φ acquires a quantum mechanical phase. The interfer- ence phenomenon based on this phase is known as the Aharonov-Bohm (AB) effect [1], and manifests itself in oscillations of the resistance of the mesoscopic ring with a period of Φ0 = h/e, where Φ0 is the flux quantum. The Aharonov-Bohm phase was later recognized as a special case of the geometric phase [2, 3] acquired by the orbital wave function of a charged particle encircling a magnetic flux line. The particle’s spin can acquire an additional geomet- ric phase in systems with spin-orbit interactions (SOI) [4, 5, 6]. The investigation of this spin-orbit (SO) in- duced phase in a solid-state environment is currently the subject of intensive experimental work [7, 8, 9, 10, 11, 12]. The common point of these experiments is the investiga- tion of electronic transport in ring-like structures defined on two-dimensional (2D) semiconducting systems with strong SOI. Electrons in InAs were investigated in a ring sample with time dependent fluctuations [7], as well as in a ring side coupled to a wire [9]. An experiment on holes in GaAs [8] showed B-periodic oscillations with a relative amplitude ∆R/R < 10−3. These observations [7, 8] were analyzed with Fourier transforms and interpreted as a manifestation of Berry’s phase. Further studies on elec- trons in a HgTe ring [10] and in an InGaAs ring network [11] were discussed in the framework of the Aharonov- Casher effect. In systems with strong SOI, an inhomogeneous, mo- mentum dependent intrinsic magnetic field Bint, perpen- dicular to the particle’s momentum, is present in the reference frame of the moving carrier [13]. The total magnetic field seen by the carrier is therefore Btot = Bext + Bint, where Bext is the external magnetic field perpendicular to the 2D system and Bint is the intrinsic magnetic field in the plane of the 2D system present in the moving reference frame (right inset Fig. 1(a)). The particle’s spin precesses around Btot and accumulates an additional geometric phase upon cyclic evolution. Effects of the geometric phases are most prominently expressed in the adiabatic limit, when the precession fre- quency of the spin around the local field Btot is much faster than the orbital frequency of the charged particle carrying the spin [4]. In this limit the ring can be consid- ered to consist of two uncoupled types of carriers with op- posite spins [14]. The total accumulated phase, composed of the AB phase and the SO induced geometric phase, is different for the two spin species, φtot = φAB ±φSO, and the magnetoresistance of the ring is obtained as the su- perposition of the oscillatory contributions from the two spin species. Such a superposition is predicted to produce complex, beating-like magnetoresistance oscillations with nodes developing at particular values of the external B- field, where the oscillations from the two spin-species have opposite phases [5]. Both h/e and h/2e peaks in the Fourier spectrum of the magnetoresistance oscilla- tions are predicted to be split in the presence of strong SOI [5, 15]. The interpretation of the split Fourier signal in Ref. [7] has been challenged [16]. The data on p-GaAs rings [8] stirred an intense discussion [17]. Our raw data di- rectly displays a beating of the h/e Aharonov-Bohm os- cillations. No Fourier transform is required to verify this effect. As additional evidence we directly measure a beat- ing of the h/2e oscillations and a pronounced and per- sistent zero field magnetoresistance minimum due to de- structive interference of time-reversed paths. The sample was fabricated by atomic force micro- scope (AFM) oxidation lithography on a p-type carbon doped (100) GaAs heterostructure, with a shallow two- dimensional hole gas (2DHG) located 45 nm below the surface [18]. An AFM micrograph of the ring structure is shown in the inset of Fig. 1(a). The average radius of the circular orbit is 420 nm, and the lithographic width of the arms is 190 nm, corresponding to an electronic width of 60−70 nm. The hole density in an unpatterned sample is 3.8×1011 cm−2 and the mobility is 200 000 cm2/Vs at a temperature of 60 mK. Therefore the Fermi wavelength is about 40 nm, and the mean free path is 2 µm. Since the circumference of the ring is around 2.5 µm, the transport through the ring is quasiballistic. From the temperature dependence of the AB oscillations we extract the phase coherence length of the holes to be Lϕ = 2 µm at a base temperature of T = 60 mK. The presence of strong spin-orbit interactions in the http://arxiv.org/abs/0704.1264v2 heterostructure is demonstrated by a simultaneous ob- servation of the beating in Shubnikov-de Haas (SdH) os- cillations and a weak anti-localization dip in the mea- sured magnetoresistance of the Hall bar fabricated on the same wafer [19]. In p-type GaAs heterostuctures, Rashba SOI is typically dominant over the Dresselhaus SOI [20]. The densities N1= 1.35×10 11 cm−2 and N2= 2.45×1011 cm−2 of the spin-split subbands, deduced from SdH oscillations, allow us to estimate the strength of the Rashba spin-orbit interaction ∆SO ≈ 0.8 meV assum- ing a cubic wave-vector dependence [13]. Due to the large effective mass of the holes, the Fermi energy in the system, EF = 2.5 meV, is much smaller than that in electron systems with the same density. The large ratio ∆SO/EF ≈ 30% documents the presence of strong SOI. We have measured the four-terminal resistance of the ring in a 3He/4He dilution refrigerator at a base temper- ature of about 60 mK with lock-in techniques. A low ac current of 2 nA and 31 Hz frequency was applied in order to prevent sample heating. Fig. 1(a) shows the magnetoresistance of the ring (fast oscillating curve, red online). The low-frequency back- ground resistance is indicated by a smooth curve (blue online). The observed Aharonov-Bohm (AB) oscillations with a period of 7.7 mT (frequency 130 T−1) correspond to a radius of the holes’ orbit of 415 nm, in excellent agreement with the lithographic size of the ring. The peak-to-peak amplitude of ∼ 200 Ω on a background of about 6 kΩ corresponds to a visibility larger than 3%. We restrict the measurements of the AB oscillations to magnetic fields in the range from -0.2 T to 0.2 T in or- der to prevent their mixing with SdH oscillations, which start to develop above 0.2 T. Throughout all measure- ments quantum point contact gates 3,4,5 and 6 are kept at the same values. Plunger gates pg1 and pg2 are set to Vpg1 = −145 mV and Vpg2 = −95 mV in the measure- ments presented in Fig. 1(a). After subtracting the low-frequency background from the raw data, a clear beating pattern is revealed in the AB oscillations with a well defined node at ∼ 115 mT [Fig. 1(b)], where a phase jump of π occurs [arrow in Fig. 2(c)]. The position of the beating node indicates the presence of two oscillation frequencies differing by 1/0.115 ≈ 9 T−1. The Fourier spectrum of the AB oscillations, taken in the symmetric magnetic field range (-0.2 T, 0.2 T), re- veals an h/e peak around 130 T−1 [Fig. 1(c)]. Zooming in on the h/e peak, a splitting into 3 peaks at the fre- quencies 127 T−1, 136 T−1 and 143 T−1 is seen. We have carefully checked that this splitting is genuine to the ex- perimental data and not a result of the finite data range, by reproducing it with different window functions for the Fourier transform. The differences of the oscillation fre- quencies agree with that anticipated from the position of the beating node in the raw data. In contrast to the h/e-periodic AB oscillations, which are very sensitive to phase changes in the ring arms, Altshuler-Aronov-Spivak (AAS) h/2e oscillations, orig- Gate1 -0.2 -0.1 0 0.1 0.2 0 200 400 frequency (1/T) 100 120 140 160 250 270 290 frequency (1/T) (c) (d) FIG. 1: (color online) (a) Measured magnetoresistance of the ring (strongly oscillating curve, red online) together with the low-frequency background resistance (smooth curve, blue on- line); Left inset: AFM micrograph of the ring with designa- tions of the in-plane gates. Bright oxide lines fabricated by AFM oxidation lithography lead to insulating barriers in the 2DHG. Right inset: Scheme of a carrier travelling around the ring in the presence of the external field Bext and SO induced intrinsic fieldBint. (b) AB oscillations obtained after subtrac- tion of the low-frequency background from the raw data. A clear beating pattern is revealed in the AB oscillations. (c) Fourier transform spectra of the AB oscillations, revealing h/e and h/2e peaks. (d) Splitting of the h/e Fourier peak. (e) Splitting of the h/2e Fourier peak. inating from the interference of time reversed paths, are expected to be more robust if the microscopic configu- ration of the arms is changed. Besides, h/2e oscillations are less susceptible to the details how the spin rotates when it enters the ring than h/e oscillations. This is due to the fact that the geometric phase accumulated along the paths contributing to the h/2e oscillations is larger than that in the case of the h/e oscillations and cannot be completely cancelled by the spin rotations in the con- tacts, as in the latter case [9]. In Fig. 1(c) we can identify B(mT) 0-10 10 20-20 B(mT) 0-10 10 20-20 Rd - Rb 0 20-20 40 60 80 B(mT) 100 120 140 160 (a) (b) Rd - Rb - Rh/e Rh/2e Rd - Rb - Rh/e Rh/2e FIG. 2: (color online) (a) Measured magnetoresistance of the ring after subtracting the low-frequency background, Rd−Rb (full line, red online), together with the filtered h/e oscilla- tions Rh/e (dashed line). (b) Difference Rd −Rb −Rh/e (full line red online) together with the inverse Fourier transform of the h/2e peak Rh/2e (dashed line). (c) Beating in filtered h/e oscillations. The width of the gray and white rectangles corresponds to the period of 7.7 mT. The arrow points to the beating node where a phase jump of π occurs. (d) Beating in filtered h/2e oscillations with arrows indicating possible nodes. the peak at about 270 T−1 in the Fourier spectrum, cor- responding to h/2e oscillations. If we zoom in on it [Fig. 1(e)], we see a splitting with the two main peaks having a separation of about 8 T−1, similar to the h/e peak split- ting. The splitting of the h/2e Fourier peak arises due to the frequency shift of the main peak by ±1/Bint [5], and the obtained splitting of 8 T−1 allows to estimate the SO induced intrinsic field to be Bint ≈ 0.25T. We now focus directly on the magnetic field-dependent resistance. In Fig. 2(a) we present the raw data after subtracting the low-frequency background (full line, red online) together with the filtered h/e oscillations (dashed line) [21]. The h/e contribution to the signal is the in- verse Fourier transform of the h/e peak in the Fourier spectrum. We will use the following notation below: Rd denotes the raw data, Rb is the low-frequency back- ground, Rh/e is the inverse Fourier transform of the h/e peak and Rh/2e is the inverse Fourier transform of the h/2e peak in the Fourier spectrum. One can see in Fig. 2(a) that Rd − Rb contains additional resistance modu- lations, beyond the h/e oscillations. In order to demon- strate that those additional features are due to h/2e oscil- lations we plot in Fig. 2(b) the difference Rd−Rb−Rh/e (full line, red online) and the curve Rh/2e obtained by inverse Fourier transform of the h/2e peak (dashed line) and find excellent agreement. We further plot in Fig. 2(d) the difference Rd − Rb − Rh/e (full line, red online), together with the filtered h/2e oscillations Rh/2e (dashed line) in a larger range of mag- netic fields. A beating-like behavior in the h/2e oscilla- tions is observed. Possible nodes develop around 40 mT, 115 mT, and 175 mT [arrows in Fig. 2(d)]. The appear- ance of these unequally spaced nodes is in agreement with the complex split-peak pattern in Fig. 1(e). In the plot of the filtered h/e oscillations [Fig. 2(c)] we notice that only the node around 115 mT is common for both, h/e and h/2e oscillations, while the other two nodes in the h/2e oscillations correspond to maxima in the beating of h/e oscillations. This kind of aperiodic modulation of the envelope function of the h/2e oscillations, rather than a regular beating, is predicted for the case of diffusive rings in the presence of Berry’s phase [5], since the latter also changes with increasing external magnetic field. The evolution of the AB oscillations upon changing plunger gate voltages Vpg1 and Vpg2 is explored in Fig. 3(a). Plunger gate voltages are changed antisymmetri- cally: Vpg1 = −120mV −V ; Vpg2 = −120 mV +V . Two distinct features are visible: there is always a local min- imum in the AB oscillations at B = 0 T, and the os- cillations experience a phase jump by π around V = 27 mV. In order to understand the origin of these two fea- tures we analyze the filtered h/e (not shown) and h/2e oscillations [Fig. 3(b)] as a function of V . The h/e os- cillations experience a phase jump of π, while the h/2e oscillations do not [Fig. 3(b)]. We have explored this behavior in several other gate configurations and always found the same result. The reason for such a behavior is that the h/e oscillations are sensitive to the phase differ- ence ∆ϕ = k1l1 − k2l2 between the two arms, which can be changed by the plunger gates, while the AAS h/2e oscillations are not. We observe a resistance minimum at B = 0 T for all gate configurations [Fig. 3(a)], which is due to a minimum at B = 0 T in the h/2e oscillations [Fig. 3(b)]. It indicates that time reversed paths of the holes’ spinors interfere destructively due to strong SOI, in contrast to n-type GaAs systems where h/2e oscillations produce a resistance maximum at B = 0T [22]. This effect has the same origin as the weak anti-localization (WAL) effect. However, the observed minimum is not caused by WAL in the ring leads, since the WAL dip in bulk 2D samples has a much smaller magnitude (less than 1Ω, [19]) than the minimum at B = 0T in the ring. The resistance minimum at B = 0T is a result of the destructive interference of the holes’ spins in the ring. The adiabatic regime is reached when ωB/ωorbit >> 1, where ωB = gµBBtot/2~ is the spin precession frequency around Btot, while ωorbit = vF /r is the orbital frequency of the holes around the ring in the ballistic regime. p- type GaAs systems have strong SOI, and therefore large Bint, which, together with a small vF (due to the large effective mass of the holes) makes p-types systems very favorable for reaching the adiabatic regime compared to other systems. Using the estimated value for Bint of 0.25T and assuming a holes’ g factor of 2, we obtain -20 -10 20100 B (mT) -20 -10 20100 B (mT) (a) (b) FIG. 3: (color online) (a) Evolution of the AB oscillations upon changing the plunger gate voltages Vpg1 = −120mV −V ; Vpg2 = −120 mV +V . (b) Filtered h/2e oscillations as a function of the plunger gate voltages, showing the local minimum at B = 0 T at all gate voltages. ωB/ωorbit ≈ 0.2 − 0.3 for the measured range of Bext up to 0.2T. Therefore the adiabatic regime is not fully reached in our measurements. There remains a pronounced discrepancy between the internal magnetic field obtained from the beating of the SdH oscillations of 7T (converting the corresponding en- ergy scale ∆SO ≈ 0.8 meV to a magnetic field via the Zeeman splitting) and the field scale of 0.25 T obtained from the beating of the AB oscillations. The latter evalu- ation is strictly valid in the diffusive regime [5] while our sample is at the crossover to the ballistic regime. It is also not clear how the limited adiabaticity in our samples will influence these numbers. In a straightforward picture one would expect that the node of the beating in the h/2e oscillations occurs at half the magnetic field as the node in the h/e oscillations since the accumulated phase difference between the two spin species should be proportional to the path length travelled in the ring. Within experimental accuracy the data in Fig. 1 (d) and (e) suggests that the splitting in the corresponding Fourier transforms is the same. In conclusion, we have measured Aharonov-Bohm os- cillations in a ring defined on a 2D hole gas with strong spin-orbit interactions. We observe a beating in the mea- sured resistance which arises from an interplay between the orbital Aharonov-Bohm and a spin-orbit induced ge- ometric phase. In addition we resolve h/2e oscillations in the ring resistance, and find that they also show a beating-like behavior, which produces a splitting of the h/2e peak in the Fourier spectrum. A resistance mini- mum at B = 0, present in all in-plane gate configura- tions, demonstrates the destructive interference of the hole spins propagating along time reversed paths. We thank Daniel Loss and Yigal Meir for stimulating discussions. Financial support from the Swiss National Science Foundation is gratefully acknowledged. [1] Y. Aharonov and D. Bohm, Phys. 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Lett. 96, 076804 (2006) [11] T. Bergsten, T. Kobayashi, Y. Sekine, and J. Nitta, Phys. Rev. Lett. 97, 196803 (2006) [12] B. Habib, E. Tutuc, and M. Shayegan, Appl. Phys. Lett. 90, 152104 (2007). [13] R. Winkler, Spin-Orbit Coupling Effects in Two- Dimensional Electron and Hole Systems, Springer Tracts in Modern Physics, Volume 191, Springer-Verlag (2003) [14] A. Stern, Phys. Rev. Lett. 68, 1022 (1992) [15] A. G. Mal’shukov, V. V. Shlyapin, and K. A. Chao, Phys. Rev. B 60, R2161 (1999) [16] H. De Raedt, Phys. Rev. Lett. 83, 1700 (1999); A. F. Morpurgo, J. P. Heida, T. M. Klapwijk, B. J. van Wees, and G. Borghs, Phys. Rev. Lett. 83, 1701 (1999) [17] A. G. Mal’shukov and K. A. Chao, Phys. Rev. Lett. 90, 179701 (2003); J. B. Yau, E. P. De Poortere, and M. Shayegan, Phys. Rev. Lett. 90, 179702 (2003); A. G. Wagh and V. C. Rakhecha, Phys. Rev. Lett. 90, 119703 (2003); J. B. Yau, E. P. De Poortere, and M. Shayegan, Phys. Rev. Lett. 90, 119704 (2003) [18] B. Grbić, R. Leturcq, K. Ensslin, D. 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0704.1265

Radio and X-ray nebulae associated with PSR J1509-5850

Astronomy & Astrophysics manuscript no. ms-revise October 31, 2018 (DOI: will be inserted by hand later) Radio and X-ray nebulae associated with PSR J1509−5850 C. Y. Hui and W. Becker Max-Planck Institut für Extraterrestrische Physik, Giessenbachstrasse 1, 85741 Garching bei München, Germany Received 11 April 2007 / Accepted 10 May 2007 Abstract. We have discovered a long radio trail at 843 MHz which is apparently associated with middle age pulsar PSR J1509−5850. The radio trail has a length of ∼ 7 arcmin. In X-rays, Chandra observation of PSR J1509−5850 reveals an associated X-ray trail which extends in the same orientation as the radio trail. Moreover, two clumpy structures are observed along the radio trail. The larger one is proposed to be the supernova remnant (SNR) candidate MSC 319.9-0.7. Faint X-ray enhancement at the position of the SNR candidate is found in the Chandra data. Key words. pulsars: individual (PSR J1509−5850)—stars: neutron—radio, X-rays: stars 1. Introduction It is generally believed that a significant fraction of the rota- tional energy of a pulsar leaves the magnetosphere in the form of a magnetized pulsar wind consisting of electromagnetic ra- diation and high energy particles. In view of this, it is energet- ically important to study the physical properties of this wind. When the relativistic wind particles interact with the shocked interstellar medium, the charged particles will be accelerated in the shock and hence synchrotron radiation from radio to X-ray is generated. In order to obtain a better understanding of the interaction nature, multiwavelength studies of the pulsar wind nebulae are deeply needed. X-ray and radio observations have recently revealed a number of pulsar wind nebulae. However, there is only a handful of shocked emission detected in both the X-ray and radio regimes (c.f. see Hui & Becker 2006 and references therein). PSR J1509−5850 was discovered by Manchester et al. (2001) in the Parkes Multibeam Pulsar Survey. The pulsar has a rotation period of P = 88.9 ms and a period derivative of Ṗ = 9.17 × 10−15 s s−1. These spin parameters imply a charac- teristic age of 1.54× 105 yrs, a dipole surface magnetic field of B⊥ = 9.14 × 10 11 G and a spin-down luminosity of 5.1 × 1035 ergs s−1 (c.f. Table 1). The radio dispersion measure yields a distance of about 3.81 kpc based on the galactic free electron model of Taylor & Cordes (1993). Using the model of Cordes & Lazio (2002) the dispersion measure based distance is es- timated to be 2.56 kpc. The proper motion of this pulsar is not yet known. Recently, a brief X-ray study of the field of PSR J1509−5850 was presented by Kargaltsev et al. (2006). The authors have reported that a trail-like pulsar wind nebula associated with PSR J1509−5850 was observed in a Chandra observation. The X-ray nebula is found to be extended in the south-west direction. Table 1. Pulsar parameters of PSR J1509−5850 (from Manchester et al. 2005) Right Ascension (J2000) 15h09m27.13s Declination (J2000) −58◦ 50′ 56.1” Pulsar Period, P (s) 0.088921760489 Period derivative Ṗ (10−15 s s−1) 9.1698 Age (105 yrs) 1.54 Surface dipole magnetic field (1012 G) 0.914 Epoch of Period (MJD) 51463 Dispersion Measure (pc cm−3) 137.7 Dispersion based distance (kpc) ∼ 2.6 − 3.8 Spin-down Luminosity (1035) ergs s−1 5.1 In this paper we report on the discovery of a possible radio counterpart of the X-ray trail associated with PSR J1509−5850 and provide a detailed X-ray analysis of the trail. In §2 we de- scribe the observations and the data analysis and in §3 we sum- marize and discuss our results. 2. Observations and data analysis PSR J1509−5850 was observed with Chandra in 2003 February 9−10 (Obs ID: 3513) with the Advanced CCD Imaging Spectrometer (ACIS). The pulsar is located on the back- illuminated (BI) ACIS-S3 chip which has a superior quantum efficiency among the spectroscopic array. Standard processed level-2 data were used. The effective exposure is about 40 ks. Chandra observation has revealed an X-ray trail associated with PSR J1509−5850. The signal-to-noise ratios for the pulsar and the trail are found to be ∼ 19 and ∼ 3 in 0.5 − 8 keV respectively. The X-ray image of the 4 × 4 arcmin field near to PSR J1509−5850 is shown in Figure 1. The binning factor of the image is 0.5 arcsec. Adaptive smoothing with a Gaussian http://arxiv.org/abs/0704.1265v2 2 C. Y. Hui and W. Becker: The pulsar wind nebula associated with PSR J1509−5850 Fig. 1. Chandra’s 4×4 arcmin view of PSR J1509−5850 and its X-ray trail in the energy band 0.3−8 keV. The pulsar position is indicated by the black cross. The white circles indicate the positions of field stars identified in the DSS image. kernel of σ < 3 arcsec has been applied to the image. The trail appears to have a length of ∼ 2 arcmin. From a Digitized Sky Survey (DSS) image, 25 bright field stars are found in the field of view of Figure 1. We subsequently identified the magnitudes of these stars from the USNO-A2.0 catalog (Monet et al. 1998), which are within the range of B ∼ 10 − 18. Their positions are plotted as white circles in Figure 1. For the spectral analysis, we extracted the spectrum of PSR J1509−5850 from a circle of 4 arcsec radius (encircled energy∼99%) centered on the pulsar position. To minimize the possible contamination from the field stars, the spectrum from the trail was extracted within a box of 25 × 95 arcsec, oriented along the direction of the trail emission. Even with this con- sideration, there are still two stars with magnitude B = 17 and B = 16.4 located on the trail (cf. Fig. 1) and unavoidably lie in the extraction region. Without the knowledge of the extinc- tions, we are not able to estimate the possible contribution in X-ray from these two stars. The background spectra were ex- tracted from the low count regions nearby. After background subtraction, there are ∼ 100 net counts and ∼ 270 net counts extracted from the pulsar and the trail in 0.5 − 8 keV respec- tively. Response files were computed by using the CIAO tools MKRMF and MKARF. The spectra were dynamically binned so as to have at least 10 counts per bin for the pulsar and 30 counts per bin for the trail. All the spectral fitting were per- formed in the energy range of 0.5 − 8 keV by using XSPEC 11.3.1. The degradation of the ACIS quantum efficiency was corrected by XSPEC model ACISABS. All the quoted errors are 1 − σ and were computed for 1 parameter in interest. For the X-ray emission from PSR J1509−5850, we found that it can be modeled with an absorbed power-law fairly well Fig. 2. Energy spectrum of the X-ray trail of PSR J1509−5850 as observed with the Chandra ACIS-S3 detector and fitted to an absorbed power-law model (upper panel) and contribution to the χ2 fit statistic (lower panel). (χν=0.68 for 8 D.O.F.). This model yields a column density of NH = 8.0 −2.1 × 10 21 cm−2, a photon index of Γ = 1.0+0.2 and a normalization at 1 keV of 5.1+1.3 −1.6 × 10 −6 photons keV−1 cm−2 s−1. The best-fitted model results in an unabsorbed flux of fX = 5.9 × 10 −14 ergs cm−2 s−1 in the energy range of 0.5 − 8 keV. The dispersion based distance implies a luminos- ity of LX = 4.8 × 10 31 and 1.0 × 1032 erg s−1 for d=2.6 and 3.8 kpc respectively. Although a blackbody model can give a compatible goodness-of-fit (χν =0.82 for 8 D.O.F.), it infers a rather high temperature (T ∼ 1.7 × 107 K) and a small pro- jected blackbody radius (R ∼ 10 m). We hence regard this model as not physically reasonable to describe the X-ray spec- trum of PSR J1509−5850. We note that the characteristic age indicates that PSR J1509−5850 belongs to the class of middle- aged pulsars. Their spectra typically consist of a soft thermal component, a harder thermal component from the heated po- lar caps as well as contribution from the non-thermal emission (cf. Becker & Aschenbach 2002). However, the small number of collected photons and the high column density does not sup- port any fitting with multicomponent models. We have tested the hypothesis that the trail emission is orig- inated from the interaction of pulsar wind and ISM by fitting an absorbed power-law model to the trail spectrum. The model yields an acceptable goodness-of-fit (χν=0.73 for 9 D.O.F.). The best fitting spectral model is displayed in Figure 2. This model yield a column density of NH = 8.2 −3.7 × 10 21 cm−2, a photon index of Γ = 1.3+0.8 −0.4 and a normalization at 1 keV of 1.9+4.3 −1.9 × 10 −5 photons keV−1 cm−2 s−1. We note that the col- umn density agrees with that inferred from the pulsar spectrum. The unabsorbed flux deduced for the best-fitted model param- eters are fX = 1.6 × 10 −13 erg s−1 cm−2 in the energy range of 0.5−8 keV. The dispersion based distance implies a luminosity of LX = 1.3×10 32 and 2.7×1032 erg s−1 for d=2.6 and 3.8 kpc, respectively. We have searched for a possible radio counterpart for the X-ray nebula with the Sydney University Molonglo Sky Survey data (SUMSS) (Bock et al. 1999). We have discovered a long C. Y. Hui and W. Becker: The pulsar wind nebula associated with PSR J1509−5850 3 Fig. 3. The 843 MHz SUMSS image of a field of 11×11 arcmin around PSR J1509-5850. The pulsar position is indicated by the black cross. The ∼ 7 arcmin long radio feature is found to have the same orientation as the X-ray trail. The contour levels are 7, 23, 39, 54 and 70 mJy/beam. Two clumps are observed along the trail. The larger clump, near to the center of this image, on the trail is identified as the SNR candidate MSC 319.9-0.7. radio trail apparently associated with PSR J1509−5850. The radio image of the 11×11 arcmin field near to PSR J1509−5850 is displayed in Figure 3. The radio feature has a length of ∼ 7 arcmin. Radio contours were calculated at the levels of 7, 23, 39, 54 and 70 mJy/beam. These contours were overlaid on the image of the Chandra ACIS-S3 chip in Figure 4. It is interesting to note that the radio trail starts exactly from the position of PSR J1509−5850 and has the same orientation as that of the X- ray trail. These facts support the interpetation that this extended radio feature is the radio counterpart of the X-ray trail and is indeed physically associated with PSR J1509−5850. There are two clumpy structures observed along the radio trail (see Figure 3). The northern clump has its emission center at RA=15h09m14.35s, Dec=−58◦ 54′ 50.7” (J2000) with a ra- dius of ∼ 1.5 arcmin. The southern clump has its emission cen- ter at RA=15h09m06.33s, Dec=−58◦ 58′ 34.7” (J2000) with a radius of ∼ 1 arcmin. While the southern clump is unidentified in SIMBAD and NED, the northern clump, which locates ∼ 4 arcmin away from PSR J1509−5850, has been proposed to be a supernova remnant candidate MSC 319.9-0.7 (Whiteoak & Green 1996). Comparing the X-ray and the radio data in Figure 4, we found that there is some faint X-ray emission near to the location of MSC 319.9-0.7. The emission does not appear to be the continuation of the trail associated with PSR J1509−5850. It cannot be excluded that this faint emission is related to MSC 319.9-0.7. However, the limited photon statistics does not allow any final conclusion. Fig. 4. The X-ray image of the Chandra ACIS-S3 chip with the radio contour lines from SUMSS data (cf. Figure 3) overlaid. The X-ray image is binned with a pixel size of 2.5 arcsec and adaptively smoothed with a Gaussian kernel of σ < 7.5 arcsec. We note that there is a faint X-ray feature near to the location of the SNR candidate MSC 319.9-0.7. Top is north and left is east. 3. Discussion & Conclusion In this paper, we report the detection of a possible radio coun- terpart of the X-ray trail associated with PSR J1509−5850 and present a first detailed X-ray study of the X-ray trail. Apart from the radio trail, we have found that there are two clumpy structures located on the trail. While the smaller one is still unidentified, the larger one, which is located ∼ 4 arcmin away from PSR J1509−5850, is identified as a SNR candidate MSC 319.9-0.7. Despite the proximity of MSC 319.9-0.7, it seems unlikely that it is the birth place of PSR J1509−5850. Assuming this shell-like SNR candidate is in a Sedov phase, the radius of the shocked shell emission can be estimated by (Culhane 1977): Rs = 2.15 × 10 5 pc (1) where t, E and n are the time after the explosion in units of years, the released kinetic energy in units of ergs and the ISM number density in units of cm−3 respectively. Taking the typical values of E = 1051 ergs and n = 1 cm−3 and t to be the charac- teristic age of PSR J1509−5850, we estimate that a SNR asso- ciated with PSR J1509−5850 should have a radius of Rs ∼ 40 pc. However, MSC 319.9-0.7 only has a radius of ∼ 1.1 − 1.7 pc for d = 2.6−3.8 kpc. Thus, the discrepancy between the ex- pected Rs and the observed value which with a factor of ∼ 30 is not likely to be reconciled by the uncertainty of the dispersion based distance. On the other hand, the characteristic age of the pulsar can be older than its actual age if its inital spin period was close to its current period. However, to reconcile such dis- crepancy would require t to be smaller by a factor of ∼ 4000 4 C. Y. Hui and W. Becker: The pulsar wind nebula associated with PSR J1509−5850 which is not likely. Moreover, associating MSC 319.9-0.7 with PSR J1509−5850 would leave the origin of the southern part of the radio trail unexplained. Thus, with the current knowledge of parameters it seems most reasonable for us to interpret MSC 319.9-0.7 as a background source. Following the discussion in Hui & Becker (2006), we ap- ply a simple one zone model (Chevalier 2000; Cheng, Tamm, & Wang 2004) to model the X-ray emission properties of the pul- sar wind nebula. Since the proper motion of PSR J1509−5850 is not yet known, we assume the pulsar is in supersonic motion on the basis that the nebula resembles a bow-shock morphol- ogy. For the supersonic motion, the termination shock radius Rts is determined by the balance of the ram pressure between the relativistic pulsar wind particles and the ISM at the head of the shock (cf. Cheng et al. 2004): Rts ≃ 2πρIS Mv2pc ∼ 3 × 1016Ė1/234 n −1/2v−1p,100cm (2) where vp,100 is the velocity of the pulsar in units of 100 km s Ė34 is the spin-down luminosity of the pulsar in units of 10 erg s−1, and n is the number density of the ISM in units of cm−3. In all the following estimation, we assume PSR J1509−5850 has a transverse velocity comparable to the average velocity, ∼ 250 km s−1, of ordinary radio pulsars (Hobbs et al. 2005). For a ISM density of 1 cm−3, equation (2) implies a termination radius of Rts ∼ 8.6 × 10 16 cm. The X-ray trail is found to be ∼ 2 arcmin long. For the dispersion based distance in the range of ∼ 2.6 − 3.8 kpc, the trail has a length of l ∼ (4.7 − 6.8) × 1018 cm. For the assumed pulsar velocity of ∼ 250 km s−1, the timescale for the passage of the pulsar over the length of its X-ray trail, tflow, is estimated to be ∼ 6000 − 8600 years. The magnetic field in the shocked region can be estimated by assuming tflow to be comparable to the synchrotron cooling timescale of electrons: τsyn = 6πmec γσT B2 ≃ 105 µG yrs (3) where γ is the Lorentz factor of the wind, taken to be 106 (cf. Cheng et al. 2004), σT is the Thompson cross section, and BµG is the magnetic field in the shocked region in unit of micro gauss. The inferred magnetic field in the shocked region is ∼ 5 − 7 µG. For comparison, the magnetic field strength in the ISM is estimated to be ∼ 2 − 6 µG (cf. Beck et al. 2003, and references therein). The X-ray luminosity and spectral index depend on the in- equality between the characteristic observed frequency νobsX and the electron synchrotron cooling frequency νc (see Chevalier 2000 and references therein): 18πemec synB3 which is estimated to be νc = (1.3 − 1.8) × 10 17 Hz. Since in general νobsX > νc, this suggests the X-ray emission is in a fast cooling regime. Electrons with the energy distribution, N (γ) ∝ γ−p, are able to radiate their energy in the trail with photon index α = (p + 2)/2. The index p due to shock accel- eration typically lies between 2 and 3 (cf. Cheng et al. 2004 and reference therein). This would result in a photon index α ≃ 2.0− 2.5. In view of the large error of the observed photon index Γ = 1.3+0.8 −0.4, we cannot firmly conclude the emission sce- nario simply based on the photon index. We note that the pho- ton index can still be possibly in the fast cooling regime within the 1σ uncertainty. With this consideration and νobsX > νc, we adopted the fast cooling scenario in the following discussion. With the assumed value p = 2.2, the calculated photon index α = 2.1 which is marginally within the 1σ uncertainty of the observed value. In a fast cooling regime, the luminosity per unit frequency is given by (cf. Cheng et al. 2004): p − 2 p − 1 )p−1 ( 4π2mec3 ts Ė 2 (5) Assuming the energy equipartion between the electron and proton, we take the fractional energy density of electron ǫe to be ∼ 0.5 and the fractional energy density of the magnetic field ǫB to be ∼ 0.01. We integrate equation (5) from 0.5 keV to 8 keV and result in a calculated luminosity of ∼ 6 × 1032 ergs s−1. With the reasonable choice of parameters stated above, the luminosity estimated by this simple model is found to be the same order as the observed value. It is obvious that the radio nebula is significantly longer than its X-ray counterpart (cf. Fig. 4). This is not unexpected. Considering a scenario of constant injection of particles with a finite synchrotron cooling time, the number of particles that can reach at a further distance from the pulsar should decrease with increasing frequency. This is because the synchrotron cooling timescale decrease with frequency. This would result in a fact that the synchrotron nebular size decreases with frequency. To further constrain the physical properties of the pul- sar wind nebula associated with PSR J1509−5850, multi- wavelength observations are badly needed. Since SUMSS data have a rather poor spatial resolution which has a typical beam size of ∼ 45 arcsec, there might be details of the nebular emis- sion remain unresolved. In particular, it is important to bet- ter resolve the nebular emission from the contribution of the SNR candidate MSC 319.9-0.7. In view of this, high resolu- tion radio observations (e.g. ATCA) are required. In the X- ray regime, although the Chandra observation has already pro- vided us with a high resolution image of the nebula, the pho- ton statistics is not sufficient to tightly constrain the spectral properties. Owing to the superior collecting power, observa- tions with XMM-Newton are expected to put a strict constraint on the emission nature of the nebula as well as the pulsar itself. Apart from the radio and X-ray observations, a complete study of pulsar wind nebula should also include TeV observa- tions (e.g. HESS). It is generally believed that the TeV pho- tons are resulted from inverse Compton scattering of soft pho- ton field by the relativistic particles in the nebulae. The seed soft photons are possibly contributed by the cosmic microwave background (Cui 2006). However, there is only a handful of pulsar wind nebulae detected in TeV regime so far (see Cui C. Y. Hui and W. Becker: The pulsar wind nebula associated with PSR J1509−5850 5 2006), a larger sample is needed for differentiating the afore- mentioned interpetation from its competing scenario (e.g. neu- tral pion decay). From the above discussion, one should note that the pul- sar’s transverse velocity is an important parameter in studying the shock physics. And hence measuring the proper motion of PSR J1509−5850 is badly needed. Moreover, although the ori- entation of the trail suggests PSR J1509−5850 is likely moving in the direction of northeast, it is not necessary for the trail to be aligned with the pulsar velocity. PSRs J2124-3358 and B2224+65 are the examples that the X-ray trails are misaligned with the direction of the pulsars’ proper motion (Hui & Becker 2006, 2007). References Becker, W., Aschenbach, B., 2002, in Neutron Stars, Pulsars and Supernova Remnants, eds. W.Becker, H.Lesch & J.Trümper, MPE-Report 278, p64, (astro-ph/0208466) Beck, R., Shukurov, A., Sokoloff, D., & Wielebinski, R. 2003, A&A, 411, 99 Bock, D. C.-J., Large, M. I., & Sadler, Elaine M. 1999, AJ, 117, 1578 Cheng, K. S., Taam, R. E., & Wang, W., 2004, ApJ, 617, 480 Chevalier, R. A. 2000, ApJ, 539, L45 Cordes, J. M., & Lazio, T. J. W., 2002, (astro-ph/0207156) Cui, W. 2006, (astro-ph/0608042) Culhane, J. L.: 1977, in D. N. Schram (ed.),Supernovae, D. Reidel Publishing Co., Dordrecht, Holland, p. 29 Hobbs, G., Lorimer, D. R., Lyne, A. G., & Kramer, M. 2005, MNRAS, 360, 974 Hui, C. Y., & Becker, W. 2006, A&A, 448, L13 Hui, C. Y., & Becker, W. 2007, A&A, in press (astro-ph/0610505v2) Kargaltsev, O., et al. 2006, AAS HEAD meeting 9 Manchester, R. N., Hobbs, G. B., Teoh, A., & Hobbs, M. 2005, AJ, 129, 1993 Manchester, R. N., et al. 2001, MNRAS, 328, 17 Monet, D. B. A., Canzian, B., Dahn, C., et al. 1998, VizieR Online Data Catalog, 1252, 0 Taylor, J. H., & Cordes, J. M 1993, ApJ, 411, 674 Whiteoak, J. B. Z., & Green, A. J. 1996, A&AS, 118, 329 http://arxiv.org/abs/astro-ph/0208466 http://arxiv.org/abs/astro-ph/0207156 http://arxiv.org/abs/astro-ph/0608042 http://arxiv.org/abs/astro-ph/0610505 Introduction Observations and data analysis Discussion & Conclusion

0704.1266

Branching fraction and charge asymmetry measurements in B to J/psi pi pi decays

BABAR-PUB-07/017 SLAC-PUB-12441 Branching fraction and charge asymmetry measurements in B → J/ψππ decays B. Aubert,1 M. Bona,1 D. Boutigny,1 Y. Karyotakis,1 J. P. Lees,1 V. Poireau,1 X. Prudent,1 V. Tisserand,1 A. Zghiche,1 J. Garra Tico,2 E. Grauges,2 L. Lopez,3 A. Palano,3 G. Eigen,4 I. Ofte,4 B. Stugu,4 L. Sun,4 G. S. Abrams,5 M. Battaglia,5 D. N. Brown,5 J. Button-Shafer,5 R. N. Cahn,5 Y. Groysman,5 R. G. Jacobsen,5 J. A. Kadyk,5 L. T. Kerth,5 Yu. G. Kolomensky,5 G. Kukartsev,5 D. Lopes Pegna,5 G. Lynch,5 L. M. Mir,5 T. J. Orimoto,5 M. Pripstein,5 N. A. Roe,5 M. T. Ronan,5, ∗ K. Tackmann,5 W. A. Wenzel,5 P. del Amo Sanchez,6 C. M. Hawkes,6 A. T. Watson,6 T. Held,7 H. Koch,7 B. Lewandowski,7 M. Pelizaeus,7 T. Schroeder,7 M. Steinke,7 W. N. Cottingham,8 D. Walker,8 D. J. Asgeirsson,9 T. Cuhadar-Donszelmann,9 B. G. Fulsom,9 C. Hearty,9 N. S. Knecht,9 T. S. Mattison,9 J. A. McKenna,9 A. Khan,10 M. Saleem,10 L. Teodorescu,10 V. E. Blinov,11 A. D. Bukin,11 V. P. Druzhinin,11 V. B. Golubev,11 A. P. Onuchin,11 S. I. Serednyakov,11 Yu. I. Skovpen,11 E. P. Solodov,11 K. Yu Todyshev,11 M. Bondioli,12 S. Curry,12 I. Eschrich,12 D. Kirkby,12 A. J. Lankford,12 P. Lund,12 M. Mandelkern,12 E. C. Martin,12 D. P. Stoker,12 S. Abachi,13 C. Buchanan,13 S. D. Foulkes,14 J. W. Gary,14 F. Liu,14 O. Long,14 B. C. Shen,14 L. Zhang,14 H. P. Paar,15 S. Rahatlou,15 V. Sharma,15 J. W. Berryhill,16 C. Campagnari,16 A. Cunha,16 B. Dahmes,16 T. M. Hong,16 D. Kovalskyi,16 J. D. Richman,16 T. W. Beck,17 A. M. Eisner,17 C. J. Flacco,17 C. A. Heusch,17 J. Kroseberg,17 W. S. Lockman,17 T. Schalk,17 B. A. Schumm,17 A. Seiden,17 D. C. Williams,17 M. G. Wilson,17 L. O. Winstrom,17 E. Chen,18 C. H. Cheng,18 A. Dvoretskii,18 F. Fang,18 D. G. Hitlin,18 I. Narsky,18 T. Piatenko,18 F. C. Porter,18 G. Mancinelli,19 B. T. Meadows,19 K. Mishra,19 M. D. Sokoloff,19 F. Blanc,20 P. C. Bloom,20 S. Chen,20 W. T. Ford,20 J. F. Hirschauer,20 A. Kreisel,20 M. Nagel,20 U. Nauenberg,20 A. Olivas,20 J. G. Smith,20 K. A. Ulmer,20 S. R. Wagner,20 J. Zhang,20 A. M. Gabareen,21 A. Soffer,21 W. H. Toki,21 R. J. Wilson,21 F. Winklmeier,21 Q. Zeng,21 D. D. Altenburg,22 E. Feltresi,22 A. Hauke,22 H. Jasper,22 J. Merkel,22 A. Petzold,22 B. Spaan,22 K. Wacker,22 T. Brandt,23 V. Klose,23 H. M. Lacker,23 W. F. Mader,23 R. Nogowski,23 J. Schubert,23 K. R. Schubert,23 R. Schwierz,23 J. E. Sundermann,23 A. Volk,23 D. Bernard,24 G. R. Bonneaud,24 E. Latour,24 V. Lombardo,24 Ch. Thiebaux,24 M. Verderi,24 P. J. Clark,25 W. Gradl,25 F. Muheim,25 S. Playfer,25 A. I. Robertson,25 Y. Xie,25 M. Andreotti,26 D. Bettoni,26 C. Bozzi,26 R. Calabrese,26 A. Cecchi,26 G. Cibinetto,26 P. Franchini,26 E. Luppi,26 M. Negrini,26 A. Petrella,26 L. Piemontese,26 E. Prencipe,26 V. Santoro,26 F. Anulli,27 R. Baldini-Ferroli,27 A. Calcaterra,27 R. de Sangro,27 G. Finocchiaro,27 S. Pacetti,27 P. Patteri,27 I. M. Peruzzi,27, † M. Piccolo,27 M. Rama,27 A. Zallo,27 A. Buzzo,28 R. Contri,28 M. Lo Vetere,28 M. M. Macri,28 M. R. Monge,28 S. Passaggio,28 C. Patrignani,28 E. Robutti,28 A. Santroni,28 S. Tosi,28 K. S. Chaisanguanthum,29 M. Morii,29 J. Wu,29 R. S. Dubitzky,30 J. Marks,30 S. Schenk,30 U. Uwer,30 D. J. Bard,31 P. D. Dauncey,31 R. L. Flack,31 J. A. Nash,31 M. B. Nikolich,31 W. Panduro Vazquez,31 P. K. Behera,32 X. Chai,32 M. J. Charles,32 U. Mallik,32 N. T. Meyer,32 V. Ziegler,32 J. Cochran,33 H. B. Crawley,33 L. Dong,33 V. Eyges,33 W. T. Meyer,33 S. Prell,33 E. I. Rosenberg,33 A. E. Rubin,33 A. V. Gritsan,34 Z. J. Guo,34 C. K. Lae,34 A. G. Denig,35 M. Fritsch,35 G. Schott,35 N. Arnaud,36 J. Béquilleux,36 M. Davier,36 G. Grosdidier,36 A. Höcker,36 V. Lepeltier,36 F. Le Diberder,36 A. M. Lutz,36 S. Pruvot,36 S. Rodier,36 P. Roudeau,36 M. H. Schune,36 J. Serrano,36 V. Sordini,36 A. Stocchi,36 W. F. Wang,36 G. Wormser,36 D. J. Lange,37 D. M. Wright,37 C. A. Chavez,38 I. J. Forster,38 J. R. Fry,38 E. Gabathuler,38 R. Gamet,38 D. E. Hutchcroft,38 D. J. Payne,38 K. C. Schofield,38 C. Touramanis,38 A. J. Bevan,39 K. A. George,39 F. Di Lodovico,39 W. Menges,39 R. Sacco,39 G. Cowan,40 H. U. Flaecher,40 D. A. Hopkins,40 P. S. Jackson,40 T. R. McMahon,40 F. Salvatore,40 A. C. Wren,40 D. N. Brown,41 C. L. Davis,41 J. Allison,42 N. R. Barlow,42 R. J. Barlow,42 Y. M. Chia,42 C. L. Edgar,42 G. D. Lafferty,42 T. J. West,42 J. I. Yi,42 J. Anderson,43 C. Chen,43 A. Jawahery,43 D. A. Roberts,43 G. Simi,43 J. M. Tuggle,43 G. Blaylock,44 C. Dallapiccola,44 S. S. Hertzbach,44 X. Li,44 T. B. Moore,44 E. Salvati,44 S. Saremi,44 R. Cowan,45 P. H. Fisher,45 G. Sciolla,45 S. J. Sekula,45 M. Spitznagel,45 F. Taylor,45 R. K. Yamamoto,45 S. E. Mclachlin,46 P. M. Patel,46 S. H. Robertson,46 A. Lazzaro,47 F. Palombo,47 J. M. Bauer,48 L. Cremaldi,48 V. Eschenburg,48 R. Godang,48 R. Kroeger,48 D. A. Sanders,48 D. J. Summers,48 H. W. Zhao,48 S. Brunet,49 D. Côté,49 M. Simard,49 P. Taras,49 F. B. Viaud,49 H. Nicholson,50 G. De Nardo,51 F. Fabozzi,51, ‡ L. Lista,51 D. Monorchio,51 C. Sciacca,51 M. A. Baak,52 G. Raven,52 H. L. Snoek,52 C. P. Jessop,53 J. M. LoSecco,53 G. Benelli,54 L. A. Corwin,54 K. K. Gan,54 K. Honscheid,54 D. Hufnagel,54 H. Kagan,54 R. Kass,54 J. P. Morris,54 A. M. Rahimi,54 J. J. Regensburger,54 R. Ter-Antonyan,54 Q. K. Wong,54 N. L. Blount,55 J. Brau,55 R. Frey,55 O. Igonkina,55 http://arxiv.org/abs/0704.1266v1 J. A. Kolb,55 M. Lu,55 R. Rahmat,55 N. B. Sinev,55 D. Strom,55 J. Strube,55 E. Torrence,55 N. Gagliardi,56 A. Gaz,56 M. Margoni,56 M. Morandin,56 A. Pompili,56 M. Posocco,56 M. Rotondo,56 F. Simonetto,56 R. Stroili,56 C. Voci,56 E. Ben-Haim,57 H. Briand,57 J. Chauveau,57 P. David,57 L. Del Buono,57 Ch. de la Vaissière,57 O. Hamon,57 B. L. Hartfiel,57 Ph. Leruste,57 J. Malclès,57 J. Ocariz,57 A. Perez,57 L. Gladney,58 M. Biasini,59 R. Covarelli,59 E. Manoni,59 C. Angelini,60 G. Batignani,60 S. Bettarini,60 G. Calderini,60 M. Carpinelli,60 R. Cenci,60 A. Cervelli,60 F. Forti,60 M. A. Giorgi,60 A. Lusiani,60 G. Marchiori,60 M. A. Mazur,60 M. Morganti,60 N. Neri,60 E. Paoloni,60 G. Rizzo,60 J. J. Walsh,60 M. Haire,61 J. Biesiada,62 P. Elmer,62 Y. P. Lau,62 C. Lu,62 J. Olsen,62 A. J. S. Smith,62 A. V. Telnov,62 E. Baracchini,63 F. Bellini,63 G. Cavoto,63 A. D’Orazio,63 D. del Re,63 E. Di Marco,63 R. Faccini,63 F. Ferrarotto,63 F. Ferroni,63 M. Gaspero,63 P. D. Jackson,63 L. Li Gioi,63 M. A. Mazzoni,63 S. Morganti,63 G. Piredda,63 F. Polci,63 F. Renga,63 C. Voena,63 M. Ebert,64 H. Schröder,64 R. Waldi,64 T. Adye,65 G. Castelli,65 B. Franek,65 E. O. Olaiya,65 S. Ricciardi,65 W. Roethel,65 F. F. Wilson,65 R. Aleksan,66 S. Emery,66 M. Escalier,66 A. Gaidot,66 S. F. Ganzhur,66 G. Hamel de Monchenault,66 W. Kozanecki,66 M. Legendre,66 G. Vasseur,66 Ch. Yèche,66 M. Zito,66 X. R. Chen,67 H. Liu,67 W. Park,67 M. V. Purohit,67 J. R. Wilson,67 M. T. Allen,68 D. Aston,68 R. Bartoldus,68 P. Bechtle,68 N. Berger,68 R. Claus,68 J. P. Coleman,68 M. R. Convery,68 J. C. Dingfelder,68 J. Dorfan,68 G. P. Dubois-Felsmann,68 D. Dujmic,68 W. Dunwoodie,68 R. C. Field,68 T. Glanzman,68 S. J. Gowdy,68 M. T. Graham,68 P. Grenier,68 C. Hast,68 T. Hryn’ova,68 W. R. Innes,68 M. H. Kelsey,68 H. Kim,68 P. Kim,68 D. W. G. S. Leith,68 S. Li,68 S. Luitz,68 V. Luth,68 H. L. Lynch,68 D. B. MacFarlane,68 H. Marsiske,68 R. Messner,68 D. R. Muller,68 C. P. O’Grady,68 A. Perazzo,68 M. Perl,68 T. Pulliam,68 B. N. Ratcliff,68 A. Roodman,68 A. A. Salnikov,68 R. H. Schindler,68 J. Schwiening,68 A. Snyder,68 J. Stelzer,68 D. Su,68 M. K. Sullivan,68 K. Suzuki,68 S. K. Swain,68 J. M. Thompson,68 J. Va’vra,68 N. van Bakel,68 A. P. Wagner,68 M. Weaver,68 W. J. Wisniewski,68 M. Wittgen,68 D. H. Wright,68 A. K. Yarritu,68 K. Yi,68 C. C. Young,68 P. R. Burchat,69 A. J. Edwards,69 S. A. Majewski,69 B. A. Petersen,69 L. Wilden,69 S. Ahmed,70 M. S. Alam,70 R. Bula,70 J. A. Ernst,70 V. Jain,70 B. Pan,70 M. A. Saeed,70 F. R. Wappler,70 S. B. Zain,70 W. Bugg,71 M. Krishnamurthy,71 S. M. Spanier,71 R. Eckmann,72 J. L. Ritchie,72 A. M. Ruland,72 C. J. Schilling,72 R. F. Schwitters,72 J. M. Izen,73 X. C. Lou,73 S. Ye,73 F. Bianchi,74 F. Gallo,74 D. Gamba,74 M. Pelliccioni,74 M. Bomben,75 L. Bosisio,75 C. Cartaro,75 F. Cossutti,75 G. Della Ricca,75 L. Lanceri,75 L. Vitale,75 V. Azzolini,76 N. Lopez-March,76 F. Martinez-Vidal,76 D. A. Milanes,76 A. Oyanguren,76 J. Albert,77 Sw. Banerjee,77 B. Bhuyan,77 K. Hamano,77 R. Kowalewski,77 I. M. Nugent,77 J. M. Roney,77 R. J. Sobie,77 J. J. Back,78 P. F. Harrison,78 T. E. Latham,78 G. B. Mohanty,78 M. Pappagallo,78, § H. R. Band,79 X. Chen,79 S. Dasu,79 K. T. Flood,79 J. J. Hollar,79 P. E. Kutter,79 Y. Pan,79 M. Pierini,79 R. Prepost,79 S. L. Wu,79 Z. Yu,79 and H. Neal80 (The BABAR Collaboration) 1Laboratoire de Physique des Particules, IN2P3/CNRS et Université de Savoie, F-74941 Annecy-Le-Vieux, France 2Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain 3Università di Bari, Dipartimento di Fisica and INFN, I-70126 Bari, Italy 4University of Bergen, Institute of Physics, N-5007 Bergen, Norway 5Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA 6University of Birmingham, Birmingham, B15 2TT, United Kingdom 7Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany 8University of Bristol, Bristol BS8 1TL, United Kingdom 9University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1 10Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom 11Budker Institute of Nuclear Physics, Novosibirsk 630090, Russia 12University of California at Irvine, Irvine, California 92697, USA 13University of California at Los Angeles, Los Angeles, California 90024, USA 14University of California at Riverside, Riverside, California 92521, USA 15University of California at San Diego, La Jolla, California 92093, USA 16University of California at Santa Barbara, Santa Barbara, California 93106, USA 17University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA 18California Institute of Technology, Pasadena, California 91125, USA 19University of Cincinnati, Cincinnati, Ohio 45221, USA 20University of Colorado, Boulder, Colorado 80309, USA 21Colorado State University, Fort Collins, Colorado 80523, USA 22Universität Dortmund, Institut für Physik, D-44221 Dortmund, Germany 23Technische Universität Dresden, Institut für Kern- und Teilchenphysik, D-01062 Dresden, Germany 24Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, F-91128 Palaiseau, France 25University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom 26Università di Ferrara, Dipartimento di Fisica and INFN, I-44100 Ferrara, Italy 27Laboratori Nazionali di Frascati dell’INFN, I-00044 Frascati, Italy 28Università di Genova, Dipartimento di Fisica and INFN, I-16146 Genova, Italy 29Harvard University, Cambridge, Massachusetts 02138, USA 30Universität Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany 31Imperial College London, London, SW7 2AZ, United Kingdom 32University of Iowa, Iowa City, Iowa 52242, USA 33Iowa State University, Ames, Iowa 50011-3160, USA 34Johns Hopkins University, Baltimore, Maryland 21218, USA 35Universität Karlsruhe, Institut für Experimentelle Kernphysik, D-76021 Karlsruhe, Germany 36Laboratoire de l’Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11, Centre Scientifique d’Orsay, B. P. 34, F-91898 ORSAY Cedex, France 37Lawrence Livermore National Laboratory, Livermore, California 94550, USA 38University of Liverpool, Liverpool L69 7ZE, United Kingdom 39Queen Mary, University of London, E1 4NS, United Kingdom 40University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom 41University of Louisville, Louisville, Kentucky 40292, USA 42University of Manchester, Manchester M13 9PL, United Kingdom 43University of Maryland, College Park, Maryland 20742, USA 44University of Massachusetts, Amherst, Massachusetts 01003, USA 45Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA 46McGill University, Montréal, Québec, Canada H3A 2T8 47Università di Milano, Dipartimento di Fisica and INFN, I-20133 Milano, Italy 48University of Mississippi, University, Mississippi 38677, USA 49Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7 50Mount Holyoke College, South Hadley, Massachusetts 01075, USA 51Università di Napoli Federico II, Dipartimento di Scienze Fisiche and INFN, I-80126, Napoli, Italy 52NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands 53University of Notre Dame, Notre Dame, Indiana 46556, USA 54Ohio State University, Columbus, Ohio 43210, USA 55University of Oregon, Eugene, Oregon 97403, USA 56Università di Padova, Dipartimento di Fisica and INFN, I-35131 Padova, Italy 57Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France 58University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 59Università di Perugia, Dipartimento di Fisica and INFN, I-06100 Perugia, Italy 60Università di Pisa, Dipartimento di Fisica, Scuola Normale Superiore and INFN, I-56127 Pisa, Italy 61Prairie View A&M University, Prairie View, Texas 77446, USA 62Princeton University, Princeton, New Jersey 08544, USA 63Università di Roma La Sapienza, Dipartimento di Fisica and INFN, I-00185 Roma, Italy 64Universität Rostock, D-18051 Rostock, Germany 65Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom 66DSM/Dapnia, CEA/Saclay, F-91191 Gif-sur-Yvette, France 67University of South Carolina, Columbia, South Carolina 29208, USA 68Stanford Linear Accelerator Center, Stanford, California 94309, USA 69Stanford University, Stanford, California 94305-4060, USA 70State University of New York, Albany, New York 12222, USA 71University of Tennessee, Knoxville, Tennessee 37996, USA 72University of Texas at Austin, Austin, Texas 78712, USA 73University of Texas at Dallas, Richardson, Texas 75083, USA 74Università di Torino, Dipartimento di Fisica Sperimentale and INFN, I-10125 Torino, Italy 75Università di Trieste, Dipartimento di Fisica and INFN, I-34127 Trieste, Italy 76IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain 77University of Victoria, Victoria, British Columbia, Canada V8W 3P6 78Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom 79University of Wisconsin, Madison, Wisconsin 53706, USA 80Yale University, New Haven, Connecticut 06511, USA (Dated: October 30, 2018) We study the decays B0 → J/ψπ+π− and B+ → J/ψπ+π0, including intermediate resonances, using a sample of 382 million BB pairs recorded by the BABAR detector at the PEP-II e+e− B factory. We measure the branching fractions B(B0 → J/ψ ρ0) = (2.7 ± 0.3 ± 0.17) × 10−5 and B(B+ → J/ψ ρ+) = (5.0± 0.7± 0.31) × 10−5. We also set the following upper limits at the 90% confidence level: B(B0 → J/ψπ+π−non- resonant) < 1.2 × 10−5, B(B0 → J/ψ f2) < 4.6 × 10 −6, and B(B+ → J/ψπ+π0 non-resonant)< 4.4×10−6. We measure the charge asymmetry in charged B decays to J/ψ ρ to be −0.11±0.12±0.08. PACS numbers: 13.25.Hw, 12.15.Hh, 11.30.Er The decay B0 → J/ψρ0 [1] can in principle be used to measure the CP violation parameter sin2β. However, the measurement is not as straightforward as for J/ψK0 [2, 3], because it involves the decay of a pseudoscalar me- son to two vector mesons, resulting in both CP -odd and CP -even final states. Furthermore, the decay can pro- ceed through either a color-suppressed tree diagram, or a penguin diagram, both shown in Fig. 1, and interfer- ence between them could result in direct CP violation [4]. Direct CP violation may also occur in B+ → J/ψρ+ de- cays, where it would manifest itself as a non-zero charge asymmetry: ACP = N(B− → J/ψρ−)−N(B+ → J/ψρ+) N(B− → J/ψρ−) +N(B+ → J/ψρ+) . (1) The large intrinsic width of the ρ meson necessitates an analysis of a significant portion of the invariant mass spectrum of the dipion system. u; ; t FIG. 1: Tree and penguin diagrams for the process B0 → J/ψ ρ0 The branching fraction for B0 → J/ψπ+π− has previ- ously been measured at BABAR to be (4.6 ± 0.7± 0.6)× 10−5 [5], including a J/ψρ0 component with a branching fraction of (1.6 ± 0.6 ± 0.4) × 10−5. This measurement used a data sample containing approximately 56 million BB pairs, which is a subset of the sample used in this ∗Deceased †Also with Università di Perugia, Dipartimento di Fisica, Perugia, Italy ‡Also with Università della Basilicata, Potenza, Italy §Also with IPPP, Physics Department, Durham University, Durham DH1 3LE, United Kingdom analysis. The charged B decay to J/ψρ+ has not pre- viously been observed, the CLEO collaboration set an upper limit B(B+ → J/ψρ+) < 7.7 × 10−4 at the 90% confidence level [6]. The data sample used here contains 382 million BB pairs collected with the BABAR detector at the PEP-II asymmetric-energy e+e− storage ring, taken at a center- of-mass (CM) energy equivalent to the mass of the Υ (4S) resonance. An additional data sample, corresponding to an integrated luminosity of 36.8 fb−1, taken at a CM en- ergy 40MeV below the Υ (4S) resonance, is used to study backgrounds from continuum qq production, where q = u, d, s, c. A detailed description of the BABAR detector can be found elsewhere [7]. Charged-particle trajectories are measured by a five-layer silicon vertex tracker (SVT) and a 40-layer drift chamber (DCH) operating in a 1.5 T solenoidal magnetic field. A detector of internally reflected Cherenkov light (DIRC) is used for charged hadron identification. Surrounding this is a CsI(Tl) elec- tromagnetic calorimeter (EMC), and finally the instru- mented flux return (IFR) of the solenoid, which consists of layers of iron interspersed with resistive plate cham- bers or limited streamer tubes. The J/ψ meson is reconstructed in decays to l+l−, where l± refers to a charged lepton, e± or µ±. Elec- trons are selected on the basis of the ratio of EMC shower energy to track momentum, and the energy pro- file of the EMC shower. For J/ψ → e+e−, an attempt is made to recover energy losses from bremsstrahlung, by looking for showers in the EMC close to those from the electron candidates. This procedure increases the selection efficiency for J/ψ → e+e− candidates by ap- proximately 30% [8]. The muon selection algorithm uses a neural network, for which the most important input is the number of interaction lengths traversed in the IFR. The lepton pairs are fitted to a common vertex and the invariant mass of the combination is required to be in the range 2.98 (3.06) to 3.14GeV/c2 for the e+e− (µ+µ−) channels. In order to reduce the background fromB0 → J/ψK∗0(K∗0 → K+π−) decays, charged pion candidates are required to satisfy stringent particle iden- tification criteria, based on combined ionization energy loss (dE/dx) in the DCH and SVT with the Cherenkov angle measured in the DIRC. All tracks are required to originate close to the interac- tion point, and to lie in polar angle ranges where particle identification efficiency is well measured. The allowed ranges correspond to the geometric acceptances of the DIRC for pions, the EMC for electrons, and the IFR for muons. Neutral pion candidates are formed by combining pairs of isolated showers in the EMC. These are required to spread over a minimum of three crystals, and to have an energy greater than 200MeV. To form a B candidate, the reconstructed J/ψ is com- bined with either a pair of oppositely charged pions, or a charged pion and a π0, and a kinematic and geomet- ric fit is used to ensure that all final state particles are consistent with coming from the same decay point. In this fit, we constrain the invariant mass of the l+l− and the γγ to have the nominal mass of the J/ψ and π0, respectively [9]. The energy difference, ∆E, between the candidate energy and the single beam energy, ECMbeam, (both in the CM frame) is expected to be close to zero for signal events, and is therefore required to be in the interval −40 to 40MeV (−60 to 80MeV) for B0 (B+) candidates, corresponding to approximately ±3σ of the ∆E resolution. Note that the range is asymmetric for B+ candidates because the π0 in the final state gives rise to a tail on the low side of the distribution, due to the EMC response to photons. For events where more than one B candidate passes the selection criteria, the candidate with the smallest value of |∆E| is chosen. The branching fraction for each signal channel is ob- tained from: × ǫsig × B(J/ψ → l+l−) , (2) where Nsig and ǫsig are the observed yield and selection efficiency, respectively, for a specific signal channel, and is the number of B meson pairs. We assume that the Υ (4S) decays equally often into neutral and charged B meson pairs. The J/ψ → l+l− branching fraction is taken to be (11.87± 0.12)% [9]. We extract the signal yields for the J/ψρ0, J/ψπ+π− non-resonant, and J/ψf2 channels by performing a fit on the sample of reconstructed B0 candidates. We also per- form a similar fit to the sample of charged B candidates in order to obtain the signal yields for the decay chan- nels B+ → J/ψρ+ and B+ → J/ψπ+π0 non-resonant. The fits are two-dimensional, extended, unbinned max- imum likelihood fits to the distributions of mES and mππ, Seven event categories are considered: (i) J/ψρ sig- nal, (ii) J/ψππ non-resonant signal, (iii) J/ψf2 signal, (iv) J/ψ K0 events, (v) background events that do not contain a J/ψ (non-J/ψ background), (vi) background events containing a J/ψ (inclusive J/ψ background), and (vii) selected background channels that have been stud- ied in more detail (exclusive J/ψ backgrounds). In the fit to neutral B candidates, the decay channels that com- prise category (vii) are J/ψK∗0, J/ψK∗+, J/ψK1(1270), J/ψK+, J/ψρ+ [10], and J/ψπ+. For the fit to charged B candidates, the exclusive J/ψ background channels are J/ψK∗0, J/ψK∗+, J/ψK1(1270), J/ψK +, J/ψK0 , and J/ψK0 . In both cases, these decay channels are not in- cluded in category (vi). Of course, categories (iii) and (iv) are only present in the fit to neutral B candidates. A probability density function (PDF) is constructed for each category, and the sum of these PDFs is used to fit the data. The likelihood function for the total sam- ple is the product of the PDF values for each candidate, multiplied by a Poisson factor: (N ′)N Pi, (3) where N and N ′ are the numbers of observed and ex- pected events, respectively, and Pi is the value of the total PDF for event i. For all event categories except for the exclusive J/ψ background, Pi is a product of one- dimensional PDFs in mES and mππ. Fig. 2 shows the mES and mππ distributions for the data, and the projections of the PDFs for each cate- gory. The functional forms of these PDFs are as follows. For the J/ψρ0, J/ψπ+π−, J/ψf2, and J/ψK compo- nents, the mES distributions are parametrized by Gaus- sian functions, all with the same values for the mean and width, which are allowed to float in the fit. In the fit to charged B candidates, a Crystal Ball function [11] is used instead for the mES distributions of the J/ψρ + and J/ψπ+π0 signal components, as the presence of a π0 in the final state gives rise to a tail on the low mass side of the peak. The J/ψ ρ signal component is modeled by a relativis- tic P -wave Breit-Wigner function [12] in mππ: Fρ(mππ) = mππΓ(mππ)P 2Leff+1 ((m2ρ −m 2 +m2ρΓ(mππ) , (4) where Γ(mππ) = Γ0 1+R2q2 . The pa- rameter q(mππ) is the pion momentum in the dipion rest frame, with q0 = q(mρ); P is the J/ψ momentum in the B rest frame; Leff is the orbital angular momentum be- tween the J/ψ and the ρ which can be 0, 1 or 2; R is the radius of the Blatt-Weisskopf barrier factor [13, 14], which is taken to be (0.5±0.5) fm, and mρ is the ρ meson mass. The mππ distribution for the J/ψππ non-resonant sig- nal is Fππ = q(mππ)P 3, the product of a three-body phase space factor q(mππ)P and a factor P 2 motivated by angular momentum conservation. For the J/ψf2 component, the mππ distribution is de- scribed by a relativistic D-wave Breit-Wigner, similar to Eq. 4, but with an extra factor (q/q0) 2 in the expression for Γ(mππ). The decays to J/ψK0 are not considered signal for this analysis. Most of them are removed by the requirement that all tracks are consistent with coming from the same vertex. The mππ distribution of the remaining J/ψK events are modeled by a narrow Gaussian function. Non-J/ψ background events are modeled by an ARGUS function [15] in mES. The mππ PDF is the sum of two Weibull functions [16], and a Breit-Wigner to de- scribe the ρ component of the continuum background. The parameters of this PDF are fixed to values obtained from fits to the J/ψ mass sidebands of the data. The mES distribution of the inclusive J/ψ background is an ARGUS function plus a Gaussian at the B mass. The width of this Gaussian is somewhat wider than that used for signal components as it represents B candidates that are not correctly reconstructed. The mππ PDF is a 4th-order polynomial. The PDF parameters for this com- ponent are fixed to values obtained by fits to a large sam- ple of B → J/ψ (→ l+l−)X Monte Carlo (MC) simulated events, with signal events and exclusive J/ψ background channels removed. Each of the exclusive J/ψ background channels is mod- eled by a two-dimensional PDF derived from the distri- bution of MC events for that decay channel. The nor- malizations of these PDFs are determined by taking into account the selection efficiency on MC simulation, and the world average branching fractions [9]. For the branching fraction fit to neutral B candidates there are twelve free parameters: the yields of the J/ψρ, J/ψππ, J/ψf2, J/ψK , and inclusive J/ψ background components, the mean and width of the Gaussian used for the signal distribution in mES, the parameters mρ, Γ0, and Leff in the ρ lineshape, and the mean and width of the mππ distribution for the J/ψK component. All other parameters are fixed, including those describing lineshape of the f2(1270), the normalizations and shapes of the exclusive J/ψ background PDFs, and the shapes of the inclusive J/ψ and non-J/ψ background PDFs. We also fix the ratio of non-J/ψ to J/ψ (inclusive plus exclu- sive) background yields to a value obtained from fitting to data in the region mES < 5.26 (i.e. lower in mass than the signal region), and extrapolated to the fit region us- ing distributions from MC simulation. The configuration for the chargedB branching fraction fit is very similar. Here, there are eight free parameters, since there are no J/ψK0 or J/ψf2 components. We find from MC simulation studies that correlations between mES and mππ give rise to small biases in the numbers of J/ψρ+ and J/ψπ+π0 non-resonant signal can- didates found in the charged B fit. The sizes of these bi- ases are evaluated by examining the distribution of resid- uals (Nobs − Ninput) for a large number of MC exper- iments, and are listed in Table I. The yields obtained from the branching fraction fit are therefore corrected to take account of this by subtracting these quantities from the fitted yields. The signal yields and statistical errors obtained from the branching fraction fits are listed in Table I. We also list the statistical significances of the observed signals, −2 ln(LNull/LMax), where LMax is the likelihood from the fit, and LNull is the value of the likelihood func- tion when the fit is performed with the signal yield con- strained to zero events. We obtain signal efficiencies using samples of MC sig- nal events, produced in monthly blocks so as to match variations in detector and background conditions. Par- ticle identification efficiency is corrected using data con- trol samples of electrons, muons, and pions. The sizes of these corrections vary with momentum and polar an- )2 (GeV/cm 5.2 5.25 5.3 )2 (GeV/cm 5.2 5.25 5.3 )2) (GeV/cπ πm( 0.5 1 1.5 20 )2) (GeV/cπ πm( 0.5 1 1.5 20 )2 (GeV/cESm 5.2 5.25 5.3 )2 (GeV/cESm 5.2 5.25 5.3 )2) (GeV/cπ πm( 0.5 1 1.5 2 )2) (GeV/cπ πm( 0.5 1 1.5 2 FIG. 2: Distributions of (a) mES and (b) mππ for B J/ψπ+π− candidates. The solid line represents the total PDF, while the other lines represent (cumulatively, from the bot- tom of the plot) non-J/ψ background, inclusive J/ψ back- ground, exclusive J/ψ background, J/ψπ+π− non-resonant signal, and J/ψ f2 signal. The points with error bars repre- sent the data and statistical errors. Plots (c) and (d) show the same distributions for B+ → J/ψπ+π0 candidates. The sharp spike in (b) corresponds to J/ψK0S events, while the broader peak is due to J/ψ ρ0 events. gle, and average corrections are about 1.5% for electrons, 5.9% for muons, and 1.8% for pions. With these correc- tions applied, about 85% (50%) of electron (muon) pairs, and about 85% of pions, satisfy their respective particle identification requirements. A small, energy-dependent correction (typically about −2% relative) is also applied to decay modes containing a π0 to account for known differences in photon detection efficiency between data and MC simulation. The corrected signal efficiencies are listed in Table I. Systematic errors on the branching fraction measure- ments arise from uncertainties on the signal efficiency, on the fitted yield, on the number of B0B0 or B+B− events in the sample, and on the J/ψ → l+l− branching fraction. The number of BB pairs is known to 1.1% accuracy, and an additional 1.6% uncertainty is assigned correspond- ing to the assumption that the Υ (4S) decays 50% of the time into B0B0 and 50% of the time into B+B− [9]. The fractional uncertainty on B(J/ψ → l+l−) is 1.0% [9]. The systematic uncertainties on the efficiency are largely due to imperfect simulation of the detector per- formance. These effects are studied using various data control samples. The largest sources of uncertainty are pion identification efficiency, a 2.0% (3.4%) relative error for charged (neutral) B decay channels, and π0 efficiency (3% for charged B decays). Tracking efficiency (1.5%) and lepton identification efficiency (1.0%) also contribute TABLE I: Signal yields, detection efficiencies, and branching fractions for the signal decay channels. The fit bias, product of secondary branching fractions (B(J/ψ → l+l−) and B(π0 → γγ)), and significances of the signals (using statistical uncertainties only) are also listed. The corrected yields are obtained by subtracting the fit bias from the fitted yields. For the yields, efficiencies, and branching fractions, the first errors are statistical and the second are systematic. For decay channels where no significant signal is observed, we quote an upper limit at the 90% confidence level. Mode Fit bias (events) Corrected yield (events) ǫ(%) Bi(%) Signif. (σ) B(×10 J/ψ ρ0 0 251.1 ± 27.5 ± 11.2 20.6 ± 0.1± 0.8 11.87 ± 0.12 13.0 2.7± 0.3± 0.2 J/ψπ+π− 0 64.5 ± 35.5 ± 7.7 20.3 ± 0.1± 0.8 11.87 ± 0.12 2.0 < 1.2 (90% C.L.) J/ψ f2 0 24.4 ± 13.8 ± 1.8 20.3 ± 0.1± 0.8 11.87 ± 0.12 2.0 < 0.46 (90% C.L.) J/ψ ρ+ −6.8± 1.1 218.5 ± 28.8± 9.5 9.7 ± 0.1 ± 0.4 11.73 ± 0.12 11.6 5.0± 0.7± 0.3 J/ψπ+π0 +8.2± 0.9 −12.7± 27.1 ± 4.7 11.9 ± 0.1± 0.5 11.73 ± 0.12 0 < 0.44 (90% C.L.) to the uncertainty on the efficiency. The polarization of the ρ in B → J/ψρ decays is unknown. We use an MC sample in which the ρ mesons are unpolarized to obtain the central value of the signal efficiency. We also evaluate the efficiency using MC data samples with different ρ po- larizations, and observe a relative variation of 2%, which is assigned as a systematic uncertainty on the branching fraction measurement. We evaluate the impact of the fit procedure by observ- ing the changes in the yields when varying the PDF pa- rameters that were fixed in the fit within their uncertain- ties. The resulting differences are added quadratically for sets of parameters that are relatively uncorrelated, and added linearly for highly correlated sets of parameters. We also repeat the fit using alternative functional forms for some PDFs, namely the shape of the inclusive J/ψ background in mππ, and the ρ lineshape, and include the resulting differences in the yield in the systematic uncer- tainty. In addition, for the J/ψρ+ and J/ψπ+π0 channels, systematic uncertainties equal to half of the bias correc- tions listed in Table I are assigned. The total systematic uncertainties on the yield vary from 1.8 events for the J/ψf2 channel, to 11.2 events for the J/ψρ 0 channel. In order to assess the charge asymmetry Aρ, we per- form a second fit to the charged B candidate sample. In this fit, all the shape parameters for the signal and back- ground components are fixed to values obtained from the branching fraction fit. This reduces the number of free parameters and improves the reliability of the fit. We in- clude terms for the asymmetries in signal and background components as follows: Pi = N (1−QiA + NNR × (1−QiA NR)PNRi (1−QiA j,i , (5) where Nρ, NNR, and N j are the yields for the J/ψρ signal, the J/ψπ+π0 non-resonant signal, and the dif- ferent background components j, respectively, Qi is the charge of the B candidate in event i, and Aρ, ANR, and j are the corresponding charge asymmetries. The asymmetry parameters for the exclusive J/ψ background channels are fixed to world average values [9]. The asym- metries for the non-J/ψ background and inclusive J/ψ background components are assumed to be the same inc = A non ≡ A bkg). This fit therefore has six free parameters: the yields of the J/ψρ+ signal, J/ψπ+π0 non-resonant signal, and inclusive J/ψ background com- ponents, and the asymmetries Aρ, ANR, and Abkg. From the charge asymmetry fit, we obtain Aρ = −0.11 ± 0.12(stat.). The signal and background yields obtained from this fit are entirely consistent with those from the branching fraction fit. A potential contribution to the systematic uncertainty on the charge asymmetry Aρ could come from differ- ent pion identification efficiencies for π+ and π−, lead- ing to different signal selection efficiencies for positively and negatively charged B candidates. Using data control samples, this effect is found to be negligible. The other sources of systematic error on the asymme- try are potential differences in the backgrounds for posi- tive and negative B candidates. The parameters describ- ing the charge asymmetries of the exclusive J/ψ back- ground channels are varied within their uncertainties [9], assuming a 10% uncertainty for the J/ψK1(1270) channel for which no measurement is available. The normaliza- tions of the exclusive background channels, and the shape parameters of the inclusive J/ψ background and non-J/ψ background components are varied in turn, and the fit is repeated. The resulting changes to the fitted value of Aρ are added in quadrature, and the total systematic uncertainty is found to be ±0.08. In summary, we measure the following branching frac- tions, where the first error in each case is statistical and the second is systematic: B(B0 → J/ψρ0) = (2.7± 0.3± 0.2)× 10−5, and B(B+ → J/ψρ+) = (5.0 ± 0.7 ± 0.3) × 10−5. The signals for B0 → J/ψf2, B 0 → J/ψπ+π− non-resonant, and B+ → J/ψπ+π0 non-resonant are not statistically significant, thus we set the following upper limits at the 90% confidence level: B(B0 → J/ψf2) < 4.6 × 10−6, B(B0 → J/ψπ+π−) < 1.2 × 10−5, and B(B+ → J/ψπ+π0) < 4.4 × 10−6. These values are cal- culated by summing the statistical and systematic un- certainties in quadrature, multiplying the result by 1.28, and adding it to the central value of the branching frac- tion. We measure the charge asymmetry defined in Eq. 1 for the decays B± → J/ψρ±, Aρ = −0.11± 0.12± 0.08. We are grateful for the excellent luminosity and ma- chine conditions provided by our PEP-II colleagues, and for the substantial dedicated effort from the comput- ing organizations that support BABAR. The collaborat- ing institutions wish to thank SLAC for its support and kind hospitality. This work is supported by DOE and NSF (USA), NSERC (Canada), IHEP (China), CEA and CNRS-IN2P3 (France), BMBF and DFG (Germany), INFN (Italy), FOM (The Netherlands), NFR (Norway), MIST (Russia), MEC (Spain), and PPARC (United Kingdom). Individuals have received support from the Marie Curie EIF (European Union) and the A. P. Sloan Foundation. [1] Charge conjugation is implied throughout this letter un- less stated otherwise. [2] BABAR Collaboration, B. Aubert et al., Phys. Rev. Lett. 94, 161803 (2005). [3] Belle Collaboration, K. Abe et al. , Phys. Rev. D 71, 072003 (2005) [Erratum-ibid. D 71, 079903 (2005)] [4] I. Dunietz, Phys. Lett. B 316, 561 (1993). [5] BABAR Collaboration, B. Aubert et at., Phys. Rev. Lett. 90, 091801 (2003). [6] CLEO Collaboration, M. Bishai et al., Phys. Lett. B 369, 186 (1996). [7] BABAR Collaboration, B. Aubert et al., Nucl. Instrum. Methods Phys. Res., Sect. A 479, 1 (2002). [8] B. Aubert et al. [BABAR Collaboration], Phys. Rev. D 65, 032001 (2002) [9] W.-M. Yao et al., J. Phys. G 33, 1 (2006). [10] Note that the J/ψ ρ+ channel which is being measured in this note is one of the exclusive J/ψ background channels in the fit to neutral B candidates. For the purpose of choosing the normalization for this PDF, we assume a value of (6.0± 6.0) × 10−5 for the branching fraction. [11] M.J.Oreglia, Ph.D Thesis, SLAC-236(1980), Ap- pendix D; J.E.Gaiser, Ph.D Thesis, SLAC-255(1982), Appendix F; T.Skwarnicki, Ph.D Thesis, DESY F31-86-02(1986), Ap- pendix E. The Crystal Ball function can be written as: CB(m) = (m−µ)2 m > µ− ασ (n/α)n exp(−α2/2) ((µ−m)/σ+n/α−α)n m < µ− ασ where µ is the mean value, σ is a measure of the width, and n and α are parameters describing the tail. [12] J. Pisut and M. Roos, N. Phys. B 6, 325 (1968). [13] J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (Wiley, New York, 1952), p. 361. [14] Values for the barrier radius can be estimated from qq meson models (S. Godfrey and N. Isgur, Phys. Rev. D 32, 189 (1985) : figure 12), and by fits to experimental data (see for example D. Aston et al., Nucl. Phys. B 296, 493 (1988)). [15] ARGUS Collaboration, H. Albrecht et al., Z. Phys. C 48, 543 (1990). The ARGUS function can be written as A(m) = y 1− y2 exp(ξ(1− y )) y < 1 A(m) = 0 y > 1 where ξ is a shape parameter, y = m/mmax and mmax is a kinematic limit, equal in this case to half the total CM energy. [16] The Weibull function can be written as W (m) = C V (m−Mon) (C−1) exp[−V (m−Mmax) where V = (C − 1)/(C(Mmax −Mon) Mmax is the position of the function maximum, Mon is the lower kinematic cut-off, and C is a shape parameter. References

0704.1267

Text Line Segmentation of Historical Documents: a Survey

Microsoft Word - likforman_et_al_sept06.doc submitted to Special Issue on Analysis of Historical Documents, International Journal on Document Analysis and Recognition, Springer, 2006. Text Line Segmentation of Historical Documents: a Survey Laurence Likforman-Sulem*, Abderrazak Zahour**, Bruno Taconet** *GET-Ecole Nationale Supérieure des Télécommunications/TSI and CNRS-LTCI, 46 rue Barrault, 75013 Paris, France email: [email protected] Phone: +33 1 45 81 73 28 Fax: +33 1 45 81 37 94 http://www.tsi.enst.fr/~lauli/ ** IUT, Université du Havre/GED, Place Robert Schuman, 76610 Le Havre, France email :{taconet|zahour}@univ-lehavre.fr Abstract There is a huge amount of historical documents in libraries and in various National Archives that have not been exploited electronically. Although automatic reading of complete pages remains, in most cases, a long-term objective, tasks such as word spotting, text/image alignment, authentication and extraction of specific fields are in use today. For all these tasks, a major step is document segmentation into text lines. Because of the low quality and the complexity of these documents (background noise, artifacts due to aging, interfering lines), automatic text line segmentation remains an open research field. The objective of this paper is to present a survey of existing methods, developed during the last decade, and dedicated to documents of historical interest. Keywords: segmentation, handwriting, text lines, Historical documents, survey 1. Introduction Text line extraction is generally seen as a preprocessing step for tasks such as document structure extraction, printed character or handwriting recognition. Many techniques have been developed for page segmentation of printed documents (newspapers, scientific journals, magazines, business letters) produced with modern editing tools [57] [38] [14] [39] [2]. The segmentation of handwritten documents has also been addressed with the segmentation of address blocks on envelopes and mail pieces [9] [10] [15][48], and for authentication or recognition purposes [53] [60]. More recently, the development of handwritten text databases (IAM database, [34]) provides new material for handwritten page segmentation. Ancient and historical documents, printed or handwritten, strongly differ from the documents mentioned above because layout formatting requirements were looser. Their physical structure is thus harder to extract. In addition, historical documents are of low quality, due to aging or faint typing. They include various disturbing elements such as holes, spots, writing from the verso appearing on the recto, ornamentation, or seals. Handwritten pages include narrow spaced lines with overlapping and touching components. Characters and words have unusual and varying shapes, depending on the writer, the period and the place concerned. The vocabulary is also large and may include unusual names and words. Full text recognition is in most cases not yet available, except for printed documents for which dedicated OCR can be developed. However, invaluable collections of historical documents are already digitized and indexed for consulting, exchange and distant access purposes which protect them from direct manipulation. In some cases, highly structured editions have been established by scholars. But a huge amount of documents are still to be exploited electronically. To produce an electronic searchable form, a document has to be indexed. The simplest way of indexing a document consists in attaching its main characteristics such as date, place and author (the so called ‘metadata’). Indexing can be enhanced when the document structure and content are exploited. When a transcription (published version, diplomatic transcription) is available, it can be attached to the digitized document: this allows users to retrieve documents from textual queries. Since text based representations do not reflect the graphical features of such documents, a better representation is obtained by linking the transcription to the document image. A direct correspondence can then be established between the document image and its content by text/image alignment techniques [55]. This allows the creation of indexes where the position of each word can be recorded, and of links between both representations. Clicking on a word on the transcription or in the index through a GUI allows users to visualize the corresponding image area and vice versa. To make such queries possible for handwritten sources of literary works, several projects have been carried out under EU and National Programs: for instance the so-called ‘philological workstation’ Bambi [6][8] and within the Philectre reading and editing environment [47]. The document analysis embedded in such systems provides tools to search for blocks, lines and words, and may include a dedicated handwriting recognition system. Interactive tools are generally offered for segmentation and recognition correction purposes. Several projects also concern printed material: Debora [5] and Memorial [3]. Partial or complete logical structure can also be extracted by document analysis and corrected with GUI as in the Viadocs project [11][18]. However, document structure can also be used when no transcription is available. Word spotting techniques [22] [55] [46] can retrieve similar words in the image document through an image query. When words of the image document are extracted by top down segmentation, which is generally the case, text lines are extracted first. Fig. 1. Examples of historical documents a) Provencal medieval manuscript. b) one page from De Gaulle’s diaries c) an ancient Arabic document from Tunisian Archives. The authentication of manuscripts in the paleographic sense can also make use of document analysis and text line extraction. Authentication consists in retrieving writer characteristics independently from document content. It generally consists in dating documents, localizing the place where the document was produced, identifying the writer by using characteristics and features extracted from blank spaces, line orientations and fluctuations, word or character shapes [43] [27] [4]. Page segmentation into text lines is performed in most tasks mentioned above and overall performance strongly relies on the quality of this process. The purpose of this article is to survey the efforts made for historical documents on the text line segmentation task. Section 2 describes the characteristics of text line structures in historical documents and the different ways of defining a text line. Preprocessing of document images (gray level, color or black and white) is often necessary before text line extracting to prune superfluous information (non textual elements, textual elements from the verso) or to correctly binarize the image. This problem is addressed in Section 3.1. In Sections 3.2-3.7 we survey the different approaches to segment the clean image into text lines. A taxonomy is proposed, listed as projection profiles, smearing, grouping, Hough-based, repulsive-attractive network and stochastic methods. The majority of these techniques have been developed for the projects on historical documents mentioned above. In Section 3.8, we address the specific problem of overlapping and touching components. Concluding remarks are given in Section 4. Fig. 2. Reference lines and interfering lines with overlapping and touching components. 2. Characteristics and representation of text lines To have a good idea of the physical structure of a document image, one only needs to look at it from a certain distance: the lines and the blocks are immediately visible. These blocks consist of columns, annotations in margins, stanzas, etc... As blocks generally have no rectangular shape in historical documents, the text line structure becomes the dominant physical structure. We first give some definitions about text line components and text line segmentation. Then we describe the factors which make this text line segmentation hard. Finally, we describe how a text line can be represented. 2.1 Definitions baseline: fictitious line which follows and joins the lower part of the character bodies in a text line (Fig. 2) median line: fictitious line which follows and joins the upper part of the character bodies in a text line. upper line: fictitious line which joins the top of ascenders. lower line: fictitious line which joins the bottom of descenders. overlapping components: overlapping components are descenders and ascenders located in the region of an adjacent line (Fig. 2). touching components: touching components are ascenders and descenders belonging to consecutive lines which are thus connected. These components are large but hard to discriminate before text lines are known. text line segmentation: text line segmentation is a labeling process which consists in assigning the same label to spatially aligned units (such as pixels, connected components or characteristic points). There are two categories of text line segmentation approaches: searching for (fictitious) separating lines or paths, or searching for aligned physical units. The choice of a segmentation technique depends on the complexity of the text line structure of the document. 2.2 Influence of author style baseline fluctuation: the baseline may vary due to writer movement. It may be straight, straight by segments, or curved. line orientations: there may be different line orientations, especially on authorial works where there are corrections and annotations. line spacing: lines that are rather widely spaced lines are easy to find. The process of extracting text lines grows more difficult as interlines are narrowing; the lower baseline of the first line is becoming closer to the upper baseline of the second line; also, descenders and ascenders start to fill the blank space left for separating two adjacent text lines (Fig. 3). insertions: words or short text lines may appear between the principal text lines, or in the margins. 2.3 Influence of poor image quality imperfect preprocessing: smudges, variable background intensity and the presence of seeping ink from the other side of the document make image preprocessing particularly difficult and produce binarization errors. stroke fragmentation and merging: punctuation, dots and broken strokes due to low-quality images and/or binarization may produce many connected components; conversely, words, characters and strokes may be split into several connected components. The broken components are no longer linked to the median baseline of the writing and become ambiguous and hard to segment into the correct text line (Fig. 3). 2.4 Text line representation separating paths and delimited strip: separating lines (or paths) are continuous fictitious lines which can be uniformly straight, made of straight segments, or of curving joined strokes. The delimited strip between two consecutive separating lines receives the same text line label. So the text line can be represented by a strip with its couple of separating lines (Fig. 4). clusters: clusters are a general set-based way of defining text lines. A label is associated with each cluster. Units within the same cluster belong to the same text line. They may be pixels, connected components, or blocks enclosing pieces of writing. A text line can be represented by a list of units with the same label. Fig. 3. The three main axes of document complexity for text line segmentation. Fig. 4. Various text line representations: paths, strings and baselines. strings: strings are lists of spatially aligned and ordered units. Each string represents one text line. baselines: baselines follow line fluctuations but partially define a text line. Units connected to a baseline are assumed to belong to it. Complementary processing has to be done to cluster non-connected units and touching components. line proximity line fluctuation writing fragmentation line proximity line fluctuation writing fragmentation 3. Text line segmentation Printed historical documents belong to a large period from 16th to 20th centuries (reports, ancient books, registers, card archives). Their printing may be faint, producing writing fragmentation artifacts. However, text lines are still enclosed in rectangular areas. After the text part has been extracted and restored, top-down and smearing techniques are generally applied for text line segmentation. A large proportion of historical documents are handwritten: scrolls, registers, private and official letters, authorial drafts. The type of writing differs considerably from one document to another. It can be calligraphed or cursive; various styles can be observed (Fig. 1). In the context of cursive handwriting, statistical information about line spacing and line orientation is hard to capture. Several techniques, which take into account handwriting and layout irregularities, as well as local and global characteristics of the text lines, have been developed 3.1 Preprocessing Text line extraction would ideally process document images without background noise and without non-textual elements; the writing would be well contrasted with as little fragmentation as possible. In reality, preprocessing is often necessary. Although preprocessing has to be accurately adapted to each document and to its characteristics, we shortly describe here some preprocessing techniques that can be performed before text line extraction. Non-textual elements around the text such as book bindings, book sides, parts of fingers (thumb marks from someone holding the book open f.i.) should be removed upon criteria such as position and intensity level. On the document itself, holes, stains, may be removed by high- pass filtering [12]. Other non-textual elements (stamps, seals) but also ornamentation, decorated initials, can be removed using knowledge about the shape, the color or the position of these elements [17]. Extracting text from figures (text segmentation) can also be performed on texture grounds [20][36] or by morphological filters [16][37]. Linear graphical elements such as big crosses (called “St Andre’s crosses”) appear in some of Flaubert’s manuscripts. Removing these elements is performed through GUI by Kalman filtering in [31]. Textual but unwanted elements such as the writing on the verso (bleed through text) may be removed by filtering and wavelet techniques [24][54][32] and by combining the verso image (the reverse side image) with the recto one (front side image). Binarization, if necessary, can be performed by global or local thresholding. Global thresholding algorithms are not generally applicable to historical documents, due to inhomogeneous background. Thus, global thresholding results in severe deterioration in the quality of the segmented document image. Several local thresholding techniques have already been proposed to partially overcome such difficulties [21]. These local methods determine the threshold values based on the local properties of an image, e.g. pixel-by-pixel or region-by- region, and yield relatively better binarization results when compared with global thresholding methods. Writing may be faint so that over-segmentation or under-segmentation may occur. The integral ratio technique [52] is a two-stage segmentation technique adapted to this problem. Background normalization [51] can be performed before binarization in order to find a global threshold more easily. 3.2 Projection–based methods Projection-profiles are commonly used for printed document segmentation. This technique can also be adapted to handwritten documents with little overlap. The vertical projection- profile is obtained by summing pixel values along the horizontal axis for each y value. From the vertical profile, the gaps between the text lines in the vertical direction can be observed (Fig. 5). yxfyprofile ),()( The vertical profile is not sensitive to writing fragmentation. Variants for obtaining a profile curve may consist in projecting black/white transitions such as in Marti and Bunke [35] or the number of connected components, rather than pixels. The profile curve can be smoothed, e.g. by a Gaussian or median filter to eliminate local maxima [33]. The profile curve is then analysed to find its maxima and minima. There are two drawbacks: short lines will provide low peaks, and very narrow lines, as well as those including many overlapping components will not produce significant peaks. In case of skew or moderate fluctuations of the text lines, the image may be divided into vertical strips and profiles sought inside each strip (Zahour et al. [58]). These piecewise projections are thus a means of adapting to local fluctuations within a more global scheme. In Shapiro et al.[49], the global orientation (skew angle) of a handwritten page is first searched by applying a Hough transform on the entire image. Once this skew angle is obtained, projections are achieved along this angle. The number of maxima of the profile give the number of lines. Low maxima are discarded on their value, which is compared to the highest maxima. Lines are delimited by strips, searching for the minima of projection profiles around each maxima. This technique has been tested on a set of 200 pages within a word segmentation task. In the work of Antonacopoulos and Karatzas [3], each minimum of the profile curve is a potential segmentation point. Potential points are then scored according to their distance to adjacent segmentation points. The reference distance is obtained from the histogram of distances between adjacent potential segmentation points. The highest scored segmentation point is used as an anchor to derive the remaining ones. The method is applied to printed records of the second World War which have regularly spaced text lines. The logical structure is used to derive the text regions where the names of interest can be found. Fig. 5. Vertical projection-profile extracted on an autograph of Jean-Paul Sartre. The RXY cuts method applied in He and Downton [18], uses alternating projections along the X and the Y axis. This results in a hierarchical tree structure. Cuts are found within white spaces. Thresholds are necessary to derive inter-line or inter-block distances. This method can be applied to printed documents (which are assumed to have these regular distances) or well separated handwritten lines. 3.3 Smearing methods For printed and binarized documents, smearing methods such as the Run-Length Smoothing Algorithm (Wong et al. [57]) can be applied. Consecutive black pixels along the horizontal direction are smeared: i.e. the white space between them is filled with black pixels if their distance is within a predefined threshold. The bounding boxes of the connected components in the smeared image enclose text lines. A variant of this method adapted to gray level images and applied to printed books from the sixteenth century consists in accumulating the image gradient along the horizontal direction (LeBourgeois [25]). This method has been adapted to old printed documents within the Debora project [26]. For this purpose, numerous adjustments in the method concern the tolerance for character alignment and line justification. Text line patterns are found in the work of Shi and Govindaraju [50] by building a fuzzy run length matrix. At each pixel, the fuzzy run-length is the maximal extent of the background along the horizontal direction. Some foreground pixels may be skipped if their number does not exceed a predefined value. This matrix is threshold to make pieces of text lines appear without ascenders and descenders (Fig. 6). Parameters have to be accurately and dynamically tuned. 3.4 Grouping methods These methods consist in building alignments by aggregating units in a bottom-up strategy. The units may be pixels or of higher level, such as connected components, blocks or other features such as salient points. Units are then joined together to form alignments. The joining scheme relies on both local and global criteria, which are used for checking local and global consistency respectively. 10 Fig. 6 Text line patterns extracted from a letter of Georges Washington (reprinted from Shi and Govindaraju [50], © [2004] IEEE). Foreground pixels have been smeared along the horizontal direction. Contrary to printed documents, a simple nearest-neighbor joining scheme would often fail to group complex handwritten units, as the nearest neighbor often belongs to another line. The joining criteria used in the methods described below are adapted to the type of the units and the characteristics of the documents under study. But every method has to face the following: - initiating alignments: one or several seeds for each alignment. - defining a unit’s neighborhood for reaching the next unit. It is generally a rectangular or angular area (Fig. 7). - solving conflicts. As one unit may belong to several alignments under construction, a choice has to be made: discard one alignment or keep both of them, cutting the unit into several parts. Hence, these methods include one or several quality measures which ensure that the text line under construction is of good quality. When comparing the quality measures of two alignments in conflict, the alignment of lower quality can be discarded (Fig. 7). Also, during the grouping process, it is possible to choose between the different units that can be aggregated within the same neighborhood by evaluating the quality of each of the so-formed alignments. 11 Fig. 7. Angular and rectangular neighborhoods from point and rectangular units (left). Neighborhood defined by a cluster of units (upright). Two alignments A and B in conflict: a quality measure will choose A and discard B (down right). Quality measures generally include the strength of the alignment, i.e. the number of units included. Other quality elements may concern component size, component spacing, or a measure of the alignment’s straightness. Fig. 8. Text lines extracted on Church Registers (reprinted from Feldbach [12] with permission from the author). Likforman-Sulem and Faure have developed in [28] an iterative method based on perceptual grouping for forming alignments, which has been applied to handwritten pages, author drafts and historical documents [29][47]. Anchors are detected by selecting connected components elongated in specific directions (0°, 45°, 90°, 125°). Each of these anchors becomes the seed of an alignment. First, each anchor, then each alignment, is extended to the left and to the right. This extension uses three Gestalt criteria for grouping components: proximity, similarity and direction continuity. The threshold is iteratively incremented in order to group components within a broader neighborhood until no change occurs. Between each iteration, alignment quality is checked by a quality measure which gives higher rates to long alignments including anchors of the same direction. A penalty is given when the alignment includes anchors of different directions. Two alignments may cross each other, or overlap. A set of 12 rules is applied to solve these conflicts taking into account the quality of each alignment and neighboring components of higher order (Fig. 14). In the work of Feldbach and Tönnies [12][13], body baselines are searched in Church Registers images. These documents include lots of fluctuating and overlapping lines. Baselines units are the minima points of the writing (obtained here from the skeleton). First basic line segments (BLS) are constructed, joining each minima point to its neighbors. This neighborhood is defined by an angular region (+-20°) for the first unit grouped, then by a rectangular region enclosing the points already joined for the remaining ones. Unwanted basic segments are found from minima points detected in descenders and ascenders. These segments may be isolated or in conflict with others. Various heuristics are defined to eliminate alignments on their size, or the local inter-line distance and on a quality measure which favours alignments whose units are in the same direction rather than nearer units but positioned lower or higher than the current direction. Conflicting alignments can be reconstructed depending on the topology of the conflicting alignments. The median line is searched from the baseline and from maxima points (Fig. 8). Pixels lying within a given baseline and median line are clustered in the corresponding text line, while ascenders and descenders are not segmented. Correct segmentation rates are reported between 90% and 97 % with adequate parameter adjustment. The seven documents tested range from the 17th to the 19th century. 3.5 Methods based on the Hough transform The Hough transform is a very popular technique [19] for finding straight lines in images. In Likforman-Sulem et al. [30], a method has been developed on a hypothesis-validation scheme. Potential alignments are hypothesized in the Hough domain and validated in the Image domain. Thus, no assumption is made about text line directions (several may exist within the same page). The centroids of the connected components are the units for the Hough transform. A set of aligned units in the image along a line with parameters (ρ, θ) is included in the corresponding cell (ρ, θ) of the Hough domain. Alignments including a lot of units correspond to high peaked cells of the Hough domain. To take into account fluctuations of handwritten text lines, i.e. the fact that units within a text line are not perfectly aligned, two hypotheses are considered for each alignment and an alignment is formed from units of the cell structure of a primary cell. 13 Fig. 9. Hypothesized cells (ρ0, θ0) and (ρ1, θ1) in Hough space. Each peak corresponds to perfectly aligned units. An alignment is composed of units belonging to a cluster of cells (the cell structure) around a primary cell. A cell structure of a cell (ρ, θ) includes all the cells lying in a cluster centered around (ρ, θ). Consider the cell (ρ0, θ0) having the greatest count of units. A second hypothesis (ρ1, θ1) is searched in the cell structure of (ρ0, θ0). The alignment chosen between these two hypotheses is the strongest one, i.e. the one which includes the highest number of units in its cell structure. And the corresponding cell (ρ0, θ0) or (ρ1, θ1) is the primary cell (Fig. 9). However, actual text lines rarely correspond to alignments with the highest number of units as crossing alignments (from top to bottom for writing in horizontal direction) must contain more units than actual text lines. A potential alignment is validated (or invalidated) using contextual information, i.e. considering its internal and external neighbors. An internal neighbor of a unit j is a within-Hough alignment neighbor. An external neighbor is a out of Hough alignment neighbor which lies within a circle of radius δj from unit j. Distance δj is the average distance of the internal neighbor distances from unit j. To be validated, a potential alignment may contain fewer external units than internal ones. This enables the rejection of alignments which have no perceptual relevance. This method can extract oriented text lines and sloped annotations under the assumption that such lines are almost straight (Fig. 10). (ρ0, θ0) (ρ1, θ1) #units (ρ0, θ0) (ρ1, θ1) #units 14 Fig. 10. Text lines extracted on an autograph of Miguel Angel Asturias. The orientations of traced lines correspond to those of the primary cells found in Hough space. The Hough transform can also be applied to fluctuating lines of handwritten drafts such as in Pu and Shi [45]. The Hough transform is first applied to minima points (units) in a vertical strip on the left of the image. The alignments in the Hough domain are searched starting from a main direction, by grouping cells in an exhaustive search in 6 directions. Then a moving window, associated with a clustering scheme in the image domain, assigns the remaining units to alignments. The clustering scheme (Natural Learning Algorithm) allows the creation of new lines starting in the middle of the page. 3.6 Repulsive-Attractive network method An approach based on attractive-repulsive forces is presented in Oztop et al. [40]. It works directly on grey-level images and consists in iteratively adapting the y-position of a predefined number of baseline units. Baselines are constructed one by one from the top of the image to bottom. Pixels of the image act as attractive forces for baselines and already extracted baselines act as repulsive forces. The baseline to extract is initialized just under the previously examined one, in order to be repelled by it and attracted by the pixels of the line below (the first one is initialized in the blank space at top of the document). The lines must have similar lengths. The result is a set of pseudo-baselines, each one passing through word bodies (Fig. 11). The method is applied to ancient Ottoman document archives and Latin texts. 15 Fig. 11. Pseudo baselines extracted by a Repulsive-Attractive network on an Ancient Ottoman text (reprinted from Oztop et al. [40] Copyright (1999) with permission from Elsevier). 3.7 Stochastic method We present here a method based on a probabilistic Viterbi algorithm (Tseng and Lee [56]), which derives non-linear paths between overlapping text lines. Although this method has been applied to modern Chinese handwritten documents, this principle could be enlarged to historical documents which often include overlapping lines. Lines are extracted through hidden Markov modeling. The image is first divided into little cells (depending on stroke width), each one corresponding to a state of the HMM (Hidden Markov Model). The best segmentation paths are searched from left to right; they correspond to paths which do not cross lots of black points and which are as straight as possible. However, the displacement in the graph is limited to immediately superior or inferior grids. All best paths ending at each y location of the image are considered first. Elimination of some of these paths uses a quality threshold T: a path whose probability is less than T is discarded. Shifted paths are easily eliminated (and close paths are removed on quality criteria). The method succeeds when the ground truth path between text lines is slightly changing along the y-direction (Fig. 12). In the case of touching components, the path of highest probability will cross the touching component at points with as less black pixels as possible. But the method may fail if the contact point contains a lot of black pixels. Fig. 12. Segmentation paths obtained by a stochastic method (reprinted from Tseng and Lee [56], Copyright (1999) with permission from Elsevier). 16 3.8 Processing of overlapping and touching components Overlapping and touching components are the main challenges for text line extractions since no white space is left between lines. Some of the methods surveyed above do not need to detect such components because they extract only baselines (3.4, 3.6), or because in the method itself some criteria make paths avoid crossing black pixels (c.f. Section 3.7). This section only deals with methods where ambiguous components (overlapping or touching) are actually detected before, during or after text line segmentation Such criteria as component size, the fact that the component belongs to several alignments, or on the contrary to no alignment, can be used for detecting ambiguous components. Once the component is detected as ambiguous, it must be classified into three categories: the component is an overlapping component which belongs to the upper (resp. lower) alignment, the component is a touching component which has to be decomposed into several parts (two or more parts, as components may belong to three or more alignments in historical documents). The separation along the vertical direction is a hard problem which can be done roughly (horizontal cut), or more accurately by analysing stroke contours and referring to typical configurations (Fig. 13). Fig. 13. Set of typical overlapping configurations (adapted from Piquin et al. [44]). The grouping technique presented in Section 3.4 detects an ambiguous component during the grouping process when a conflict occurs between two alignments [28] [29]. A set of rules is applied to label the component as overlapping or touching. The ambiguous component extends in each alignment region. The rules use as features the density of black pixels of the component in each alignment region, alignment proximity and contextual information (positions of both alignments around the component). An overlapping component will be assigned to only one alignment. And the separation of a touching component is roughly performed by drawing a horizontal frontier segment. The frontier segment position is decided by analysing the vertical projection profile of the component. If the projection profile includes two peaks, the cut will be done middle way from them, as in Figure 14. Else the component will be cut into two equal parts. 17 Fig. 14. Touching component separated in a ‘Lettre de Remission’. In Likforman-Sulem et al. [30], touching and overlapping components are detected after the text line extraction process described in Section 3.5. These components are those which are intersected by at least two different lines (ρ,θ) corresponding to primary cells of validated alignments. In Zahour et al. [58][59], the document page is first cut into eight equal columns. A projection-profile is performed on each column. In each histogram, two consecutive minima delimit a text block. In order to detect touching and overlapping components, a k-means clustering scheme is used to classify the text blocks so extracted into three classes: big, average, small. Overlapping components necessarily belong to big physical blocks. All the overlapping cases are found in the big text blocks class. All the “one line” blocks are grouped in the average block text class. A second k-means clustering scheme finds the actual inter-line blocks; put together with the “one line” block size, this determines the number of pieces a large text block must be cut into (cf. Fig. 16). A similar method such as the one presented above is applied to Bangla handwriting Indian documents in Pal and Datta [41]. The document is divided into vertical strips. Profile cuts within each strip are computed to obtain anchor points of segmentation (PSLs) which do not cross any black pixels. These points are grouped through strips by neighboring criteria. If no segmentation point is present in the adjacent strip, the baseline is extended near the first black pixel encountered which belongs to an overlapping or touching component. This component is classified as overlapping or touching by analysing its vertical extension (upper, lower) from each side of the intersection point. An empirical rule classifies the component. In the touching case, the component is horizontally cut at the intersection point (Fig. 15). Fig.15. Overlapping components separated (circle) and touching component separated into two parts (rectangle) in Bangla writing (from Pal and Datta [41], © [2003] IEEE). 18 Some solutions for separation of units belonging to several text lines can be found also in the case of mail pieces and handwritten databases where efforts have been made for recognition purposes [44] [7]. In the work of Piquin et al. [44], separation is made from the skeleton of touching characters and the use of a dictionary of possible touching configurations (Fig. 13). In Bruzzone and Coffetti [7], the contact point between ambiguous strokes is detected and processed from their external border. An accurate analysis of the contour near the contact point is performed in order to separate the strokes according to two registered configurations: a loop in contact with a stroke, or two loops in contact. In simple cases of handwritten pages (Marti and Bunke [35]), the center of gravity of the connected component is used either to associate the component to the current line or to the following line, or to cut the component into two parts. This works well if the component is a single character. It may fail if the component is a word, or part of a word, or even several words. 3.9 Non Latin documents The inter-line space in Latin documents is filled with single dots, ascenders and descenders. The Arabic script is connected and cursive. Large loops are present in the inter-line space and ancient Arabic documents include diacritical points [1]. In the Hebrew squared writing, the baseline is situated on top of characters. Documents such as decorated Bibles, prayer books and scientific treatises include diacritical points which represent vowels. Ancient Hebrew documents may include decorated words but no decorated initials as there is no upper/lower case character concept in this script. In the alphabets of some Indian scripts (like Devnagari, Bangla and Gurumukhi), many basic characters have an horizontal line (the head line) in the upper part [42]. In Bangla and Telugu text, touching and overlapping occur frequently [23].To date, the published studies on historical documents concern Arabic and Hebrew. Work about Chinese and Bangla Indian writings on good quality documents have been already mentioned in Sections 3.7 and 3.8: they should be also suitable to ancient documents as they include local processing. 3.9.1 Ancient Arabic documents Figure 1 is a handwritten page extracted from a book of the Tunisian National Library. The writing is dense and inter-line space is faint. Several consecutive lines are often connected by one character at least, and the overlapping situations are obvious. Baseline waving produces various text orientations. The method developed in Zahour et al. [59] begins with the detection of overlapping and touching components presented in §3.8, using a two-stage clustering process which separates big blocks including several lines into several parts. Blocks are then linked by neighborhood using the y coordinates. Figure 16 shows line separators using the clustering technique recursively, as described in Section 3.8. 19 Fig. 16. Text line segmentation of the ancient Arabic handwritten document in Fig. 1. 3.9.1 Ancient Hebrew documents The manuscripts studied in Likforman-Sulem et al. [27], are written in Hebrew, in a so-called squared writing as most characters are made of horizontal and vertical strokes. They are reproducing the biblical text of the Pentateuch. Characters are calligraphed by skilled scribes with a quill or a calamus. The Scrolls, intended to be used in the synagogue, do not include diacritics. Characters and words are written properly separated but digitization make some characters touch. Cases of overlapping components occur as characters such as Lamed, Kaf, and final letters include ascenders and descenders. Since the majority of characters are composed of one connected component, it is more convenient to perform text line 20 segmentation from connected components units. Fig. 17 shows the resulting segmentation with the Hough-based method presented in Section 3.5. Fig. 17. Text line segmentation of a Hebrew document (Scroll). 21 Table 1. Text line segmentation methods suitable for historical documents Authors Description Line Description Writing Type Units Suitable for Project/ Documents [Antonacopoulos and Karatzas, 2004] projection profiles linear paths Latin printed pixels separated lines Memorial/person al records(World War II) [Calabretto and Bozzi, 1998] projection profiles (gray level image) linear paths cursive handwriting pixels separated lines Bambi/italian manuscripts (16th century) [Feldbach and Tönnies, 2001] grouping method baselines cursive handwriting minima points fluctuating lines Church registers (18th, 19th century) [He and Downton, 2003] projections (RXY cuts) linear paths Latin printed and handwriting pixels separated lines Viadocs/ Natural History Cards [Lebourgeois et al., 2001] smearing (accumulated gradients) clusters Latin printed pixels separated lines Debora/books (16th century) [Likforman-Sulem and Faure, 1994] grouping strings Latin handwriting connected components fluctuating lines Philectre/ authorial manuscripts [Likforman-Sulem et al., 1995] Hough transform, (hypothesis- validation scheme) strings Latin handwriting connected components different straight line directions Philectre/ authorial manuscripts, manuscripts of the 16th century [Oztop et al., 1997] repulsive - attractive network baselines Arabic and Latin handwriting pixels (gray levels) fluctuating lines (same size) ancient Ottoman documents [Pal and Datta, 2003] piecewise projections piecewise linear paths Bangla handwriting segmentatio n points overlapping/ touching lines Indian handwritten documents [Pu and Shi, 1998] Hough transform (moving window) clusters Latin handwriting minima points fluctuating lines handwritten documents [Shapiro et al., 1993] projection profiles linear paths Latin handwriting pixels skewed separated lines handwritten documents [Shi and Govindaraju, 2004] smearing (fuzzy run length) cluster Latin handwriting pixels straight touching lines Newton, Galileo manuscripts [Tseng and Lee, 1999 ] stochastic (probabilistic Viterbi algorithm) non linear paths Chinese handwriting pixels overlapping lines handwritten documents [Zahour et al., 2004] piecewise projection and k-means clustering piecewise linear paths Arabic handwriting text blocks overlapping/ touching lines. ancient Arabic documents 22 4. Discussion and concluding remarks An overview of text line segmentation methods developed within different projects is presented in Table 1. The achieved taxonomy consists in six major categories. They are listed as: projection-based, smearing, grouping, Hough-based, repulsive-attractive network and stochastic methods. Most of these methods are able to face some image degradations and writing irregularities specific to historical documents, as shown in the last column of Table 1. Projection, smearing and Hough-based methods, classically adapted to straight lines and easier to implement, had to be completed and enriched by local considerations (piecewise projections, clustering in Hough space, use of a moving window, ascender and descender skipping), so as to solve some problems including: line proximity, overlapping or even touching strokes, fluctuating close lines, shape fragmentation occurrences. The stochastic method (achieved by the Viterbi decision algorithm) is conceptually more robust, but its implementation requires great care, particularly the initialization phase. As a matter of fact, text-line images are initially divided into mxn grids (each cell being a node), where the values of the critical parameters m and n are to be determined according to the estimated average stroke width in the images. Representing a text line by one or more baselines (RA method, minima point grouping) must be completed by labeling those pixels not connected to, or between the extracted baselines. The recurrent nature of the repulsive-attractive method may induce cascading detecting errors following a unique false or bad line extraction. Projection and Hough-based methods are suitable for clearly separated lines. Projection-based methods can cope with few overlapping or touching components, as long text lines smooth both noise and overlapping effects. Even in more critical cases, classifying the set of blocks into “one line width” blocks and “several lines width” blocks allows the segmentation process to get statistical measures so as to segment more surely the “several lines width” blocks. As a result, the linear separator path may cross overlapping components. However, more accurate segmentation of the overlapping components can be performed after getting the global or piecewise straight separator, by looking closely at the so crossed strokes. The stochastic method naturally avoids crossing overlapping components (if they are not too close): the resulting non linear paths turn around obstacles. When lines are very close, grouping methods encounter a lot of conflicting configurations. A wrong decision in an early stage of the grouping results in errors or incomplete alignments. In case of touching components, making an accurate segmentation requires additional knowledge (compiled in a dictionary of possible configurations or represented by logical or fuzzy rules). Concerning text line fluctuations, baseline-based representations seem to fit naturally. Methods using straight line-based representations must be modified as previously to give non linear results (by piecewise projections or neighboring considerations in Hough space). The more fluctuating the text line, the more refined local criteria must be. Accurate locally oriented processing and careful grouping rules make smearing and grouping methods convenient. The stochastic methods also seem suited, for they can generate non linear segmentation paths to separate overlapping characters, and even more to derive non linear cutting paths from touching characters by identifying the shortest paths. 23 Pixel based methods are naturally robust at dealing with writing fragmentation. But, as a consequence of writing fragmentation, when units become fragmented, sub-units may be located far from the baseline. Spurious characteristic points are then generated, disturbing alignment and implying a loss of accuracy, or more, a wrong final representation. Quantitative assessment of performance is not generally yielded by the authors of the methods; when it is given, this is on a reduced set of documents. As for all segmentation methods, ground truth data are harder to obtain than for classification methods. For instance the ground truth for the real baseline may be hard to assess. Text line segmentation is often a step in the recognition algorithm and the segmentation task is not evaluated in isolation. To date, no general study has been carried out to compare the different methods. Text line representations differ and methods are generally tuned to a class of documents. Analysis of historical document images is a relatively new domain. Text line segmentation methods have been developed within several projects which perform transcript mapping, authentication, word mapping or word recognition. As the need for recognition and mapping of handwritten material increases, text line segmentation will be used more and more. Contrary to printed modern documents, a historical document has unique characteristics due to style, artistic effect and writer skills. There is no universal segmentation method which can fit all these documents. The techniques presented here have been proposed to segment particular sets of documents. They can however be generalized to other documents with similar characteristics, with parameter tuning that depends on script size, stroke width and average spacing. The major difficulty consists in obtaining a precise text line, with all descenders and ascenders segmented for accessing isolated words. 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0704.1268

On the interpretation of muon-spin-rotation experiments in the mixed state of type-II superconductors

On the interpretation of muon-spin-rotation experiments in the mixed state of type-II superconductors I. L. Landau a,b H. Keller a aPhysik-Institut der Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland bInstitute for Physical Problems, 117334 Moscow, Russia Abstract We argue that claims about magnetic field dependence of the magnetic field pene- tration depth λ, which were made on the basis of muon-spin-rotation (µSR) studies of some superconductors, originate from insufficient accuracy of theoretical models employed for the data analysis. We also reanalyze some of already published exper- imental data and demonstrate that numerical calculations of Brandt [E.H. Brandt, Phys. Rev. B 68, 54506 (2003)] may serve as a reliable and powerful tool for the analysis of µSR data collected in experiments with conventional superconductors. Furthermore, one can use this approach in order to distinguish between conventional and unconventional superconductors. It is unfortunate that these calculations have practically never been employed for the analysis of µSR data. Key words: type-II superconductors, mixed state, unconventional superconductivity, muon-spin-rotation experiments, magnetic field penetration depth PACS: 74.70.Dl, 74.25.Op, 74.25.Ha,76.75.+i 1 Introduction Muon-spin-rotation (µSR) experiments in the mixed state of type-II supercon- ductors provide unique information about superconducting properties of the investigated sample. An important advantage of this method is that muons probe the bulk of the sample and therefore, the results are not distorted by possible imperfections of the sample surface. At the same time, in order to extract quantitative results from µSR measurements, a detailed model of the magnetic field distribution in the mixed state is needed. As well as we are Preprint submitted to Elsevier 2 November 2018 http://arxiv.org/abs/0704.1268v2 aware, only the Ginzburg-Landau (GL) theory [1] of the Abrikosov vortex lat- tice [2] is developed to such a level [3,4,5]. As was recently demonstrated, if an adequate model is available, not only the magnetic field penetration depth λ but also the upper critical field Hc2 can be found from µSR data collected in different applied magnetic fields [6]. It has to be remembered, however, that theoretical calculations of Refs. [3,4,5] are related to superconductors with one and isotropic energy gap only. This is why, this kind of analysis should be used with extreme caution in the case of unconventional superconductors, in which the applicability of theoretical models is not obvious. We also point out a very interesting and promising approach which was de- veloped in Refs. [7,8,9,10]. In these works a microscopic theory was used for calculation of the mixed state parameters. An important advantage of this ap- proach is that the results are not limited to conventional superconductors and it can be used at temperatures well below Tc both for s- and d-wave pairing. In recent years, µSR measurements were widely used for studying of different unconventional superconductors such as high-Tc materials, MgB2 and oth- ers. Some very interesting results were obtained. It was demonstrated that in some cases the magnetic field penetration depth λ and the superconducting co- herence length ξ, evaluated from µSR measurements, depend on the applied magnetic field (see, e.g., [11,12,13,14,15,16,17,18,19]). This result, however, contradicts the GL theory, which was used as a basis for the data analysis. This contradiction is a clear sign that the corresponding models are not ade- quate for describing the magnetic field distribution in the mixed state of these compounds and rises the question about physical meanings of λ(H) and ξ(H) obtained in such a way. As we argue below, magnetic field dependences of λ and ξ cannot be obtained from µSR experiments if the conventional GL the- ory or the London model were employed for the analysis of experimental data. Moreover, because in the mixed state the superconducting order parameter is not spatially uniform, there is no reasonable way to define either λ or ξ. In other words, the physical meanings of magnetic field dependences of λ and ξ, evaluated from µSR data, are quite different from traditional definitions of these two lengths. This circumstance was recognized in Refs. [20,21,22] where it was pointed out that λ(H), evaluated in such a way, represents some fit- parameter rather than the magnetic field penetration depth. We underline that the same should also be addressed to ξ(H) dependences. In the following section, in order to avoid confusion, we shall use λ0 and ξ0 to denote values λ and ξ for H → 0. 2 Conventional superconductors Superconductors with s-pairing and one energy gap we shall consider as con- ventional, independent of their pairing mechanism. Because the GL theory is traditionally used for analyses of µSR data, we limit our consideration to this theory. The magnetic field penetration depth λ0 together with the zero-field coher- ence length ξ0 represent two fundamental lengths of the GL theory. If their values for some particular temperature T are known, one can calculate the GL parameter κ(T ) = λ0(T )/ξ0(T ), (1) the thermodynamic critical magnetic field Hc(T ) = 2πλ0(T )ξ0(T ) , (2) the upper critical field Hc2(T ) = 2κHc(T ) = , (3) the lower critical field Hc1(T ) = ln κ(T ) + α(κ)√ 2κ(T ) Hc(T ) = [ln κ(T ) + α(κ)] with α(κ) = 0.49693 + exp[−0.41477− 0.775 lnκ− 0.1303(lnκ)2] [5] Further- more, in the case of conventional superconductors, any characteristics of the sample for any value of an applied magnetic field may also be calculated and expressed via λ0 and ξ0. Very detailed numerical calculations of different pa- rameters of the mixed state for a very wide range of κ and for magnetic fields ranging from Hc1 to Hc2 are presented in Ref. [5]. Muons probe the distribution of the magnetic induction in the sample. In high- κ superconductors and low magnetic inductions B, contributions of vortex cores can be neglected (London limit) and the distribution of the magnetic induction around a single vortex line may be written as B(r) = K0(r/λ0), (5) where r is the distance from the vortex center, Φ0 is the magnetic flux quantum and K0 is the modified Bessel function. Because Eq. (5) is obtained from the London theory, it gives an unphysical divergence of B at r = 0. In order to improve Eq. (5), an appropriate cutoff has to be introduced [23,24,25]. It should be remembered, however, that the results of Refs. [23,24,25] can be considered as sufficiently accurate in low magnetic fields H ≪ Hc2 only. If this condition is not satisfied, numerical solution of the GL equations must be used for a reliable analysis µSR data. The magnetic induction distribution may be calculated as a linear superposition of inductions created by different vortices (see, for instance, Ref. [25]). By measuring muon relaxation rates, one obtains the distribution of the mag- netic induction P (B) experimentally, which allows to calculate the variance of the magnetic induction B2(r)−B2 , (6) where · · · = (1/V ) · · · d3r means spatial averaging over superconductor of volume V . If the distribution of the magnetic induction around vortices is known, σ can also be calculated theoretically. According to [5] σ = F (κ,B/Bc2)/λ , (7) where the parameter F depends on κ and B/Bc2. If the value of F is known, λ0 may straightforwardly be evaluated. In the case of κ ≫ 1 and b ≪ 1, F ≈ 0.061Φ0. In other situations, reliable results can be obtained from Ref. [5]. Eq. (7) may also be written as σ = (2πHc2/Φ0)F (κ,B/Bc2)/κ 2. This representation may be convenient if evaluation of κ is preferable. While the zero-field value of λ enters the theory, the actual magnetic field penetration depth is field dependent. According to the original Ginzburg and Landau publication [1], if the magnetic field is parallel to the sample surface, λ(H) = λ0 [1 + f(κ)H/Hc] . (8) The function f(κ) grows monotonically with κ in such a way that for κ ≪ 1 f(κ) ∼ κ/4 2 and f(∞) = 0.125 [1]. Taking into account Eq. (1), we see that even the magnetic field dependence of λ may be expressed via λ0 and ξ0. The λ(H) dependence arises due to suppression of the order parameter |ψ| by the applied magnetic field. In bulk type-II superconductors, Eq. (8) is applicable in the Meissner state only, i.e., in magnetic fields H < Hc1. If H ≥ Hc1, the magnetic field penetrates into the bulk of the sample forming a lattice of Abrikosov vortices. 0 0.5 1.0 10-5 10-4 10-3 10-2 10-1 100 B/Bc2 B/Bc2 κ = 200 κ = 20 κ = 5 Ref. 3 κ = 2 Fig. 1. F multipied by (1− 0.069/κ2) as a function of B/Bc2 according to [5]. The dashed line shows the F (B/Bc2) according to interpolation formula proposed in [3]. The horizontal line corresponds to F = 0.061Φ0. The inset shows the same curves on linear scales. If spatial variations of the order parameter can be neglected, the magnetic induction decays exponentially on the flat surface of the sample. In the case of cylindrical geometry (around vortices), the same decay is described by the Bessel function (see Eq. (5)). Considering µSR experiments in the mixed sate of type-II superconductors, we can use Eq. (5) if the total volume of vortex cores is negligibly small compared to the volume of the sample, i.e., κ ≫ 1 and B ≪ Bc2. If one or both of these conditions is not satisfied, spatial variations of the order parameter have to be taken into account. Numerical calculations of Brandt are shown in Fig. 1. If κ ≫ 1, the function F (B/Bc2) is practically independent of κ. As may be seen in Fig. 1, F remains magnetic field dependent even for very small values of B/Bc2. The maximum on the F (B/Bc2) dependence at B/Bc2 ≈ 0.17/κ1.2 is the obvious consequence of the fact that σ vanishes at B/Bc2 → 0. At very low magnetic inductions F is proportional to (B/Bc2) [5]. It must be clearly understood that, although the applied magnetic field influ- ences properties of the sample both in the Meissner and the mixed states, the physics of this influence is completely different. In the Meissner state, super- currents are induced in the surface layer of the sample. The density of these currents is proportional to H and they depress the order parameter, which leads to an increase of λ (see Eq. (8)). Because the reduction of the order parameter |ψ| is small (|∆ψ| ≪ |ψ|), one may still introduce the magnetic field penetration depth in its traditional way. In the mixed state, the situation is completely different. Because there are no currents, which are proportional to H , the absolute value of the applied magnetic field is irrelevant. Only the distance between vortices given by the magnetic induction B is important. At magnetic inductions B . 0.1Bc2, over- lapping of vortex cores may be neglected and vortex properties are indepen- dent of the applied magnetic field [5]. Only the vortex density is changed. At higher magnetic inductions, vortex cores overlap and not only the vortex den- sity, but also properties of individual vortices are magnetic field dependent. Because the local value of the magnetic field penetration depth is inversely pro- portional to the modulus of the order parameter |ψ|, there is no much sense to introduce any unique value of λ corresponding to each particular magnetic field. The correct approach is to calculate some measurable quantities theo- retically and compare them with experimental results. In this way, however, only the zero-field value of λ can be evaluated. If the value of λ resulting from the analysis of experimental data depends on the applied magnetic field, it means that the theory, which was employed for the analysis, does not describe the actual experimental situation and the approach to the analysis should be reconsidered. If the magnetic field dependence of F is not taken into account or accounted for incorrectly, the analysis of muon depolarization rates would result in some effective λeff , which is magnetic field dependent. The knowlege of λeff(H), however, does not represent any particular interest. This is why it is important to use reliable models of the mixed sate in order to evaluate λ0. As is well known, the GL theory is formally applicable at temperatures close to Tc only. This is why quantitative applicability of theoretical calculations to the analysis of experimental data at temperatures well below Tc is not obvious. However, as it was recently demonstrated, the magnetic field dependence of σ at T → 0 can be very well fitted by calculations of Brandt with two fit- parameters λ0 and Hc2 [6]. Moreover, the value ofHc2, evaluated in such a way, coincides with the result of magnetization measurements. We consider this as a proof that the theoretical σ(H) dependence calculated in framework of the GL theory can indeed be used for quantitative analysis of isothermal experimental data even at temperatures T ≪ Tc. Below we reconsider several experimental µSR studies and demonstrate that their results may perfectly be described by the conventional GL theory although the magnetic field dependence of λ was claimed in some of the original publications. 3 Unconventional superconductors It must be remembered that calculations of Brandt [5], which we have dis- cussed above, are valid for conventional superconductors only. There are no reasons to believe that the vortex core structure should be the same in two- gap superconductors or in superconductors with nodes in the order parameter. Furthermore, one may assume that the influence of the vortex core region on the distribution of the magnetic induction should be even stronger than in the case of conventional superconductors. This is why, if calculations of [5] or any other calculations based on the conventional GL theory are used for the analysis of µSR data collected in different magnetic fields, it would produce an unphysical λ(H) dependence. Although this result does not mean any spe- cial behavior of the magnetic field penetration depth, it should be considered as interesting. Indeed, if the conventional GL theory cannot describe the re- sults of µSR experiments and all other possibilities for this disagreement are excluded, 1 one may conclude that this superconductor is unconventional. Superconductors with d-pairing as well as two-gap superconductors are more complex than conventional ones. For instance, two lengths ξ0 and λ0 are insuf- ficient for their characterization and some additional information is needed. At present, there is no experimentally proven theory of the mixed state in uncon- ventional superconductors. In this sense, magnetic field dependences of muon relaxation rates cannot be interpreted quantitatively without some additional assumptions. At the same time, one can try to obtain F (B/Bc2) experimen- tally in order to compare results for different superconducting materials. Un- fortunately, concerning high-Tc superconductors, the Hc2(T ) curves are not yet reliably established. Interesting theoretical approaches for interpretation of the µSR experiments in the case of d-pairing was developed in [7,8,9,10,26,27,28,29]. In [26,27,28,29] was convincingly argued that because of the nodes of the order parameter, the electrodynamics of the mixed state becomes nonlocal. This nonlocality effectively increases the vortex core radius and changes the distribution of the magnetic induction around vortices (see Fig. 6 of Ref. [28]). If this effect is not taken into account, the magnetic field penetration depth evaluated from µSR experiments will be overestimated and magnetic field dependent. The distortion of the results is very clearly demonstrated in Fig. 4 of Ref. [28]. In order to correct the results, the function λeff(B)/λ0 was introduced [28]. Using this function, which is an analog of F (B/Bc2), one can evaluate the magnetic field penetration depth λ0. At the same time, in high κ superconductors and at low magnetic inductions, the total volume of vortex cores is small and contribution of vortex cores cannot considerably change the muon signal. In this case, one may use F = 0.061Φo for evaluation of λ0 also in unconventional superconductors. Because λ ∼ 1/ σ, the resulting error is not expected to be big. This means that, if an experimental σ(H) dependence is available, extrapolation of σ(H) (or 1/ 1 For instance, the traditional analysis cannot be used in the case of polycrystalline samples of anisotropic superconductors. to H = 0 gives more reliable values of λ0. 4 Analysis of experimental results In this section, in order to simplify notation, we shall use λ and ξ without indexes, having in mind the magnetic field penetration depth and the super- conducting coherence length as they are introduced in the GL theory. 4.1 RbOs2O6, Cd2Re2O7, PrOs4Sb12. Experimental σ(H) data for a polycrystalline sample of RbOs2O6 are shown in Fig. 2(a). Because RbOs2O6 is an isotropic superconductor, using of such samples is justified. This sample was investigated in [30] and experimental data were analyzed by employing of an interpolation formula proposed in [3]. Because this formula deviates significantly from more accurate numerical calculations (see Fig. 1), we reanalyze these data using calculations of Ref. [5]. As may be seen in Fig. 2(a), experimental data-points for H > 2 kOe can very well be fitted by the theoretical σ(B/Bc2) curve. This fit results in Hc2 = (67 ± 10) kOe and λ = (220 ± 5) nm (the value of λ = 260 nm was obtained in the original publiction). Because the value of Hc2 is obtained by the extrapolation of the σ(H) curve to σ = 0, the corresponding error margins are large. It is important to emphasize that Hc2, evaluated in such a way, is in agreement with the Hc2(T ) curve presented in [30]. This agreement together with sufficiently high quality of fitting strongly supports our analysis. We have chosen a high field part of the experimental σ(H) curve for the analysis because in higher magnetic fields F (B/Bc2) is independent of κ (see Fig. 1). In principle, analyzing the low field part of the curve, the value of κ may straightforwardly be evaluated. This, however, is not always feasible. As was already mentioned, the correct parameter is not H but the magnetic induction B. The value of B determines intervortex distances and all other characteristics of the mixed state. In magnetic fields H ≫ Hc1, the difference (H − B) ≪ H and one can use H instead of B. In low fields, however, the equilibrium value of B is considerably smaller thanH and the actual difference (H − B) depends on pinning and on the demagnetizing factor of the sample. Furthermore, in low fields, the magnetic induction is nonuniform throughout the sample if its shape is not ellipsoidal. In polycrystalline samples, the sit- uation complicates even further. Indeed, in such samples, some vortices may go along intergrain boundaries, which can significantly influence the magnetic induction distribution. This is the reason that we do not speculate on the low-field behavior of the σ(H) curve. 0.1 1 10 T = 1.6 K λ = (220 5) nm+— Hc2 = (67 10) kOe+— RbOs2O6 0 1 2 3 4 5 6 H (kOe) T = 0.1 K λ = (830 40) nm+— Hc2 = (5.75 1) kOe+— Cd2Re2O7 λ = (318 4) nm+— Hc2 = (21 4) kOe+— T = 0.1 K PrOs4Sb12 Fig. 2. σ(H) data for three different superconducting compounds. The solid lines represent the theoretical σ(B/Bc2) curve fitted to data-points. Only the data marked by closed symbols were used for fitting. The resulting values of λ and Hc2 are indicated in the figure. (a) RbOs2O6 sample studied in [30]. The vertical dashed line indicates the value of Hc1. (b) Cd2Re2O7 sample studied in [31]. The dashed line represents a linear approximation to a high field part of the theoretical σ(H) curve. (c) PrOs4Sb12 studied in [32]. Similar results for a Cd2Re2O7 sample studied in [31] are shown in Fig. 2(b). For the reasons explained above, we disregard the lowest field data-point. Again in this case, data can be very well fitted with the GL theory, providing Hc2 = (5.75±1) kOe in agreement with the original data (see [31]). The value of λ = (830 ± 40) nm is also close to the result λ = 750 nm of Ref. [31]. We also note that approximation of experimental σ(H) data-ponts by a linear dependence, as it was done in [31] and some other publications, is unjustified. As may be seen in Figs. 1 and 2(b), the theoretical σ(B) curves are not at all linear. Fig. 2(c) presents σ(H) data for a heavy-fermion superconductor PrOs4Sb12 [32]. We do not discuss here different features of this rather unusual supercon- ductor but limit ourselves to one simple question whether the σ(H) depen- dence for this compound can be described by the conventional GL theory. As may be seen in Fig. 2(c) (see also Fig. 4 of Ref. [32]), the values of σ(1kOe) and σ(2kOe) practically coincide. It was assumed in Ref. [32] that a change of vortex lattice symmetry or some other important changes of the vortex structure, which occur in magnetic fields aboveH = 1 kOe, may be responsible for such a behavior. This explanation seems to be plausible and we, as a precaution, do not use the highest field data-point in the analysis. The solid line in Fig. 2(c) represents the best fit of the theoretical σ(H) curve to the data collected in magnetic fields 0.2 kOe≤ H ≤ 1 kOe. Quite amazingly, the resulting value of Hc2 = 21 kOe practically coincides with Hc2 = 22.2 kOe obtained in Ref. [33] from resistivity measurements. The value of λ = (318±4) nm is also close to the result λ = 290 nm of the original publication. Table 1 RbOs2O6 (1.6K) Cd2Re2O7 (0.1K) PrOs4Sb12 (0.1K) Hc2 (kOe) 67 ± 10 5.75 ± 1 21.4 ± 4 κ 34± 1 35± 3 25.5 ± 2.5 λ (nm) 220 ± 5 830 ± 40 318 ± 4 ξ (nm) 7± 0.4 24± 3.5 12.5 ± 1.5 The main characteristics of the superconducting compounds, resulting from our analysis of the µSR data published in Refs. [30,31,32], are listed in Ta- ble 1. We emphasize that all parameters were evaluated by fitting of σ(H) data-points with the theoretical σ(B/Bc2) dependence calculated in [5]. In all cases, the values of Hc2 practically coincide with results of independent measurements. 4.2 CeRu2[18]. Calculation of σ(T,H) considered above is not the only way of analysis of µSR experiments in the mixed state of type-II superconductors. A different method was employed in Refs. [11,12,13,14,15,16,17,21,22]. In this approach, the dis- tribution of local fields (the Fourier transform of the muon precession signal) P (B) was directly analyzed by comparing with corresponding theoretical cal- culations. In real experiments, however, the P (B) line is usually different from theoretical predictions. This difference is expected. Indeed, the calculations are made for a perfect sample and for a perfect vortex lattice. All imperfections, which cannot be avoided in experiments, influence the P (B) curves. This is why, in order to approximate experimental data with theoretical calculations, some gaussian smearing factor is introduced. In such a way, satisfactory agree- ment between the theory and experiments can be achieved. This is justified if it is a priori known that the spatial distribution of the magnetic induction around vortex lines is in agreement with the theory, which is used for the cal- culations. If it is not the case, introducing of additional Gaussian relaxation may mask the disagreement and provide misleading results. For some type-II superconductors, the P (B) curves for different values of the applied magnetic field are available in the literature. This allows to calculate σ(H) and to employ the same kind of the analysis as was used above. Below we present the results of such analysis for single crystals of CeRu2 and vanadium. Fig. 3(a) shows σ as a function of H for a CeRu2 sample experimentally investigated in Ref. [18]. The difference to the results displayed in Fig. 2 is that broadening of the P (B) line resulting from other sources of field inhomogeneity was not accounted for. In this case σ, evaluated from µSR experiments may be written as σ2sc + σ bg, (9) where σsc and σbg are the mixed state and background contributions, respec- tively. As may be seen in [18], σbg is not small and cannot be evaluated from the data presented in the publication with sufficient accuracy. This is the reason that we introduce σbg as an additional adjustable parameter. Because experimental data are insufficient for evaluation of λ, Hc2 and σbg together, we take the value of Hc2 from the original publication. 2 We also note that H = 40 kOe is the only data point corresponding to the peak-effect region 2 There is some confusion in Ref. [18]. While Fig. 2 provides Hc2 = 50 kOe, the value of Hc2 evaluated from Fig. 1 is closer to 45 kOe. Taking into account that the resulting λ is not very sensitive to some variation of the assumed Hc2 value, we have chosen Hc2 = 47.5 kOe for the analysis. 0 10 20 30 40 50 H (kOe) 1 10 100 CeRu2 λ = (167 3) nm+— T = 2 K Fig. 3. σ (upper panel) and σsc = σ2 − σ2 (lower panel) versus H for a CeRu2 sample studied in [18]. The solid lines represent the theoretical curves calculated as explained in the text. The chosen value of Hc2 = 47.5 kOe is indicated in the figures by vertical lines. Only data-points shown by closed symbols were used for evaluation of σbg and λ. (see Fig. 1 in [18]). Because the origin of this effect is not yet established, we exclude the corresponding data point from the analysis. As may be seen in Fig. 3(a), all data-points for H ≤ 30 kOe can be fairly well fitted by the theory, providing λ = (167 ± 3) nm and σbg = (8.4 ± 1) G. The magnetic field dependence of σsc = σ2 − σ2bg is shown in Fig. 3(b). Using the σsc(H) plot presented in Fig. 3(b), we can calculate λ for each of the data-points. Such calculations were made for two different values of Hc2 and they are presented in Fig. 4. As was expected, the absolute value of λ is practically independent of the chosen value of Hc2. One can also see that, contrary to claims of Ref. [18], there is no any noticeable dependence of λ on 0 10 20 30 H (kOe) Hc2 = 45 kOe Hc2 = 50 kOe CeRu2 T = 2 K Fig. 4. λ as a function of H calculated assuming Hc2 = 45 kOe and Hc2 = 50 kOe. At the same time, the value of σ(40kOe) deviates quite significantly from the theoretical curve (see Fig. 3). 3 If this deviation is not an experimental error, it means that the distribution of the magnetic induction in the case of the peak-effect is rather different in comparison with the conventional mixed state. However, one should be extremely careful with such conclusions. In the case of the peak-effect, the value of σ is rather sensitive even to insignificant variations of H (see Fig. 13(d) in [18]). In this situation, ∼ 10−5H change of the external field may explain the difference between σ(40kOe) and the theoretical curve. 4.3 Vanadium single crystal [22] We discuss experimental data of Ref. [22] in some detail in order to demon- strate general problems of interpretation of µSR experiments in the case of low-κ superconductors. We also discuss some typical errors that can be found in the literature. Vanadium is one of the very few pure metals, which displays type-II super- conductivity at all temperatures. Superconducting characteristics of vanadium have been rather well investigated (see, for instance, [34,35,36,37]). Although vanadium has a cubic (bcc) structure, Hc2 depends on the orientation of the applied magnetic field [35]. According to [35], the value of Hc2 along [111] direction is approximately 10% higher than that for [001]. 3 σ(40kOe) is larger than the corresponding theoretical value. The higher σ means smaller λ. This conclusion is just opposite to that made in the original publication. T (K) Ref. 34, H || [110] Ref. 35, H || [110] Ref. 35, H || [111] 0 1 2 3 4 5 Ref. 36, H || [491] Ref. 37, H || [491] Vanadium Fig. 5. Hc2(T ) for three different orientations of vanadium single crystals The dependences Hc2(T ) for three different orientations are shown in Fig. 5. As may be seen, results of different studies are in excellent agreement. The value of Tc may be evaluated as Tc = (5.40± 0.05) K [34,35,36,37]. While Hc2 is orientation dependent, its normalized temperature dependence is practically universal [35]. This is illustrated in Fig. 6 where Hc2(T )/(TcdHc2/dT )T=Tc is plotted versus T/Tc. The results of different works, presented in such a way, nicely collapse onto a single curve. We note that the temperature dependence of Hc2 is somewhat different from predictions of Helfand-Werthamer (HW) theory [38]. Vanadium is a low-κ material with κ(0K) = 1.5 for a pure sample investigated in [34]. This circumstance adds some peculiarities to the mixed state and its description. First, the condition H ≪ Hc2 is not satisfied even in magnetic fields down to Hc1, i.e., the London approach, in which vortices are considered as independent, is inapplicable in the entire range of magnetic fields. In this situation, the actual magnetic induction distribution in the sample strongly depends on spatial variations of the order parameter, and the accuracy of the 0.0 0.2 0.4 0.6 0.8 1.0 t = T/Tc HW theory Ref. 36, H || [491] Hc2(0K) = 2.83 kOe Ref. 37, H || [491] Hc2(0K) = 2..78 kOe Ref. 34, H || [110] Hc2(0K) = 3 kOe Ref. 35, H || [110] Hc2(0K) = 3 kOe Ref. 35, H || [111] Hc2(0K) = 3.14 kOe Vanadium Fig. 6. The normalized upper critical field of vanadium single crystals as a function of t = T/Tc. The solid line represents the HW theory [38]. corresponding calculations plays a crucial role. Second, if κ ∼ 1, the condition λ ≫ ξP (ξP = 0.74ξ(0K) is the Pippard coherence length) is not satisfied at low temperatures and electrodynamics become nonlocal, i.e., quantitative applicability of the GL theory at T ≪ Tc is questionable. Furthermore, the results of Refs. [34,35,36] clearly demonstrate that superconducting properties of vanadium at T ≪ Tc cannot be described by the GL theory and a more complex approach is necessary. At the same time, as we argue below, experi- mental σ(H) curves are close to theoretical predictions of Brandt [5] and can be used for evaluation of the magnetic field penetration depth. The values of σ were calculated using the P (B) curves presented in Ref. [22]. Clearly visible peaks arising from muons stopped outside the sample were approximated by Gaussians and subtracted from the data. The resulting values of σ are plotted in Fig. 7(a). 4 Our analysis gives λ = (49 ± 1.5) nm and Hc2 = (3.8 ± 0.15) kOe. The estimation of λ is in very good agreement, with λ = 50 nm, which may be calculated from Hc2(T ) and κ(T ) curves experimentally measured in [34]. The value of Hc2 evaluated above is just 10% below of Hc2 = 4.2 kOe provided in the original publication [22]. We note that there are no estimations of experimental uncertainty for Hc2 in [22]. One can assume that Hc2(0.02K) was obtained by extrapolation of higher temperature data and the corresponding error margins are considerable. We also note that Hc2(0.02K) = 4.2 kOe is well above earlier results (see Fig. 5). Partly this difference may be explained 4 Because the demagnetizing factor of the sample is close to 1, one can safely assume H = B for all considered magnetic fields. 2 3 4 Vanadium single crystal, H [111]|| T = 0.02 K Hc2 = (3.8 0.15) kOe+— λ = (49 1.5) µm+— H (kOe) Fig. 7. (a) σ versus H for a vanadium single crystal studied in [22]. The solid line represent a fit with the Brandt theory [5]. The resulting values of λ and Hc2 are indicated in the figure. by the fact that the sample that we are discussing here was substantially less pure than those of Refs. [34,35,36,37]. However, such a significant increase of Hc2 seems to be unlikely. Furthermore, as we show below, the value of Hc2, evaluated by the analysis of the temperature dependence of muon relaxation rates, agrees better with the estimate of Hc2 presented in Fig. 7 than with the value given in [22]. As was already mentioned, experimental results presented in [34] allow for evaluation both Hc2(T ) and λ(T ) dependences. Using these data, we can also obtain the expected value of σ for any magnetic field and temperature. Such results for H = 1.6 kOe are plotted in Fig. 8 for comparison with the µSR data of Ref. [22]. As may be seen in Fig. 8, the two σ(T ) curves are similar. In order to emphasize this similarity, we approximate both data sets by the same functional dependence (see Fig. 8). Considering the results presented in Fig. 8, one can conclude that the sample investigated in [22] has indeed a somewhat higher Hc2. The value of Tc(H) may straightforwardly be evaluated from σ(T ) data as the value of T , at which σ vanishes. Such estimate gives Hc2 = 1.6 kOe at T = 3.38 K. 5 Using this value of Hc2(3.38K) and the normalized Hc2(T ) curve presented in Fig. 6, we can evaluate Hc2(0) = (3.4 ± 0.25) kOe, which is in reasonable agreement with the estimate made from the analysis of σ(H) data (see Fig. 7). 6 5 This is well below the value of Tc(1.6kOe) = 3.65 K provided in [22]. 6 We use Tc = 5.4 K, as it follows from earlier measurements (the same value is provided in reference data of Goodfellow Ltd.), assuming that Tc = 5.2 K given in [22], is a misprint. If, however, we except Tc = 5.2 K, Hc2(0) = (3.65 ± 0.2) kOe, 0 1 2 3 λ according to Ref. 34 σ according to Ref. 34 σ according to Ref. 22 T (K) 2 (µ Vanadium Fig. 8. Temperature dependence of σ(1.6kOe) for a vanadium single crystal studied in [22]. The temperature dependences of σ and λ−2 (right y-axis), evaluated from experimental data of [34], are shown for comparison. The solid and the dashed lines are the guides to the eye. The dotted line (f ′(T )) is obtained by scaling of the solid line (f(T )), i.e., f ′(T ) = 1.29f(1.09T ). Our results presented in this section are rather different from the conclusions of Ref. [22]. First and foremost, as it is clearly demonstrated in Fig. 7, the magnetic field dependence of σ is very close to the result of the GL theory. Our value of λ for T = 0, which is in excellent agreement with measurements of Ref. [34], is about 1.5 times smaller than the result of [22] for H = 1.6 kOe. The temperature dependence of 1/λ2 calculated according to Ref. [34] is also plotted in Fig. 8. As may be seen, while σ vanishes at T = 3.12 K, the value of λ(3.12K) remains almost the same as at T = 0. In other words, σ vanishes at Tc(H) not because of the divergence of λ but because the coefficient F in Eq. (7) vanishes at this temperature. It seems important to emphasize that Fig. 10 of Ref. [22] is based on an obvious misunderstanding. 7 There exists no theory that predicts divergence of λ at Tc(H). The reference on theoretical calculations of Mühlschlegel [39], given in [22], is misleading. Indeed, the thermodynamical consideration of Ref. [39] is based on the fact that the difference between free energies of the normal and the superconducting states per unit of volume can be written as H2c /8π. The same difference can also be written as ncp∆ where ncp is be the density of Cooper pairs and ∆ is the equilibrium (zero-field) superconducting energy gap. Using this, one obtains ncp(T ) and λ(T ). Nothing in this consideration can be used to justify Fig. 10 of Ref. [22]. We also note that the temperature variation ofHc2 should be taken into account if the temperature dependence of λ is evaluated from measurements in fixed magnetic fields. It can be neglected the value practically coinciding with the result of Fig. 7. 7 A similar plot one can also find in [30] only if the condition H ≪ Hc2 is satisfied at all temperatures. In fact, good agreement with the theory, demonstrated in Figs. 7 and 8, is rather surprising. As was already mentioned, it is well established that vana- dium does not obey the GL theory at T ≪ Tc [34,35,36].The most probable is that the distribution of the magnetic induction in the sample (P (B)) is different from theoretical predictions, while σ, as a more integral character- istic of this distribution, remains practically the same. This assumption can also explain the difference between our results and those of Ref. [22]. Indeed, the analysis of P (B) functions, carried out in [22], resulted in an unphysical magnetic field dependence of λ, which clearly demonstrates the inapplicability of the GL theory to this analysis. It is important to underline that, although the distribution of the magnetic induction in the sample cannot be described by the GL theory in low-κ type-II superconductors at T ≪ Tc, the σ(H) curves can still be used for evaluation of the magnetic field penetration depth, as it is proven by a very close agreement between our value of λ(0K) = 49± 1.5 nm and 50 nm calculated from results of Ref. [34]. We also note that at temperatures closer to Tc the GL theory should be applicable and both analyses should result in the same λ values. 4.4 YNi2B2C [14] In order to demonstrate that in some cases the GL theory cannot describe µSR data, we consider a study of a borocarbide superconductor YNi2B2C [14]. Rare-earth nickel borocarbide superconductors attracted a lot of attention dur- ing the past decade. Already the very first studies of YNi2B2C demonstrated a pronounced positive curvature of the Hc2(T ) curve, indicating unconventional superconductivity [40,41]. Similar conclusions were made from specific heat data [40,41]. Although YNi2B2C has been extensively studied, the nature of this unconventionality is still under discussion. While Refs. [42,43,44,45,46,47] provide evidences of point nodes in the superconducting gap function, other works point out on multiband superconductivity [48,49,50,51]. The distinction between these two possibilities is sometimes difficult to make. For instance, as was recently demonstrated, specific heat data may be fitted equally well by nodal and two-gap models [52]. Experimental results of Ref. [14] are plotted in Fig. 9 as σ versus H . The value of Hc2(3K) = 70 kOe for this particular crystal is given in [14]. As may clearly be seen in Fig. 9, σ(H) data-points cannot be fitted with the theory if the entire range of magnetic fields is considered. Because there are sufficient experimental evidences that YNi2B2C is an unconventional superconductor [42,43,44,45,46,47,48,49,50,51,52], disagreement between the GL theory and 10 100 H (kOe) λ = 123 nm YNi2B2C T = 3 K λ = 90 nm Fig. 9. σ as a function of H for a YNi2B2C sample studied in Ref. [14]. The solid lines are the theoretical σ(H) dependencies calculated for Hc2 = 70 kOe and for two different values of λ. experimental data is expected. We also note that in YNi2B2C a transition from a triangular to a square vortex lattice was observed [53,54]. However, because this transition occurs in lower magnetic fields, it cannot have any influence on µSR data presented in Fig. 9. 8 While the totality of data cannot be fitted with the theory, both high-field and low-fild results may amazingly well be approximated with two different theo- retical curves, corresponding to two different λ values (Fig. 9). Unfortunately, insufficient number of data-points does not allow to make unambiguous con- clusions on this matter, however, if this behavior will be confirmed by a more detailed study, it may be considered as a rather interesting result, indicating two gap superconductivity. In low magnetic fields H ≪ Hc2, most of muons stopped outside vortex cores, i.e., the magnetic induction distribution in vortex core regions is not very important for µSR data. In this case, the difference between conventional and two gap superconductors should not be significant and the resulting σ(H) curves can be close in these two cases. In higher magnetic fields, as it was established in studies of MgB2, supercon- ductivity in one of two bands is completely suppressed and the superconduc- tor behaves itself as a one gape superconductor but with a smaller number of Cooper pairs [55,56,57]. This can explain the fact that the two data points for 8 As was demonstrated in [10], although the magnetic induction distributions for square and triangular lattices are quite different, σ(H) remains practically the same in both cases. H ≥ 30 kOe follow a standard theoretical curve with a higher value of λ (see Fig. 9). The quantity 1/λ2 is proportional to the density of Cooper pairs. If two gap superconductivity is assumed, the values of the magnetic field penetration lengths, evaluated from low- and high-field data, allows evaluation of relative weights of two superconducting bands. Such estimate gives 54% and 46% for stronger and weaker gaps, respectively. These values are noticeably different from the result 71% and 29% obtained in Ref. [52]. At present, however, it is too early to discuss such differences. Two data points in the high-field range part of the curve (see Fig. 9) are clearly insufficient in order to make any definite conclusion about superconductivity in YNi2B2C. 5 Conclusion. In this work, we applied numerical calculations of Brandt [5] for the analysis of µSR experiments carried out in the mixed state of several superconducting compounds. It turned out that this approach may serve as a very powerful tool for the interpretation of µSR experiments. If the magnetic field dependences of muon depolarization rates are available, not only λ but also Hc2 can reliably be evaluated. We show that in the most of considered cases the magnetic field dependences of σ may very well be described by a single and temperature independent λ. In contrast to approximate analytical models, Ref. [5] provides precise numer- ical solutions of 2-dimensional GL equations for different values of κ (0.85 ≤ κ ≤ 200) and for magnetic fields ranging from Hc1 to Hc2. Using these so- lutions, different characteristics of the mixed state, including the σ(B/Bc2) dependences for various κ values, were calculated. As well as we are aware, these calculations provide the best available description of the magnetic induc- tion distribution in the mixed state of conventional type-II superconductors. We also note that numerical calculations of σ(H) are available since 1997 (see Fig. 3 in [4]). In spite of this, for some mysterious reasons, these theoretical calculations have practically never been used for the analysis of µSR data. We also argued that the magnetic field dependence of λ can never be obtained from analyses of experimental data collected in the mixed state. Indeed, be- cause the local value of λ is inversely proportional to the absolute value of the superconducting order parameter, one cannot introduce any single value of λ in the mixed state. Calculations of Brandt [5] represent the conventional GL theory and their validity for the description of unconventional superconductors is question- able. In fact, there are no reasons to believe that the conventional GL theory can quantitatively describe either two-gap superconductors or superconduc- tors with nodes of the order parameter and one should expect disagreement between Brandt’s theory and experimental results in the case of unconven- tional superconductors, as it is demonstrated in Fig. 9. We demonstrated that in conventional superconductors, the results of µSR experiments may be used for the evaluation of both λ and Hc2. If applicability of the conventional GL theory is questionable, the knowledge of Hc2 is of primary importance. Disagreement between the values of Hc2 resulting from µSR data and independent measurements may be considered as convincing evidence that this particular superconductor is unconventional. We are grateful to R. 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Introduction Conventional superconductors Unconventional superconductors Analysis of experimental results RbOs2O6, Cd2Re2O7, PrOs4Sb12. CeRu2kado. Vanadium single crystal son06 YNi2B2C ohi Conclusion. References

0704.1269

Phase Transitions in the Coloring of Random Graphs

Phase Transitions in the Coloring of Random Graphs Lenka Zdeborová1 and Florent Krza̧ka la2 LPTMS, UMR 8626 CNRS et Univ. Paris-Sud, 91405 Orsay CEDEX, France PCT, UMR Gulliver 7083 CNRS-ESPCI, 10 rue Vauquelin, 75231 Paris, France We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study of the space of proper colorings (solutions). We show that for a fixed number of colors and as the average vertex degree (number of constraints) increases, the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses. First, at the clustering transition, the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard. Afterward, the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one. Another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state. Eventually, above the coloring threshold, no more solutions are available. We compute all the critical connectivities for Erdős-Rényi and regular random graphs and determine their asymptotic values for large number of colors. Finally, we discuss the algorithmic consequences of our findings. We argue that the onset of computational hardness is not associated with the clustering transition and we suggest instead that the freezing transition might be the relevant phenomenon. We also discuss the performance of a simple local Walk-COL algorithm and of the belief propagation algorithm in the light of our results. PACS numbers: 89.20.Ff, 75.10.Nr, 05.70.Fh, 02.70.-c I. INTRODUCTION Graph coloring is a famous yet basic problem in combinatorics. Given a graph and q colors, the problem consists in coloring the vertices in such a way that no connected vertices have the same color [1]. The celebrated four- colors theorem assures that this is always possible for planar graphs using only four colors [2]. For general graphs, however, the problem can be extremely hard to solve and is known to be NP-complete [3], so that it is widely believed that no algorithm can decide in a polynomial time (with respect to the size of the graph) if a given arbitrary instance is colorable or not. Indeed, the problem is often taken as a benchmark for the evaluation of the performance of algorithms in computer science. It has also important practical application as timetabling, scheduling, register allocation in compilers or frequency assignment in mobile radios. In this paper, we study colorings of sparse random graphs [4, 5]. Random graphs are one of the most fundamental source of challenging problems in graph theory since the seminal work of Erdős and Rényi [6] in 1959. Concerning the coloring problem, a crucial observation was made by focusing on typical instances drawn from the ensemble of random graphs with a given average vertex connectivity c, as c increases a threshold phenomenon is observed. Bellow a critical value cs a proper coloring of the graph with q colors exists with a probability going to one in the large size limit, while beyond cs it does not exist in the same sense. This sharp transition also appears in other Constraint Satisfaction Problems (CSPs) such as the satisfiability of Boolean formulae [1]. The existence of the sharp COLorable/UNCOLorable (COL/UNCOL) transition was partially [85] proven in [7], and computing rigorously its precise location is a major open problem in graph theory. Many upper and lower bounds were established [8, 9, 10, 11, 12, 13, 14, 15] for Erdős-Rényi and regular random graphs. It was also observed empirically [16, 17] that deciding colorability becomes on average much harder near to the coloring threshold cs than far away from it. This onset of computational hardness cannot be explained only by the simple fact that near to the colorable threshold the number of proper colorings is small [18]. Some progress in the theoretical understanding has been done by the analysis of search algorithms [19, 20]. For the coloring problem, it was proven [21] that a simple algorithm q-colors almost surely in linear time random graphs of average connectivity c ≤ q log q − 3q/2 for all q ≥ 3 (see [21] for references on previous works). For 3-coloring the best algorithmic lower bound is c = 4.03 [10]. An important and interesting open question [22] is the existence of an ǫ > 0 and of a polynomial algorithm which q-colors almost surely a random graph of connectivity c = (1 + ε)q log q for arbitrary large q. The sharp coloring threshold and the onset of hardness in its vicinity has also triggered a lot of interest within the http://arXiv.org/abs/0704.1269v2 COL/UNCOLRigidityClustering Condensation cd cc cr cs Average connectivity FIG. 1: A sketch of the set of solutions when the average connectivity (degree) is increased. At low connectivities (on the left), all solutions are in a single cluster. For larger c, clusters of solutions appear but the single giant cluster still exists and dominates the measure. At the dynamic/clustering transition cd, the phase space slits in an exponential number of clusters. At the condensation/Kauzmann transition cc, the measure condenses over the largest of them. Finally, no solutions exist above the COL/UNCOL transition cs. The rigidity/freezing transition cr (which might come before or after the condensation transition) takes place when the dominating clusters start to contain frozen variables (dominating clusters is a minimal set of clusters such that it covers almost all proper colorings). The clusters containing frozen variables are colored in black and those which do not are colored in grey. statistical physics community following the discovery of a close relation between constraint satisfaction problems and spin glasses [23, 24]. In physical terms, solving a CSP consists in finding groundstates of zero energy. The limit of infinitely large graphs corresponds to the thermodynamic limit. In the case of the q−coloring problem, for instance, one studies the zero temperature behavior of the anti-ferromagnetic Potts model [25]. Using this correspondence, a powerful heuristic tool called the cavity method has been developed [24, 26, 27, 28]; it allows an exact analytical study of the CSPs on sparse random graphs. Unfortunately, some pieces are still missing to make the cavity method a rigorous tool although many of its results were rigorously proven. The cavity method is equivalent to the famous replica method [29] (and unfortunately for clarity, it has also inherited some of its notations, as we shall see). The cavity method and the statistical physics approach have been used to study the q-coloring of random graphs in [30, 31, 32]. The coloring threshold cs was calculated [30], the self-consistency of the solution checked [31] and the large q asymptotics of the coloring threshold computed [31]. These results are believed to be exact although proving their validity rigorously remains a major subject in the field. Nevertheless —as the results obtained for the random satisfiability problem [28, 33, 34]— they agree perfectly with rigorous mathematical bounds [8, 9, 10, 11, 12, 13, 15], and with numerical simulations. The coloring threshold is thus fairly well understood, at least at the level of cavity method. A maybe more interesting outcome of the statistical physics analysis of the CSPs was the identification of a new transition which concerns the structure of the set of solutions, and that appears before the coloring threshold [24, 28, 35]: while at low connectivities all solutions are in a single pure state (cluster) [86], the set of solutions splits in an exponential number of different states (clusters) at a connectivity strictly smaller than cs. Roughly speaking, clusters are groups of near-by solutions that are in some sense disconnected from each other. Recently, the existence of the clustered phase was proven rigorously in some cases for the satisfiability problem [36, 37]. A major step was made by applying the cavity equations on a single instance: this led to the development of a very efficient message- passing algorithm called Survey Propagation (SP) that was originally used for the satisfaction problem in [24] and later adapted for the coloring problem in [30]. Survey propagation allows one to find solutions of large random instances even in the clustered phase and very near to the coloring threshold. Despite all this success, the cavity description of the clustered phase was not complete in many aspects, and a first improvement has been made with the introduction of a refined zero temperature cavity formalism that allows a more detailed description of the geometrical properties of the clusters [38, 39]. We pursue in this direction and give for the first time a detailed characterization of the structure of the set of solutions. We observed in particular that the clustering threshold was not correctly computed, that other important transitions were overlooked and the global picture was mixed up. The corrected picture that we describe in this paper is the following: when the connectivity is increased, the set of solutions undergoes several phase transitions similar to those observed in mean field structural glasses (we sketch these successive transitions in fig. 1). First, the phase space of solutions decomposes into an exponential number of states which are entropically negligible with respect to one large cluster. Then, at the clustering threshold cd, even this large state decomposes into an exponential number of smaller states. Subsequently, above the condensation threshold cc, most of the solutions are found in a finite number of the largest states. Eventually, the connectivity cs is reached beyond which no more solutions exist. Another important transition, that we refer to as rigidity, takes place at cr when a finite fraction of frozen variables appears inside the dominant pure states (those containing almost all the solutions). All those transitions are sharp and we computed the values of the corresponding critical connectivities. A nontrivial ergodicity breaking takes place at the clustering transition, in consequence uniform sampling of solutions becomes hard. On the other hand, clustering itself is not responsible for the hardness of finding a solution. Moreover, until the condensation transition many results obtained by neglecting the clustering effect are correct. In particular for all c < cc: i) the number of solutions is correctly given by the annealed entropy (and, for general CSP, by the replica symmetric entropy), and ii) simple message passing algorithm such as Belief Propagation (BP) [40, 41] converges to a set of exact marginals (i.e. the probability that a given node takes a given color). In consequence we can use BP plus a decimation-like strategy to find proper colorings on a given graph. Finally we give some arguments to explain why the rigidity transition is a better candidate for the onset of computational hardness in finding solutions. Our results are obtained within the one-step replica symmetry breaking approach, and we believe (and argue partially in section IV), that our results would not be modified by considering further steps of RSB (as opposed to previous conclusions [31]). A shorter and partial version of our results, together with a study of similar issues in the satisfiability problem, was already published in [42]. We refer to [43] for a detailed discussion of the satisfiability problem. The paper is organized as follows: In section II, we present the model. In section III, we introduce the cavity formalism at the so-called replica symmetric level, and discuss in detail why and where this approach fails. In section IV we take into account the existence of clusters of solutions and employ the so called one-step replica-symmetry breaking formalism to describe the properties of clusters. The results for several ensembles of random graphs are then presented in section V. We finally discuss the algorithmic implications of our findings in section VI and conclude by a general discussion. Some appendixes with detailed computations complete the paper. II. THE MODEL A. Definition of the model For the statistical physics analysis of the q−coloring problem [30, 32, 44] we consider a Potts [25] spin model with anti-ferromagnetic interactions where each variable si (spin, node, vertex) is in one of the q different states (colors) s = 1, . . . , q. Consider a graph G = (V , E) defined by its vertices V = {1, . . . , N} and edges (i, j) ∈ E which connect pairs of vertices i, j ∈ V ; we write the Hamiltonian as H({s}) = (i,j)∈E δ(si, sj) . (1) With this choice there is no energy contribution for neighbors with different colors, but a positive contribution otherwise. The ground state energy is thus zero if and only if the graph is q-colorable. The Hamiltonian leads to a Gibbs measure [45] over configurations (where β is the inverse temperature) : µ({s}) = 1 e−βH({s}) , (2) In the zero temperature limit, where β → ∞, this measure becomes uniform over all the proper colorings of the graph. B. Ensembles of Random Graphs We will consider ensembles of graphs that are given by a degree distribution Q(k). The required property of Q(k) is that its parameters should not depend on the size of the graph. All the analytical results will concern only very large sparse graphs (N → ∞). Provided the second and higher moments of Q(k) do not diverge, such graphs are locally tree-like in this limit [4, 5]. More precisely, call a d-neighborhood of a node i the set of nodes which are at distance at most d from i. For d arbitrary but finite the d-neighborhood is almost surely a tree graph when N → ∞. This property is connected with the fact that the length ℓ of the shortest loops (up to a finite number of them) scales with the graph size as ℓ ∼ log(N). We will consider the two following canonical degree distribution functions: (i) Uniform degree distribution, Q(k) = δ(k− c), corresponding to the c-regular random graphs, where every vertex has exactly c neighbors. (ii) Poissonian degree distribution, Q(k) = e−cck/k!, corresponding to the Erdős and Rényi (ER) random graphs [6]. A simple way to generate graphs with N vertices from this ensemble is to consider that each link is present with probability c/(N − 1). The binomial degree distribution converges to the Poissonian in the large size limit. It will turn out that the cavity technics simplify considerably for the regular graphs. However, ER graphs have the advantage that their average connectivity is a real number that can be continuously tuned, which is obviously very convenient when one wants to study phase transitions. It is thus useful to introduce a third ensemble, which still has the computational advantage of regular graphs, but that at the same time gives more freedom to vary the connectivity: (iii) bi-regular random graphs, where nodes with connectivity c1 are all connected to nodes with connectivity c2, and vice-versa. There are thus two sets of nodes with degree distributions Q(k) = δ(k − c1) and Q(k) = δ(k − c2). Notice that these graphs are bipartite by definition and therefore have always a trivial 2-coloring which we will have to dismiss in the following. This can be easily done within the cavity formalism (it is equivalent to neglecting the ordered “crystal” phase in glass models [46]). In the first two cases, the parameter c plays the role of the average connectivity, c = kQ(k). In the cavity approach, a very important quantity is the excess degree distribution Q1(k), i.e. the distribution of the number of neighbors, different from j, of a vertex i adjacent to a random edge (ij): Q1(k) = (k + 1)Q(k + 1) . (3) This distribution remains Poissonian for Erdős-Rényi graphs, whereas the excess degree is equal to c− 1 in the case of regular graphs. In the bi-regular case, there are two sets of nodes with excess degrees c1 − 1 and c2 − 1. III. THE CAVITY FORMALISM AT THE REPLICA SYMMETRIC LEVEL We start by reviewing the replica symmetric (RS) version of the cavity method [26, 27] and its implications for the coloring problem. In the last part of the section we show when, and why, the RS approach fails. A. The replica symmetric cavity equations The coloring problem on a tree is solved exactly by an iterative method called the belief propagation algorithm [40] (note some boundary conditions have to be imposed, otherwise a tree is always 2-colorable) that is equivalent to the replica symmetric cavity method [41]. At this level, the method is in fact a classical tool in statistical physics to deal with that tree structure that dates back to the original ideas of Bethe, Peierls and Onsager [47]. It allows one to compute the marginal probabilities that a given node takes a given color and, in the language of statistical physics, observables like energy, entropy, average magnetization, etc. The applicability of the method goes however beyond tree graphs and we will discuss when it is correct for random locally tree-like graphs in section III C. ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ���k k2 ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ������ ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ����� ������������ FIG. 2: Iterative construction of a tree by adding a new Potts spin. Let us now describe the RS cavity equations. Denote ψi→jsi the probability that the spin i has color si when the edge (ij) is not present and consider the iterative construction of a tree in fig. 2. The probability follows a recursion: ψi→jsi = k∈i−j e−βδsiskψk→isk = k∈i−j 1 − (1 − e−β)ψk→isi . (4) where Z 0 is a normalization constant (the cavity partition sum) and β the inverse temperature. The notation k ∈ i − j means the set of neighbors of i except j. The normalization Zi→j0 is related to the free energy shift after the addition of the node i and the edges around it, except (ij), as 0 = e −β∆F i→j . (5) In the same way, the free energy shift after the addition of the node i and all the edges around it is e−β∆F = Zi0 = 1 − (1 − e−β)ψk→is . (6) The Zi0 is at the same time the normalization of the total probability that a node i takes color si (the marginal of i): ψisi = 1 − (1 − e−β)ψk→isi . (7) The free energy shift after the addition of an edge (ij) is e−β∆F 0 = 1 − (1 − e−β) ψj→is ψ s . (8) The free energy density in the thermodynamic limit is then given by (see for instance [26]) f(β) = ∆F i − ∆F ij  . (9) Note that this relation for the free energy is variational, i.e. that if one differentiates with respect to β, then only the explicit dependence needs to be considered (see [26] for details). The energy (the number of contradictions) and the corresponding entropy (the logarithm of the number of solutions with a given energy) densities can be then computed from the Legendre transform − βf(β) = −βe+ s(e) , (10) were f = F/N , e = E/B and s = S/N are intensive variables. The learned reader will notice that in some previous works using the cavity method [30, 31], these equations were often written for a different object than the probabilities ψ. Instead, the so-called cavity magnetic fields h and biases u where considered. The two approaches are related via −βhi→jsi ≡ Zi→j0 ψi→jsi ≡ k∈i−j −βuk→isi . (11) Strictly speaking the ψ are “cavity probabilities” while h and u are “cavity fields”, however ψ are sometimes also referred to as cavity fields (or messages) in the literature, and the reader will thus forgive us if we do so. Note that each of the two notations is suitable for performing the zero temperature limit in a different way: in the first one we fix the energy zero and we obtain the zero temperature BP recursion which gives the marginal probabilities for each variable, while in the second case, we obtain a simpler recursion called Warning Propagation (49) that deals with the energetic contributions but neglects the entropic ones [28, 30]. This is the origin of the discrepancy between the RS results of refs. [32] and [30]. As we shall see, the differences between these two limits will be an important point in the paper. B. Average over the ensemble of graphs and the RS solution To compute (quenched) averages of observables over the considered ensemble of random graphs given by the degree distribution Q(k) we need to solve the self-consistent cavity functional equation P(ψ) = Q1(k) dψiP(ψi)δ(ψ −F({ψi})), (12) where Q1(k) is the distribution of the number of neighbors given that there is already one neighbor, and the function F({ψi}) is given by eq.(4). Beware that ψ here is a q-components vector while we omit the vector notation to lighten the reading. This equation is quite complicated since the order parameter is nontrivial, but we can solve it numerically using the population dynamics method described in [26, 27]. Throughout this paper we will search only for the color symmetric functions P(ψ), i.e. invariant under permutation of colors. Clearly with this assumption we might miss some solutions of (12). Consider for example q > 2 colors and the ensemble of random bi-regular graphs. Since every bipartite graph is 2-colorable there are q(q−1)/2 corresponding color asymmetric solutions for P(ψ). For the ensemble of random graphs considered here, we later argue that this assumption is however justified. Another important observation is that for regular graphs the equation (12) crucially simplifies: the solution factor- izes [48] in the sense that the order parameter ψ is the same for each edge. This is due to the fact that, locally, every edge in such a regular graph has the same environment. All edges are therefore equivalent and thus the distribution P(ψ) has to be a delta function. For the bi-regular graphs, the solution of (12) also factorizes, but the two sets of nodes of connectivity c1 and c2 (each of them being connected to the other) have to be considered separately. It is immediate to observe that P(ψ) = δ(ψ−1/q) (i.e. each of the q components of each cavity field ψ equals 1/q), is always a solution of (12). By analogy with magnetic systems we shall call this solution paramagnetic. Numerically, we do not find any other solution in the colorable phase. For regular random graphs the paramagnetic solution is actually the only factorized one. The number of proper colorings predicted by the RS approach is thus easy to compute. Since all messages are of the type P(ψ) = δ(ψ − 1/q), the free energy density simply reads − βfRS = log q + 1 − 1 − e . (13) The entropy density at zero temperature thus follows sRS = log q + 1 − 1 . (14) It coincides precisely with the annealed (first moment) entropy. We will see in the following that, surprisingly, the validity of this formula goes well beyond the RS phase (actually until the so-called condensation transition). C. Validity conditions of the replica symmetric solution We used the main assumption of the replica symmetric approach when we wrote eq. (4): we supposed that the cavity probabilities ψk→isi for the neighbors k of the node i are “sufficiently” independent in absence of the node i, because only then the joint probability factorizes. This assumption would be true if the lattice were a tree with non-correlated boundary conditions, but loops, or correlations in the boundaries, may create correlations between the neighbors of node i (in absence of i) and the RS cavity assumption might thus cease to be valid in a general graph. The aim here is to precise and quantify this statement both from a rigorous and heuristic point of view. 1. The Gibbs measure uniqueness condition Proving rigorously the correctness of the RS cavity assumption for random graphs is a crucial step that has not yet been successfully overcome in most cases. The only success so far was obtained by proving a far too strong condition: the Gibbs measure uniqueness [49, 50, 51, 52]. Roughly speaking: the Gibbs measure (2) is unique if the behavior of a spin i is totally independent from the boundary conditions (i.e. very distant spins) for any possible boundary conditions. More precisely, let us define {sl} colors of all the spins at distance at least l from the spin i. The Gibbs measure µ is unique if and only if the following condition holds for every i (and in the limit N → ∞) {sl},{s |µ(si|{sl}) − µ(si|{s′l})| 0 , (15) where the average is over the ensemble of graphs. In [50, 51], it was proven that the Gibbs measure in the coloring problem on random regular graphs is unique only for graphs of degree c < q. To the best of our knowledge, this has not been computed for Erdős-Rényi graphs (later in this section, we argue on the basis of a physical argument that it should be c < q − 1 in this case). 2. The Gibbs measure extremality condition In many cases, the RS approach is observed to be correct beyond the uniqueness threshold. It was thus suggested in [53] (see also [42]) that the Gibbs measure extremality provides a proper criterion for the correctness of the replica symmetric assumption. Roughly speaking, the difference between uniqueness and extremality of a Gibbs measure is that although there may exist boundary conditions for which the spin i is behaving differently than for others, such boundary conditions have a null measure if the extremality condition is fulfilled. Formally (and keeping the notations from the previous section), the extremality corresponds to µ({sl}) |µ(si|{sl}) − µ(si)| 0 . (16) In mathematics the “extremal Gibbs measure” is often used as a synonym for a “pure state”. Recently, the authors of [53] provided rigorous bounds for the Gibbs measure extremality of the coloring problem on trees. There exist two heuristic equivalent approaches to check the extremality condition. In the first one, one studies the divergence of the so called “point-to-set” correlation length [54, 55]. The second one is directly related to the cavity formalism: one should check for the existence of a nontrivial solution of the one-step replica symmetry breaking equations (1RSB) at m = 1 (see section V). Both these analogies were remarked in [53] and exploited in [42]. We will show in section V that the extremality condition ceases to be valid at the clustering threshold cd, beyond which the 1RSB formalism will be needed. 3. The local stability: a simple self-consistency check A necessary, simple to compute but not sufficient, validity condition for the RS assumption is the non-divergence of the spin glass susceptibility (see for instance [56]). If it diverges, a spin glass transition occurs, and the replica symmetry has to be broken [29]. The local stability analysis thus gives an upper bound to the Gibbs extremality condition, which remarkably coincides with the rigorous upper bound of [57]. This susceptibility is defined as χSG = 〈sisj〉2c . (17) The connectivities above which it diverges at zero temperature can be computed exactly within the cavity formalism (we refer to appendix A for the derivation). It follows for regular and Erdős-Rényi graphs: RS = q 2 − 2q + 2 , cERRS = q2 − 2q + 1 , (18) while the stability of the bi-regular graphs of connectivities c1, c2 is equivalent to regular graphs with c = 1 + (c1 − 1)(c2 − 1). Note that for regular and ER graphs the RS instability threshold is in the colorable phase only for q = 3. Indeed the 5-regular graphs are 3-colorable [31] (and rigorous results in [13, 14]) and exactly critical since c RS(3) = 5, and for ER graphs the COL/UNCOL transition appears at cs ≈ 4.69 [30] while cERRS (3) = 4. This means that the replica symmetry breaking transition appears continuously at the point cRS so that above it the RS approach is not valid anymore. For all q ≥ 4, however, the local stability point is found beyond the best upper bound on the coloring threshold for both regular and ER graphs. In this case, the extremality condition will not be violated by the continuous mechanism, but we will see that, instead, a discontinuous phase transition, as happens in mean field structural glasses, will take place. Interestingly enough, a similar computation can be made for the ferromagnetic susceptibility χF = 〈sisj〉c (see again appendix A). It diverges at c = q for regular graph and c = q− 1 for Erdős-Rényi graphs. This divergence (called modulation instability in [56]) would announce the transition towards an anti-ferromagnetic ordering on a tree, which is however incompatible with the frustrating loops in a random graph (although such order might exist on the bi-regular graphs). This is precisely the solution we dismiss when considering only the color symmetric solution of (12). Note however that the presence of this instability shows that the problem ceases to a have a unique Gibbs state (although it is still extremal) as for some specific (and well-chosen) boundary conditions, an anti-ferromagnetic solution may appear. Indeed it coincides perfectly with the rigorous uniqueness condition c = q for regular graph, and suggests strongly that the uniqueness threshold (or at least an upper bound) for Erdős-Rényi graphs is c = q− 1. IV. ONE-STEP REPLICA SYMMETRY BREAKING FRAMEWORK So far we described the RS cavity method for coloring random graphs and explained that the extremality of the Gibbs measure gives a validity criterion. We now describe the one-step replica symmetry breaking cavity solution [26, 27]. In this approach, the non-extremality of the Gibbs measure is cured, by decomposing it into several parts (pure states, clusters) in such a way that within each of the states the Gibbs measure becomes again extremal. This decomposition has many elements/states, not just a finite numbers like the q-states of the usual ferromagnetic Potts models. It is actually found that the number of pure states is growing exponentially with the size of the system. Let us define the state-entropy function Σ(f) —called the complexity— which is just the logarithm of the number of states with internal free energy density f , i.e. N (f) = exp[NΣ(f)]. In the glass transition formalism, this complexity is usually referred to as the configurational entropy. Dealing with exponentially many pure states is obviously a nightmare for all known rigorous approaches to the thermodynamic limit. The heuristic cavity method overcomes this problem elegantly, as was shown originally in the seminal work of [26, 27]. Another very useful intuition about the 1RSB cavity method comes from the identification of states α with the fixed points {ψ} of the belief propagation equations (4). The goal is thus to compute the statistical properties of these fixed points. Each of the states is weighted by the corresponding free energy (9) to the power m, where m is just a parameter analogous to the inverse temperature β (in the literature m is often called the Parisi replica symmetry breaking parameter [29, 58]). The probability measure over states {ψ} is then µ̃({ψ}) = Z0({ψ}) e−βmNf({ψ}) , (19) where Z1 is just the normalization constant. To write the analog of the belief propagation equations we need to define the probability (distribution) P i→j(ψi→j) of the fields ψi→j . This can be obtained from those of incoming fields as P i→j(ψi→jsi ) = k∈i−j dψk→isi P k→i(ψk→isi )δ(ψ −F({ψk→isi })) δ(ψ −F) . (20) The function F is given by eq. (4) and the delta function ensures that the set of fields ψi→j is a fixed point of the belief propagation (4). The re-weighting term takes into account the change of the free energy of a state after the addition of a cavity spin i and its adjacent edges except (ij), as defined in eq. (5). This term appears for the same reason as a Boltzmann factor e−βδsi,sk in eq. (4): it ensures that the state α is weighted by (Zα) m in the same way a configuration {s} is weighted by e−βH({s}) in (2). Finally Zi→j1 is a normalization constant. In the second line of (20) we introduced an abbreviation that will be used from now on to make the equations more easily readable. The notation POP comes from “population dynamics” which refers to the numerical method we use to solve eq. (20). The probability distribution P (ψ) can be represented numerically by a set of fields taken from P (ψ), and then the probability measure P (ψ)dψ becomes uniform sampling from this set, for more details see appendix D. We define the “replicated free energy” and compute it in analogy with eq. (9) as Φ(β,m) ≡ − 1 log(Z1) = ∆Φi −  , (21) where e−βm∆Φ e−βm∆F , e−βm∆Φ e−βm∆F . (22) Putting together (19) and (21) we have Z1 = e −βmNΦ(β,m) = e−βmNf({ψ}) = df e−N [βmf(β)−Σ(f)], (23) where the sum over {ψ} is over all states (or BP fixed points). In the interpretation of [59] m is the number of replicas of the system, thus the name “replicated free energy” for Φ(β,m). Note that we are using the word “replica” only to refer to the established terminology as no replicas are needed within the cavity formalism. From the saddle point method, it follows that the Legendre transform of complexity function Σ(f) gives the replicated free energy Φ(m) − βmΦ(β,m) = −βmf(β) + Σ(f). (24) Notice that this equation is correct only in the highest order in the system size N , i.e. in densities and at the thermodynamic limit. From the properties of the Legendre transform we have Σ = βm2∂mΦ(β,m) , f = ∂m[mΦ(β,m)] , βm = ∂fΣ(f). (25) Thus, from eq.(21), the free energy reads f(β) = ∆F ie−βm∆F e−βm∆F ∆F ije−βm∆F e−βm∆F . (26) When the parameter m is equal to one (the number of replicas is actually one in the approach of [59]), then −βΦ(β, 1) = −βf(β) + Σ(f) reduces to the usual free energy function considered in the RS approximation Φ(β, 1) = e− Σ + s = e− Tstot , (27) where s in the internal entropy density of the corresponding clusters and stot the total entropy density of the system. A. Analyzing the 1RSB equations Combining (21) and (26) we can compute Σ and f for each value m, that gives us implicitly Σ(f). To compute the thermodynamic observables in the model we have to minimize the total free energy ftot = f(β) − Σ/β over such values of f where the complexity Σ(f) is non-negative (so that the states exist in the thermodynamic limit). The minimum of the total free energy corresponds to a value of parameter m = m∗ and states with the free energy f∗ dominate the thermodynamics. Three different cases are then observed: a) If there is only the trivial (replica symmetric) solution at m = 1, then the Gibbs measure (2) is extremal and the replica symmetric approach is correct. If at the same time a nontrivial solution exists for some m 6= 1, then the clusters corresponding to this solution have no influence on the thermodynamics. b) If there is a nontrivial solution at m = 1 with a positive complexity, then m∗ = 1 minimizes the total free energy. The system is in a clustered phase with an exponential number of dominating states. c) If however the complexity is negative at m = 1, then the corresponding states are absent with probability one in the thermodynamic limit. Instead the total entropy is dominated by clusters corresponding to m∗ such that Σ(m∗) = 0: the system is in a condensed phase. Note that the condition Σ(m∗) = 0 corresponds to the maximum of the replicated free energy (21). The transitions between these cases are well known in structural glass phenomenology where they appear when the temperature is lowered [60, 61]. The transition from the paramagnetic phase to the clustered one is usually referred to as the dynamical transition [62] or the clustering transition. It is not a true thermodynamic transition as the total free energy of the system at m∗ = 1 is still equal to the replica symmetric one (9) (see appendix C) and thus is an analytical function of connectivity. However, the phase space is broken into exponentially many components and, as a consequence, the dynamics fall out-of-equilibrium beyond this transition. The second transition from the clustered to the condensed phase is, however, a genuine thermodynamic transition (the free energy has a discontinuity in the second derivative at cc) and is called the replica symmetry breaking transition, or the static glass transition. At this point the measure condenses into few clusters, and we shall call it the condensation transition. In structural glasses, it corresponds to the well known Kauzmann transition [63]. The sizes of the clusters in the condensed phase follow the so called Poisson-Dirichlet process which is discussed shortly in appendix B. The procedure to compute the replicated free-energy (21) and the related observables was described above for a single large random graph. To compute the averages over the ensemble of random graphs, we need to solve an equation analogous to eq. (12) P [P (ψ)] = Q1(k) dP i(ψi)P [P i(ψi)] δ(P (ψ) −F2({P i(ψi)})), (28) where the functional F2 is given by eq. (20). Solving this equation for a general ensemble of random graphs and a general parameter m is a numerically quite tedious problem. In the population dynamics algorithm [26, 27] we need to deal with a population of populations of q-components fields. It is much more convenient to look at the ensemble of random regular graphs where a factorized solution P [P (ψ)] = δ(P (ψ) − P0(ψ)) must exists. Then we are left with only one functional equation (20). Before discussing the zero temperature limit, we would like to point out that there exists another very important case in which eq. (28) simplifies. For m = 1, as first remarked and proved in [53], when dealing with the problem of reconstruction on trees, the equations can be written (and numerically solved) in a much simpler way. We again refer to the appendix C for details. Especially for the Poissonian random graphs this simplification is very useful. B. Zero temperature limit We now consider the zero temperature limit β → ∞ of the 1RSB equations (24)-(28) to study the coloring problem. In most of the previous works [27, 28, 30] the energetic zero temperature limit was employed. The β → ∞ limit of eq. (24) was taken in such a way that mβ = y remains constant. The replicated free energy (24) then becomes − yΦe(y) = −ye+ Σ(e) . (29) It is within this approach that the survey propagation (SP) algorithm was derived. The connectivity at which the complexity function Σ(e = 0) becomes negative is the coloring threshold. Above this connectivity Σ(e) was used to compute the minimal number of violated constraints (the ground state energy). The reweighting in eq. (20) becomes e−y∆E , and when y → ∞ all the configurations with positive energy are forbidden. In this paper we adopt the entropic zero temperature limit, suggested originally in [38, 39]. The difference in the two approaches was already underlined in sec. III A. We want to study the structure of proper colorings, i.e. the configurations of zero energy, and we thus fix the energy to zero. Then we obtain the entropy by considering −βf = s and introduce a free entropy —or in replica term a “replicated entropy”— as Φs(m) = −βmΦ(β,m)|β→∞. Eq. (24) then becomes Φs(m) = ms+ Σ(s). (30) The belief propagation update (4) becomes ψi→jsi = k∈i−j 1 − ψk→isi , (31) while the 1RSB equation (20) keeps the same expression (and thus the same computational complexity). The partition sum Z0 in (2) becomes in this limit the number of proper colorings or solutions. The clusters are now sets of such solutions, and are weighted by their size to the power m. The free entropy Φs(m) is then computed as Φs(m) = ∆Φis − ∆Φijs  , (32) where ∆Zi and ∆Zij are given by eqs. (5) and (6). The analysis from the previous section is valid also for the entropic zero temperature limit. The information extracted from the number of clusters of a given size, Σ(s), is one of the main results of this paper and will be discussed and interpreted in section V. C. The role of frozen variables In this section we discuss the presence and the role of the frozen variables and explain the connection between the energetic and the entropic zero temperature limits. This allows us to revisit (and extend) the survey propagation equations. Remember that the components of the cavity field ψi→jsi are the probabilities that the node i takes the color si when the constraint on the edge ij is not present. In the zero temperature limit we can classify them in two categories: (i) A hard field corresponds to the case when all components of ψi→j are zero except one, s. Then only that color is allowed for the spin i, in absence of edge (ij). (ii) A soft field corresponds to the case when more than one component of ψi→jsi is nonzero. The variable i is thus not frozen in absence of edge (ij), and the colors of all the nonzero components are allowed. This distinction is also meaningful for the full probabilities ψisi (4), if ψ is a hard field then the variable i is frozen. In the colorable region there cannot exist a finite fraction of frozen variables (even if we consider properly the permutational symmetry of colors) since by adding a link the connectivity changes by 1/N but the probability of becoming uncolorable would be finite. On the contrary, in the 1RSB picture, we observe that a finite fraction of variables can be frozen within a single cluster. In other words, in all the solutions that belong to this given cluster a finite fraction of variables can take one color only. By adding a link into the graph, the connectivity grows by 1/N , and there is a finite probability that a cluster with frozen variables disappears. The distinction between hard and soft fields is useful not only for the intuition about clusters, but also for the analysis of the cavity equations and it also leads to the survey propagation algorithm. The distribution of fields over states P i→j(ψi→j) (20) can be decomposed into the hard- and soft-field parts P i→j(ψi→j) = ηi→js δ(ψ i→j − rs) + ηi→j0 P̃ i→j(ψi→j) , (33) where P̃ i→j is the distribution of the soft fields and the normalization is s=0 η s = 1. Interestingly, the presence of frozen variables in the entropically dominating clusters is connected to the divergence of the size of average minimal rearrangement [55, 64]. Precisely, choose a random proper coloring {s} and a random node i in the graph. The average minimal rearrangement is the Hamming distance to the nearest solution in which node i has a color different from si averaged over the nodes i, the proper colorings, and graphs in the ensemble. Another interesting role of the frozen variables arises within the whitening procedure, introduced in [65] and studied, between others, for the satisfiability problem in [66, 67]. This procedure is equivalent to the warning propagation (WP) update (49) which we outlined in sec. III A. Whitening is able to identify if a solution belongs to a cluster with frozen variables or not. Particularly, the result of the whitening is a set of hard cavity fields. Since the survey propagation algorithm is computing statistics over the states that contain hard fields, then the solution found after decimating the survey propagation result should a priori also contain hard fields. However, recent works show that if one applies the whitening procedure starting from solutions found by SP on large graphs, whitening converges every time to the trivial fixed point (see detailed studies for K-SAT in [66, 67]). A possible solution to this apparent paradox is discussed in sec. VI. 1. Hard fields in the simplest case, m = 0 Let us now consider the survey propagation equations originally derived in [30] from the energetic zero temperature limit (29) when y → ∞. For simplicity we will write them only for the 3-coloring. We consider the 1RSB cavity equation (20) for m → 0, then the reweighting factor (Zi→j0 )m is equal to zero when the arriving hard fields are contradictory, and equal to one otherwise. The update of probability ηs that a field is frozen in direction s is then written from eq.(20): ηi→js = k∈i−j(1 − ηk→is ) − k∈i−j(η 0 + η p ) + k∈i−j η k∈i−j(1 − ηk→ip ) − k∈i−j(η 0 + η p ) + k∈i−j η . (34) In the numerator there is a telescopic sum counting the probability that color s and only color s is not forbidden by the incoming fields. In the denominator the telescopic sum is counting the probability that there is at least one color which is not forbidden. If we do not want to actually find a proper coloring on a single graph but just to compute the replicated free energy/entropy, we can further simplify eq. (34) by imposing the color symmetry. Indeed, the probability that in a given state a field is hard in direction of a color s has to be independent of s (except s = 0 which corresponds to a soft field). Then (34) becomes, now for general number of colors q: ηi→j = w({ηk→i}) = l=0 (−1)l k∈i−j 1 − (l + 1)ηk→i l=0 (−1)l k∈i−j [1 − (l + 1)ηk→i] . (35) Note that since ∂Σ(s)/∂s = −m, the value m = 0 corresponds to the point where the function Σ(s) has a zero slope. If a nontrivial solution of (35) exists, then Σ(s)|m=0 is the maximum of the curve Σ(s) and is counting the total log-number of clusters of size s, which is due to the exponential dependence, also the total log-number of all clusters, regardless their size. There are two points that we want to emphasize: • Suppose that a nontrivial solution of (35) exists, i.e. many clusters exist and their number can be computed with the energetic zero temperature limit calculations. Then the clusters might be very small and contain very few solution in comparison to bigger less numerous clusters; or in comparison to a giant single cluster which might still exist. This situation cannot be decided by the energetic formalism that weights clusters equally independently of their size. • Suppose, on contrary, that a nontrivial solution of (35) does not exist. It might still well be that many clusters exist, but the Σ(s) curve has no part with zero slope. We will see that these two cases are actually observed. The energetic method, that can locate the coloring threshold and from which the survey propagation can be derived, is therefore not a good tool to study the clustering transition. 2. Generalized survey propagation recursion Let us compute how the fraction of hard fields η evolves after one iteration of equation (20) at general m. There are two steps in each iterations of (20). In the first step, η iterates via eq. (35). In the second step we re-weight the fields. Writing P hardm (Z) the —unknown— distribution of the reweightings Z m for the hard fields, one gets ηi→j = dZ P hardm (Z)Z mw({ηk→i}) = w({η k→i}) dZ P hardm (Z)Z w({ηk→i}) hard. (36) A similar equation can formally be written for the soft fields 1 − qηi→j = 1 − qw({η k→i}) soft. (37) Writing explicitly the normalization N , we finally obtain the generalized survey propagation equations: ηi→j = w({ηk→i}) qw({ηk→i}) + [1 − qw({ηk→i})] r({ηk→i}) , with r({η k→i}) = Zmsoft Zmhard . (38) In order to do this recursion, the only information needed is the ratio r between between soft- and hard-field reweight- ings, which is in general difficult to compute since it depends on the full distribution of soft fields. There are two cases where eq. (38) simplifies so that the hard-field recursion become independent from the soft-field distribution. The first case is, of course, m = 0 then r = 1 independently of the edge (ij), and the equation reduces to the original SP. The second case arise for m = 1, where one can use the so-called reconstruction formalism and obtain again a closed set of equations. The computation is done in appendix C, and the SP equations at m = 1 read ηi→js = (−1)l s1,...,sl 6=s k∈i−j 1 − q q − 1 ηk→isα  . (39) It would be interesting in the future to use eq. (38) in an algorithm to find proper graph colorings, as it has been done with the original SP equation [30]. As an approximation one might also use a value r independent of the edge (ij), but different from one. For the purpose of the present work, it is important to notice that it is also possible to use eq. (38) in the population dynamics to simplify the numerical evaluation of the 1RSB solution by separating the hard-field and the soft-field contributions. Indeed, it gives the exact density of hard fields provided the ratio r is calculated, which is doable numerically (see appendix D). This allows us to monitor precisely the hard-field density and only the soft-field part is given by the population dynamics. This turns out to greatly improve the precision of the numerical solution of the cavity equations and to considerably fasten the code. 3. The presence of frozen variables A natural question is: “When are the hard fields present?” or more precisely: “When does eq. (38) have a nontrivial solution η > 0?” First notice that in order to constrain a node into one color, one needs at least q− 1 incoming fields that forbids all the other colors. It means that function w({ηk→i}) defined in eq. (35) is identically zero for k < q− 1 and might be non-zero only for k ≥ q − 1, where k is the number of incoming fields. In the limit r → 0 (which corresponds to m → −∞) eq. (38) gives η = 1/q if w({ηk→i}) is positive, and η = 0 if w({ηk→i}) is zero. Updating eq. (38) on a given graph, from initial conditions η = 1/q everywhere, is equivalent to recursive removing of all the nodes of connectivity smaller than q. This shows that the first nontrivial solution with hard fields exists if and only if the q-core [68] of the graph is extensive. For regular graphs it is simply at connectivity c = q while for Erdős-Rényi graphs these critical connectivities can be computed exactly and read, for small q, c3 = 3.35, c4 = 5.14, c5 = 6.81 [68]. Indeed we see that the first nontrivial solution to the 1RSB equation appears much before those of the original SP equation at m = 0. On a regular graph, the equations further simplify as η factorizes (is edge independent) and follows a simple self-consistent equation η = w(η) qw(η) + [1 − qw(η)] r . (40) This equation can be solved for every possible ratio r so that for all c ≥ q, we can compute and plot the curve η(r). We show the results in fig. 3 for different numbers of colors q. On this plot, we observe that η = 1/q, as predicted, for r = 0. It then gets smaller for larger value of the ratio and, at a critical value rcrit, the solution disappears discontinuously and only the (trivial) solution η = 0 exists. The values rcrit correspond to a critical value of mr. For all m > mr no solution containing frozen variables can exist. 0.65 0.75 0.85 0.95 0 0.5 1 1.5 2 2.5 3 3.5 4 3-coloring of regular graphsc=3 SP, general SP m=0 SP m=1 0.65 0.75 0.85 0.95 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 4-coloring of regular graphs c=6 c=7 SP, general SP m=0 SP m=1 FIG. 3: The lines are solutions of eq. (40) and give the total fraction qη of hard fields for a given value of ratio r = Zmsoft/Z for q = 3 (left) and q = 4 (right) in regular random graphs. There is a critical value of the ratio (full point) beyond which only the trivial solution η = 0 exists. Note that the solutions at m = 0 and m = 1 only exist for a connectivity large enough. D. Validity conditions of the 1RSB solution Now that we have discussed in detail the 1RSB formalism the next question is: “Is this approach correct?” To answer this question, one has to test if the Gibbs measure is extremal within the thermodynamically dominating pure states. This is equivalent to checking if the two-step Replica Symmetry Breaking (2RSB) solution is non trivial. Computing explicitly the 2RSB solution is however very complicated numerically, especially for Erdős-Rényi graphs. Instead, the local stability of the 1RSB solution towards 2RSB should be checked, in analogy with the RS stability in sec. III C 3. It is indeed a usual feature in spin glass physics to observe that the 1RSB glass phase become unstable at low temperatures towards a more complex RSB phase and this phenomenon is called the Gardner transition [69]. To perform the stability analysis [31, 34, 56, 70], one first writes the 2RSB recursion —where the order parameter is a distribution of distributions of fields on every edge P1(P2(ψ))— and then two types of 1RSB instabilities have to be considered depending on the way the 2RSB arises from the 1RSB solution. The first type of instability —called states aggregation— corresponds to δ(P (ψ)) → P1(P2(ψ)) while the second type —called states splitting— corresponds to P (δ(ψ)) → P1(P2(ψ)). A complete stability analysis is left for future works, but it is worth to discuss the relevance of the results derived over the last few years [31, 34, 70]. The 1RSB stability was studied for the coloring problem in [31] but only for the energetic zero temperature limit (29). In this case the parameter y = βm is conjugated to the energy. The results derived in [31] —as well as those previously derived for other problems [34, 70]— thus concern only the clusters of sizes corresponding to m = 0 at zero (for y = ∞) or at positive (for finite y) energy. The main result of [31, 34, 70] was that the 1RSB approach was stable in vicinity of the coloring threshold cs. As we shall see the clusters corresponding to m = 0 are those dominating the total entropy at the coloring threshold and as a consequence its location is thus exact within the cavity approach. The states of the lowest energy (the ground states) in the uncolorable phase also correspond to m = 0, and thus the conclusions of [31, 34, 70] concerning the uncolorable phase are also correct. In particular, a Gardner transition towards further steps of RSB appears in the uncolorable phase beyond a connectivity denoted cG in [31]. On the other hand in the colorable phase the stability of the entropically dominating clusters that correspond to m > 0 should be investigated. Some more relevant information can be, however, already drawn from known results. It was indeed found that the 1RSB approach at m = 0 is type I stable for all y, and type II stable for all y > yI in vicinity of the coloring threshold. These results concerns the states of positive energy, but keeping in mind the interpretation of y as a slope in T,m diagram, we see that the clusters of zero energy corresponding to small but nonzero positive m and zero temperature are also stable with respect to both types of stabilities. Near the colorable threshold, the value of m∗ which describes the dominating clusters is close to zero and as a consequence all the dominating clusters are 1RSB stable in vicinity of the coloring threshold. Far from the coloring threshold, the stability analysis of [31, 34] is irrelevant. In particular, the predictions of a full-RSB colorable phase made in [31, 34, 70] is not correct. Quite the contrary, our preliminary results indicate that all the dominating clusters are 1RSB stable for q > 3. The 3−coloring is however a special case as the clustering transition is continuous. Although the type II instability seems irrelevant in this case as well, we cannot at the moment dismiss a type I instability close to the clustering transition. Indeed the entropically relevant clusters correspond to values of m∗ close to one in this case, and it is simple to show that the clusters at m = 1 are type I unstable in the case there is a continuous transition: this is because the type I stability is equivalent to the convergence of the 1RSB update on a single graph. Since for m = 1 the averages of the 1RSB fields satisfy the RS belief propagation equations, and since we know from the RS stability analysis in section III C 3 that those equations do not converge in the RS unstable region (i.e. for c > cRS = 4 in 3-coloring of Erdős-Rényi graphs), it then follows that the 3-coloring is unstable against state aggregation at m = 1 for all connectivities c > 4. Therefore, it is possible that the 1RSB result for 3-coloring are only approximative for what concerns the number and the structure of solutions close to the clustering transition (note that the critical values for the phase transition are however correct and do not depend on that). This, and related issues [71], will be hopefully clarified in future works. To conclude, we believe that all the transition points we discuss in this paper (and those computed in the K-SAT problem in [42]) as well as the overall picture, are exact and would not be modified by considering further steps of replica symmetry breaking. V. THE COLORING OF RANDOM GRAPHS: CAVITY RESULTS We now solve the 1RSB equations, discuss and interpret the results. We solve the equation (28) by the population dynamics technique, the technical difficulties and the precision of the method are discussed in appendix D. Let us stress at this point that the correctness of eq. (28) is guaranteed only in the limit of large graphs (N → ∞), unfortunately the cavity method does not give us any direct hint about the finite graph-size corrections. We start by the results for the regular random graphs, then consider ensemble of bi-regular graphs and after that we turn towards Erdős-Rényi graphs. Finally, we discuss the limit of large number of colors. A. Regular random graphs Let us fix the number of colors q, vary the connectivity, and identify successively all the transitions that we shall encounter. For the sake of the discussion, we choose as a typical example the 6-coloring and we discuss later in details the cases, for different number of colors, where some transitions are missing or are arriving in a different order. We solved the 1RSB equation (20) for regular graphs, where the distribution P i→j(ψ) is the same for every edge (ij) (see appendix D) and plot the curves for Σ(s) we obtained doing so in fig. 4. We now describe the phase space of solutions when the connectivity is increased: 1) At very low connectivities c < q, only the paramagnetic RS solution is found at all m. i.e. P (ψ) = δ(ψ − 1/q). The phase space is made of a single RS cluster. 2) For larger connectivities c ≥ q, we saw in section IV C 3 that the 1RSB equations start to have nontrivial solutions with hard fields in an interval [−∞,mr]. Interestingly, another nontrivial solution, without hard fields, can be found numerically in an interval [ms,∞], and we shall call this one the soft-field solution. As the connectivity -0.04 -0.02 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 q=4, c=9 q=5, c=14 q=5, c=13 -0.05 0.05 0.15 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.1 0.2 0.3 FIG. 4: Complexity as a function of the internal state entropy for the q-coloring problem on random regular graphs of connectivity c. The full line corresponds to the clusters where a finite fraction of hard fields (frozen variables) is present and the dotted line to the clusters without hard fields. The circle signs the entropically dominating clusters. Left: (q = 4, c = 9) is in the clustered phase; (q = 5, c = 13) is in a simple replica symmetric phase and (q = 5, c = 14) is in the condensed clustered phase. Right: results for 6-coloring for connectivities 17 (RS), 18 (clustered), 19 (condensed) and 20 (uncolorable). For 4-,5- and 6-coloring all the smaller connectivities are in the RS phase while all the larger one are uncolorable. increases, we find that mr increases while ms decreases, so that the gap [mr,ms] where no nontrivial solution exists it getting continuously smaller. However, there is no nontrivial solution at m = 1 for connectivities smaller then cd (see fig. 4 for the example of the 6-coloring at c = 17). This means that the Gibbs measure (2) is still extremal. In other words the large RS state still exists and is entropically dominant (its entropy (14) is noted by a circle in fig. 4). Despite the fact that an exponential number of clusters of solutions exist and that the SP equations converge to a nontrivial result, a random proper coloring will almost surely belong to the large RS cluster. 3) If the connectivity is increased at and above the clustering threshold cd, a nontrivial solution with positive complexity Σ is found at m = 1. In fig. 4, we see that this happens at cd = 18 for the regular 6-coloring. At this point, the RS Gibbs measure (2) ceases to be extremal and the single large RS cluster splits into exponentially numerous components. To cover almost all proper colorings we need to consider exponentially many clusters N ∼ eNΣ(m∗=1). The probability that two random proper colorings belong to the same cluster is going exponentially to zero with the system size. The connectivity cd is thus the true clustering (dynamic) transition. This is not, however, a thermodynamic phase transition because the 1RSB total entropy reduces to the RS entropy (14) at m = 1 which is analytical in c. Thus the RS approach gives a correct number of solution and correct marginals as long as the complexity function at m = 1 is non-negative. 4) For even larger connectivities c ≥ cc, the complexity at Σ(m = 1) becomes negative, e.g. cc = 19 for 6-coloring. It means that the clusters corresponding to m = 1 are absent with probability one. The total entropy is then smaller than the RS/annealed one and is dominated by clusters corresponding to m∗ < 1 such that Σ(m∗) = 0. The ordered weights of the entropically dominating clusters follow the Poisson-Dirichlet process (explained in appendix B). As a consequence, the probability that two random proper colorings belong to the same cluster is finite in the thermodynamic limit. Another way to describe the situation is that the entropy condenses into a finite number of clusters. This condensation is a true thermodynamic transition, since the total entropy is non- analytical at cc (there is a discontinuity in its second derivative with respect to connectivity). The condensation is analogous to the static (Kauzmann) glass transition observed in mean field models of glasses [60, 61]. 5) For connectivities c ≥ cs (cs = 20 for 6-coloring) even the maximum of the complexity Σ(m = 0) becomes negative. In this case proper colorings are absent with probability going to one exponentially fast with the size of the graph, and we are in the uncolorable phase. It is useful to think of the growing connectivity as additions of the constraints into a fixed set of nodes. From this point of view the set of solutions which exists at connectivity c gets smaller when new edges are introduced and the connectivity increased. This translates into the cartoon in the introduction (fig. 1) where all the successive transitions are represented. Finally, another important transition has to be considered: 6) There is a connectivity cr beyond which the measure is dominated by clusters that contain a finite fraction of frozen variables. For the regular 6−coloring, cr = 19. We refer to this as the rigidity transition, by analogy with [72, 73]. The presence or the absence of hard fields inside a given cluster is crucial: if a cluster contains only soft fields, then after the addition of a small but finite fraction of new constraints, its size will get smaller (or it will split). If, however, a cluster contains a finite fraction of frozen variable, then after adding a small but finite fraction of links the cluster will almost surely disappear. Since the connectivities of regular graphs are integer numbers, we define the dynamical threshold cd as the smallest connectivity where a nontrivial 1RSB solution exists at m = 1, the condensation transition cc as the smallest connec- tivity where complexity at m = 1 is negative, cr the smallest connectivity where hard fields are present at m ∗ and the coloring threshold cs as the first uncolorable case. The scenario described here is observed for all cases of the regular ensemble, although, since connectivities are integer, the transitions are not very well separated at small q. We summarize the results in table I. Note that for q > 3, the local RS stability discussed in section III C 3 is irrelevant in the colorable regime. The only subtle case being for 3−coloring of 5-regular graphs where the RS solution is only marginally stable, i.e. the spin glass correlation function goes to zero only algebraically instead of exponentially (from this point of view c = 5 would correspond to the critical point well known in the second order phase transitions). More interesting cases will arise in the other ensembles of random graphs. q cSP [31] cd [53] cr cc cs [31] 3 5 5+ - 6 6 4 9 9 - 10 10 5 13 14 14 14 15 6 17 18 19 19 20 7 21 23 - 25 25 8 26 29 30 31 31 9 31 34 36 37 37 10 36 39 42 43 44 20 91 101 105 116 117 q c m∗ mr ms 5 3 RS+ 0.12 1.2(1) 4 8 RS -0.03 3.4(1) 4 9 1 0.41 0.41 5 12 RS -0.02 3.7(1) 5 13 RS 0.20 2.0(1) 5 14 0.50 0.90 0.90 6 16 RS -0.02 4.3(1) 6 17 RS 0.05 3.2(1) 6 18 1 0.40 0.40 6 19 0.92 0.96 0.96 7 21 RS 0.01 4.7(1) 7 22 RS 0.17 3.2(1) 7 23 1 0.60 0.60 7 24 1 0.95 0.95 TABLE I: Left: The transition thresholds for regular random graphs: cSP is the smallest connectivity with a nontrivial solution at m = 0; the clustering threshold cd is the smallest connectivity with a nontrivial solution at m = 1; the rigidity threshold cr is the smallest connectivity at which hard fields are present in the dominant states, the condensation cc is the smallest connectivity for which the complexity at m = 1 is negative and cs the smallest UNCOL connectivity. Note that 3−coloring of 5−regular graphs is exactly critical for that cd = 5 +. The rigidity transition may not exist due to the discreteness of the connectivities. Right: Values of m∗ (corresponding to the dominating clusters), and in the range of [−∞,mr] the hard-field solution exists, in the range [ms,∞] the soft-field solution exists. B. Results for the bi-regular ensemble The bi-regular ensemble allows us to fine-tune the connectivity while preserving the factorization of the 1RSB solution, which is crucial for the numerical precision. It is actually more correct to say that the solution is “bi- factorized”, as all the messages going from the nodes with connectivity c1 to c2 are the same and the other way around. The bi-regular ensemble allows us to describe with large precision two interesting cases, which reappear in the Erdős-Rényi ensemble and which are not present in the regular ensemble (again, due to the discrete nature of the connectivity). Let us remind here that bipartite graphs are always 2-colorable, but we consider only the color symmetric cavity solutions and that is why we get a nontrivial result from this ensemble. -0.02 -0.01 0.01 0.02 0.03 0.04 0.05 0.06 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 4-coloring of bi-regular graphs (5-21) -0.01 0.01 0.02 0.03 0.04 0.05 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 (4,36) (4,39) (4,42) (4,45) 4-coloring of bi-regular graphs 0.003 0.006 0.33 0.34 (4,39) FIG. 5: The complexity as a function of entropy for 4-coloring or bi-regular graphs. Left: 5-21-bi-regular graph, an example where the entropy is dominated by clusters with soft fields while the gap in the curve Σ(s) still exists. Right: 4-c-bi-regular graphs for c=36, 39, 42, 45. In all these cases the replica symmetric solution is locally unstable. In the dependence Σ(s) we see an unphysical branch of the complexity which is zoomed in the inset for c = 39. In fig. 5, the left picture is the result for the complexity as a function of entropy Σ(s) for 4-coloring of 5-21-bi-regular graphs. The replica symmetric solution on this case is locally stable. We see clearly the gap between the hard-field and the soft-field solution, and yet we are already beyond the clustering transition cd; actually the system is in the condensed phase. This example is similar to what happens for the 4-coloring of Erdős-Rényi graphs. The second interesting case, the right hand side of fig. 5, is given by the results for Σ(s) for the 4-coloring of 4-c-bi-regular graphs, which are RS unstable for c > 28. Both the clustering and the condensation transitions coincide with the RS instability cd = cc = 28. The survey propagation equations have a nontrivial solution starting from cSP = 37. The rigidity transition is at cr = 49. Finally the coloring threshold is cs = 57. Qualitatively, the results for this 4-c-bi-regular ensemble are the same as those for the 3-coloring of Erdős-Rényi random graphs. We see that for c ≤ 42 the gap between the hard-field (full line) and soft-field (dotted line) solution exists. For m > ms there is a non-physical nontrivial soft-field solution, the convex part of the line in the figure, zoomed in the inset. It means that for m < mr we actually can find two solutions depending if we start or not with a population containing enough hard fields. The unphysical branch survives even when the gap [mr,ms] closes, see the example of c = 45 in the figure. We would like to stress at this point the enormous similarity of the soft-field part of the curve Σ(s) to the one in fig. 4 in ref. [35]. Actually the variational results of [35] should be very precise and relevant near to the continuous clustering transition (this is also case for the 3-coloring of Erdős-Rényi graphs or for 3-SAT). C. Results for Erdős-Rényi random graphs For Erdős-Rényi random graphs obtaining the solution of eq. (28) is computationally more involved as the solution is no longer factorized. In the population dynamics a population of populations has to be updated, which is numerically possible only for small populations, and so one has to be careful that the finite population-size corrections are small enough, see details in appendix D. However, all the computations can be done with the same computational complexity as for the regular graphs for m = 0, the energetic zero temperature limit (section IV C 1), and for m = 1 (appendix C). That is enough to obtain the SP, clustering, condensation and COL/UNCOL transitions (from which the first and last one was computed in [30]). We can also compute exactly when hard fields appear for m = 1, eq. (C9), this transition is further studied in [64]. Finally, using the generalized survey propagation equation introduced in section IV C 2, the rigidity transition can be computed quite precisely. 1. The general case for q > 3, discontinuous clustering transition The phase transitions in q-coloring of random Erdős-Rényi graphs for q > 3 are qualitatively identical to those discussed in the case of random regular graphs. We plot the results for the total entropy (number of solutions) and complexity (number of clusters which dominate the entropy) in the 4− and 5− coloring in fig. 6. At the clustering transition cd the complexity becomes discontinuously positive, the large RS cluster suddenly splits in an exponential number of smaller ones. The total entropy Σ∗ + s∗ is given by the RS formula (14) up to the condensation transition cc. At the condensation transition the complexity of the dominating clusters becomes zero, the total entropy stot = s∗ < sRS is given by the point where Σ(s ∗) = 0. The function stot(c) is non-analytical at the point cc, it has a discontinuity in the second derivative. At the coloring threshold cs all the clusters of solutions disappear, note, however, that the total entropy of the last existing clusters is strictly positive (about a half of the total entropy at the condensation transition). That means that the COL/UNCOL transition is not only sharp but also discontinuous in terms of entropy of solutions. Note that the positive entropy has two contributions: the trivial and smaller one coming from presence of leaves and other small subgraphs, and the nontrivial and more important one connected with the fact, that the ground state entropy is positive, even in the uncolorable phase or for the random regular graphs. Finally we located the rigidity transition, when frozen variables appears in the dominating clusters. For 3 ≤ q ≤ 8 this transition appears in the condensed phase. As the number of colors grows it approaches the clustering transition. All the four critical values cd, cr, cc and cs are summarized in table II, values of cSP and cr(m = 1) are given for comparison. 0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.18 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9 Σ*(c) cd cc cs Poissonian graphs, q=4 0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.18 12.8 12.9 13 13.1 13.2 13.3 13.4 13.5 13.6 13.7 Σ*(c) cd cc cs Poissonian graphs, q=5 FIG. 6: The 1RSB total entropy and complexity of the dominating clusters for 4- and 5-coloring of Erdős-Rényi random graphs. The complexity jumps discontinuously at the clustering transition cd while the total entropy stays analytical. The complexity disappears at the condensation transition cc causing a non-analyticity in the total entropy. Finally the total entropy discontinuously disappears at the coloring threshold. Dashed is the RS entropy left for comparison. 2. The special case of 3−coloring, continuous clustering transition The only case which is left to be discussed is the 3-coloring of Erdős-Rényi graphs. It is different from q > 3 because the replica symmetric solution is locally unstable in the colorable phase (see section III C 3). The extremality condition underlying the RS assumption ceases to be valid because of the mechanism discussed in section III C 3, with a divergence of the spin glass correlation length: the main difference with the previous cases is therefore that the clustering transition is continuous and coincide with the condensation transition. However, the phenomenology does not differ too much from the other cases: cRS = cd = cc = 4; the phase where the entropy is dominated by exponential number of states is thus missing and the complexity corresponding to m = 1 is always negative (see fig. 7 left together with the dependence of the total entropy on the connectivity). Note that the curves Σ(s) for the 3-coloring have been already studied in [38, 39] where the authors considered however only the range of connectivities c = [4.42, 4.69] = [cSP, cs]. q cd cr cc cs cSP cr(m=1) 3 4 4.66(1) 4 4.687(2) 4.42(1) 4.911 4 8.353(3) 8.83(2) 8.46(1) 8.901(2) 8.09(1) 9.267 5 12.837(3) 13.55(2) 13.23(1) 13.669(2) 12.11(2) 14.036 6 17.645(5) 18.68(2) 18.44(1) 18.880(2) 16.42(2) 19.112 7 22.705(5) 24.16(2) 24.01(1) 24.455(5) 20.97(2) 24.435 8 27.95(5) 29.93(3) 29.90(1) 30.335(5) 25.71(2) 29.960 9 33.45(5) 35.658 36.08(5) 36.490(5) 30.62(2) 35.658 10 39.0(1) 41.508 42.50(5) 42.93(1) 35.69(3) 41.508 TABLE II: Critical connectivities cd (dynamical, clustering), cr (rigidity, rearrangments), cc (condensation, Kauzmann) and cs (COL/UNCOL) for the phase transitions in the coloring problem on Erdős-Rényi graphs. The connectivities cSP (where the first non trivial solution of SP appears) and cr(m=1) (where hard fields appear at m = 1) are also given. The error bars consist of the numerical precision on evaluation of the critical connectivities by the population dynamics technique, details are given in appendix D. All the results derived for the 4-coloring of 4-c-bi-regular bipartite graphs are quantitatively valid also here. We are thus not surprised by the fact that in interval c = [4, 4.42] the survey propagation algorithm gives us a trivial result: simply the maximum of the curve Σ(s) does not exist yet there is no nontrivial solution at m = 0. Yet, the entropy is dominated by finite number of largest clusters which do not contain hard fields. The two solutions (hard-field and soft-field) join at a connectivity around 4.55. Finally at cr = 4.66 the hard fields arrive to the dominating states (and in consequence to all others). 0.05 0.15 0.25 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Σm=1(c) cd=cc cs Poissonian graphs, q=3 0 0.2 0.4 0.6 0.8 1 (c-cc)/(cs-cc) 0 0.2 0.4 0.6 0.8 1 (c-cc)/(cs-cc) large q FIG. 7: Left: The total entropy for 3-coloring of Erdős-Rényi random graphs. The dashed line is the replica symmetric (also the annealed) entropy, left for comparison. The complexity at m = 1 is shown, it is negative for c > 4, however, for connectivity near to four it is very near to zero. Right: The values of parameter m∗ (Σ(m∗) = 0) as a function of connectivity for q = 3, 4, 5 and in the large q limit. The connectivity c is rescaled as (c− cc)/(cs − cc). It is striking that for q > 3 the curves are so well fitted by the large q limit one. We are even not able to see the difference due to the error bars which are roughly of the point size. 3. The overlap structure We now give some results about the overlap structure in the random coloring to elaborate the intuition about clusters. First, consider marginal probabilities ψi,αsi within a cluster α. Note that due to the color symmetry there exist another q! − 1 clusters different only in the permutation of colors. We define the intra-cluster overlap of two solutions (averaged over states) as 〈(ψi,αsi ) 2〉α . (41) In the paramagnetic phase δ = 1/q, otherwise we have to compute it from the fixed point of equation (20). The overlap between two solutions which lie in two clusters, which differ just by permutation π of colors is δj = δ j − 1 q − 1 + q − j q(q − 1) , (42) where j is the number of fixed positions in the permutation π (in particular δq = δ, and δ1 = 1/q). In fig. 8 we show the overlap structure for 3- and 4-coloring. The probabilities that two random solutions have one of the overlaps can be computed from the Poisson-Dirichlet process described in appendix B, in fact this is not a self-averaging quantity [29]. 0.25 0.75 4.8 4.687 4.4 4.2 4 3.8 3.6 connectivity cd=cc cs 0.25 0.75 9 8.9 8.8 8.6 8.46 8.35 8.2 8 connectivity cd cc cs FIG. 8: Left: Overlaps structure in 3-coloring of random graphs as a function of connectivity. The intra-cluster overlap (upper curve) grows continuously from 1/3 at the clustering transition c = 4. In the figure from up there are δ = δ3, δ1 and δ0. Right: Overlaps structure in 4-coloring of random graphs as a function of connectivity. The intra-cluster overlap (upper curve) jumps discontinuously from 1/4 at the clustering transition c = 8.35. The probability that two random solutions belong to the same cluster, however, is zero between the clustering and condensation transition [8.35, 8.46]. In the figure from up there are δ = δ4, δ2, δ1 and δ0. D. Large q Asymptotics We give here the exact analytical large q expansion of the previous results. In the asymptotic computations the regular and Erdős-Rényi ensembles are equivalent (the corrections are of smaller order in q that the orders we give). We refer to the appendix E for the explicit derivation of the formulae. At large q a first set of transitions arises for connectivities scaling as q log q: cSP = cr(m = 0) = q [log q + log log q + 1 − log 2 + o(1)] , (43) cr(m = 1) = q[log q + log log q + 1 + o(1)]. (44) cSP was already computed in [31] and cr is the rigidity transition. The clustering transition has to appear before the rigidity one cd < cr. For all the finite q cases we looked at, cd was between cSP and cr. A second set of transitions arises for connectivities scaling as 2q log q: cc = 2q log q − log q − 2 log 2 + o(1) , (45) cs = 2q log q − log q − 1 + o(1) . (46) The condensation thus appears very close the COL/UNCOL transition and both are very far from the clustering and rigidity transitions (those are on a half way in the phase diagram). We show also in appendix E that for connectivity c = 2q log q − log q + α, one has 2qs(m) ≃ 2m log 2 , (47) 2qΣ(m) ≃ 2m − 2 −m2m log 2 − α . (48) Since the RS free energy is correct until cc, which differers just by constant from cs, that means that for all connec- tivities bellow cc the number of solutions is correctly given by the replica symmetric entropy (14). Indeed, the value s(m = 1) can be obtain by a large expansion of eq. (14). In fig. 9 we plot the complexity of dominating clusters Σ∗ = Σ(m∗), the total entropy stot = Σ∗ + s∗, and the physical value of m∗ as a function of connectivity c = 2q log q − log q+ α. Note that the properly scaled values of the total number of solutions at cc and cs, and the values cc, cs themselves, are already very close to those at q = 3, 4, 5 (see figs. 6 and fig. 7 left). The closeness is particularly striking for the values m∗ for q = 4 and q = 5 (see fig. 7 right). These formulae show that in the large q limit, near to the coloring threshold, it is the number of clusters which change with connectivity (i.e. α), and not their internal entropy (size). In the leading order, adding constraints near to the COL/UNCOL transition thus destroys clusters of solutions, but do not make them smaller: this is due to the fact that these clusters are dominated by frozen variables so that adding a link kills them most of the time. We also computed the entropy value at the condensation transition, and found s(m = 1) = log 2/q. The entropy of the last cluster (exactly at the COL/UNCOL transition) is s = log 2/2q. 0 0.5 1 1.5 2 α=-1.75 αg=-2log2 α=-1.2 αq=-1 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 2q Σ*(α) αg αq FIG. 9: Analytical result for the large q asymptotics close to the COL/UNCOL transition for c = 2q log q − log q + α. Left: (rescaled) complexity versus (rescaled) internal entropy for different connectivities. The condensation transition appears for α = −2 log 2. The maximum of the complexity becomes zero at cs for α = −1. Right: Total entropy (s tot), complexity (Σ∗) and the parameter m∗ versus α. Notice how the values for the total number of solution are already very close to those for finite low q in figs. 6, 7. VI. ALGORITHMIC CONSEQUENCES In this section we give some algorithmic consequences of our findings. First, we discuss the whitening procedure. We then introduce a random walk algorithm adapted from the Walk-SAT strategy and study its performance. We show in particular that the clustering/dynamical transition does not correspond to the onset of hardness in the problem and argue that it is instead the rigidity transition. Finally, we discuss the performance of the belief propagation algorithm in counting and finding solutions, and show that is works much better than previously anticipated. A. The whitening procedure The whitening procedure as introduced in [65] can distinguish between solutions which belong to a cluster containing hard fields and those which do not. Generally whitening is equivalent to the warning propagation (version of belief propagation which distinguish only if a field is hard or not). Warning propagation for coloring was derived in [30]. Let us call ui→j = (1, 0, 0, . . . , 0) the hard field in the direction of the first color, i.e. in absence of node j the node i takes only the first color in all the colorings belonging to the cluster in consideration, and similarly for other colors. Denote ui→j = (0, 0, 0, . . . , 0) if ψi→j is not frozen in the cluster, we say that the oriented edge i→ j is then “white”. The update for u’s follows from (4) ui→js = min k∈i−j uk→ir + δr,s − min k∈i−j uk→ir  . (49) To see if a solution {si} belongs to a cluster with frozen variables or not we initialize warning propagation with ui→js = δs,si , and update iteratively according to (49) until a fixed point is reached (the update every time converge, because starting from a solution we are only adding white edges). In the fixed point or all edges are white, then the solution {si} does not belong to a frozen cluster, or some of the edges stay colored (non-white), then the solution {si} belongs to a frozen cluster. Note that in the K-SAT problem (but not in general), whitening is equivalent to a more intuitive procedure, where the directed edged are not considered [66, 67]. We wish to offer here an explanation of a paradox observed in [66, 67]. The SP algorithm gives information on the frozen variables in the most numerous clusters (m = 0). Yet, the solutions which are found by the standard implementation (decimation and SP plus Walk-SAT) do not belong to clusters with frozen variables, since they always give a trivial whitening result (all directed edges are white)[66, 67]. We suggest that the decimation strategy drives the system towards a solution belonging to a large cluster, which does not contain frozen variables. In this case, it is logical that the result of the whitening is trivial, as it is observed. We believe this is reason why no nontrivial whitenings are observed so far in the study of the K-SAT problem on large graphs. Note that beyond the rigidity transition this argument does not work anymore, since there all the clusters (for all m such that Σ(m) > 0) contain frozen variables. More precisely, for q ≥ 9 we could in principle end up in soft-clusters even beyond the rigidity transition (since that one concerns only the dominant states), if this is possible is let for further investigation. Interestingly, in the coloring problem we have not been able to find solutions beyond the rigidity transition even with survey propagation algorithm (compare cr with the performance of SP in [30]). Further, more systematic, investigations have to be done about these issues, employing other strategies for the use of the survey propagation equations (for example the reinforcement [74]). B. A Walk-COL algorithm to color random graph In this paper, we have computed the correct clustering transition cd for the random coloring problem. Beyond this transition, Monte Carlo algorithms are proven not to reach equilibrium as their time of equilibration diverges [55, 62]. It was often claimed, or assumed, that this point corresponds to the onset of hardness of the problem. However, the fact that the physical dynamics does not equilibrate just means that the complete set of solutions will not be correctly sampled —indeed Monte-Carlo experiments clearly display slow relaxation [75]— but not that no solutions can be eventually found. This simple fact explains the results of [32] where a simple annealing procedure was shown to 3-color a ER graph beyond cd = 4. In this section, we use a local search strategy which does not satisfy the detailed balance condition. Therefore, we do not expect to be able to find typical solutions, however it might be possible to find some solutions to the problem. The Walk-COL algorithm [87] is a simple adaptation of the celebrated Walk-SAT [76]. More precisely, we adapted the method designed for satisfiability in [77]. Given a graph, and starting from an initial random configuration, we recursively apply the following procedure: 1) Choose at random a spin which is not satisfied (i.e. at least one of its neighbors has the same color). 2) Change randomly its color. Accept this change with probability one if the number of unsatisfied spin has been lowered, otherwise accept it with probability p. 3) If there still are unsatisfied nodes, go to step 1) unless the maximum running time is reached The probability p has to be tuned in each different case for a better efficiency of the algorithm. Typically, values between 0.01− 0.05 give good results. We shall now briefly discuss the performance of the algorithm, to illustrate the two following points: (a) When the phase space is RS, we observe that Walk-COL finds a solution in linear time. (b) Even in the “complex” phase for c > cd, the algorithm can find in some cases solutions in linear time. Concerning the first point, we tested the algorithm in the RS phase of regular random graphs for q = 3, 4, 5, 6, 7. In all these cases, we were able to color in linear time all the graphs of connectivities that correspond to a replica symmetric solution. In particular, the cases (q = 3, c = 5), (q = 5, c = 13), (q = 6, c = 17), (q = 7, c = 21), (q = 7, c = 22) are found to be colorable with the Walk-COL algorithm even if a nontrivial solution to the SP equations exists. Concerning the second point, we considered the 3− and 4−coloring of Erdős-Rényi random graphs. The results are shown in fig. 10 where the percentage of unsatisfied spins versus the number of attempted flips (averaged over 5 different realizations) divided by N is plotted. We observe that the curves corresponding to different values of N superpose quite well (and that actually the results for N = 2 · 105 are systematically lower than those for N = 5 · 104) so that an estimation of the time needed to color a graph can be obtained. The connectivities of these graphs are beyond the dynamical transition (cd = 4 for 3-coloring and cd = 8.35 for 4-coloring). It would be interesting to systematically test Walk-COL, as it has been done for Walk-SAT in [77], to derive the precise connectivity at which it ceases to be linear. 1e-05 1e-04 0.001 0.01 100 1000 10000 100000 1e+06 c=4.1 c=4.3 c=4.4 c=4.5 3-coloring N=50 000 N=200 000 1e-05 1e-04 0.001 0.01 100 1000 10000 100000 1e+06 c=8.0 c=8.3 c=8.4 c=8.54-coloring N=50 000 N=200 000 FIG. 10: Performance of the Walk-COL algorithm in coloring random graphs for 3−coloring (left) and 4−coloring (right). We plot the rescaled time (averaged over 5 instances) needed to color a graph of connectivity c. The strategy allows one to go beyond the clustering transition (cd = 4 for 3-coloring and cd = 8.35 for 4-coloring) in linear time with respect to the size of the graph. Already these results show that the dynamical transition is not a problem for the algorithms. This can also be observed in a number of numerical experiments for the satisfiability [77, 78] and the coloring [73, 79] problems. We believe, however, that the rigidity transition plays a fundamental role for the average computational complexity. A first argument for this is that, for large graphs, it seems that all the known algorithms are only able to find solutions with a trivial whitening, i.e. solutions that belong to clusters without hard fields. Beyond the rigidity transition however, the clusters without hard fields become very rare (in the sense that the dominating clusters and all the smaller, more numerous ones, contain hard fields). For q ≥ 9 maybe the connectivity where hard fields appear in clusters corresponding to Σ(m) = 0, m > 1 should be considered. This suggests that the known algorithms will not be able to find a solution beyond this point. A second argument is the following: local search algorithms are either attracted into a solution or stucked in a metastable state. These metastable states, in order to be able to trap the dynamics, have to contain a finite fraction of hard fields. Given an algorithm, determining which of these two situations happens is not only a question of existence of states, but also a question of basins of attraction and a theoretical analysis of such basins is a very difficult task so that the precise analysis of the behavior of local algorithms remains a hard problem. However, the metastable states are known, from the cavity formalism, to be much more numerous than the zero-energy states. Moreover the basin of attraction of a zero-energy state that contains hard fields does not probably differ much from those of the metastable state (while, on the other hand, the basin of attraction of a zero-energy state which does not contain hard fields might be slightly different and arguably relatively larger). It thus seems to us reasonable that local algorithms will get trapped by the metastable states beyond the rigidity transition. A similar conclusion was reached recently in [73] where the recursive implementation of the Walk-COL algorithm 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 cd=cc cr cs N=2000 N=4000 N=8000 N=16000 N=32000 8 8.2 8.4 8.6 8.8 9 cd cc cr csN=2000 N=4000 N=8000 N=16000 N=32000 FIG. 11: Performance of the BP algorithm plus decimation in coloring random graphs for 3−coloring (left) and 4−coloring (right). The strategy described in the text allows to color random graphs beyond the clustering and even the condensation transitions. was studied and found to be somehow simpler to analyze. Again the strategy was found to be efficient (with linear time with respect to the size of the graph) beyond cd but bellow cr. The precise algorithmic implications of the rigidity transition thus require further investigation, maybe in the lines of [73, 78]. C. A belief propagation algorithm to color random graph Another consequence of our results, that we already discussed shortly in [42], is that the standard belief propagation (BP) algorithm gives correct marginals until the condensation transition. It is actually a simple algebraic fact that the 1RSB approach at m = 1 gives the same results for the marginals (average probabilities over all clusters) as the simple RS approach (see appendix C). Moreover the log-number of solutions in clusters corresponding to m = 1 is also equal to the RS one. This suggests to use the BP marginals (as was already suggested in [40]) and a decimation procedure to find proper colorings. Compared with the SP algorithm which has computational complexity proportional to q! (factorial) the BP is only q. We have seen moreover that for large numbers of colors, the condensation point is very close to the COL/UNCOL transition, so that BP could be used in a large interval of connectivities. As a simple application, we tested how the straightforward implementation of the BP algorithm plus decimation allows one to find solutions of 3− and 4−coloring of random Erdős-Rényi graphs. Note that in the 3−coloring the clustered but not condensed phase is missing, so the argumentation above does not concern this case. The algorithm works by iterating the following procedure: (i) Run BP on the graph for a given number l of iterations. (ii) Consider the most biased variable, and color it with its most probable color. Two problems have to be mentioned. The first one is rather trivial: since at the beginning all colors are symmetric, the first color had to be put at random. The second one is more serious and concerns the convergence of BP. Indeed, we saw that there is local instability in the BP (replica symmetric) equations at connectivity c = 4 for the 3−coloring of random graphs, so that the BP equations do not converge. This seems to be a problem restricted to the 3−coloring, but even in the case of 4− or more coloring, the BP equations do not converge on the decimated graph when a finite fraction (typically few percent) of variables is fixed. The reason or that is to be understood. Nevertheless, since we merely want to design an effective tool to solve the coloring problem, we choose to avoid this problem by fixing the number of iteration l at each step and thus ignore the non-convergence. We tried the method on both the 3− and 4−coloring and obtained unexpectedly good results. We used the following protocol in the code: We first try to find a solution with l = 10. If we do not succeed, we restart with l = 20 and once more with l = 40. We tried that on 10 different samples for different connectivities. The probability to find a proper coloring with these conditions is shown in fig.11. We thus observe that the BP strategy is able to find solutions, even beyond the condensation transition. This shows clearly that the decimation procedure is a nontrivial one, and that the problem is not really hard in that region of connectivities. Note that the SP algorithm plus decimation has been shown to work in the 3−coloring very well until about 4.60 [30]: our results are thus very close to those obtained using SP. This rises again the question of the rigidity transition cr = 4.66, which might also be problematic for the decimated survey propagation solver. VII. CONCLUSIONS Let us summarize the results. They are perhaps best illustrated looking back to the cartoon in fig. 1, where the importance of the size of clusters is evidenced. We find that the set of solutions of the q-coloring problem undergoes the following transitions as the connectivity is increased: (i) At low connectivity, c < cd, many clusters might exist but they are very small and the measure over the set of solutions is dominated by the single giant cluster described by the replica symmetric approach. (ii) Only at the dynamical transition cd the giant cluster decomposes abruptly into an exponentially large number of clusters (pure states). For connectivities cd < c < cc, the measure is dominated by an exponential number of clusters. Yet, the total number of solutions is given by the replica symmetric entropy (14), and the marginals are given by the fixed point of the replica symmetric equations (belief propagation) (4). Starting from this transition the uniform sampling of solution becomes hard. (iii) At connectivity cc the set of solutions undergoes a condensation transition, similar to the one appearing in mean field spin glasses. In the condensed phase the measure is dominated by finite number of the largest clusters. The total entropy is strictly smaller than the replica symmetric one and has a discontinuity in the second derivative at cc. (iv) When connectivity cs is reached, no more clusters exist: this is the COL/UNCOL transition. Note that the entropy of last existing clusters is strictly positive, and not given only by the contribution of the isolated nodes, leaves and other small subgraphs, the COL/UNCOL transition is thus discontinuous in entropy. This picture is very similar to the well-known scenario of the glass transition in temperature, with the dynamical and glass (Kauzmann) transition [61]. In some cases, the main one being the 3-coloring of Erdős-Rényi graphs, the clustering and the condensation transition merge and a continuous transition take place at cd = cc, which then coincide with the local instability of the replica symmetric solution. Interestingly the variational approach of [35] is very precise near to the continuous clustering transition. Since the 3-SAT problem behaves in the same manner, this solves the apparent contradictions between the results of [35] and [24]. In addition to the transitions describing the organization of clusters, another important phenomenon concerning the internal structure of clusters takes place. A finite fraction of frozen variables can appear in the clusters (a frozen variable takes the same color in all the solutions that belong to the cluster). We found that the fraction of such variables in each cluster undergoes a first order transition and jumps from zero to a finite fraction at a connectivity that depends on the size of the cluster. In particular: (v) There exists a critical connectivity cr (rigidity/freezing) at which the thermodynamically relevant clusters — those that dominate the Gibbs measure— start to contain a finite fraction of frozen variables. The results above were obtained within the 1RSB scheme, but should not change when considering further steps of RSB (an exception might be the 3-coloring near to the clustering transition). We discussed some algorithmic consequences of these transitions. First, the belief propagation algorithm is efficient in counting solution and estimating marginals until the condensation transition. More interestingly, it can also be used, just like survey propagation, together with a decimation procedure in order to find solutions as we numerically demonstrated. Secondly, the dynamical transition is not the one at which simple algorithms fail as we illustrated using the Walk-COL strategy. For the 3-coloring of ER graph, there is even a rigorous proof of algorithmic performance beyond cd = 4 and until c = 4.03 [10]. We argued that, instead, the rigidity phenomenon is responsible for the onset of computational hardness. This is a major point that we hope to see more investigated in the future. Our study opens a way to many new and promising investigations and developments. For instance, we wrote the equivalent of the survey propagation equations for general value of m, which has a particularly simple form for m = 1 (39). It would be interesting to use these equations to find solutions. The behavior at finite temperature and the performance of the annealing procedure are also of interest. It would furthermore be interesting to re-discuss other finite connectivity spin glass models like for instance the lattice glass models [56] in the light of our findings. The stability towards more steps of replica symmetry breaking, or the super-symmetric approach [71], should be further investigated. Finally, it would be interesting to combine the entropic and energetic approach to investigate the frozen variables in the meta-stable states. We hope that our results will stimulate the activity in these lines of thoughts. Acknowledgments We thank Jorge Kurchan, Marc Mézard, Andrea Montanari, Federico Ricci-Tersenghi, Guilhem Semerjian and Riccardo Zecchina for cheerful and very fruitful discussions concerning these issues. The numerical computations were done on the cluster EVERGROW (EU consortium FP6 IST) at LPTMS, Orsay, and on the cluster DALTON at ESPCI, Paris. This work has been partially supported by EVERGROW (EU consortium FP6 IST). APPENDIX A: STABILITY OF THE PARAMAGNETIC SOLUTION In this appendix, we show how to compute the stability of the paramagnetic solution towards the continuous appearance of a 1RSB solution. This happens, as usual for continuous transition, when the spin glass correlation length, or equivalently, the spin glass susceptibility, diverges. Obviously, the presence of the diverging correlation length invalidate the premise of the RS cavity method. Recall that the spin glass susceptibility is defined as χSG = 〈sisj〉2c . (A1) and can be rewritten for the present purpose as χSG = γdE(〈s0sd〉2c) , (A2) where we consider the average over graphs, in the thermodynamic limit, where spins s0 and sd are at distance d. The factor γd stands for the average number of neighbors at distance d, when d≪ logN . Assuming that the limit for large d of the summands in (A2) exists (with the limit N → ∞ performed first), we relate it to the stability parameter : λ = lim E(〈s0sd〉2c) . (A3) Then the series in (A2) is essentially geometric, and converges if and only if λ < 1. Using the fluctuation-dissipation theorem we relate the correlation 〈s0sd〉c to the variation of magnetization in s0, caused by an infinitesimal field in sd. Finally, using the fact that we perform the large-N limit first, the variation above is dominated by the direct influence through the length-d path between the two nodes, and this induces a “chain” relation: if the path involves the nodes (d, d− 1, . . . , 0) we have E(〈s0sd〉2c) = C · ∂ψa→0 ∂ψb→d = C · E ∂ψl→l−1 ∂ψl+1→l . (A4) The stability parameter of the paramagnetic solution of the cavity equations towards small perturbations can be computed from the following Jacobian T τσ = ∂ψ1→0τ ∂ψ2→1σ , (A5) which gives the infinitesimal probability that a change in the input probability ψ2→1σ will change the output probability ψ1→0τ . The index RS says that the expression has to be evaluated at the RS paramagnetic solution. This matrix has only two different entries, all the diagonal elements are equal, and all the non-diagonal elements are also equal. As an immediate consequence all Jacobians commute and are thus simultaneously diagonalizable so that it will be sufficient to study the effect after one cavity iteration (one step in the chain). The matrix T has only two distinct eigenvalues, ∂ψ2→11 ∂ψ2→12 ∂ψ2→11 + (q − 1) ∂ψ ∂ψ2→12 The second eigenvalue corresponds to the homogeneous eigenvector (1, 1, ..., 1) and describes a fluctuation changing all ψ2→1τ , τ = 1, ..., q, by the same amount, and maintains the color symmetry. It is thus not likely to be the relevant one and we will see that indeed λ2 = 0. The first eigenvalue, however, is (q − 1)-fold degenerate and its eigenvectors are spanned by (1,−1, 0, ..., 0), (0, 1,−1, 0, ..., 0), . . . , (0, ..., 0, 1,−1). The corresponding fluctuations explicitly break the color symmetry, and are in fact the critical ones. Using the cavity recursion (4), the two derivatives simply read ∂ψ2→12 = (1 − e−β) (ψ 1−(1−e−β)ψ21 ∂ψ2→11 = (1 − e−β) (ψ1→01 ) 1−(1−e−β)ψ2→11 1−(1−e−β)ψ2→11 so that the values of the two eigenvalues evaluated at the RS solution, where all ψ are equal to 1/q, are 1 − q 1−e−β , λ2 = 0. (A8) The stability parameter (A3) is thus λ = γλ21 and the critical temperatures bellow which the instability sets in are T regc (q, c) = −1/log 1 − q√ c− 1 + 1 , TERc (q, c) = −1/log 1 − q√ . (A9) For regular and Erdős-Rényi graphs respectively. Thus at zero temperature the critical connectivities reads RS stab = q 2 − 2q + 2 , cERRS stab = q2 − 2q + 1. (A10) These results coincide perfectly with the numerical simulations of the cavity recursion of [32]. The analytical ex- pressions equivalent to (A9) were in fact first obtained in [57] in the context of the reconstruction problem on trees as an upper bound for the Gibbs measure extremality, and its connection with the statistical physics approach was explained in [53]. The case of bi-regular random graphs can be easily understood by noticing that two recursions should be considered, one with γ = c1−1, and one with γ = c2−1. As a consequence, the stability point is equivalent in this case to the one of a regular random graph with an effective connectivity equal to c = 1 + (c1 − 1) (c2 − 1). Another instability appears when γ|λ1| > 1. This has been refered to as the modulation instability in [56]. Actually, this is the continuous instability towards the appearance of the anti-ferromagnetic order. Since at zero temperature λ1 = (1 − q)−1, then for connectivities larger than cmod = q for random regular graphs (and cmod = q − 1 for Erdős- Rényi) the paramagnetic solution becomes unstable towards the anti-ferromagnetic order. However, this is correct if we study a tree with some given (and well chosen) boundary condition, but as noted in [56], the anti-ferromagnetic solution in impossible on random graphs because of the existence of frustrating loops of arbitrary length. The cavity equations (4) can actually never converge towards an anti-ferromagnetic solution of a random graph. Instead, when iterating, the fields oscillate between different solutions (thus the name modulation). In other words, although on a random tree with special boundary conditions there exists for c > cmod a nontrivial solution to the cavity recursion (for the Gibbs state is no longer unique (15)), this solution does not exist on a random graph (and the Gibbs state is still extremal (16)). Note that this instability could anyway be a source of numerical problems that can be overcome considering that the distribution of cavity fields P(ψ) over the ensemble of random graphs has to be symmetric in the color permutation. Another possibility is to randomly mix the new and old populations in the population dynamics so that the anti-ferromagnetic oscilations are destroyed. APPENDIX B: THE RELATIVE SIZES OF CLUSTERS IN THE CONDENSED PHASE In this section, we introduce the Poisson-Dirichlet point process and we shortly review some of its important properties. We also sketch its deep connection with the size of clusters in the condensed phase. Poisson-Dirichlet (PD) point process is a set of points {xi}, i = 1, . . . ,∞ such that x1 > x2 > x3 > . . . and i=1 xi = 1. To construct these points we consider a Poisson process {yi}, i = 1, . . . ,∞ of intensity measure y−1−m , 0 < m∗ < 1 (note that this measure is not a probability measure). We order the sequence {yi} in such a way that y1 > y2 > y3 > . . . and define the PD point process as i=1 yi . (B1) If we identify the parameter m∗ with the value for which the complexity is zero Σ(m∗) = 0 then yi is proportional to the number of solutions in cluster i (or to e−βF for non-zero temperature), and xi is the size on that cluster relative to the total number of solutions. This connection was (on a non-rigorous level) understood in [80], for more mathematical review see [81]. Note that that due to the permutation symmetry in graph coloring there are every time q! copies of one clusters (different in the color permutation). To get feeling about the PD statistics let us answer in fig. 12 to the following question: Given the value m∗ how many clusters do we need to cover fraction r of solutions, in other words what is the smallest k such that i=1 xi > r? The mathematical properties of the PD process are very clearly reviewed in [82]. To avoid confusion, note at this point that the PD process we are interested in is the PD(m∗, 0) in the notation of [82]. In the mathematical literature, it is often referred to the PD(0, θ) without indexing by the two parameters. Let us remind two useful results. Any moment of any xi can be computed from the generating function E[exp (−λ/xi)] = e−λφm∗(λ)i−1ψm∗(λ)−i , (B2) where λ ≥ 0 and the functions φm∗ and ψm∗ are defined as φm∗(λ) = m e−λxx−1−m dx , (B3) ψm∗(λ) = 1 +m (1 − e−λx)x−1−m dx . (B4) Another relation is that the ratio of two consequent points Ri = xi+1/xi, i = 1, 2, . . . is distributed as im In particular its expectation is E[Ri] = im ∗/(1 + im∗) and the random variables Ri are mutually independent. We used these relation to obtain data in figure 12. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 FIG. 12: The sketch of size of the largest clusters for given value of parameter m∗. The lower curve is related to the average size of the largest clusters as 1/E[1/x1] = 1−m ∗. The following curves are related to the size of i largest clusters, their distances are E[Ri]E[Ri−1] . . .E[R1](1 −m APPENDIX C: THE 1RSB FORMALISM AT m = 1 AND THE RECONSTRUCTION EQUATIONS In this appendix we discuss the considerable simplification of eq. (28) that is obtained by working directly at m = 1. This was first remarked and proved in [53] when dealing with the tree reconstruction problem (for a discussion of a case where the RS solution is not paramagnetic see [43]). We first introduce the probability distribution of fields (20) averaged over the graph P (ψ) ≡ dP P [P (ψ)]P (ψ) = Q1(k) dψi P (ψi) δ ψ −F({ψi}) Z0, (C1) where Z1 is computed from (20) as dψi P i(ψi)Z0 = 1 − ψi , (C2) where ψ = dψ P (ψ)ψ. Generally, ψ is a solution of the RS equation (12), which is easily seen from (20), (28). Since the RS solution is the paramagnetic one ψ = 1/q the form of Z1 is particularly simple. In the next step, we want to get rid of the term Z0 in eq. (C1). We thus introduce q distribution functions P s P s(ψ) = qψsP (ψ) . (C3) It is then easy to show that if ψ = 1/q then P s(ψ) satisfies P s(ψ) = Q1(k) s1...sk π(si|s) ψ −F({ψi}) dψiP si(ψ i) , (C4) where π(si|s) = 1 − (1 − e−β)δ(si, s) q − (1 − e−β) . (C5) We solve eq. (C4) by population dynamics. In order to do this, one needs to deal with q populations of q-component fields, and to update them according to (C4). Is is only a functional equation and not a double-functional as the general 1RSB equation (28). Moreover the absence of the reweighting term Zm0 simplifies the population dynamics algorithm significantly. Finally, it is important to note that the computational complexity here is the same as the one for regular and ER random graphs. A crucial theorem is also proven in [53]: the population dynamics of eq. (C4) has a nontrivial solution if and only if it converge to a nontrivial solution starting from initial conditions: r (ψs) = δ(r, s). (C6) This shows that when a paramagnetic solution is found, then no other solutions exist. Similar manipulations allow us to obtain the replicated free energy (21) which is equal in this case to the replica symmetric free energy (9), and the free energy (26) inside the corresponding states as − βf(β) = s1...sk π(si|s) logZi0 dψiP si(ψ i) , (C7) s1,s2 π(s1|s2) logZ120 dψ1dψ2 P s1(ψ 1)P s2(ψ where the normalization factors Zi0, Z 0 are defined by (6) and (8). The complexity follows from (24). Since the replicated free energy Φ(β, 1) is equal, according to (27), to the total free energy, we showed the statement used several time in the paper, i.e. the total free energy (entropy) at m = 1 is equal to the replica symmetric free energy Another important point is that one can write the recursion separating the hard and soft fields. In general, at zero temperature, we can write the distribution P q(ψs) in eq. (C4) as P r(ψ) = µr,sδ(ψs − 1) + (1 − µr,s)P̃r(ψ). (C8) Plugging this to eq. (C4) and taking into account the initial condition (C6) and color symmetry, we see that µq,s = qηδ(q, s), where η satisfies Q1(k) (−1)m q − 1 1 − mq q − 1η . (C9) On ER graphs, the sum can be performed analytically and one finds 1 − e− . (C10) This equation can be solved iteratively starting from ηinit = 1/q. It is a very simple equation, as the one obtained for m = 0, which gives us a very efficient way to compute the fraction of hard fields at m = 1 for both regular and ER graphs. Indeed η is larger that zero only above a certain average connectivity cr(m = 1). APPENDIX D: NUMERICAL METHODS In this section, we detail the numerical methods we used to solve the 1RSB equations (20,28), and the procedures used to generate the data. We use a population dynamics method, as introduced in [26, 27], and model the distribution P i→j(ψi→j) by a population of N vectors ψi→j . To compute P i→j(ψi→j) knowing the P k→i(ψk→i) for all incoming k we perform the 1RSB recursion in eq. (20) in two steps: (i) first we compute the new vectors ψi→j using the simple RS recursion in eq. (31) (this is the iterative step) and (ii) we take into account the weight (Z m for each of the vectors (this is the reweighting step). For the reweighting we tried different strategies, two of them perform very well. a) For every field ψ in the population, we keep its weight Z0. We then compute the cumulative distribution of weights Z0 and sample uniformly the incoming fields. Using dichotomy we generate a random fields with its proper weight in O(log(N)) steps. A complete iteration thus takes O(N logN) steps. b) We compute N new vectors and then we make a new population when we clone some of them while erasing others so that in this new population each field is present according to its weight (in principle, one can even change the size of the population, although we have not implemented this strategy). This second approach can be implemented in linear time (generating an ordered list of random numbers is a linear problem, see [83]), but is a bit less precise as we introduce redundancy in the population. We finally choose to use the second strategy, as we observed that it performs almost as good at the first one (for a given size of population) while it was much faster, so that, for given computer time, it allows a better representation of the population. We also force the population to be color-symmetric by adding a random shift of colors in the incoming messages. This is needed in order to avoid the anti-ferromagnetic solution. The learned reader will notice that this is equivalent to solving a disordered Potts glass instead of a anti-ferromagnet model. Indeed the fact that an Ising anti-ferromagnet on a random graph is equivalent to an Ising spin glass was already noticed [84]. Another important issue is the presence of hard fields. In fig. 13 are histograms of the first component of the vectors in the population for 3- and 4-coloring of 5- and 9-regular random graphs respectively. It is interesting to see how they peak around fractional values due to the presence of hard fields (see the three upper one). Maybe even more interesting are the lower one where no hard fields are present. However, since there are soft fields with values 1 − ǫ, where ǫ can be almost arbitrary small, one cannot see from these picture the absence of frozen variables. For the case c = 9, q = 4,m = 0.8 for instance, the presence of the quasi-hard fields makes the distribution clearly concentrate on values around one, zero and half (note however that the amplitude —on a logarithmic scale— is far less important). 1e-05 1e-04 0.001 0.01 0 0.2 0.4 0.6 0.8 1 c=5, q=3, m=0 1e-05 1e-04 0.001 0.01 0 0.2 0.4 0.6 0.8 1 c=5, q=3, m=2 1e-05 1e-04 0.001 0.01 0 0.2 0.4 0.6 0.8 1 c=9, q=4, m=0 1e-05 1e-04 0.001 0.01 0 0.2 0.4 0.6 0.8 1 c=9, q=4, m=0.4 1e-05 1e-04 0.001 0.01 0 0.2 0.4 0.6 0.8 1 c=9, q=4, m=0.8 1e-05 1e-04 0.001 0.01 0 0.2 0.4 0.6 0.8 1 c=9, q=4, m=1.6 FIG. 13: Histograms of the first component of the cavity field ψ1, i.e. the probability that a node takes color one. Notice the logarithmic y-scale. Frozen fields (ψ1 = 1) are present in the solution for the three upper cases; there are delta-peaks on 0,1,1/2 and other simple fractions depending on q. Notice that even when frozen fields are not present, there are many almost frozen fields (the distributions only concentrate around 0,1,1/2 and other fractions). The quasi-hard fields are therefore very hard to distinguish numerically from the true hard ones. This is evidenced on fig. 14 where we plot the fraction of hard fields computed using the expression (38) together with a numerical estimate made by population dynamics without the separate hard/soft implementation. We show that the fraction of fields of value 1 − ǫ is not zero in regions where we know that there are no hard fields even for ǫ = 10−20. This demonstrates the presence of quasi-hard fields, with ǫ going to zero as the critical m is approached. This transition is further studied in [64]. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 c=9, q=4 mrε=10-4 ε=10-8 ε=10-12 ε=10-16 ε=10-20 FIG. 14: Fraction of the hard and the quasi-hard cavity fields qη (a field is quasi-hard if ψ > 1−ǫ) in the 4−coloring of 9−regular graph. The bold line is obtained with the analytical computation of the fraction of hard fields and the dot corresponds to the threshold mr. An important simplification of the 1RSB update (20) arises when we consider the soft and the hard fields separately. The fraction of hard fields can be computed using the generalized SP equation (38), provided the ratio Zmsoft/Z hard is computed. This considerably reduced the size of the population as only soft field has to be kept in memory. Another way to further speed up the code is to generate directly soft fields with a uniform measure instead of waiting for them to come out from the 1RSB recursion. Indeed they might be quite rare in the region of small m and one can spend a considerable amount of time before being able to sample them correctly. Generating the soft fields with a uniform weight turns out to be rather easy using the following method: (i) Choose two random colors q1 and q2. (ii) Perform the usual recursion (31) in order to have a new vector but forbid incoming hard fields to q1 and q2. (iii) To obtain a uniform soft field generator, the resulting field should be weighted by 1/ where s is the number of non-null components in the vector. This is specially useful in the case of Erdős-Rényi graphs. The formula for the free entropy Φs(m) (32) also simplifies in this case. Consider a given site i; the site free entropy term can be split into three parts when (i) the field is hard, (ii) the total field is soft and (iii) the field is contradictory. Φis = log phard(Zhard)m + psoft(Zsoft)m , (D1) where phard/psoft are the probabilities that the total field is frozen/soft, and are given by the SP recursion. Indeed the probability that the total field is not contradictory (phard + psoft) is the denominator in eq. (35) while phard is the numerator of eq. (35). The link part can also be simplified using the fact that contradictions arise when two incoming frozen messages of the same color are chosen, so that Φijs = log pno contr(Zno contr)m , (D2) where pno contr is simply (1 − qηi→jηj→i). For m = 0 the formula further simplify as Zmhard = Z soft = Z no contr = 1 so that Φis = log (−1)l 1 − (l + 1)ηk→i , (D3) Φijs = log 1 − qηi→jηj→i . (D4) This is precisely what was obtained within the energetic cavity approach in [30]. The numerical population dynamics implementation with mixed hard/soft strategy is therefore as precise as it could be since we obtain the exact evaluation in the m = 0 case. This simple computation also demonstrates how one can recover the energetic zero temperature limit from the generic formalism. Finally, we obtain the function Φs(m). We fit this function using an ansatz Φs(m) = a + b2 m + c3m . . . and then perform the Legendre transform to obtain the entropy and complexity. It is also possible to compute directly the complexity from the population data using the expression of the derivative of the potential directly in the code. Both methods lead to very good results. We show an example of the raw data and their fit in fig. D, where the data have been obtained with relatively small population (N = 5000) but where the mixed strategy separating the hard and soft fields have been used. For the purely soft-field branch, we used N = 50000. It took few hours up to few days to generate these curves on present Intel PCs. -0.02 0.02 0.04 0.06 0.08 0.12 0.14 0.16 -1 -0.5 0 0.5 1 1.5 2 Φ(x) for q=6, c=19 -0.08 -0.06 -0.04 -0.02 0.02 0 0.05 0.1 Σ(s) for q=6, c=19 0.05 0.15 0.25 -2 -1 0 1 2 Φ(x) for q=4, c=9 -0.12 -0.08 -0.06 -0.04 -0.02 0.02 0.04 0 0.05 0.1 0.15 Σ(s) for q=4, c=9 FIG. 15: The numerical results for the free entropy (30) and its fit with a function a+b2x +c3x... for the 6-coloring of 19-regular graphs and the 4-coloring of 9-regular graphs. Circles give the analytical results at m = 0 and m = 1. On the right parts, we present the complexity versus internal entropy with the numerical points and the Legendre transform of the fit of the free entropy. The analytical result for Σmax is also shown. In the case of bi-regular random graphs, one needs two different populations: one for the fields going from nodes with connectivity c1 and one for the fields going from nodes with connectivity c2. Then each iteration for population 1 (resp. 2) should be performed using as incoming messages the vectors of population 2 (resp. 1.). Again, one can separately perform the recursion for the hard-field fractions in both population. The case of Erdős-Rényi random graphs is more involved, as one needs a large number Npop of populations, each of them of size N . In this case, using the separate hard/soft fields implementation and the formulae (D1,D2) for complexity is crucial, as it allows a good precision even for smaller population sizes. We used typically 2Npop/c ≈ (1 − 3) · 103 and N ≈ (1 − 3) · 102. The error bars in table II are computed from several independent runs of the population dynamics. In each case we were able to make the equilibration times and the population sizes large enough such that by doubling the time or the population size we did not observed any significant systematic changes in the average results. APPENDIX E: HIGH-q ASYMPTOTICS The quenched averages in the large q limit are the same for the regular and Erdős-Rényi graphs and we thus consider directly the regular ensemble of connectivity c = k + 1. The appearance of a nontrivial 1RSB solution for m = 0, which correspond to cSP, was already computed in [31] and reads cr(m = 0) = cSP = q [log q + log log q + 1 − log 2 + o(1)] , (E1) ηd(m = 0) = 1 − 1 log q log q , (E2) while the coloring threshold is [31] cs = 2q log q − log q − 1 + o(1). (E3) We now show how the connectivity where a solution with hard fields at m = 1 first appears, and how the complete free entropy Φs(m) (30) can be computed close to the COL/UNCOL transition 1. The appearance of hard fields at m = 1 We first show that the correct scaling for the appearance of hard fields at m = 1 is k = q[log q + log log q + α]. (E4) and compute the value of α. In the order O(q) we can write also k = (q − 1)[log(q − 1) + log log(q − 1) + α]. The starting point is the equation (C9), with Q1(x) = δ(x − k). In the large q limit the fraction of hard fields is µ(q, k) = qη(q, k) = 1− θ(q, k), where θ(q, k) = o(1) is the fraction of soft fields. We check self-consistently at the end of the computation that only the two first terms of (C9) are important. Then we have µ(q, k) = 1 − (q − 1) 1 − 1 q − 1µ(q, k) = 1 − (q − 1)e− kµ(q,k) q−1 . (E5) A self-consistent equation for θ(q, k) follows log(q − 1)θ(q, k) = (q − 1)θ(q,k)e−α , (E6) which is solved by θ(q, k) = γ(α)/ log(q − 1) where γ(α)e−γ(α) = e−α . (E7) The maximum of the left hand side is 1/e for γ(α) = 1. It means that a solution of (E7) exists for α > 1. Finally the hard fields appear in the 1RSB solution for m = 1 at connectivity cr(m = 1) = q[log q + log log q + 1 + o(1)]. (E8) The clustering transition cd should be between cSP and cr(m = 1) as this is what we observed for finite q. We see that cSP and cr(m = 1) differs only in the third order and both are very far from the coloring threshold and also from the condensation transition as we show in section E 2. It would be interesting to compute a large q expansion of the connectivity at which the hard fields appears in all the clusters (for all m such that Σ(m) > 0). Together with our conjecture about rigidity being responsible for the computational hardness that might give a hint about the answer on the long-standing question [22]: “Is there a polynomial algorithm and ǫ such that the algorithm would color random graphs of average connectivity (1 + ǫ)q log q for all large q?” 2. The condensation transition To compute the large-q asymptotic of the condensation transition, we first need to derive the large-q expansion of the free entropy (30) in the connectivity regime c = 2q log q. Let us show self-consistently that the following scaling is relevant for the condensation transition in the large q limit cs = 2q log q − γ log q + α , (E9) , (E10) and compute the constants γ, α, B. Using the above scaling, the function w(η) (35) is dominated by the first two terms in numerator and denominator, and reads in the first two leading orders 1 − qw(η) ≈ qe so that w(η) = log q (E11) independently of γ, α, and B. To take into account the reweighting we expand eq. (38) in the two leading orders log q . (E12) Note that almost all the incoming fields are hard, i.e. have one component of value 1. Since there are on average only 2B log q incoming soft fields, the leading order of the hard-field reweighting (the normalization in eq. (4)) is different from 1 with a probability only O(log q/q). Similarly, almost all the soft fields have two nonzero and equal components. The normalization in eq. (4) is thus almost surely 2, thus the average reweighting factor of the soft fields is Zms = 2 m + O log q . (E13) Finally, log q . (E14) Therefore the constant B in (E10) is B = 2m/2, independently of γ and α. The computation of the complexity requires the next order in the hard-field reweighting. Indeed the normalization in (4) might not be 1 but 1/2; and this happens when there is a soft field arriving of the color corresponding to the hard field in consideration. The probability of this event is 2c(1−qη) = O( q log q ). The hard-field reweighting is thus 1 − 2c (1 − qη) 2c (1 − qη) log q . (E15) We now expand the replicated free energy (30) in the large q limit and regime (E9). Remind that from (6, 8) Φs(m) = log (Z m − c . (E16) The averages are over the population in the sense of (21). The site free energy is the logarithm of the average of the total field normalization. This average can be split into three parts when (i) the total field is a hard field, (ii) the total field is a soft field and (iii) the total field is contradictory (and its normalization zero). The probability that the total field is not contradictory is the denominator in eq. (35) g(η) = (−1)l l + 1 [1 − (l + 1)η]c, (E17) where again only the first two terms are relevant in the expansion. The site free energy is then log (Zi0) m = log g(η) + log q w(η)Zm + (1 − q w(η))Zms ≃ log q(1 − η)c − q(q − 1) (1 − 2η)c + log 1 − 1 − 2c (1 − qη) 1 − 1 .(E18) where q [1 − η]c + q (q − 1) [1 − 2η]c = log q + c log [1 − η] + log 1 − q − 1 1 − 2η 1 − η (E19) ≈ log q + c log 1 − 1 + log 1 − 1 + o(1/q) . (E20) To compute the link contribution in (E16) we need to consider two fields ψi→js and ψ s and to compute 0 = 1 − ψi→js ψ s . (E21) There are three different cases: 1. Two hard fields are chosen, then Z 0 = 0 with probability qη 2 (this is of order 1/q) and Z 0 = 1 with probability q(q − 1)η2 (this is of order 1). 2. Two soft fields are chosen then Z 0 = 1 with probability (1 − qη)2 (this is of order 1/q2), all other situations being O(1/q3). Let us remind that the dominant soft fields are two-component of type (1/2, 1/2, 0, 0, . . . ). 3. One hard and one soft field is chosen, then Z 0 = 1 with probability 2η(1 − qη)(q − 2)/q, and Z 0 = 1/2 with probability 4η(1 − qη) (this is of order 1/q2). On average, one thus obtain for the link contribution = log 1 − qη2 − 4η (1 − qη) 1m + 4η (1 − qη) 1 . (E22) Putting together the two pieces (E18) and (E22), expanding η according to (E14) and considering only the highest order in c, we can finally write the free energy as Φs(m) = log q − 2m − 2 . (E23) The internal entropy s(m) and the complexity Σ = Φs(m) −ms(m) are then s(m) = ∂Φs(m) 2m log 2 , (E24) Σ(m) = log q − c 2m − 2 −m 2m log 2 , (E25) and the complexity is thus zero for cΣ=0 = 2q log q − log q − 2 + 2m [1 −m log 2] + o(1). In particular, one has for the coloring and the condensation thresholds cΣ=0(m = 0) = 2q log q − log q − 1 + o(1) , (E26) cΣ=0(m = 1) = 2q log q − log q − 2 log 2 + o(1) . (E27) For connectivity c = 2q log q − log q + α, one gets 2qs(m) ≃ 2m log 2 , (E28) 2qΣ(m) ≃ 2m − 2 −m2m log 2 − α . (E29) [1] S. 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Probab. 25, 855-900 (1997). [83] L. Devroye, Non-Uniform Random Variate Generation, Springer-Verlag, New York, 1986 [84] T. Castellani, F. Krzakala and F. Ricci-Tersenghi; Eur. Phys. J. B 47, 99 (2005) [85] More precisely [7] proves the existence of a sharp threshold for a possibly size dependent sequence of thresholds, whose convergence is not proven. [86] In this paper we use the words ”cluster” and ”state” as synonyms. [87] A slightly different adaptation, closer to [76], was performed few years ago by Andrea Pagnani and Martin Weigt (private communication). Introduction The Model Definition of the model Ensembles of Random Graphs The cavity formalism at the replica symmetric level The replica symmetric cavity equations Average over the ensemble of graphs and the RS solution Validity conditions of the replica symmetric solution The Gibbs measure uniqueness condition The Gibbs measure extremality condition The local stability: a simple self-consistency check One-step replica symmetry breaking framework Analyzing the 1RSB equations Zero temperature limit The role of frozen variables Hard fields in the simplest case, m=0 Generalized survey propagation recursion The presence of frozen variables Validity conditions of the 1RSB solution The coloring of random graphs: cavity results Regular random graphs Results for the bi-regular ensemble Results for Erdos-Rényi random graphs The general case for q>3, discontinuous clustering transition The special case of 3-coloring, continuous clustering transition The overlap structure Large q Asymptotics Algorithmic consequences The whitening procedure A Walk-COL algorithm to color random graph A belief propagation algorithm to color random graph Conclusions Acknowledgments Stability of the paramagnetic solution The relative sizes of clusters in the condensed phase The 1RSB formalism at m=1 and the reconstruction equations Numerical Methods High-q asymptotics The appearance of hard fields at m=1 The condensation transition References

0704.1270

Core-Corona Separation in Ultra-Relativistic Heavy Ion Collisions

Core-Corona Separation in Ultra-Relativistic Heavy Ion Collisions Klaus Werner∗ SUBATECH, University of Nantes – IN2P3/CNRS– EMN, Nantes, France Abstract: Simple geometrical considerations show that the collision zone in high energy nuclear collisions may be divided into a central part (“core”), with high energy densities, and a peripheral part (“corona”), with smaller energy densities, more like in pp or pA collisions. We present calcula- tions which allow to separate these two contributions, and which show that the corona contribution is quite small (but not negligible) for central collisions, but gets increasingly important with de- creasing centrality. We will discuss consequences concerning results obtained in heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Super Proton Synchrotron (SPS). Nuclear collisions at the Relativistic Heavy Ion Col- lider (RHIC) are believed to provide sufficiently high en- ergy densities to create a thermalized quark-gluon fireball which expands by developing a strong collective radial flow [1, 2, 3, 4]. However, not all produced hadrons par- ticipate in this collective expansion: the peripheral nu- cleons of either nucleus essentially perform independent pp or pA-like interactions, with a very different particle production compared to the high density central part. For certain observables, this “background” contribution spoils the “signal”, and to get a detailed understanding of RHIC and SPS data, we need to separate low and high density parts. In order to get quantitative results, we need a simula- tion tool, and here we take EPOS [5], which has proven to work very well for pp and dAu collisions at RHIC. EPOS is a parton model, so in case of a AuAu collision there are many binary interactions, each one represented by a parton ladder. Such a ladder may be considered as a lon- gitudinal color field, conveniently treated as a relativistic string. The strings decay via the production of quark- antiquark pairs, creating in this way string fragments – which are usually identified with hadrons. Here, we mod- ify the procedure: we have a look at the situation at an early proper time τ0, long before the hadrons are formed: we distinguish between string segments in dense areas (more than ρ0 segments per unit area in given transverse slices), from those in low density areas. We refer to high density areas as core, and to low density areas as corona. In figure. 1, we show an example (randomly chosen) of a semi-peripheral (40-50%) AuAu collisions at 200 GeV (cms), simulated with EPOS. There is always a contribution from the low density area, but much more importantly, as discussed later, the importance of this contribution depends strongly on par- ticle type and transverse momentum. For central colli- sions, the low density contribution is obviously less im- portant, for more peripheral collisions this contribution will even dominate. We adopt the following strategy: the low density part will be treated using the usual EPOS particle production which has proven to be very successful in pp and dAu scattering (the peripheral interactions are essentially pp -6 -4 -2 0 2 4 6 40-50% FIG. 1: A Monte Carlo realization of a semi-peripheral (40- 50%) AuAu collision at 200 GeV (cms). We show string seg- ments in the core (full gray circles) and the corona (open circles). The big circles are put in just to guide the eye: they represent the two nuclei in hard sphere approximation.We consider a projection of segments within z = ±0.4 fm to the transverse plane (x,y). or pA scatterings). For the high density part, we sim- ply try to parameterize particle production, in the most simple way possible (it is not at all our aim to provide a microscopic description of this part). In practice, we consider transverse slices characterized by some range in η = 0.5 ln(t+z)/(t−z). String segments in such a slice move with rapidities very close to η. We subdivide a given slice into elementary cells, count the number of string segments per cell, and determine such for each cell whether it belongs to the core or the corona (bigger or smaller than the critical density ρ0). Con- nected cells (closest neighbors) in a given slice are con- sidered to be clusters, whose energy and flavor content are completely determined by the corresponding string segments. Clusters are then considered to be collectively expanding: Bjorken-like in longitudinal direction with in addition some transverse expansion. We assume par- ticles to freeze out at some given energy density εFO, having acquired at that moment a collective radial flow. The latter one is characterized by a linear radial rapidity profile from inside to outside with maximal radial rapid- ity yrad. In addition, we impose an azimuthal asymme- try, being proportional to the initial spatial eccentricity y2 − x2 y2 + x2 , with a proportionality factor fecc. By imposing radial flow, we have to rescale the http://arxiv.org/abs/0704.1270v1 cluster mass M as M → M × 0.5 y2rad/(yrad sinh yrad − cosh yrad + 1), in order to conserve energy. Hadronization then occurs according to covariant phase space, which means that the probability dP of a given final state of n hadrons is given speciesα d3pi gi si (2πh̄)32Ei δ(M − ΣEi) δ(Σ~pi) δf,Σfi , with pi = (Ei, ~pi) being the four-momentum of the i-th hadron, gi its degeneracy, and fi its quark flavor con- tent (u− ū,d− d̄...). The number nα counts the number of hadrons of species α. The term M/εFO is the clus- ter proper volume. We use a factor si = γs ±1 for each strange particle (sign plus for a baryon, sign minus for a meson), with γs being a parameter. We believe that si mimics final state rescattering, but for the moment we can only say that this factor being different from unity improves the fit of the data considerably. The whole procedure perfectly conserves energy, momentum, and flavors (microcanonical procedure). So the core definition and its hadronization are param- eterized in terms of few global parameters (in brackets the values): the core formation time τ0 (1 fm), the core formation density ρ0 (2/fm 2), the freeze out energy den- sity εFO(0.22GeV/fm 3), the maximum radial flow yrad (0.75+0.20log( s/200GeV)), the eccentricity coefficient fecc (0.45), and the factor γs (1.3). At RHIC energies, the final results are insensitive to variations of τ0: even changes as big as a factor of 2 do not affect the results. This is a nice feature, indicating that the very details of the initial state do not matter so much. We call these pa- rameters “global”, since they account for all observables at all possible different centralities and all energies. In the following, we are going to discuss results, all obtained with the above set of parameters. Our hadronization of the core part is certainly moti- vated by the remarkable success of statistical hadroniza- tion models [6] and blast-wave fits [7, 8]. We use co- variant statistical hadronization, whereas usual the non- covariant procedure is employed, but the difference is minor. We also impose a collective flow, with an as- sumed flow profile, as in the blast wave fit. So the gen- eral ideas are the same. However, a really new aspect is the possibility of making a “global fit”, considering all energies, centralities, and colliding systems with one set of parameters. In the above-mentioned models one has a set of fit parameters for each of these possibili- ties. An important new aspect is also the separation of a (collectively behaving) core and a corona contribution, which seems to be very important for understanding the centrality dependence of hadron yields. Finally, our sta- tistical hadronization is based on initial energy densities provided by a parton model (EPOS), which works well 0 0.5 1 1.5 2 2.5 3 mt-m EPOS 1.30 ____ core 0-5% AuAu 200GeV _ _ _ pp 200GeV FIG. 2: Invariant yields 1/2πmt dn/dydmt of pions and lamb- das, for the core contribution corresponding to a central (0- 5%) AuAu collision (full lines) and proton-proton scattering (dashed lines). The core spectra are divided by the number of binary collisions. EPOS 1.30 ..... 0-5% _ _ _ 40-50% ____ 70-80% 0 1 2 mt-m (GeV) 0 1 2 3 mt-m (GeV) FIG. 3: The relative contribution of the core (core/(core+corona)) as a function of the transverse mass for different hadrons (π, K, p, Λ) at different centralities. for pp and dAu scattering. This fixes the overall multi- plicity already within 10%, flow and freeze out condition have only a minor effect on this quantity. All the discussion of heavy ion data will be based on the interplay between core and corona contributions. To get some feeling, we first compare in fig. 2 the mt spec- tra of pions and lambdas from the core in central (0- 5%) AuAu collisions with the corresponding spectra in pp scattering (which is qualitatively very similar to the corona contribution). The core spectra are divided by the number of binary collisions. We observe several re- markable features: the shapes of the pion and lambda curves in pp are not so different, whereas there is much more species dependence in the core spectra, since the heavier particles acquire large transverse momenta due to the flow effect. One observes furthermore that the 0 50 100 150 200 250 300 350 400 participant number Np EPOS 1.30 FIG. 4: Rapidity density dn/dy per participant as a function of the number of participants (Np) in Au+Au collisions at 200 GeV (RHIC) for π−, K−, p̄, Λ̄, Ξ̄+. We show data (points) [9, 10] together with the full calculation (full lines) and just the core part (dotted lines). yields for the two spectra in pp are much wider spread than the ones from the core; in particular, pion produc- tion is suppressed in the core hadronization compared to pp, whereas lambda production is favored. All this is quite trivial, but several “mysteries” discussed in the lit- erature (and to be discussed later in this paper) are just due to this. In fig. 3, we plot the relative contribution of the core (relative to the complete spectrum, core + corona) as a function of mt−m, for different particle species. For cen- tral collisions, the core contribution dominates largely, whereas for semi-central collisions (40-50%) and even more for peripheral collisions the core contribution de- creases, giving more and more space for the corona part. Apart of these general statements, the precise mt depen- dence of the relative weight of core versus corona depends on the particle type. We are now ready to investigate data. In fig. 4, we plot the centrality dependence of the particle yield per participant (per unit of rapidity) in Au+Au collisions at 200 GeV (RHIC), for π+, K+, p, Λ̄, Ξ̄+: we show data [9, 10] together with the full calculation (quite close to the data), but also indicating the core contribution. In fig. 5, we show the corresponding results for Pb+Pb colli- sions at 17.3 GeV (SPS). Concerning the SPS results, we consider dn/dy/Np in case of Ks, Λ̄, and Ξ̄ +, whereas we have 4π multiplicities per participant in case of π− and K−(for simulations and data). Whereas central collisions are always clearly core dominated, the core contributes less and less with decreasing centrality. The difference between solid and dotted curves (in other words: the importance of the corona contribution) is bigger at the SPS compared to RHIC, and it is bigger for light parti- cles compared to heavy ones. For example there is a big corona contribution for pions and a very small one for Ξ̄ 0 50 100 150 200 250 300 350 400 participant number EPOS 1.30 FIG. 5: Multiplicity per participant as a function of the num- ber of participants (Np) in Pb+Pb collisions at 17.3 GeV (SPS) for π−, K−, Ks, Λ̄, Ξ̄ +. We show data (points) [11, 12, 13, 14] together with the full calculation (full lines) and just the core part (dotted lines). 0 50 100 150 200 250 300 350 400 participant number Np EPOS 1.30only core π+ K+ p Λ at 200 GeV π-(4π) K-(4π) Ks Λ at 17.3 GeV FIG. 6: Multiplicity per participant as a function of Np for only the core part. We show results for π−, K−, p̄, Λ̄, Ξ̄+ in Au+Au collisions at 200 GeV (dotted lines), and for π−, K−, Ks, Λ̄, Ξ̄ + in Pb+Pb collisions at 17.3 GeV (dashed lines). particles. Also the strength of the centrality dependence depends on the hadron type: for example Ξ̄+particles show a stronger centrality dependence than pions. It seems that the centrality dependence is essentially deter- mined by relative importance of the corona contribution: the less the corona contributes, the more the yield varies with centrality. To further investigate the connection between relative corona weight and centrality dependence, we plot in fig. 6 the centrality dependence of multiplicities per partici- pant for different hadrons, at 200 GeV (RHIC) and 17.3 GeV (SPS), for the core contribution. We observe two universal curves, one per energy. So for a given energy, the core contributions for all the different hadrons show the same centrality dependence. This proves that the different centrality dependencies for the different hadron 0 0.5 1 1.5 2 2.5 3 EPOS 1.30 0-5% AuAu 200GeV FIG. 7: Nuclear modification factors in central AuAu colli- sions at 200 GeV. Lines are full calculations, symbols rep- resent data [9, 10]. We show results for pions (dashed line; triangles), protons (full line; circles), and lambdas (dashed- dotted line; squares). species are simply due to different core-corona weights. For example the fact that Ξ̄ particles show a stronger cen- trality dependence than pions is simply due to the fact that the former ones have less corona admixture than the latter ones. Lets us come to pt spectra. We checked all available pt data (π +, K+, p, Λ̄, Ξ̄+, for pt ≤ 5GeV), and our combined approach (core + corona) describes all the data within 20%. Lacking space, we just discuss a (typical) ex- ample: the nuclear modification factor (AA/pp/number of collisions), for π+, p, Λ̄ in central AuAu collisions at 200 GeV, see fig. 7. For understanding these curves, we simply have a look at fig. 2, where we compare the core contributions from AuAu (divided by the number of bi- nary collisions) with pp. Since for very central collisions the core dominates largely, the ratio of core to pp (the solid lines divided by the dotted ones in fig. 2) corre- sponds to the nuclear modification factor. We discussed already earlier the very different behavior of the core spectra (flow plus phase space decay) compared to the pp spectra (string decay): pions are suppressed, whereas heavier particles like lambdas are favored. Or better to say it the other way round: the production of baryons compared to mesons is much more suppressed in string decays than in statistical hadronization. This is why the nuclear modification factor for lambdas is different from the one for pions. So what we observe here is nothing but the very different behavior of statistical hadroniza- tion (plus flow) on one hand, and string fragmentation on the other hand. This completely statistical behavior indicates that the low pt partons get completely absorbed in the core matter. The Rcp modification factors (central over peripheral) are much less extreme than RAA, since peripheral AuAu collisions are a mixture of core and corona (the latter one being pp-like), so a big part of the effect seen in RAA is simply washed out. To summarize: we have discussed the importance of separating core and corona contributions in ultra- relativistic heavy ion collisions. The core-corona sepa- ration is realized based on the determination of string densities at an early time. Particle production from the corona is done as in proton-proton scattering, whereas the core hadronization is parameterized in a very sim- ple way, imposing radial flow. The corona contribution is quite small (but not negligible) for central collisions, but gets increasingly important with decreasing central- ity. The core shows a very simple centrality dependence: it is the same for all hadron species, at a given bombard- ing energy. The fact that the centrality dependence of the total hadron yield is strongly species dependent, is simply due to the fact that the relative corona contribu- tion depends on the hadron type. ∗ Electronic address: [email protected] [1] I.Arsene et.al, BRAHMS Collaboration, Nucl. Phys. A757, 1-27, (2005) [2] K. Adcox et al., PHENIX Collaboration, Nucl. Phys. A757, 184-283 (2005) [3] B.B. Back, PHOBOS Collaboration, Nucl. Phys. A 757, 28 (2005) [4] J. Adams et al., STAR Collaboration, Nucl. Phys. A757, 102-183 (2005) [5] K. Werner, F.M. Liu, T. Pierog, Phys. Rev. C 74, 044902 (2006), hep-ph/0506232 [6] P. Braun-Munzinger, I. Heppe, J. Stachel, Phys. Lett B465 (1999) 15; F. Becattini, J. Manninen , M. Gazdz- icki , Phys.Rev. 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Lett. 93 (2004) 022302 http://arxiv.org/abs/hep-ph/0506232 http://arxiv.org/abs/hep-ph/0511092 http://arxiv.org/abs/hep-ph/0511094 http://arxiv.org/abs/nucl-ex/0405024

0704.1271

Numerical Evaluation of Six-Photon Amplitudes

arXiv:0704.1271v2 [hep-ph] 23 Jul 2007 Preprint typeset in JHEP style - PAPER VERSION Numerical Evaluation of Six-Photon Amplitudes Giovanni Ossola∗, Costas G. Papadopoulos† Institute of Nuclear Physics, NCSR ”DEMOKRITOS”, 15310 Athens, Greece. Roberto Pittau‡ Dipartimento di Fisica Teorica, Univ. di Torino and INFN, sez. di Torino, Italy. Abstract: We apply the recently proposed amplitude reduction at the integrand level method, to the computation of the scattering process 2γ → 4γ, including the case of a massive fermion loop. We also present several improvements of the method, including a general strategy to reconstruct the rational part of any one-loop amplitude and the treatment of vanishing Gram-determinants. Keywords: NLO Computations, Hadronic Colliders, Standard Model, QCD. ∗e-mail: [email protected] †e-mail: [email protected] ‡e-mail: [email protected] http://arxiv.org/abs/0704.1271v2 Contents 1. Introduction 1 2. The method and the computation of the rational terms 2 3. Dealing with numerical instabilities 8 4. Results and comparisons 11 5. Conclusions 14 A. Computing the extra-integrals 15 B. The general basis for the 2-point functions 17 1. Introduction In the last few years a big effort has been devoted by several authors to the problem of an efficient computation of one-loop corrections for multi-particle processes. This is a problem relevant for both LHC and ILC physics. In the case of QCD, the NLO six gluon amplitude has been recently obtained by two different groups [1], and, in the case of e+e− collisions, complete EW calculations, involving 5-point [2] and 6-point [3] loop functions are available at the cross section level. The used techniques range from purely numerical methods to an- alytic ones, also including semi-numerical approaches. For analytical approaches, the main issue is reducing, using computer algebra, generic one-loop integrals into a minimal set of scalar integrals (and remaining pieces, the so called rational terms), mainly by tensor re- duction [4–7]. For multi-particle processes though this method becomes quite cumbersome because of the large number of terms generated and the appearance of numerical insta- bilities due to the zeros of Gram-determinants. On the other hand, several numerical or semi-numerical methods aim for a direct numerical computation of the tensor integrals [8]. Although purely numerical methods can in principle deal with any configuration of masses and also allow for a direct computation of both non-rational and rational terms, their applicability remains limited due to the high demand of computational resources and the non-existence of an efficient automation. In a different approach, the one-loop amplitude rather than individual integrals are evaluated using the unitarity cut method [9], which relies on tree amplitudes and avoids the computation of Feynman diagrams. In another development, the four-dimensional unitarity cut method has been used for the calculation of QCD amplitudes [10], using – 1 – twistor-based approaches [11]. Moreover, a generalization of the the unitarity cut method in d dimensions, has been pursued recently [12]. Nevertheless, in practice, only the part of the amplitude proportional to the loop scalar functions can be obtained straightforwardly. The remaining piece, the rational part, should then be reconstructed either by using a direct computation based on Feynman diagrams [13–15] or by using a bootstrap approach [16]. Furthermore the complexity of the calculation increases away from massless theories. In a recent paper [17], we proposed a reduction technique for arbitrary one-loop sub- amplitudes at the integrand level by exploiting numerically the set of kinematical equations for the integration momentum, that extend the quadruple, triple and double cuts used in the unitarity-cut method. The method requires a minimal information about the form of the one-loop (sub-)amplitude and therefore it is well suited for a numerical implementation. The method works for any set of internal and/or external masses, so that one is able to study the full electroweak model, without being limited to massless theories. In this paper, we describe our experience with the first practical non-trivial imple- mentation of such a method in the computation of a physical process: namely 2γ → 4γ, including massive fermion loops. For the massless case, there are a few results available in the literature. Analytical expressions were first presented by Mahlon [18] some time ago, however his results do not cover all possible helicity configurations. More recently the complete set of six-photon amplitudes was computed numerically by Nagy and Soper [19]. Very recently the same results were also obtained by Binoth et al. [20], that also provide compact analytical expressions. In section 2, we recall the basics of our method and, in particular, we show how the knowledge of the rational terms can be inferred, with full generality, once the coefficients of the loop functions have been determined. In section 3, we outline our solution to cure the numerical inaccuracies related to the appearance of zeros of Gram-determinants. We explicitly illustrate the case of 2-point amplitudes, that we had to implement to deal with the process at hand. In section 4, we present our numerical results. For massless fermion loops we compare with available results. Moreover, since we are not limited to massless contributions, we also present, for the first time, results with massive fermion loops. Finally, in the last section, we discuss our conclusions and future applications. 2. The method and the computation of the rational terms The starting point of the method is the general expression for the integrand of a generic m-point one-loop (sub-)amplitude [17] A(q̄) = D̄0D̄1 · · · D̄m−1 , D̄i = (q̄ + pi) 2 −m2i , p0 6= 0 , (2.1) where we use a bar to denote objects living in n = 4 + ǫ dimensions, and q̄2 = q2 + q̃2 1. In the previous equation, N(q) is the 4-dimensional part of the numerator function of 1q̃2 is ǫ-dimensional and (q̃ · q) = 0. – 2 – the amplitude 2. N(q) depends on the 4-dimensional denominators Di = (q+ pi) 2 −m2i as follows N(q) = i0<i1<i2<i3 d(i0i1i2i3) + d̃(q; i0i1i2i3) i 6=i0,i1,i2,i3 i0<i1<i2 [c(i0i1i2) + c̃(q; i0i1i2)] i 6=i0,i1,i2 i0<i1 b(i0i1) + b̃(q; i0i1) i 6=i0,i1 [a(i0) + ã(q; i0)] i 6=i0 + P̃ (q) Di . (2.2) Inserted back in Eq. (2.1), this expression simply states the multi-pole nature of any m- point one-loop amplitude, that, clearly, contains a pole for any propagator in the loop, thus one has terms ranging from 1 to m poles. Notice that the term with no poles, namely that one proportional to P̃ (q) is polynomial and vanishes upon integration in dimensional regularization; therefore does not contribute to the amplitude, as it should be. The coefficients of the poles can be further split in two pieces. A piece that still depend on q (the terms d̃, c̃, b̃, ã), that vanishes upon integration, and a piece that do not depend on q (the terms d, c, b, a). Such a separation is always possible, as shown in Ref. [17], and, with this choice, the latter set of coefficients is therefore immediately interpretable as the ensemble of the coefficients of all possible 4, 3, 2, 1-point one-loop functions contributing to the amplitude. Once Eq. (2.2) is established, the task of computing the one-loop amplitude is then reduced to the algebraical problem of determining the coefficients d, c, b, a by evaluating the function N(q) a sufficient number of times, at different values of q, and then inverting the system. That can be achieved quite efficiently by singling out particular choices of q such that, systematically, 4, 3, 2 or 1 among all possible denominators Di vanishes. Then the system of equations is solved iteratively. First one determines all possible 4-point functions, then the 3-point functions and so on. For example, calling q±0 the 2 (in general complex) solutions for which D0 = D1 = D2 = D3 = 0 , (2.3) (there are 2 solutions because of the quadratic nature of the propagators) and since the functional form of d̃(q; 0123) is known, one directly finds the coefficient of the box diagram containing the above 4 denominators through the two simple equations N(q±0 ) = [d(0123) + d̃(q 0 ; 0123)] i 6=0,1,2,3 0 ) . (2.4) 2If needed, the ǫ-dimensional part of the numerator should be treated separately, as explained in [21]. – 3 – This algorithm also works in the case of complex denominators, namely with complex masses. Notice that the described procedure can be performed at the amplitude level. One does not need to repeat the work for all Feynman diagrams, provided their sum is known: we just suppose to be able to compute N(q) numerically. As a further point notice that, since the terms d̃, c̃, b̃, ã still depend on q, also the separation among terms in Eq. (2.2) is somehow arbitrary. Terms containing a different numbers of denominators can be shifted from one piece to the other in Eq. (2.2), by relaxing the requirement that the integral over the terms containing q vanishes. This fact provides an handle to cure numerical instabilities occurring at exceptional phase-space points. In Section 3 we will show in detail such a mechanism at work for the 2-point part of the amplitude. The described procedure works without any modification in 4 dimensions. However, even when starting from a perfectly finite tensor integral, the tensor reduction may even- tually lead to integrals that need to be regularized. A typical example are the rank six 6-point functions contributing to the scattering 2γ → 4γ we want to study. Such tensors are finite, but tensor reduction iteratively leads to rank m m-point tensors with 1 ≤ m ≤ 5, that are ultraviolet divergent when m ≤ 4. For this reason, we introduced, in Eq. (2.1), the d-dimensional denominators D̄i, that differs by an amount q̃ 2 from their 4-dimensional counterparts D̄i = Di + q̃ 2 . (2.5) The result of this is a mismatch in the cancellation of the d-dimensional denominators of Eq. (2.1) with the 4-dimensional ones of Eq. (2.2). The rational part of the amplitude comes from such a lack of cancellation. In [17] the problem of reconstructing this rational piece has been solved by looking at the implicit mass dependence in the coefficients d, c, b, a of the one-loop functions. Such a method is adequate up to 4-point functions; for higher-point functions the dependence becomes too complicated to be used in practice. In addition, it requires the solution of further systems of linear equations, slowing down the whole computation. For those reasons, we suggest here a different method. One starts by rewriting any denominator appearing in Eq. (2.1) as follows , with Z̄i ≡ . (2.6) This results in A(q̄) = D0D1 · · ·Dm−1 Z̄0Z̄1 · · · Z̄m−1 . (2.7) Then, by inserting Eq. (2.2) in Eq. (2.7), one obtains A(q̄) = i0<i1<i2<i3 d(i0i1i2i3) + d̃(q; i0i1i2i3) D̄i0D̄i1D̄i2D̄i3 i 6=i0,i1,i2,i3 – 4 – i0<i1<i2 c(i0i1i2) + c̃(q; i0i1i2) D̄i0D̄i1D̄i2 i 6=i0,i1,i2 i0<i1 b(i0i1) + b̃(q; i0i1) D̄i0D̄i1 i 6=i0,i1 a(i0) + ã(q; i0) i 6=i0 + P̃ (q) Z̄i . (2.8) The rational part of the amplitude is then produced, after integrating over dnq, by the q̃2 dependence coming from the various Z̄i in Eq. (2.8). It is easy to see what happens, for any value of m, by recalling the generic q dependence of the spurious terms. In the renormalizable gauge one has [17] P̃ (q) = 0 , ã(q; i0) = ã µ(i0; 1)(q + pi0)µ , b̃(q; i0i1) = b̃ µ(i0i1; 1)(q + pi0)µ + b̃ µν(i0i1; 2)(q + pi0)µ(q + pi0)ν , c̃(q; i0i1i2) = c̃ µ(i0i1i2; 1)(q + pi0)µ + c̃ µν(i0i1i1; 2)(q + pi0)µ(q + pi0)ν , + c̃µνρ(i0i1i1; 3)(q + pi0)µ(q + pi0)ν(q + pi0)ρ , d̃(q; i0i1i2i3) = d̃ µ(i0i1i2i3; 1)(q + pi0)µ . (2.9) Eq. (2.9) simply states the fact that ã(q; i0) and d̃(q; i0i1i2i3) are at most linear in (q+pi0), b̃(q; i0i1) at most quadratic, and c̃(q; i0i1i2) at most cubic. The tensors denoted by (· · · ; 1), (· · · ; 2) and (· · · ; 3) stand for the respective coefficients. We will also make use of the fact that, due to the explicit form of the spurious terms [17] c̃µν(i0i1i2; 2) gµν = 0 , c̃µνρ(i0i1i2; 3) gµν = c̃ µνρ(i0i1i2; 3) gµρ = c̃ µνρ(i0i1i2; 3) gνρ = 0 and b̃µν(i0i1; 2) gµν = 0 . (2.10) The necessary integrals that arise, after a change of variable q → q − pi0 , are of the form (n;2ℓ) s;µ1···µr ≡ dnq q̃2ℓ qµ1 · · · qµr D̄(k0) · · · D̄(ks) , with D̄(ki) ≡ (q̄ + ki) 2 −m2i , ki ≡ pi − p0 (k0 = 0) , (2.11) where we used a notation introduced in [22] and r ≤ 3. Such integrals (from now on called extra-integrals) have dimensionality D = 2(1+ ℓ− s)+ r and give a contribution O(1) only when D ≥ 0, otherwise are of O(ǫ). This counting remains valid also in the presence of infrared and collinear divergences, as explained, for example, in Appendix B of [22] and in [14]. – 5 – We also note that, since all Z̄i are a-dimensional, the dimensionality D of the extra- integrals generated through Eq. (2.8) does not depend on m. We list, in the following, all possible contributions, collecting the computational details in Appendix A. Contributions proportional to d(i0i1i2i3) In this case r = 0. All extra-integrals are therefore scalars with D = −4 and do not contribute. Contributions proportional to d̃µ(i0i1i2i3; 1) In this case r = 1. All extra-integrals are therefore rank one tensors with D = −3 and do not contribute. Contributions proportional to c(i0i1i2) In this case r = 0 with D = −2 and no contribution O(1) is developed. Contributions proportional to c̃µ(i0i1i2; 1) Here r = 1 and D = −1. Therefore, once again, there is no contribution. Contributions proportional to c̃µν(i0i1i2; 2) Now r = 2 with D = 0 and a finite contribution is in principle expected, generated by extra-integrals of the type I(n;2(s−2))s;µν . (2.12) Nevertheless, such contribution is proportional to gµν [22]. Therefore, due to Eq. (2.10), it vanishes. Contributions proportional to c̃µνρ(i0i1i2; 3) Now r = 3 and D = 1. The contributing extra-integrals are of the type I(n;2(s−2))s;µνρ , (2.13) and one easily proves that the contributions O(1) are always proportional to gµν or gµρ or gνρ. Therefore, thanks again to Eq. (2.10), they vanish. Contributions proportional to b(i0i1) Those are the first non vanishing contributions. The relevant extra-integrals have r = 0 and D = 0 I(n;2(s−1))s , with 2 < s ≤ m − 1. They have been computed, for generic values of s, in [22] (see also Appendix A) I(n;2(s−1))s = −iπ s(s− 1) +O(ǫ) . (2.14) – 6 – Contributions proportional to b̃µ(i0i1; 1) In this case the relevant extra-integrals are 4-vectors with D = 1 I(n;2(s−1))s;µ with 2 < s ≤ m− 1 . A computation for generic values of s gives I(n;2(s−1))s;µ = iπ (s + 1)s(s − 1) (kj)µ +O(ǫ) . (2.15) Contributions proportional to b̃µν(i0i1; 2) The relevant extra-integrals are now rank two tensor with D = 2 I(n;2(s−1)s;µν with 2 < s ≤ m− 1 . They read I(n;2(s−1))s;µν = −2iπ (s+ 2)(s + 1)s(s− 1) (kj)µ(kj)ν + i 6=j (kj)µ(ki)ν + O(gµν) +O(ǫ) . (2.16) The gµν part is never needed because b̃ µν(i0i1; 2) gµν = 0, according to Eq. (2.10). Contributions proportional to a(i0) They involve scalar extra-integrals with D = 2 I(n;2s)s , with 1 < s ≤ m− 1 . One computes I(n;2s)s = −2iπ (s+ 2)(s + 1)s k2j + i 6=j (kj · ki) + (m2j − k + O(ǫ) . (2.17) Contributions proportional to ãµ(i0; 1) This last category involves extra-integrals with r = 1 and D = 3 I(n;2s)s;µ , with 1 < s ≤ m− 1 . One obtains I(n;2s)s;µ = iπ (s+ 3)(s + 2)(s + 1)s k2j (kj)µ + 2 i 6=j k2j (ki)µ + 2(kj · ki)(kj)µ – 7 – i 6=j ℓ 6=i (kj · ki)(kℓ)µ + (s+ 3) (m2j − k j )(kj)µ i 6=j (m2j − k j )(ki)µ +O(ǫ) . (2.18) To conclude, the set of the five formulas in Eqs. (2.14)-(2.18) allows one to compute the rational part of any one-loop m-point (sub-)amplitude, once all the coefficients of Eq. (2.2) have been reconstructed. 3. Dealing with numerical instabilities In this section we show how to handle, in the framework of the method illustrated in the previous section, the simplest numerical instability appearing in any one-loop calculation, namely that one related to the tensor reduction of 2-point amplitudes in the limit of vanishing Gram-determinant 3. This situation is simple enough to allow an easy description, but the outlined strategy is general and not restricted to the 2-point case. We start from the integrand of a generic 2-point amplitude written in the form A(q̄) = D̄0D̄1 , (3.1) in which we supposeN(q) at most quadratic in q. Our purpose is dealing with the situation in which k21 ≡ (p1−p0) 2 = 0 exactly (that always occur in processes with massless external particles), as well as to set up an algorithm to write down approximations around this case with arbitrary precision. According to Eq. (2.2), we can write an expansion for N(q) as follows: N(q) = [b(01) + b̃(q; 01)] + [a(0) + ã(q; 0)]D1 + [a(1) + ã(q; 1)]D0 . (3.2) If the Gram-determinant of the 2-point function is small, the reduction method introduced in [17] cannot be applied, because the solution for which D0 = D1 = 0, needed to determine the coefficients b and b̃, becomes singular4, in the limit of k21 → 0, when adding the requirement dnq b̃(q; 01) = 0 . (3.3) Then, we must consider two separate cases: k21 → 0 , but k 1 6= 0 , k21 → 0 , because k 1 = 0 . (3.4) 3In this case the Gram-determinant is simply the square of the difference between the momenta of the two denominators. 4Such a solution goes like 1/k21 . – 8 – The former situation may occur because of the Minkowskian metric, while the latter takes place at the edges of the phase-space, where some momenta become collinear. In the first case one can still find a solution for which D0 = D1 = 0 by relaxing the further requirement of Eq. (3.3). Such a solution is given in Appendix B and goes like 1/(k1.v), where v is an arbitrary massless 4-vector, therefore is never singular in the first case of Eq. (3.4). The price to pay is that new non zero integrals appear of the type 5 [(q + p0) · v] D̄0D̄1 with j = 1, 2 and v2 = 0 . (3.5) What has been achieved with this new basis is then moving part of the 1-point functions to the 2-point sector, in such a way that combinations well behaved in the limit k21 → 0 appear. The fact that solutions exist to the condition D0 = D1 = 0, still allows one to find the coefficients of such integrals (together with all the others). This solves the first part of the problem, namely reconstructing N(q) without knowing explicitly its analytic structure, but one is left with the problem of computing the new 2-point integrals. In the following, we present our method to determine them at any desired order in k21. Let us first consider the case j = 1 in Eq. (3.5). The contribution O(1) can be easily obtained from the observation that6 (q · v)(q · k1) D̄(k0)D̄(k1) = O(k21) , (3.6) as it is evident by performing a tensor decomposition. On the other hand, by reconstructing denominators, one obtains (q · k1) D̄(k1)− D̄(k0) (q · k1) + , (3.7) f = m21 − k 0 . (3.8) Eq. (3.7), inserted in Eq. (3.6) gives the desired expansion in terms of loop functions with less points but higher rank, in agreement with well know results [23,24] (q · v) D̄(k0)D̄(k1) dnq (q · v) D̄(k1) D̄(k0) 2(q · k1) +O(k21) .(3.9) Expansions at arbitrary orders in k21 can be obtained in an analogous way from the two following equations: (q · k1) D̄(k1)− D̄(k0) i+j=p−1 (q · k1) (q · v)(q · k1) D̄(k0)D̄(k1) = O(k 1 ) . (3.10) 5Since v2 = 0 they still fulfill the third one of Eqs. (2.10), therefore, even in this case, terms O(gµν) can be neglected in Eq. (2.16). 6From now on, we shift the integration variable: q̄ → q̄ − p0. The definition of the new resulting denominators is given in Eq. (2.11). – 9 – To deal with the case j = 2 in Eq. (3.5) one starts instead from the equation (q · v)2(q · k1) D̄(k0)D̄(k1) = O(k 1 ) . (3.11) This procedure breaks down when the quantity f vanishes. In this case a double expansion in k21 and f can still be found in terms of derivatives of one-loop scalar functions. We illustrate the procedure for the case j = 1 of Eq. (3.5). Our starting point is now the equation D̄(k0) = D̄(k1)− 2(q · k1) + f . (3.12) By multiplying and dividing by D̄(k0) one obtains (q · v) D̄(k0)D̄(k1) (q · v) D̄(k0)2D̄(k1) D̄(k1)− 2(q · k1) + f (q · v) D̄(k0)2 (q · v)(q · k1) D̄(k0)2D̄(k1) +O(f) . (3.13) Applying once more Eq. (3.12) to the last integral gives (q · v)(q · k1) D̄(k0) 2D̄(k1) (q · v)(q · k1) D̄(k0) 3D̄(k1) D̄(k1)− 2(q · k1) + f (q · v)(q · k1) D̄(k0)3 (q · v)(q · k1) D̄(k0)3D̄(k1) +O(f) . (3.14) Since the last integral in the previous equation is O(k21), the final result reads (q · v) D̄(k0)D̄(k1) (q · v) D̄(k0)2 (q · v)(q · k1) D̄(k0)3 +O(k21) +O(f) . (3.15) In a similar fashion, expansions at any order can be obtained. We now turn to the second case of Eq. (3.4), namely k 1 → 0. In this case no solution can be found to the double cut equation D(k0) = D(k1) = 0 . (3.16) The reason is that now D(k1) and D(k0) are no longer independent: D(k0) = D(k1) + f +O(k1) , (3.17) and clearly no q exists such that the two denominators can be simultaneously zero. Notice that this also implies that one cannot fit separately the coefficients of the 2-point and 1- point functions in Eq. (3.2). This results is a singularity 1/(k1 · v) in the system given of Appendix B and we should change our strategy. We than go back to Eq. (3.1) and split the amplitude from the beginning by multiplying it by D̄(k0)− D̄(k1) 2(q · k1) , (3.18) – 10 – resulting to A(q̄) = A(1)(q̄) +A(2)(q̄) +O(k1) , (3.19) A(1)(q̄) = D̄(k1) , A(2)(q̄) = − D̄(k0) . (3.20) Now the two amplitudes A(1,2) can be reconstructed separately, without any problem of vanishing Gram-determinant. Notice also that corrections at orders higher than O(k1) are perfectly calculable by inserting again Eq. (3.18) in the term O(k1) of Eq. (3.19). Once again, when f → 0, double expansions in k1 and f can be obtained involving derivatives of scalar loop functions by using Eq. (3.12). For example, at the zeroth order in k1 and at the first one in f , one gets A(q) = D̄(k0)D̄(k1) D̄(k0)2D̄(k1) D̄(k1)− 2(q · k1) + f D̄(k0) D̄(k0) 3D̄(k1) D̄(k1)− 2(q · k1) + f +O(k1) D̄(k0) D̄(k0) +O(k1) +O(f 2) . (3.21) This last case exhausts all possibilities. The same techniques can be applied for higher-point functions. For example, in the case of a 3-point function, instead of k1, one introduces the 4-vector sµ = det (k2 · k1) (k2 · k2) , (3.22) with the properties s · k2 = 0 , s 2 ∝ ∆(k1, k2) , (k1 · s) ∝ ∆(k1, k2) , (3.23) where ∆(k1, k2) is the Gram-determinant of the two momenta k1 and k2. Then, instead of Eq. (3.6) one has, for example, (q · v)(q · s)2 D̄(k0)D̄(k1)D̄(k2) = O(∆(k1, k2)) . (3.24) As before, ∆(k1, k2) can vanish either because s 2 = 0 or sµ = 0 and the two cases should be treated separately. 4. Results and comparisons We started by checking our implementation of the rational terms. For 4-point functions up to rank four, we reproduced the results obtained with the alternative technique illustrated in [17]. Furthermore, we reproduced the rational part of the full 2γ → 2γ amplitude given – 11 – in [25]. We also checked with an independent calculation [26] the rational terms coming from all of the 6-point tensors up to rank six. Finally, we computed the rational piece of the whole 2γ → 4γ amplitude by summing up all 120 contributing Feynman diagrams and finding zero, as it should be [14]. As a first test on full amplitudes, we checked our method by reproducing the contribu- tion of a fermion loop to the 2γ → 2γ process. This result is presented in Eqs. (A.18)-(A.20) of Ref. [25], for all possible helicity configurations. We are in perfect agreement with the analytic result, in both massless and massive cases. The next step was the computation of the 2γ → 4γ amplitude with zero internal mass7, finding the results given in Fig. 1 and Fig. 2. It should be mentioned that our results are obtained algebraically, so there is no integration error involved. In Fig. 1, we reproduce 0 0.5 1 1.5 2 2.5 3 10000 15000 20000 25000 Figure 1: Comparison with Fig. 5 of Ref. [19]. Helicity configurations [+ +−−−−] and [+ −−++−] for the momenta of Eq. (4.1), represented by black dots and gray diamonds re- spectively, and comparison with the analytic result of Ref. [18] (continuous line). the results presented by Nagy and Soper [19] and very recently also by Binoth et al. [20]. We employ the same values of the external momenta as in Fig. 5 of Ref. [19], namely the following selection of final state three-momenta {~p3, ~p4, ~p5, ~p6}: ~p3 = (33.5, 15.9, 25.0) , ~p4 = (−12.5, 15.3, 0.3) , ~p5 = (−10.0,−18.0,−3.3) , ~p6 = (−11.0,−13.2,−22.0) . (4.1) After choosing the incoming photons such that they have momenta ~p1 and ~p2 along the z-axis, we present in the plot the amplitude obtained by rotating the final states of an- gle θ about the y-axis. This is done for both helicity configurations [+ +−−−−] and 7We thank Andre van Hameren for providing us with his program to compute massless one-loop scalar integrals. – 12 – 0 0.5 1 1.5 2 2.5 3 10000 12000 14000 16000 18000 Figure 2: Helicity configurations [+ +−−−−] and [+ +−−+−] for the momenta of Eq. (4.2), represented by black dots and gray diamonds respectively, and comparison with the analytic result of Ref. [18] (continuous line). [+ −−++−]. In the same plot also appears the analytic results for the configuration [+ +−−−−] obtained by Mahlon [18]. In Fig. 2, we use a different set of external mo- menta. Starting from the following choice of {~p3, ~p4, ~p5, ~p6}: ~p3 = (−10.0,−10.0,−10.0) , ~p4 = (12.0,−15.0,−2.0) , ~p5 = (10.0, 18.0, 3.0) , ~p6 = (−12.0, 7.0, 9.0) (4.2) we proceed as in the previous case. The results for the amplitudes are plotted in Fig. 2 for the helicity configurations [+ +−−−−] and [+ +−−+−]. It is known that the six- photons amplitude vanish for the helicity configurations [+ + ++++] and [+ + +++−], we checked this result for both choices of the external momenta. Finally, using the external momenta of Eq. (4.1), we computed the amplitude introducing a non-zero mass mf for the fermions in the loop 8. The results are plotted in Fig. 3, for the three cases mf = 0.5 GeV, mf = 4.5 GeV and mf = 12 GeV. The code we prepared for producing the results presented in this section is written in FORTRAN 90. Even if we did not spend too much effort in optimizations, it can compute about 3 phase-space points per second, when working in double precision. All figures in this section are actually produced by using double precision, but, to perform a realistic integration, we still need quadruple precision, that slows down the speed by about a factor 60. We are working in implementing the expansions presented in the previous section with the aim of being able to perform a stable integration over the full phase space, that is a “proof of concept” for any method. 8We used here the scalar one-loop functions provided by FF [27]. – 13 – 0 0.5 1 1.5 2 10000 15000 20000 25000 30000 Figure 3: Helicity configuration [+ +−−−−] for the momenta of Eq. (4.1) for different values of the fermion mass in the loop: mf = 0.5 GeV (diamond), mf = 4.5 GeV (gray box) and mf = 12 GeV (black dots). The continuous line is the result for the massless case. 5. Conclusions Computing the massless QED amplitude for the reaction 2γ → 4γ, although still unob- served experimentally, is a very good exercise for checking new methods to calculate one- loop virtual corrections. Such a process posses all complications typical of any multi-leg final state, for example a non trivial tensorial structure, but also keeps, at the same time, enough simplicity such that compact analytical formulas can still be used as a benchmark. However, it is oversimplified in two respects. Firstly, the amplitude it is completely mass- less. Secondly, the amplitude is cut constructible, namely does not contain any rational part. In the most general case of one-loop calculations, the presence of both internal and external masses prevents from obtaining compact analytical expressions. Then one has to rely on other computational techniques. For example, it is known that cut-constructible amplitudes can be obtained through recursion relations. But, even then, the presence of rational parts usually requires a separate work. For such reasons, it would be highly advisable to have a method not restricted to massless theories, in which moreover both cut-constructible and rational parts can be treated at the same time. Such a method has been introduced recently in Ref. [17] and, in this paper, we applied it to the computation of the six-photon amplitude in QED, giving also results for the case with massive fermions in the loop. We also showed in detail how the rational part of any m-point one-loop amplitude is intimately connected with the form of the integrand of the amplitude. Once this integrand is numerically computable, cut- constructible and rational terms are easily obtained, at the same time, by solving the same system of linear equations. This is a peculiar property of our method, that we tested in the actual computation of the six-photon amplitude. In practice, we did not use the additional – 14 – information on its cut-constructibility and verified only a-posteriori that the intermediate rational parts, coming from all pieces separately, drop out in the final sum. Finally, we presented all relevant formulas needed to infer the rational parts from the integrand of any m-point loop functions, in the renormalizable gauges. In addition, we presented, by analyzing in detail the 2-point case, an idea to cure the numerical instabilities occurring at exceptional phase-space points, outlining a possible way to build up expansions around the zeroes of the Gram-determinants. Having been able to apply our method to the computation of the massive six-photon amplitude, we are confident that our method can be successfully used for a systematic and efficient computation of one-loop amplitudes relevant at LHC and ILC. Acknowledgments We thank Andre van Hameren for numerical comparisons and Zoltan Nagy and Pier- paolo Mastrolia for interesting discussions. G.O. acknowledges the financial support of the ToK Program “ALGOTOOLS” (MTKD-CT-2004-014319). C.G.P’s and R.P.’s re- search was partially supported by the RTN European Programme MRTN-CT-2006-035505 (HEPTOOLS, Tools and Precision Calculations for Physics Discoveries at Colliders). The research of R.P. was also supported in part by MIUR under contract 2006020509 004. Appendices A. Computing the extra-integrals In this appendix, we compute the extra-integrals listed in Section 2. Since a contribution O(1) can only develop for non-negative dimensionality D, the integrand in the Feynman parameter integral is always polynomial. First we decompose the integration as follows dnq̄ = d4q dǫµ (q̃2 = −µ2) , (A.1) then, after using Feynman parametrization and performing first the integral over dǫµ and then that one over d4q, one derives, for the extra-integrals of Eqs. (2.14)-(2.18) I(n;2(s−1))s = −iπ 2Γ(s− 1) [dα]s +O(ǫ) , I(n;2(s−1))s;µ = iπ 2Γ(s− 1) [dα]s (Ps)µ +O(ǫ) , I(n;2(s−1))s;µν = −iπ 2Γ(s− 1) [dα]s (Ps)µ(Ps)ν +O(gµν) +O(ǫ) , I(n;2s)s = −iπ 2Γ(s) [dα]s Xs +O(ǫ) , I(n;2s)s;µ = iπ 2Γ(s) [dα]s Xs(Ps)µ +O(ǫ) , (A.2) – 15 – where [dα]s = dα0 · · · dαs δ(1 − αj) , Xs = P αjkj , M j − k j ) , (k0 = 0) . (A.3) In the following, we compute, as an illustrative example, the first three integrals of Eq. (A.2). The remaining two can be obtained analogously. We start by changing the integration vari- ables as follows: α1 = ρ1ρ2 · · · ρs α2 = ρ1ρ2 · · · ρs−1(1− ρs) α3 = ρ1ρ2 · · · ρs−2(1− ρs−1) αs = ρ1(1− ρ2) α0 = (1− ρ1) , (A.4) so that [dα]s = dρ2 · · · dρs ρ (s−1) (s−2) 2 · · · ρs−1 , (A.5) from which one trivially obtains the first integral I(n;2(s−1))s = −iπ 2Γ(s− 1) Γ(s+ 1) +O(ǫ) . (A.6) To compute the second integral an integration over (Ps)µ in needed. Since the integrand is symmetric when interchanging all ki, we concentrate on the coefficient of, say, k1. Since dρ2 · · · dρs ρ (s−1) (s−2) 2 · · · ρs−1 α1k1µ = k1µ dρ2 · · · dρs ρ (s−1) 2 · · · ρ s−1ρs = k1µ Γ(s+ 2) , (A.7) the final result reads I(n;2(s−1))s;µ = iπ 2Γ(s− 1) Γ(s+ 2) (kj)µ +O(ǫ) . (A.8) To compute the third integral we need to integrate over the product (Ps)µ(Ps)ν . Once again, given the symmetry of the problem, we can focus on the two contributions proportional to – 16 – k1µk1ν and k1µk2ν . The first one gives dρ2 · · · dρs ρ (s−1) (s−2) 2 · · · ρs−1 α 1k1µk1ν = k1µk1ν dρ2 · · · dρs ρ (s+1) 2 · · · ρ = k1µk1ν Γ(s+ 3) , (A.9) and the second reads dρ2 · · · dρs ρ (s−1) (s−2) 2 · · · ρs−1 α1α2k1µk2ν = k1µk2ν dρ2 · · · dρs ρ (s+1) 2 · · · ρ s−1ρs(1− ρs) = k1µk2ν Γ(s+ 3) . (A.10) Summing up all of the possibilities one obtains I(n;2(s−1))s;µν = −2iπ 2Γ(s− 1) Γ(s+ 3) (kj)µ(kj)ν + i 6=j (kj)µ(ki)ν + O(gµν) +O(ǫ) . (A.11) B. The general basis for the 2-point functions In this appendix, we solve the problem of reconstructing the coefficients of the 2-point part of the integrand of any amplitude A(q̄) = D̄0D̄1 , (B.1) by assuming N(q) at most quadratic in q and k1 ≡ (p1 − p0) 6= 0. In particular also the case of vanishing k21 is included. First, we introduce a massless arbitrary 4-vector v, such that (v · k1) 6= 0, that we use to rewrite k1 in terms of two massless 4-vectors (we also take ℓ2 = 0) k1 = ℓ+ α v , (B.2) giving γ ≡ 2 (k1 · v) = 2 (ℓ · v) and α = . (B.3) Then, we introduce two additional independent massless 4-vectors ℓ7,8 defined as 7 = < ℓ|γ µ|v] , ℓ 8 =< v|γ µ|ℓ] , (B.4) – 17 – for which one finds (ℓ7 · ℓ8) = −2γ , (B.5) and we decompose qµ + p 0 in the basis of k1, v, ℓ7 and ℓ8 qµ = −p 0 + yk 1 + yvv µ + y7ℓ 7 + y8ℓ 8 , (B.6) so that N(q) takes the form N(q) = b+ b̂0[(q + p0) · v] + b̂00[(q + p0) · v] 2 + b̃11[(q + p0) · ℓ7] + b̃21[(q + p0) · ℓ8] + b̃12[(q + p0) · ℓ7] 2 + b̃22[(q + p0) · ℓ8] + b̃01[(q + p0) · ℓ7][(q + p0) · v] + b̃02[(q + p0) · ℓ8][(q + p0) · v] +O(D0) +O(D1) . (B.7) Notice that, because of the identity 2 (q · k1) = D1 −D0 + (d1 − d0) , with di = m i − p i , (B.8) any term proportional to [(q + p0) · k1] either contributes to the constant term b or it is included in the terms O(D0,1) we are neglecting 9. The same happens for the combination [(q + p0) · ℓ7][(q + p0) · ℓ8]. To be able to determine all of the coefficients appearing in Eq. (B.7), disentangling completely the contributions O(D0,1), we look for a q that fulfill the requirement D0 = D1 = 0 . (B.9) For a q written as in Eq. (B.6) this implies the system y7y8 = Fy d1 − d0 − 2yk , (B.10) where Fy = − m20 − y (d1 − d0) + y . (B.11) It is convenient to introduce two classes of solutions. In the first class, that we call q±yk, we take y fixed and choose y7 = ±e iπ/k. In the second class, that we call q′± , we take y fixed but choose y8 = ±e iπ/k. The coefficients b, b̃11, b̃21, b̃12 and b̃22 can be obtained by evaluating Eq. (B.7) at the values q±01 , q 02 , q 03 , (B.12) q′±01 , q 02 , q 03 . (B.13) 9We suppose to determine them at a later stage of the calculation. – 18 – In the first case, the coefficients read b0 = b , b1 = −2γb̃21 , b2 = 4γ 2b̃22 , b−1 = −2γF0b̃11 , b−2 = 4γ 2F 20 b̃12 , (B.14) b±1 = − T−(q1)± iT −(q2) T+(q1) + T +(q2) b±2 = T+(q1)− T +(q2) − e±2iπ/3(T+(q3)− b0) 1− e∓2iπ/3 , (B.15) and where T±(qk) ≡ N(q+0k)±N(q . (B.16) In the second case, one obtains instead b′0 = b , b 1 = −2γb̃11 , b 2 = 4γ 2b̃12 , b −1 = −2γF0b̃21 , b −2 = 4γ 2F 20 b̃22 , (B.17) b′±1 = − T−(q′1)± iT −(q′2) b′0 = T+(q′1) + T +(q′2) b′±2 = T+(q′1)− T +(q′2) − e±2iπ/3(T+(q′3)− b 1− e∓2iπ/3 , (B.18) and where T±(q′k) ≡ N(q′+0k)±N(q . (B.19) The reason why we have chosen two sets of solutions is that, in some special kinematical configurations, F0 can vanish. Therefore, numerical stable solutions are obtained by taking b̃21 and b̃22 from Eq. (B.14), and b̃11 and b̃12 from Eq. (B.17), while b is well defined in both cases. The coefficients b̂0 and b̂00 can be determined, in terms of additional solutions of the kind q± and q±σ1, by defining the combinations S(q) ≡ N(q)− b− b̃11[(q + p0) · ℓ7]− b̃21[(q + p0) · ℓ8] − b̃12[(q + p0) · ℓ7] 2 − b̃22[(q + p0) · ℓ8] U(λ) ≡ S(q+λ1) + S(q , (B.20) as the two solutions of the system . 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0704.1272

Dynamics of shear homeomorphisms of tori and the Bestvina-Handel algorithm

Dynamics of shear homeomorphisms of tori and the Bestvina-Handel algorithm Tali Pinsky and Bronislaw Wajnryb Abstract Sharkovskii proved that the existence of a periodic orbit of period which is not a power of 2 in a one-dimensional dynamical system implies existence of infinitely many periodic orbits. We obtain an analog of Sharkovskii’s theorem for periodic orbits of shear homeomorphisms of the torus. This is done by obtaining a dynamical order relation on the set of simple orbits and simple pairs. We then use this order relation for a global analysis of a quantum chaotic physical system called the kicked accelerated particle. 1. Introduction Given a dynamical system (X, f), a key question is which periodic orbits exist for this system. Since periodic orbits are in general difficult to compute, we would like to have the means to deduce their existence without having to actually compute them. Sharkovskii addressed the dynamics of continuous maps on the real line. He defined an order C on the natural numbers, Sharkovskii’s or- der (see [14]), and proved that the existence of a periodic orbit of a certain period p implies the existence of an orbit of any period q C p. We say the q orbit is forced by the p orbit. This offers the means of showing the existence of many orbits if one can find a single orbit of “large” period. For a dynamical system depending on a single parame- ter, if periodic orbits appear when we change the parameter, they must appear according to the Sharkovskii’s order. Hence, Sharkovskii’s the- orem gives the global structure of the appearance of periodic orbits for one dimensional systems. Ever since the eighties there has been inter- est in obtaining analogs for Sharkovskii’s theorem for two dimensional systems (see [4] and [16]). A homeomorphism of a torus is said here to be of shear type if it is isotopic to one Dehn twist along a single closed curve. Let h be a shear homeomorphism, and let x be a periodic orbit of h. We can then define the rotation number of x, see discussion in Section 2. Thus, a rational number in the unit interval [0, 1) is associated to each orbit. We consider orbits up to conjugation: orbits (x, f) and (y, g) are similar (of the same type) if there exists a homeomorphism h of the torus T 2 such that h is isotopic to the identity, h takes orbit x onto orbit y and hfh−1 is isotopic to g rel y. We define below a specific family of periodic orbits we call simple orbits. In this family there is a unique element up to similarity corresponding to each rotation number; hence they can be specified by their rotation numbers. We emphasize it is not true in general that an orbit of a shear homeomorphism is characterized by its rotation number. Simple orbits are analyzed in Section 2. As it turns out (see Remark in section 2), one simple orbit is indeed simple and does not force the existence of any other orbit. More generally a periodic orbit is of twist type if it does not force the existence of any orbit of different type with the same rotation number. It is tempting to conjecture that the simple orbits are the only orbits of twist type, but Lemma 2.4 shows that this is false. We give there an example of an orbit of twist type which is not simple. This example also shows that a periodic orbit with a given rotation number does not necessarily force a simple orbit of the same rotation number. We turn in Section 3 to analyze pairs of orbits. Two coexisting simple periodic orbits can form a simple pair and these are considered. The pairs do force some more interesting dynamics, as follows. We denote the integers by letters p, q and the rational numbers by r, s, t possibly with indices. For a pair of simple orbits of rotation numbers q1 and q2 to constitute a simple pair, it is necessary that the rotation numbers be Farey neighbors, i.e. |p2q1 − p1q2| = 1. We denote such a pair of rational numbers by q1 We now define an order relation on the following set P of rational numbers and pairs in the unit interval, P = {r|r ∈ Q ∩ [0, 1)} ∪ {r ∨ s|r, s ∈ Q ∩ [0, 1)}. Define the order relation on P to be r ∨ s < t⇔ t ∈ [r, s] r1 ∨ r2 < s1 ∨ s2 ⇔ s1, s2 ∈ [r1, r2]. where we denote by [r1, r2] the interval between r1 and r2, regardless of their order. Theorem 1.1. The order relation < on P describes the dynamical forcing of simple periodic orbits. Namely, the existence of a simple pair of periodic orbits with rotation numbers r∨s in P implies the existence of all simple orbits and simple orbit pairs of rotation numbers smaller than r ∨ s according to this order relation . This is the main result of this paper and the proof is completed in Section 4. The idea of the proof is as follows. Consider the torus punc- tured on one or more periodic orbits of a homeomorphism h. Then h induces an action on this punctured torus, and on its (free) fundamen- tal group. Now apply the Bestvina-Handel algorithm to this dynamical system. The idea of using the Bestvina-Handel algorithm was used by Boyland in [6] and he describes the general approach in [5]. In our case, after puncturing out a simple pair of orbits, applying the algo- rithm yields an isotopic homeomorphism which is pseudo-Anosov. The algorithm also offers a Markov partition for this system and we use the resulting symbolic representation to find that there are periodic orbits of each rotation number between the pair of numbers we started with. Then we directly analyze the structure of the pseudo-Anosov representative to show that all these orbits are in fact simple orbits. Furthermore any two of them corresponding to rotation numbers which are Farey neighbors form a simple pair. Finally we establish the iso- topy stability of these orbits using results of Asimov and Franks [2] and Hall [11]. Thus, the orbits exist for any homeomorphism for which the simple pair exists, and are forced by it. One should compare this result with a very strong theorem of Doeff (see Theorem 3.6), where existence of two periodic orbits of different periods for a given shear homeomorphism h implies existence of peri- odic orbit of every intermediate rotation number. However an explicit description of these orbits is not given, while our stronger assumptions imply existence of simple periodic orbits and simple pairs of orbits. It may be true that the existence of any two periodic orbits with differ- ent rotation numbers implies the existence of a simple orbit with any given intermediate rotation number, but we feel that the evidence is not strong enough to make a conjecture either way, in particular in view of Lemma 2.4. Even more difficult question is to determine whether there exists a simple pair of periodic orbits in the situation of Doeff Theorem. First one should try to find a pair of simple orbits which is not a simple pair while the rotation numbers are Farey neighbors, but it is very difficult to understand the geometry of the pseudo-Anosov homeomorphism which arises in this situation. This research was originally motivated by a question we were asked by Professor Shmuel Fishman: Is there a topological explanation for the structure of appearance of accelerator modes in the kicked particle system. In section 5 we give a description of the kicked particle system. This system turns out to be described precisely by a family of shear homeomorphisms of the torus. The existence of accelerator modes is equivalent to existence of periodic orbits. The global structure of this system is given by the order relation in Theorem 1.1, while it cannot be directly computed due to the complexity of the system. The authors would like to thank Professor Shmuel Fishman for offer- ing valuable insights, Professor Italo Guannieri for some critical advice, and Professor Philip Boyland for many indispensable conversations. 2. Simple orbits Let {x1, ....., xN} be a set of points belonging to one or more pe- riodic orbits for a homeomorphism f of a surface S. The dynamical properties of this set of orbits are captured by the induced action of f on the complement S0 = S \ {x1, ...., xN} in a sense that will shortly become clear. Choose any graph G which is a deformation retract of the punctured surface S0. A homeomorphism of S0 then induces a map on G. The converse is also true: a given map of G determines a homeo- morphism of S0 up to isotopy. Therefore we specify the periodic orbits we analyze in terms of the action on a graph which is a deformation retract of the surface after puncturing out the orbit. Denote by GN the graph obtained by attaching N small loops to the standard unit circle at the points exp(2πj i), j = 0, . . . , N − 1. Definition 2.1. We call a periodic orbit x = {x1, . . . , xN} for a shear homeomorphism f on the two-torus a simple orbit if the following hold. (1) There can be found a graph G which is a deformation retract of T0 = T 2 \ x as on Figure 1 such that G is homeomorphic to We call the loop in G corresponding to the unit circle the hor- izontal loop and the loops attached to it the vertical loops. (2) There exists a homeomorphism f̃ of T0 isotopic to f rel x (i.e. the isotopy is fixed on x) so that a neighborhood of G is invari- ant under f̃ and the induced action on G satisfies: (a) There exists a fixed number k ∈ {0, .., N − 1} such that each vertical loop is mapped k loops forwards (clockwise along the unit cir- cle) to another vertical loop. (b) The horizontal loop is mapped to itself with one twist around one of the vertical loops. Figure 1. A standard graph for a simple orbit Figure 2. The action on a standard graph for a simple orbit Remark. A homeomorphism h for which we are given a simple periodic orbit must be of shear type as we can deduce from the action on the homology of the non-punctured torus. For a shear homeomorphism there exists a basis for the first homol- ogy for which the induced map is represented by the matrix From here on we refer to any two axes given by an homology basis that gives us the above representation as standard axes. The horizontal loop and one of the vertical loops in a graph for a simple orbits constitute a standard basis. Definition 2.2. Let h be a shear homeomorphism, and let ĥ be a lift of h to the universal cover (a plane). For any periodic point x of h of period p, ĥp maps any lift x̂ of x the same integer number q along the horizontal axis away from x̂, in a standard choice of axis (and x̂ is possibly mapped some integer number along the vertical axis as well). We can then define the rotation number of x to be ρ(x) = q mod 1. The rotation number does not depend on the lift ĥ of h. (The rotation number is often define relative to the given lifting of the homeomorphism h and is not computed modulo 1, but we want it to depend only on the orbit and not on the lifting. In particular we want a simple orbit to have well defined rotation number independent of the lifting of h.) Remark. In the case of a homeomorphism isotopic to a Dehn twist on a torus, which is our interest here, it can be easily shown that the abelian Nielsen type equals exactly the rotation number defined above. There exists a simple orbit for any given rational rotation number r ∈ [0, 1), and it is unique up to similarity. Denote the similarity class by r̂. In the following we use the word vertical to describe the y axis, in a standard choice of axis for f (the direction along which the twist is made). Lemma 2.3. Let x be a periodic orbit for a shear-type homeomorphism f of T2 for which there exists a family of vertical loops such that they bound a set of annuli each containing one point of the periodic orbit, and this family is invariant under a homeomorphism f̃ isotopic to f rel x. Then x is a simple orbit. Proof. Choose a vertical loop l of the invariant family. f is orientation preserving, and so is f̃ . The first loop to the right of l is therefore mapped to the first loop to the right of f̃(l). Hence, the vertical loops in the invariant family are all mapped the same number of loops to the right. Now we have to find a horizontal line with the desired image. We write the invariant family of loops as {li} i=1, where p = period(x), ordered along the horizontal axis. We choose another family of vertical loops {mi} i=1, such that mi is contained in the annulus between li and li+1 (lp+1 = l1), and passes through the periodic point xi also contained in this annulus. Choose a point a1 6= x1 on m1. f can be adjusted in such a way that the new homeomorphism f̃ leaves both families of vertical loops invariant, and in addition, so that a1 be a periodic point of f̃ with period p. We denote the orbit of a1 by {ai} i=1 where ai ∈ mi for 1 ≤ i ≤ p. Choose a line segment n1 connecting a1 to a2, so it crosses the annulus between m1 and m2 from side to side. We choose nj+1 to be the line segment f̃ j(n1) for 1 ≤ j ≤ p − 2. The boundary points of nj and nk coincide whenever they lie on the same vertical loop. Now, we look at the horizontal loop n = j=1 nj. Each segment of n is mapped exactly to the next segment, except np−1 which is mapped into the annulus between m1 and m2. Since the mapping class group of an annulus is generated by a twist with respect to any loop going once around the annulus we may assume, that np is mapped to n1 plus a number of twists along such a loop. On the other hand we know that f(n) is homotopic to itself plus one twist in the negative direction (on the closed torus), so f̃ maps n to itself plus one negative twist along this loop. By further adjustment of f̃ we may assume the twist is made along l2. Thus the union of the vertical family {li} with n chosen as above constitute a graph showing x to be a simple orbit. � The Thurston-Nielsen classification theorem, see [7], states that any homeomorphism f on a closed connected oriented surface of negative Euler characteristic is isotopic to a homeomorphism f̃ which is (1) pseudo Anosov, or (2) of finite order, or (3) reducible. where a homeomorphism φ is called of finite order if there exists a natural number n such that φn = id. A homeomorphism φ is called pseudo-Anosov if there exists a real number λ > 0 and a pair of transverse measured foliations (Fu, µu) and (F s, µs) with φ(Fu, µu) = (Fu, λµu) and φ(F s, µs) = (F s, 1 µs). A homeomorphism φ on a sur- face M is called reducible if there exists a collection of pairwise disjoint simple closed curves Γ = {Γ1, ...,Γk} in int(M) such that φ(Γ) = Γ and each component of M \Γ has a negative Euler characteristic. The representative f̃ in the isotopy class of f which is of one of the three forms above is called the Thurston-Nielsen canonical form of f . When the surface has a finite number of punctures and φ permutes the punctures then the same is true except that in the case of pseudo- Anosov map we treat the punctures as distinguished points (there is a unique way to extend a homeomorphism to the distinguished points) and we allow an additional type of singularities of the measured foli- ations, the 1-prong singularities at the distingushed points (See [11], and section 0.2 of [3]). Of course homeomorphism f is reducible with respect to a simple orbit since it contains an invariant family of loops {li} and the comple- ment of the invariant family consists of punctured annuli (which have negative Euler characteristic). Remark. A homeomorphism f with a simple orbit x can be constructed in such a way that x is the only periodic orbit of f . The invariant set of vertical loops is evenly spaced with the distance between the consecutive loops equal to 1/p. The loops are moved by q/p to the right and by fixed irrational number downward. Punctures (the points of the periodic orbit) are also evenly spaced and have the same height. The vertical lines containing punctures are moved by q/p to the right. The punctures keep their height and all other points of the loop move a little downwards. Every other vertical line is moved to another vertical line by a little more than q/p to the right (not all lines by the same distance). Example 1. Not every periodic orbit for a shear homeomorphism is reducible. Consider the homeomorphism h described on Figure 3. It takes the graph on the left of Figure 3 to the graph on the right and is a shear homeomorphism. It has a periodic orbit of order 2, shown on the pictures, with rotation number 1/2 and it is pseudo-Anosov in the complement of the orbit. Figure 3. Lemma 2.4. There exists an orbit of twist type for a shear homeomor- phism of the torus which is not simple. Proof. We construct an example of an orbit of length 4 with the rota- tion number 1/2. It cannot be a simple orbit and yet we prove it does not force the existence of any periodic orbit not similar to itself, and is thus of twist type. Such examples may be known, possibly considered for a different phenomena. We include it here in order to show the independence of our results. We represent the torus as the unit square with the opposite sides identified. The points A1, A2, A3, A4 of the orbit are spaced evenly on the horizontal middle line with the x-coordinate 1/8, 3/8, 5/8, 7/8. We split the square into 2 equal parts U1 and U2 by the vertical line x = 1/2. Homeomorphism h translates U1 to the right to U2. Vertical lines go to vertical lines, lines x = 0 and x = 1/2 move downward by an irrational number α < 1/40 and the movement is damped out to the horizontal translation for t < α and t > 1/2− α, so the other vertical 1 2 3 4 5 6 7 Figure 4. Some vertical lines in U2 � �' $ '' %%%� ' %' %�� %' %' %� 1 2 3 4 5 6 7 Figure 5. The foliation in U2 realizing a non-simple orbit of twist type. lines are translated horizontally by 1/2. In particular A1 moves to A3 and A2 moves to A4. The restriction of h to U2 is defined in two steps. The second step simply translates U2 horizontally by 1/2 to the right (which is the same as the translation by 1/2 to the left). The first step is isotopic to the half-twist along the segment connecting A3 and A4, followed by the Dehn twist with respect to the right side (right boundary of the cylinder). In particular it switches A3 and A4. We shall prove that we can construct such h for which h2 has no fixed points and therefore h has no periodic orbit of length 2 and in particular no simple orbit with the rotation number 1/2. We describe the first step of h restricted to U2. Figure 4 shows some vertical lines in U2, the big dots show the points A3 and A4 of the periodic orbit. Figure 5 shows their images under the first step. In these pictures U2 is represented as a square, to make more space, but in the reality the base has length 1/2 and the height is equal to 1. Line x = 1/2 is mapped to itself and moves downward by α. The near by vertical lines (for t < 1/2+1/30) are moved to the vertical lines and to the right where line x = 1/2+1/30 is moved to the line x = 1/2+1/20. For t ∈ (1/2 + 1/30, 1−1/20) the line x = t is moved to a curve Lt and for t ∈ (1−1/20, 1) the line x = t moves to a vertical line to the right of it and downward, to get the full Dehn twist plus a movement downward by α when we get to the line x = 1. For t ∈ (1/2+1/30, 1−1/20) curve Lt starts at a point on the top side to the right of x = t, it moves to the left, then to the right, then to the left again and ends at the bottom side (exactly below its starting point). In particular each vertical line meets Lt in at most two points. Some lines Lt are shown on Figure 5. We may arrange it in such a way that there exist t0, t1 such that 1/2 < t0 < t1 < 1 and the line x = t: is disjoint from Lt, and lies on the left side of Lt when t < t0; meets Lt at one point for t = t0; meets Lt at two points when t0 < t < t1; meets Lt at one point for t = t1; is disjoint from Lt and lies on the left side of Lt when t1 < t < 1. We get a new trivial foliation of the annulus U2. In step 1 we map the vertical foliation onto the new foliation Lt. We can further change the first step moving each leave Lt along itself to reach the following goal. Let Pt, Qt denote the intersection points of x = t with Lt, Pt lies below Qt (the points coincide for t0 and t1). For t = t0 the line x = t meets Lt in one point Pt. We may assume that the image of Pt in Lt lies in the part below Pt. Then for the nearby leave the images of both points Pt and Qt lie in the lower part of Lt below the point Pt (see the small dots on the first curve in Figure 5). The images of Pt and Qt lie further away from each other when we move to the right (see the small dots on the second curve). The third line passes through A3, its image Lt passes through A4 and the images of Pt and Qt lie on different sides of A4 along the third curve. Next the upper point Qt moves backwards along Lt and when we reach the fourth line of Figure 4 (also shown on Figure 5) it coincides with the point Pt on Lt. Next the image of Qt lies inside the arc of Lt between the points Pt and Qt and when we reach the fifth line on Figure 4, which passes through the point A4, then the curve Lt passes through A3 and the image of Qt on Lt lies above A3 (see Figure 5). When we move further to the right the image of the point Qt moves again forward towards the image of Pt and at the line number 6 on Figure 4 the image of Qt again coincides with Pt at the intersection of x = t with Lt. Next the images of Pt and Qt move further down and gets close together and when t = t1 we have one intersection point Pt and its image lie below Pt along Lt. Step 1 has no fixed points. Step 2 translates U2 to U1. We now consider the homeomorphism h2. We start with U1. Any point on x = 0 and x = 1/2 moves down by 2α. Any point with x ∈ [0, 1/30] moves to a point with a bigger x-coordinate. Any point with x ∈ [1/30, 9/20] moves horizontaly by 1/2 then we apply step 1, which has no fixed points, and then the point moves again horizontaly by 1/2 so it comes to a new point. Any point with x ∈ [9/20, 1/2] moves to a point with a bigger x-coordinate. For points in U2 the situation is similar. Any point with x ∈ (1/2, 16/30] moves to a point with a bigger x-coordinate. Any point with x ∈ [16/30, 19/20] moves under the first step to a new point with the x- coordinate in [11/20, 29/30] and then moves horizontaly twice by 1/2. Finally any point with x ∈ [19/20, 1] moves to a point with a bigger x-coordinate. Homeomorphism h2 has no fixed points and h has no periodic points of order 2. We now show that there exists a homeomorphism f isotopic to h in the complement of the orbit A1, A2, A3, A4, which has only periodic orbits similar to this orbit and periodic orbits of order 2. We consider parts U1 and U2 as before. The restriction of f to U1 translates it hor- izontaly by 1/2. In U2 we choose two circles with center (3/4, 1/2) and radius 1/7 and 1/6 respectively. We rotate the interior of the smaller circle by 180 degrees. The rotation is damped out to the identity at the outer circle and the intermediate circles are moving out towards the outer circle. The exterior of the outer circle with x < 19/20 is pointwise fixed. The lines with x > 19/20 move to the right and down to get the full Dehn twist when we get to the line x = 1. The second step of f restricted to U2 translates it horizontally by 1/2. Now each point inside the smaller circle, different from its center (which has pe- riod 2), belongs to an orbit similar to A1, A2, A3, A4. Points between the circles and points with x ∈ (19/20, 1) are not periodic and other points in U2 have period 2 and the same is true for the corresponding points in U1. Therefore the orbit A1, A2, A3, A4 does not force any periodic orbit not similar to itself. � 3. Simple orbit pairs Let x and y be two coexisting simple periodic orbits, for a homeo- morphism f of T2 (f must be of shear type), with rotation numbers q1 and q2 respectively. Assume p1 > p2, i.e., y has lesser period than x. Definition 3.1. We call the pair of orbits a s imple pair if • We can find an embedded graph G in their complement home- omorphic to Gp1 as on Figure 6. Figure 6. Each component in the complement of the graph is a topolog- ical rectangle which contains exactly one point of orbit x and at most one point of orbit y. • The homeomorphism f acts on this graph in the following way: each vertical loop except one moves to another vertical loop, there is one vertical loop denoted l such that f(l) is a vertical loop m plus a small loop around one point of the (shorter) y orbit, in a rectangle adjacent to line m on the right (as on Figure 7) or on the left, and the horizontal line is mapped to itself plus a twist in the negative direction around f(l), as on Figure 7. The graph which appears in Definition 3.1 divides the torus T2 into p1 rectangles. The homeomorphism f moves each vertical loop the same Figure 7. distance, say k rectangles, to the right except for the small additional loop for line l. Let R0 be the rectangle adjacent to m in which the small loop in the image of the graph occurs. R0 must contain exactly one point of each orbit. We denote these points x0 and y0 respectively. Under p1 iterations of f the point x0 runs q1 times around the whole torus, that is q1p1 rectangles to the right. So, p1k = q1p1 and k = q1. The point y0 is mapped to itself after p2 iterations.Under each itera- tion the image of y0 is mapped k (= q1) rectangles to the right, except the last iteration under which it is moved an additional rectangle to the right or left. Altogether it has moved p2q1± 1 rectangles. At the same time, it is mapped around the torus q2 times, hence q2p1 rectangles. This means p2q1 ± 1 = q2p1. Thus for a simple pair of periodic orbits the rotation numbers q1 and q2 are Farey neighbors and the additional loop is on the right (as on Figure 7) if and only if q2 . Denote by r̂ ∨ ŝ the similarity class of a simple pair corresponding to a pair of Farey neighbors r ∨ s. Consider again the points x0 and y0 in the rectangle R0. Continue the notation to all the points of x and y by xi = f i(x0) and yi = f i(y0). We draw a small loop around each of the points of y. The union of these loops will be the peripheral subgraph P for the Bestvina-Handel algorithm, since we may assume the union of these loops to be f - invariant. Now we consider separately two cases. Case 1 will be the case in which m is the left boundary curve of R0, while in case 2 it is the right boundary (in other words in the first case q1 and in the second case q1 ). Choose some point on the loop around y0 and connect it, by a curve l0, to a point on the section of the horizontal line in R0, in case 1 from below the segment and in case 2 from above. Figure 8. Then f(l0) is a curve connecting the loop around y1 and the corre- sponding horizontal segment. We denote it by l1, and do the same for each yi. After adding the above edges to the graph case 1 is topologi- cally as in Figure 8. The inclusion G ↪→ S0 is a homotopy equivalence (where S0 is the punctured torus). We know the action of f on all edges of G except for the curve lq1−1 connecting yq1−1 and the horizontal segment in the corresponding rectangle. It’s image is a curve connecting the horizontal segment in the rectangle adjacent to m which is not R0 to the loop around y0. This image might wind around a disk containing y0 and xq1−1 as in Figure 9 Figure 9. The graph and its image for case 2 are exactly the same except that the loops are connected to the horizontal segments from above. We shall prove in Proposition 3.2 that we may assume that the image of the segment lq1−1 has no winding. Hence we draw from now on the graph images without winding, and we may assume the graphs given in Figure 10 also have an invariant neighborhood by a further isotopy of f . Figure 10. The standard graph for a simple pair, case 1 on the left and case 2 on the right. The action of f (up to isotopy) on this graph is given by one of the actions on Figure 11, drawn in some regular neighborhood of the graph, where each vertical loop moves q1 loops to the right. Figure 11. The action on a standard graph for a simple pair, for both cases respectively. Proposition 3.2. Let {x, y} be a simple pair with the graph as on Figure 8 and with windings as on Figure 9. Then there may be chosen a different invariant graph, which also makes {x, y} a simple pair, whose image is without winding. Proof. To simplify the picture we prove the proposition for rotation numbers 1 and 1 . The general proof proceeds in the same way. We start by looking at a simple pair {x, y} of rotation numbers 1 and 1 a homeomorphism f with the corresponding invariant graph G given so that the action on it is without any twists, as on the left side of Fig- ures 10 and 11. We now choose a different system of curves (a different graph), in a small neighborhood of G as in Figure 12, which will serve as a new graph for the pair. Figure 12. Solid lines are the new vertical loops and dashed lines are the new di- agonal segments like in figure 10. The horizontal loop consists of the dashed lines and the long pieces of the solid lines. To move from left to right along the horizontal line, move along a dashed line and turn to the left when meeting a solid line. Continue up along a vertical loop and then along the next dashed line. We add to this graph the periph- eral subgraph and the connecting segments and get the graph H as on Figure 13. It is clear that topologically the graph H has the same form as the graph on Figure 10 and that it has an invariant neighborhood. The reader can check (using the precise knowledge of the image of each edge of the original graph) that the action of f on the graph H has the properties required from a simple pair. Each vertical loop is mapped onto another vertical loop except for one loop l for which f(l) is equal to a loop m plus a loop around the next periodic point yp of the shorter orbit. The horizontal loop is mapped onto itself plus a negative Dehn twist along f(l). Consider the image s of the segment which connects the horizontal loop to the periodic point yp−1. When the action has no twist then s moves along the horizontal loop in its Figure 13. positive direction until it meets the original segment connecting to yp and then it follows along the segment. However in our case s goes first backwards along the horizontal loop than moves in the counterclockwise direction along the boundary of the ”rectangle” adjacent to the vertical loop m and finally follows the horizontal loop and the segment to yp. This means that the action f on the graph H has one positive twist. We proved that a simple pair for a shear homeomorphism with a given graph and a given action without twists can be given another graph which also describes it as a simple pair and the action on the new graph has one positive twist. This process is reversible. Therefore, by induction, we can add or remove any number of twists using a suitable graph. This implies Proposition 3.2. � Hence for a simple pair the action on a spine is given by Figures 10 and 11. We can now apply the Bestvina-Handel algorithm (see [3]), endowing a neighborhood of G with a fibered structure in the natural way. The algorithm specifies a finite number of steps which we apply to the graph G, altering G together with the induced action on it, but without changing the isotopy class of f on T2 \ (x ∪ y). When the algorithm terminates, it gives a new homeomorphism f̃ which is the Thurston Nielsen canonical form of f . For simple pairs, the action in each of the two cases above is easily seen to be tight, as no edge backtracks and for every vertex there are two edges whose images emanate in different directions. The action has no invariant non-trivial forest or nontrivial invariant subgraph and the graphs have no valence 1 or 2 vertices. This is the definition in [3] for an irreducible map on a graph. Definition 3.3. Assuming g, the induced map on the graph itself, does not collapse any edges, there is an induced map Dg, the derivative of g, defined on {(v, e)|v is a vertex of G, e is an oriented edge emanating from v} by Dg(v, a) = (g(v), b) where b is the first edge in the edge path g(a) which emanates from g(v). Definition 3.4. We say two elements (v, a) and (v, b) in L corresponding to the same vertex v are equivalent if they are mapped to the same element under D(gn) for some natural n. The equivalence classes are called gates The gates in each of the cases above are given by Figure 14, indicated there by small arcs. There is no edge which g sends to an edge path Figure 14. which passes through one of the gates - enters the junction through one arm of the gate and exits through the other. Such an irreducible map is efficient. i.e., this is an end point of the algorithm. Now, since there are edges mapped to an edge path longer than one edge, we arrive at our next theorem. Theorem 3.5. A homeomorphism f of the two torus for which a simple pair of periodic orbits exists is isotopic to a pseudo-Anosov homeomor- phism relative to this pair of orbits. Let f be a shear type homeomorphism of the torus, and fix a lift f̃ of f . Define the lift rotation number of a point x ∈ T2 to be ρ(x, f̃) = limn→∞ (f̃n(x̂)− x̂)1 for any lift x̂ of x, when the limit exists, where the subscript 1 denotes the projection to the horizontal axis. Define the rotation set ρ(f̃) of f̃ to be the set of accumulation points of{ (f̃n(x̂)− x̂)1 |x̂ ∈ R2and n ∈ N Then, the above theorem follows from the following much more gen- eral theorem by Doeff, see [9] and [10]. Theorem 3.6. (Doeff) Let h be a shear type homeomorphism of T2, and fix a lift h̃ of h. If h has two periodic points x and y with ρ(x, h̃) 6= ρ(y, h̃) then h is pseudo-Anosov relative to x and y. Furthermore, the closure of the rotation set is a compact interval, and any rational point r in the interior of this interval corresponds to a periodic point x ∈ T2 with ρ(x, h̃) = r. In particular, Doeff proves existence of two periodic orbits of dif- ferent rotation numbers implies existence of an orbit for any rational rotation number between these two. But he does not give any charac- terization of these orbits. Example 1 shows two different orbits, both with rotation number equal to 1/2, one of which is pseudo-Anosov, and the other reducible. Thus the rotation number does not give much information about the orbit and in this sense this theorem does not give a satisfactory dynamical understanding of what is happening in regions of coexistence of orbits. In contrast with Doeff’s general theo- rem, we get results for a very specific family of periodic orbits, but for this family we are able to give exactly the orbits forced by others, as we show in section 4. In our case the canonical form f̃ of f we get by applying the Bestvina- Handel algorithm is a pseudo Anosov homeomorphism. When this is the case, the algorithm gives a canonical way of endowing a regular neighborhood S0 of G with a rectangle decomposition {R1, ..., RN}. The decomposition is a Markov partition for the homeomorphism PSL. A Markov partition for a dynamical system offers a symbolic repre- sentation for the system in the following way. Let ΣN be the subset of the full N -shift (the set of bi-infinite series on N symbols), where N is the number of rectangles in the decomposition. Let Σ be a subset of ΣN defined by Σ = {s = (..., sn, sn+1, ....) : Rsn ∩ f̃−1Rsn+1 6= ∅} On Σ we naturally define a dynamical system with the operator of the right shift denoted by σ, and (Σ, σ) is called the subshift corresponding to the dynamical system. Σ can be completely described by stating which transitions k → m for k,m ∈ {1, ...., N} are allowed (i.e., for which k,m, f̃−1Rm ∩ Rk 6= ∅). See [1] for the definitions and for a proof that in this case we can define a map π : Σ→ S0 by π : s 7→ n=0 f̃ nRs−n ∩ .... ∩ f̃−nRsn which satisfies the following properties: • πσ = f̃π, • π is continuous, • π is onto. We take here the set of sequences with the Tichonoff topology. Thus a periodic point in the symbolic dynamical system which is just a peri- odic sequence corresponds to a periodic point in the original dynamical system. To obtain the Markov partition in our case as in [3], we thicken the edges of the graph to rectangles. In particular, the rectangles can be glued directly to each other without any junctions. This can be done in a smooth way, endowing S0 with a compact metric space structure by giving a length and width to each rectangle, consistently. Each edge of the standard graph for the pair (figure 10) corresponds to one rec- tangle, except the edge which is mapped to the loop around y0. This edge we divide in two (this is necessary to avoid having a rectangle intersecting twice an inverse image of another rectangle). Now we have edges of 7 different types on the graph. The vertical loops of the graph consist of long edges we denote as A edges, and short edges we call B edges. The loops around the points of the y orbit and vertical seg- ments connecting the loops to the diagonal edges we call C’s and D’s respectively. In rectangles which contain two punctures and therefore two diagonal edges we call the upper ones L edges and the lower ones K edges in the first case, and the lower ones L edges, upper ones K edges, in the second case. The last type of edges are diagonals of once punctured rectangles, these we call M edges. Next, we label the rectangles in order to have explicitly the transition rules: • For 0 ≤ i ≤ p2 − 1 denote the rectangle corresponding to the D edge connecting the loop around yi to the diagonal by ri+1. Denote the rectangle corresponding to the C edge which is the loop around yi by rp2+i+1. • For 1 ≤ i ≤ p2, denote the rectangles corresponding to the L and K edges connected to ri by r2p2+i and r3p2+i respectively. Figure 15. • Denote the rectangle corresponding to the A edge belonging to the vertical line we referred to as m by r4p2+1 and the B edge which is part of the same line m as r4p2+p1+1. • For the vertical line f i(m) denote it’s A and B rectangles by r4p2+1+i and r4p2+p1+1+i respectively for all 1 ≤ i ≤ p1 − 2 • For the vertical line fp1−1(m), denote it’s A edge as by r4p2+p1 . There are two rectangles corresponding to the B edge as ex- plained above, denote the lower one by r4p2+2p1 and the upper one by r4p2+2p1+1. • Label the p1− p2 remaining rectangles corresponding to the M edges by starting with the first of these to the right of m, and then continuing by the order along the horizontal axis, denoting them by r4p2+2p1+2, ... ,r3p1+3p2+1 Finally, we can look at the diagram in figure 16, showing the set of rectangles and transitions in this Markov partition which we now use. A periodic symbolic sequence of allowed transitions gives as ex- plained a periodic point in the original dynamical system. Therefore by this diagram we can easily find other periodic orbits on the torus that must exist for f . We will later prove that these orbits are in fact simple, but this will require some more work. Hence, by this diagram we prove only existence of orbits with specified rotation num- bers. For every pair (n,m) of natural numbers, n,m 6= 0, by starting from rp1+4p2+1, going n times around the first loop in the diagram {rp1+4p2+1, . . . , r2p1+4p2}, then going m-1 times around the second loop {r1, . . . , rp2} (and skipping it if m=1) and then returning through the final sequence {r2p2+1, . . . , r3p2} to rp1+4p2+1, we get a periodic symbolic allowed sequence, and so a new periodic orbit we denote On,m. These Figure 16. Some of the rectangles in the Markov par- tition, where the arrows denote allowed transitions be- tween them symbolic sequences are all different and hence so are the periodic orbits. We look at a point p ∈ On,m such that p is in the rectangle rp1+4p2+1. For the first n ·p1 iterations of p, corresponding to each time the upper loop in the diagram appears in the symbolic sequence of p, the images are contained in the B edges. The vertical loops are mapped under f retaining the same ”horizontal distance” from the periodic points from the x orbit to their left. So, p is mapped a total distance of n · q1 along the horizontal axis under fn·p1 . Similarly, point fnp1(p), which lies in the rectangle r1 corresponding to D edge, is mapped a distance q2 along the horizontal axis under each iteration of fp2 , for every occurrence of the second loop in the symbolic sequence of p. This is because the D edges retain their distance from the y orbit points below them. The final sequence in the symbolic representation of p until the return to the first loop also corresponds to the horizontal distance q1. These last points of the periodic orbit lie in rectangles corresponding to L edges. So p is mapped a horizontal distance of nq1 + mq2 under f np1+mp2 . Hence, the new orbit On,m has rotation number nq1+mq2 np1+mp2 See [12] for a proof that any two Farey neighbors span this way all rational numbers between them, that is all rationals between q1 and q2 are of the form nq1+mq2 np1+mp2 . So we found a periodic orbit of any rational number between the two original rotation numbers q1 and q2 Note we have found these simple periodic orbits for the Thurston- Nielsen canonical form of the homeomorphism f we started with. It remains to relate these periodic orbits to the periodic orbits of f itself. Recall the following definition from [2]. A periodic point x0 ∈ S of period p for homeomorphism f0 is called unremovable if for each given homomorphism f1 with ft : f0 ' f1 there is a periodic point x1 of period p for f1and an arc γ : [0, 1] → S with γ(0) = x0, γ(1) = x1 and γ(t) is a periodic point of period p for ft. It was proven by Asimov and Franks in [2] that every periodic orbit of a pseudo-Anosov diffeomorphism is unremovable. Thus orbits found for the pseudo-Anosov representative exist for any other homeomorphism in its isotopy class. This yields all these periodic orbits exist for the original homeomorphism f as well. Thus we get theorem 3.6 for our specific case: Theorem 3.7. If there exists a simple pair of orbits for a homeomor- phism f of the torus of abelian Nielsen types s and t which are Farey neighbors, there exists a periodic orbit for f with abelian Nielsen type equal to r for every rational number r between s and t. 4. The order relation For any simple pair q̂1 ∨ q̂2 , The orbit O1,1 out of the family of new orbits we constructed above has rotation number equal exactly to q1+q2 p1+p2 . This orbit corresponds to the symbolic sequence rp1+4p2+1 → rp1+4p2+2 → ....→ r2p1+4p2 → r2p2+1 → ....→ r3p2 → rp1+4p2+1 as in the diagram in figure 16. So we have a list of rectangles, each containing exactly one periodic point from the new orbit O1,1. We denote the point of O1,1 that is in a rectangle rj by oj. Graphically, assuming the first case map, when we draw the rectangle decomposition corresponding to the standard graph as in figure 10 we get Figure 17. We will now show that for any simple pair q̂1 ∨ q̂2 the orbit O1,1 is a simple orbit, and forms a simple pair with each periodic orbit of the pair, that is with q̂1 and q̂2 . For the first assertion, we define Figure 17. The intermediate orbit denoted by black circles. The gray areas are junction, which can be deleted, and the rectangles can be glued directly to one another. a family of vertical loops as follows: we choose a vertical loop that crosses both rectangles corresponding to the m line and passes to the right of the periodic point op1+4p2+1. Denote this loop by A. It is shown graphically in figure 18. All its images under f until the p1st iteration are exactly of the same form, as the rectangles are simply mapped to the right without changing their forms. Its p1st image is the first time it returns to the same rectangles, and is determined by the images of the corresponding vertical edges of the graph. These images are shown in figure 11. We use the fact f preserves orientation to determine the relation between the image of the curve and the points of O1,1. We denote this image by B. It is shown in figure 18. By similar considerations, knowing the rectangles containing B in the original picture (Figures 10 and 11) the rectangle adjacent to fp1−1(m) on the left contains a point of the y-orbit and therefore contains a rectangle of type L of Markov partition. This rectangle contains the point o3p2 of the new orbit. Line A lies to the right of m and of op1+4p2+1 therefore fp1−1(A) lies to the right of o3p2 and to the right of o2p1+4p2 . It follows that the line B = fp1(A) lies to the right of op1+4p2+1 and to the right of o2p2+1, as shown on Figure 18. Also B can be isotoped to the right of A relative to the points of the new periodic orbit. Next p2 − 1 iterations of f translate A and B and whole rectangle adjacent to m on the right to the right. We arrive at the rectangle adjacent to Figure 18. The intermediate orbit with the family of vertical loops l on the left containing point o3p2 of the new orbit. The point o2p1+4p2 lies in a rectangle on the line l. The loop fp2−1(B) lies to the right of o3p2 and to the left of o2p1+4p2 therefore the loop C = f p2(B) lies to the right of op1+4p2+1 and to the left of o2p2+1, as shown on Figure 18. Also p2 iterations of f take line m to a distance p2q1 = p1q2 − 1 rectangles to the right, which means one rectangle to the left of line m. Since A and B are to the right of the point op1+4p2+1 in line m and since this point moves to the leftmost point in the new periodic orbit shown on Figure 18, loop C must be to the right of it, as in Figure 18. The point o2p1+1 may be above or below the loop C but this does not change the discussion bellow. Note that if we disregard the orbits x and y of the original pair we can isotop C to A relative the points of the new orbit. This shows that the new orbit is a simple periodic orbit of length p1 + p2, by Lemma 2.3. Now we fill in the x orbit (the longer orbit) and consider a torus punctured at the y orbit and the new orbit together. We have the family of vertical loops f i(A) and the action on it is exactly as in the condition for a simple pair as the loop C can be isotoped to A plus a loop around y0. We choose a horizontal loop as the loop D on Figure 19. Then its image D′ is as shown on Figure 19. The image has the required properties. The orbit y together with the new orbit form a simple pair for the homeomorphism f . Next we fill in the y orbit and leave punctures at the x orbit and the new orbit. We choose the initial vertical loop A differently, as Figure 19. in Figure 20. This loop A is one rectangle to the right of m plus a loop on the left. After k = p2 − 1 iterations of f it will move to line l which is q1 rectangles to the left of m. Indeed it will move to q1(p2 − 1) + 1 = p1q2 − q1 rectangles to the right of m which means q1 rectangles to the left. The loop fk(A) looks like the loop A and lies to the left of the point r3p2 and to the left of the point o2p1+4p2 . Next iteration of f takes it to a curve which looks like f(l) but lies to the left of op1+4p2+1 and to the left of o2p2+1. Since we filled the point y0 we can isotop this loop to a vertical loop near m, which passes to the left of op1+4P2+1. Subsequent iterations translate it to the right and f p1(A) is equal to curve B on Figure 20 Next p2− 1 iterations will take B to the loop near l which lies to the left of o2p1+4p2 . Next iteration of f takes this loop to a loop similar to f(l), but lies to the left of o2p2+1. Since the points of y-orbit are filled we can isotop it to the loop C on Figure 20. It can be further isotoped, relative to the x-orbit and the new orbit, to the loop A plus a small loop around x0 to the left of A. If we choose the same horizontal loop as in the previous case , with the same image as before, we get the required action of f for a simple pair consisting of the x-orbit and the new orbit. Now we can continue by the same analysis for each of these two simple pairs, finding their Farey intermediate to be a simple orbit as well that forms a simple pair with each of them, and so on. It remains to prove the persistence of all these simple pairs under isotopies. For this, recall The following theorem from [11]. Figure 20. Theorem 4.1. (Hall) Let S be a closed surface and let A be a finite subset of S. Let f be a homeomorphism of S which leaves A invari- ant. Let p = x1, . . . , xk be a finite collection of periodic points for f which are essential, uncollapsible, mutually non-equivalent and non- equivalent to points of A. Then the collection p is unremovable, which means that for every homeomorphism g isotopic to f rel A there exists an isotopy ft rel A and paths xi(t) in S such that f0 = f , f1 = g, xi(0) = xi, xi(t) is a periodic point of ft of period equal exactly to the period of xi. (This theorem is a generalization of the main result of Asimov and Franks in [2] to several periodic orbits. In fact this generalization was mentioned in [2] as a remark with a hint of a proof.) Recall also that if f is pseudo-Anosov in the complement of A then it is condensed and by [6] Lemma 1 and Theorem 2.4 each periodic point is uncollapsible and essential and points from different orbits are non-equivalent and points disjoint from A are not equivalent to points of A. Corollary 4.2. : Let T be a torus and let A be a finite subset of T . Let f be a shear-type homeomorphism of T which is pseudo-Anosov in the complement of A. Let g be a homeomorphism of T isotopic to f in the complement of A. If x is a simple periodic orbit for f then there exists a simple periodic orbit z for g with ρ(z) = ρ(x). If x, y is a simple pair of periodic orbits for f , one or both disjoint from A, then there exists a simple pair of periodic orbits z, w for g with ρ(z) = ρ(x) and ρ(w) = ρ(y). Proof. Chose points x1 and y1 from the orbits x and y . By Theorem 4.1 there exists an isotopy ft and paths x1(t) and y1(t) such that x1(0) = x1, y1(0) = y1, x1(t) is a periodic point of ft of a fixed order p for all t and y1(t) is a periodic point of ft of a fixed order q for all t and y1(t) = y1(0) for all t if y(0) ∈ A. For a given t all points in the orbits of x1(t) and y1(t) for f are distinct, they form a braid with p+q strands. They move when t changes and their movement can be extended to an ambient isotopy ht which is fixed on A. Then ht(f i(x1(0)) = f t (x1(0)) and ht(f i(y1(0)) = f t (y1(0)). Consider isotopy Ft = h t ftht. We have i(x1(0)) = f i+1(x1(0)) and Ft(f i(y1(0)) = f i+1(y1(0)) so Ft is fixed on the orbits x and y. In particular x and y form a simple pair of periodic orbits for F1 (or x forms a simple periodic orbit for F1 if there is no y). But F1 = h 1 gh1 so h1(x) and h1(y) form a simple pair of periodic orbits for g. This concludes the proof of Theorem 1.1. 5. Global analysis of the kicked accelerated particle system The physical system called the kicked accelerated particle consists of particles that do not interact with one another. They are subject to gravitation and so fall downwards, and are kicked by an electro- magnetic field, i.e., the electro magnetic field is turned on for a very short time once in a fixed time interval. This electromagnetic field is a sine function of the height of the particle, hence the particles are kicked upwards or downwards by different amounts, depending on their po- sition at the time of a kick. For a short review of the results for this system see [8]. Experiments of this system were conducted by the Ox- ford group, see [17], and the system was found to show a phenomena that is now called ”quantum accelerator modes”: as opposed to the natural expectation that particles fall with more or less the gravita- tional acceleration, it was found that a finite fraction of the particles fall with constant nonzero acceleration relative to gravity, as can be seen in Figure 21 This is a truly quantum phenomenon having no counterpart in the classical dynamics. A theoretical explanation for this phenomenon was given by Fishman, Guanieri and Rebuzzini in [13], and it establishes a correspondence between accelerator modes of the physical system, and periodic orbits of the classical map Figure 21. Accelerator modes Experimental Data (taken from Oberthaler, Godun, d’Arcy, Summy and Burnett, see [17]) showing the number of atoms with specified momentum relative to the free falling frame as the system develops in time (the numbers on the y axis represents time by the number of kicks, while the z coordinate is proportional to the number of atoms) J + k̃sin(θ + J) + Ω θ + J mod2π(1) Where the J coordinate corresponds to the particles momentum, and θ to its coordinate. This map is of shear type, and the acceleration for a periodic orbit with rotation number q is given by − Ω(2) Hence, by analyzing the structure of existence of periodic orbits for the classical map above, we would be able to find which modes should be expected for which values of the parameters k and Ω. We remark that actual experimental observation also requires stability of the pe- riodic orbits. It is important to stress here that since these parameters correspond to the kick strength and the time interval between kicks they can be controlled in the experiments as we wish, so results ob- tained for this system can be tested experimentally. When one plots the numerical results describing which periods exist for different values of k and Ω one gets an extremely complicated figure, see figure 22. Figure 22. Tongues of periodic orbits An exact mathematical analysis of this system is extremely compli- cated. Perturbative methods have been used in [13] to analyze the ex- istence of these ”tongues” of periodic orbits in the region where k → 0, as well as giving estimates on their widths. Look at the map f given by (1) in regions where Ω is equal q for some rational number q in the unit interval, and small k. For a small enough k it can be seen both from the numerical results shown graphically in figure 22 and from perturbative arguments that in the above region a periodic orbit with period p exists. For small k the periodic points of this orbit must be pretty much equally spaced along the J axis, and we can choose (for k small enough) a family of vertical loops that are equally spaced at distance exactly Ω apart, and each is at distance at least, say, 3k from any of the periodic points. The image of a loop parameterized by Γ1(θ) = is given by( J0 + ksin(J0 + θ) + Ω J0 + θ J0 + ksin(θ ′) + Ω and so is very close (for small k) to another loop of the chosen family. It follows that there exists a map f̃ isotopic to f rel the orbit which keeps this family of curves invariant, and so, by Lemma 2.3, all the periodic orbits seen in the tips of the tongues in Figure 22 are simple orbits. Note the rotation number of each of these orbits is equal exactly to the value of Ω in the tip of the tongue (k = 0) as for very small k the J coordinate increases by an almost fixed value, close as we wish to Ω. And, by equation (2) the rotation number q is related to the acceleration of the corresponding acceleration mode by α = q 2π − Ω So the topological meaningful numbers here are in fact also the ones with physical significance. While Ω changes through the region in which this periodic orbit exists, q is of course a topological invariant and therefore fixed. Hence the acceleration vanishes on the line with fixed Ω in the middle of each tongue, and changes signs when one crosses this line. This was measured experimentally in [15]. For any other point higher in the tongue which we can reach by an isotopy along which the periodic orbit exists, we also have the orbit is a simple orbit. We will assume, as is very natural and was checked numerically for many cases, that the orbits remain simple throughout the region of each tongue. In some of the cases for which we drew a portrait of the phase space, we found that the fact the homeomorphism is isotopic to one which is reducible rel the periodic orbit is realized by the physical map itself, as seen in Figure 23. Here the phase space is truly divided into pieces. Each of the annuli in this decomposition is mapped to another, and returns to itself with one twist after p iterations of f . Therefore every periodic orbit must have a period which is a multiple of p. On the other hand, when an annulus is mapped to itself with one twist under an area preserving map (here under fp), every rotation number in the unit interval exists for it (here we mean the standard annulus rotation number measuring the rotations around the annulus), and so every period exists, as for every rational number n there is a periodic point of order m which rotates n times around the annulus before it returns to itself. This yields that for such a point in the parameter space, exactly all periods that are multiples of p exist. It is our belief that this situation is typical Figure 23. Phase portrait for a two-orbit Drawn for k = 1 2π and Ω = π, the two-orbit which is clearly seen is a stable orbit with two stable neighborhoods drawn. There is another two-orbit present, at which the arrows point, and it is the stable and unstable manifolds for this unstable orbit which divide the phase space into non intersecting regions which do not mix. for the center of each tongue, that is for Ω = 2π q . At other points, namely in all point we have numerically checked outside the center of the tongue, orbits of coprime lengths may exist simultaneously. We believe that the coexisting orbits whose rotation numbers are Farey neighbors form a simple pair together, as in the example on Figure 24, which shows a simple pair of orbits with rotation numbers 1/3 and 1/2 found in the physical system. This coexistence happens at a point in Figure 22 for which two tongues intersect. We assume that the same orbit persists through- out the tongue, and therefore we have at such a point two coexisting simple orbits. We believe that in all points of intersecting tongues coming from k = 0 and Ω1 = , Ω2 = which are Farey neighbors, p1 > p2, the coexisting orbits form a simple pair. Theorem 3.7 therefore implies that there are infinitely many periodic orbits for the parameters at a region of intersection of two such tongues, with rotation numbers equal to all rational numbers between the ones Figure 24. A pair of coexisting orbits in the physical system Drawn with a collection of curves on the torus and their images, which show this is a simple pair. of these two tongues. If we assume all these simple orbits present also come from tongues, this yields that each rational tongue between q1 and q2 intersects each of these two tongues lower (along the k axis) than they intersect each other. In other words, following a path from a tip of a tongue upwards in the tongue, if it intersects a Farey neighbor tongue we know it intersects earlier all tongues of rational numbers between them. This determines the global structure appearing in Figure 22 of all accelerator modes in the physical system, as Sharkovskii’s theorem determines it for one dimensional systems. References [1] R. L. Adler, Symbolic dynamics and Markov partitions, Bulletin of the AMS 35 N1 (1998), 1-56. [2] D. Asimov and J. Franks, Unremovable closed orbits, Geometric Dynamics, Lecture Notes in Mathematics 1007 (1983), 22-29. [3] M. Bestvina and M. Handel, Train-tracks for surface homeomorphisms, Topol- ogy 34 (1992), 109-140. [4] P. Boyland, An analog of Sharkovskiis theorem for twist maps, Hamiltonian Dynamical systems,Contemporary Mathematics 81 (1988), 119133. [5] P. Boyland, Topological methods in surface dynamics, Topology and it’s appli- cations 58 (1994), 223-298. [6] P. Boyland, Isotopy stability for dynamics on surfaces, Geometry and topology in dynamics, Contemp. Math. 246 (1999), 17-45. [7] A.J. Casson, S.A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, Cambridge University Press, 1988. [8] M.B. d’Arcy, G.S. Summy, S. Fishman and I. Guarneri, Novel Quantum Chaotic Dynamics in Cold Atoms, Physica Scripta 69 (2004), 25-31. [9] E. Doeff, Rotation measures for homeomorphisms of the torus homotopic to a Dehn twist, Ergod. Theor. Dynam. Syst. 17 (1997), 1-17. [10] E. Doeff and M. Misiurewicz, Shear rotation numbers, Nonlinearity 10 (1997), 1755-1762. [11] T. Hall, Unremovable periodic orbits of homeomorphisms, Math. Proc. Camb. Phil. Soc. 110 (1991), 523-531. [12] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Clarendon Press, Oxford, 1979. [13] S. Fishman, I. Guanieri and L. Rebuzzini, J. Stat. Phys. 110 (2003), 911; S. Fishman, I. Guarneri and L. Rebuzzini, Phys. Rev. Lett. 89 (2002), 84101-1- 4; I. Guarneri, L. Rebuzzini and S. Fishman, Arnol’d Tongues and Quan- tum Accelerator Modes, submitted for publiaction in Nonlinearity, (quant- ph/0512086) [14] M. Misiurewicz, Rotation Theory in: Online Proceedings of the RIMS Work- shop on ”Dynamical Systems and Applications: Recent Progress”. [15] Z-Y. Ma, M.B. d’Arcy and S. Gardiner, Phys. Rev. Lett. 93 (2004), 164101-1-4. [16] T. Matsuoka, Braids of periodic points and a 2-dimensional analogue of Sharkovskiis ordering, World Sci. Adv. Ser. in Dynamical Systems 1 (1986), 5872. [17] M.K. Oberthaler, R.M. Godun, M.B. d’Arcy, G.S. Summy, and K. Burnett, Phys. Rev. Lett. 83 (1999), 4447-4451. Tali Pinsky Department of Mathematics The Technion 32000 Haifa, Israel e-mail: [email protected] Bronislaw Wajnryb Department of Mathematics Rzeszow University of Technology ul. W. Pola 2, 35-959 Rzeszow, Poland e-mail: [email protected] http://arxiv.org/abs/quant-ph/0512086 http://arxiv.org/abs/quant-ph/0512086 Abstract 1. Introduction 2. Simple orbits 3. Simple orbit pairs 4. The order relation 5. Global analysis of the kicked accelerated particle system References

0704.1273

Coniveau over $p$-adic fields and points over finite fields

CONIVEAU OVER p-ADIC FIELDS AND POINTS OVER FINITE FIELDS HÉLÈNE ESNAULT Abstract. If the ℓ-adic cohomology of a projective smooth variety, defined over a p-adic field K with finite residue field k, is supported in codimension ≥ 1, then any model over the ring of integers of K has a k-rational point. Version française abrégée. Soit X une variété projective et absolument irréductible sur un corps local K. Rappelons qu’un modèle de X/K sur l’anneau de valuation R de K est un morphisme X → SpecR projectif et plat, tel que (X → SpecR)⊗K = (X → SpecK). Nous considérons la cohomologie ℓ-adique H i(X̄) à coefficients dans Qℓ. Dire qu’elle est supportée en codimension 1 signifie que toute classe dans H i(X̄) a une restriction nulle dans H i(Ū), où U ⊂ X est un ouvert non vide. Le but de cette note est de prouver le théorème suivant. Théorème: Soit X une variété projective lisse et absolument irréductible sur un corps local K de caractéristique 0 et à corps résiduel fini k. On suppose que la cohomologie ℓ-adique H i(X̄) est supportée en codimension ≥ 1 pour tout i ≥ 1. Soit X /R un modèle. Alors il existe un morphisme projectif surjectif σ : Y → X de R-schémas tel que |Y(k)| ≡ 1 modulo |k|. On en déduit immédiatement le corollaire suivant. Corollaire: Sous les hypothèses du théorème, tout modèle X /R possède un point k-rationnel. Pour ce qui concerne l’existence du point k-rationnel, ceci affranchit [6, Theo- rem 1.1], (qui est vrai aussi si K est de caractéristique p > 0), de l’hypothèse de régularité sur le choix du modèle X , qui était utilisée pour pouvoir appliquer le théorème de pureté de Gabber [7]. Pour ce faire, nous montrons que d’avoir des singularités quotient est suffisant, de même que pour l’étude de l’application de spécialisation. Nous appliquons alors la version plus précise du théorème de Jong ainsi qu’elle est exposée dans [2]. 1. Introduction Let X be a projective, absolutely irreducible variety defined over a local field K with finite residue field k. Recall that a model of X/K on the valuation ring R Date: April 6, 2007. Partially supported by the DFG Leibniz Preis. http://arxiv.org/abs/0704.1273v1 POINT 2 of K is a flat projective morphism X → SpecR such that (X → SpecR)⊗K = (X → SpecK). We consider ℓ-adic cohomology H i(X̄) with Qℓ-coefficents. One defines the first coniveau level N1H i(X̄) = {α ∈ H i(X̄), ∃ divisor D ⊂ X s.t. 0 = α|X\D ∈ H i(X \D)}.(1.1) As H i(X̄) is a finite dimensional Qℓ-vector space, one has by localization ∃D ⊂ X s.t. N1H i(X̄) = Im (X̄) → H iX̄) ,(1.2) where D ⊂ X is a divisor. One says that H iX̄) is supported in codimension 1 if N1H i(X̄) = H i(X̄). This definition is general, but has good properties only if X is irreducible and smooth or has only very mild singularities. In [6, Theorem 1.1] it is shown that if X/K is smooth, projective, absolutely irreducible over a local field K with finite residue field k, and if ℓ-adic cohomology H i(X̄) is supported in codimension ≥ 1 for all i ≥ 1, then any regular model X /R of X/K has the property X (k) ≡ 1 mod |k|.(1.3) The purpose of this note is to drop the regularity assumption if K has character- istic 0. Theorem 1.1. Let X be a smooth, projective, absolutely irreducible variety de- fined over a local field K of characteristic 0 with finite residue field k. Assume that ℓ-adic cohomology H i(X̄) is supported in codimension ≥ 1 for all i ≥ 1. Let X be a model of X over the ring of integers R of K. Then there is a projective surjective morphism σ : Y → X of R-schemes such that |Y(k)| ≡ 1 mod |k|. As an immediate corollary, one obtains Corollary 1.2. Under the assumptions of the theorem, every model X /R has a k-rational point. The regularity of the model X in the proof of [6, Theorem 1.1] (which is shown also when K has characteristic p > 0) was used to apply Gabber’s purity theorem [7]. We show that for the piece of regularity one needs, it is enough to have quotient singularities. Likewise, for the properties needed on the specialization map, quotient singularities are good enough. The more careful use of de Jong’s theorem as exposed in [2] allows then to conclude. Acknowledegment: This note relies on de Jong’s fundamental alteration theorems. T. Saito suggested to us the use of them in the shape formulated in [2]. We thank him for this, and for many subsequent discussions on the subject. We exposed a weaker version of Theorem 1.1 at the conference in honor of S. Bloch in Toronto in March 2007. Discussions with him, A. Beilinson and L. Illusie contributed to simplify our original exposition. POINT 3 2. Proof of Theorem 1.1 Let K be a local field of characteristc 0 with finite residue field k. Let R ⊂ K be its valuation ring. Let X → SpecR be an integral model of a projective variety X → SpecK. We do not assume here that X is absolutely irreducible, nor do we assume that X/K is smooth. Then by [2, Corollary 5.15], there is a diagram SpecR (2.1) and a finite group G acting on Z over Y with the properties (i) Z → SpecR and Y → SpecR are flat, (ii) σ is projective, surjective, and birational, (iii) Y is the quotient of Z by G, (iv) Z is regular. So Y → SpecR is not quite a model of X → SpecK, but is close to it. We show in the sequel that σ in (2.1) does it in Theorem 1.1. Set Y = Y ⊗K, Z = Z ⊗K. For an open U ⊂ X let us set YU = U ×X Y, ZU = U ×X Z. Let us assume now that X/K is smooth. This implies that H i(Ū) σ∗ inj −−−→ H i(YU).(2.2) Moreover, one has a trace map from Y to X H i(YU) (trace)Y/X // H i(Ū)(2.3) which splits σ∗ in (2.2). Let i ≥ 1 and let D ⊂ X be a divisor such that (X̄) ։ H i(X̄) and such that σ|X\D : Y \σ −1(D) → X \D is an isomorphism. Then (2.3) yields the commutative diagram H i(Ȳ ) (trace)Y/X // H i(Y \ σ−1(D)) H i(X̄) // H i(X \D) (2.4) and we conclude X/K smooth =⇒ N1H i(X̄) = H i(X̄) ⊂ N1H i(Y ) = H i(Y ).(2.5) We endow all schemes considered (which are R-schemes) with the upper subscript u to indicate the base change ⊗RR u or ⊗KK u, where Ku ⊃ K is the maximal POINT 4 unramified extension, and Ru ⊃ R is the normalization of R in Ku. Likewise, we write ? to indicate the base change ⊗RR̄, ⊗KK̄, ⊗kk̄, where K̄ ⊃ K, k̄ ⊃ k are the algebraic closures and R̄ ⊃ R is the normalization of R in K̄. We consider as in [6, (2.1)] the F -equivariant exact sequence ([5, 3.6(6)]) . . . → H i −→ H i(B̄) = H i(Yu) −−→ H i(Y u) → . . . ,(2.6) where F ∈ Gal(k̄/k) is the geometric Frobenius, and B = Y ⊗ k. One has Claim 2.1. The eigenvalues of the geometric Frobenius F ∈ Gal(k̄/k) acting on H i(Xu) and on H i(Y u) lie in q · Z̄ for all i ≥ 1. Proof. For H i(Xu), this is [6, Theorem 1.5(ii)]. One has H i(Ȳ ) = H i(Z̄)G, thus in particular, π∗ : H i(Ȳ ) → H i(Z̄) is injective. By (2.5) one has H i(Ȳ ) π∗ inj −−−→ N1H i(Z̄).(2.7) Since K has characteristic 0, and Z is regular by (iv), Z is smooth. Thus we can apply again [6, Theorem 1.5(ii)]. This finishes the proof. Claim 2.2. The eigenvalues of the geometric Frobenius F ∈ Gal(k̄/k) acting on ι(H i (Yu)) ⊂ H i(B̄) lie in q · Z̄ for all i ≥ 1. Proof. By (iii), one has H i (Yu) = H i (Zu)G ⊂ H i (Zu), where C = π−1(B). Since by (iv), Z is regular, we can apply [6, Theorem 1,4], which is a consequence of Gabber’s purity theorem [7], to conlude. Proof of Theorem 1.1. Claims 2.1 and 2.2 together with (2.6) show that the eigen- values of F acting on H i(B̄) lie in q · Z̄ for all i ≥ 1. We apply the Lefschetz trace formula |B(k)| = TrF |H∗(B̄). As B is absolutely connected and defined over k, F |H0(B̄) = Identity. By the discussion, one has |B(k)| ∈ N ∩ (1 + q · Z̄) ⊂ 1 + q · Z. � 3. Remarks Starting from Theorem 1.1, and Corollary 1.2, we may ask what happens if K has equal characteristic p > 0 and whether or not the congruence of the theorem is true on all models. We have no counter-examples for either question. What K is concerned, characteristic 0 is used in the proof of Claim 2.1: if K has characteristic p > 0, we only know that Z is regular, thus we can’t apply immediately [6, Theorem 1.5(ii)]. Going up to a strict semi-stable model does not help as for this, one has to ramify R and one loses regularity of Z and Z. What the POINT 5 congruence is concerned, instead of going to one birational model Y (or birational up to some inseprable extension in characteristic p > 0), one should go up to a hypercover built out of such Y . In doing Deligne’s construction of hypercovers with resolutions of singularities being replaced by de Jong’s morphisms of the type σ in (2.1), one creates components which do not dominate X , the cohomology of which is very hard to control. So one perhaps loses the coniveau property. References [1] de Jong, A. J.: Smoothness, semi-stability and alterations, Publ. Math. IHES 83 (1996), 51-93. [2] de Jong, A. J.: Families of curves and alterations, Ann. Inst. Fourier 47 no2 (1997), 599-621. [3] Deligne, P.: Théorème d’intégralité, Appendix to Katz, N.: Le niveau de la cohomologie des intersections complètes, Exposé XXI in SGA 7, Lect. Notes Math. vol. 340, 363-400, Berlin Heidelberg New York Springer 1973. [4] Deligne, P.: Théorie de Hodge III, Publ. Math. IHES 44 (1974), 5-77. [5] Deligne, P.: La conjecture de Weil, II. Publ. Math. IHES 52 (1981), 137-252. [6] Esnault, H.: Deligne’s integrality theorem in unequal characteristic and rational points over finite fields, with an appendix with P. Deligne, Annals of Mathematics 164 (2006), 715-730. [7] Fujiwara, K.: A Proof of the Absolute Purity Conjecture (after Gabber), in Algebraic Ge- ometry 2000, Azumino, Advanced Studies in Pure Mathematics 36 (2002), Mathematical Society of Japan, 153-183. Universität Duisburg-Essen, Mathematik, 45117 Essen, Germany E-mail address : [email protected] 1. Introduction 2. Proof of Theorem ?? 3. Remarks References

0704.1274

Parametric Learning and Monte Carlo Optimization

Keywords: Monte Carlo Optimization, Black-box Optimization, Parametric Learning, Automated Annealing, Bias-variance-covariance Parametric Learning and Monte Carlo Optimization David H. Wolpert [email protected] MS 269-1, Ames Research Center Moffett Field, CA 94035. Dev G. Rajnayaran [email protected] Department of Aeronautics and Astronautics, Durand Rm. 158, 496 Lomita Mall, Stanford, CA 94305. Abstract This paper uncovers and explores the close relationship between Monte Carlo Optimization of a parametrized integral (MCO), Parametric machine-Learning (PL), and ‘blackbox’ or ‘oracle’-based optimization (BO). We make four contributions. First, we prove that MCO is mathematically identical to a broad class of PL problems. This identity potentially provides a new application domain for all broadly applicable PL techniques: MCO. Second, we introduce immediate sampling, a new version of the Probability Collectives (PC) algorithm for blackbox optimization. Immediate sampling transforms the original BO problem into an MCO problem. Accordingly, by combining these first two contributions, we can apply all PL techniques to BO. In our third contribution we validate this way of improving BO by demonstrating that cross-validation and bagging improve immediate sampling. Finally, conventional MC and MCO procedures ignore the relationship between the sample point locations and the associated values of the integrand; only the values of the integrand at those locations are considered. We demonstrate that one can exploit the sample location information using PL techniques, for example by forming a fit of the sample locations to the associated values of the integrand. This provides an additional way to apply PL techniques to improve MCO. 1. Introduction This paper uncovers and explores some aspects of the close relationship between Monte Carlo Optimization of a parametrized integral (MCO), Parametric machine Learning (PL), and ‘blackbox’ or ‘oracle-based’ optimization (BO). We make four primary contributions. First, we establish a mathematical identity equating MCO with PL. This identity poten- tially provides a new application domain for all broadly-applicable PL techniques, viz., Our second contribution is the introduction of immediate sampling. This is a new version of the Probability Collectives (PC) approach to blackbox optimization. PC encom- passes Estimation of Distribution Algorithms (EDAs)(De Bonet et al., 1997; Larraaga and Lozano, 2001; Lozano et al., 2005) and the Cross Entropy (CE) method (Rubinstein and Kroese, 2004) as special cases. However PC is broader and more fully motivated. This means it uncovers (and overcomes) formal shortcomings in those other approaches. In the immediate sampling version of PC the original BO problem is transformed into an MCO problem. In light of our first contribution, this means we can apply PL to immediate sampling. In this way all PL techniques — including cross-validation, bagging, boosting, active learning, stacking, and others — can be applied to blackbox optimization. In our third contribution we experimentally explore the power of this identity between MCO and PL. In these experiments we demonstrate that cross-validation and bagging improve the performance of immediate sampling blackbox optimization. In particular, in these experiments we show that cross-validation can be used to adaptively set an ‘annealing schedule’ for blackbox optimization using immediate sampling without any extra calls to the oracle. In some cases, we show that this adaptively formed annealing schedule results in better optimization performance than any exponential annealing schedule.1 Finally, conventional MC and MCO procedures ignore the relationship between the sample point locations and the associated values of the integrand. (Only the values of the integrand at the sample locations are considered by such algorithms.) We end by exploring ways to use PL techniques to exploit the information in the sample locations, for instance, by Bayesian fitting of a surface from the sample locations to the associated values of the integrand. This constitutes yet another way of applying PL to MCO in general, and therefore to BO in particular. 1.1 Background on PL, MCO, Blackbox Optimization, and PC We begin by sketching the four disciplines discussed in this paper: 1. A large number of parametric machine-learning problems share the following two properties. First, the goal in these problems is to find a set of parameters, θ, that minimizes an integral of a function that is parametrized by θ. Second, to help us find that θ we are are given samples of the integrand. These problems reduce to a sample-based search for the θ that we predict minimizes the integral. We will refer to problems of this class as Parametric Learning (PL) problems. An example of PL is parametric supervised learning, where we want to find an optimal predictor or regressor zθ that minimizes the associated expected loss, dx dy P (x)P (y | x)L[y, zθ(x)], where x’s are inputs and y’s are outputs. We do not, however, know P (x)P (y | x). Instead, we are provided a training set of samples of P (x)P (y | x). The associated PL problem is to use those samples to estimate the optimal θ. 2. MCO is a technique for solving problems of the form argminφ dw U(w, φ) (see Ermoliev and Norkin, 1998). MCO starts by replacing that integral with an importance-sample generated estimate of it. That estimate is a sum parametrized by φ. In MCO one searches for the value φ that minimizes this sum; the result of this search is one’s estimate of the φ that optimizes the original integral. 3. Blackbox optimization algorithms are ways to minimize functions of the form G : X → R when one does not actually know the function G. Such algorithms work by an iterative process in which they first select a query x ∈ X, and then an ‘oracle’ returns to the algorithm a (poten- tially noise-corrupted) value G(x), and no other information, in particular, no gradient infor- mation. The difference between one blackbox optimization algorithm and another is how they select each successive query based on the earlier responses of the oracle. Examples of blackbox optimization algorithms are genetic algorithms (Mitchell, 1996), simulated annealing (Kirk- patrick et al., 1983), hill-climbing algorithms, Response-Surface Methods (RSMs) (Myers and Montgomery, 2002), and some forms of Sequential Quadratic Programming (SQP) (Gill et al., 1981; Nocedal and Wright, 1999), Estimation of Distribution Algorithms (EDAs)(De Bonet et al., 1997; Larraaga and Lozano, 2001; Lozano et al., 2005), tabu search, the Cross Entropy (CE) method (Rubinstein and Kroese, 2004), and others. 1. Since they are special cases of PC, presumably we could similarly apply PL techniques to improve EDA’s or the CE method. 4. PC is a set of techniques that can be used for blackbox optimization. Broadly speaking, PC works by transforming a search for the best value of a variable x into a search for the best probability distribution over the variable, q(x) (see Wolpert et al., 2006; Macready and Wolpert, 2005; Wolpert, 2003, 2004a; Bieniawski and Wolpert, 2004; Antoine et al., 2004; Lee and Wolpert, 2004). Once one solves for the optimal q(x), inversion to get the optimal x for the original search problem is stochastic; one simply samples q. As described below, PC has many practical strengths, and is related to RSMs, EDAs, and the CE method. 1.2 Roadmap of This Paper We make four primary contributions: 1. Sec. 2 begins with a detailed review of MCO and PL. Conventional analysis of Monte Carlo estimation involves a bias-variance decomposition of the error of the estimator of a particular integral. We show that for MCO, a full analysis requires more than simply extending such bias-variance analysis separately to each of the estimators given by the separate φ’s. Moments coupling the errors of the estimators for the separate φ’s must also be taken into account. How should we do that? To answer this, we note that in a different context, the techniques of PL take such coupling moments into account, albeit implicitly. This leads us to explore the relation between MCO and PL. This in turn leads to our first major contribution, the proof that MCO is identical to PL. This contribution means that one can apply all PL techniques, for instance, cross- validation, bagging, boosting, stacking, active learning and others, to MCO. Such PL-based MCO (PLMCO) provides a new way of minimizing potentially high-dimensional parametrized integrals. Experimentally testing the utility of applying PL to MCO requires an MCO application do- main. Here we choose the domain of blackbox optimization. To establish how blackbox optimization is an application domain for MCO requires our second contribution, as follows. 2. We start in Sec. 3 by presenting an overview of previous versions of the blackbox optimization approach of PC. We then make our second contribution in the following section, where we introduce immediate sampling, a new version of PC that overcomes some of the limitations of previous versions. These first two contributions are combined by the fact that immediate sampling is a special case of MCO. The resultant identity between PL and immediate sampling means that, in principle, any PL technique can be applied to blackbox optimization. In particular, regular- ization, cross-validation, bagging, active learning, boosting, stacking, kernel machines, and others, can be ‘cut and paste’ to do blackbox optimization. This use of PL for blackbox optimization constitutes a new application domain for PL. In Sec. 4.5 we present some concrete instances of how to modify immediate sampling to use PLMCO rather than conventional MCO. It is important to note that when applied (via im- mediate sampling) to blackbox optimization, these PL techniques do not require additional calls to the oracle. For example, using cross-validation to set regularization parameters in immediate sampling (the equivalent of an annealing schedule in SA) does not involve running the entire blackbox optimization algorithm with different regularization schedules. As an- other example, using bagging in immediate sampling does not mean running the optimization algorithm multiple times based on different subsets of the sample points found so far. 3. Our third contribution is to experimentally demonstrate in Sec. 5 that PLMCO substantially outperforms conventional MCO when used this way for blackbox optimization. We are partic- ularly interested in blackbox optimization problems where calls to the oracle are the primary expense. Accordingly, non-oracle, ‘offline’ computation is considered free. So in our experi- ments we compare algorithms based on the values of G found by the algorithms versus the associated number of calls to the oracle2. In particular, we show that bagging and cross- validation leads to faster blackbox optimization on two well-known benchmark problems for continuous nonconvex optimization. It should be emphasized that these experiments are not intended to investigate whether PLMCO applied to immediate sampling is superior to other blackbox optimization algorithms. Rather their purpose is to investigate whether one can indeed leverage the formal connection between PL and MCO to improve immediate sampling. Accordingly, these experiments are on toy domains, and we do not compare performance with other blackbox optimization algo- rithms. We leave such comparisons to future papers. 4. In estimating the value of an integral based on random samples of its integrand, conventional MC and MCO techniques ignore how the locations of the sample points are related to the associated values of the integrand. Such techniques concentrate exclusively on those sample values of the integrand that are returned by the oracle. However, one can use the sample locations and associated integrand values to form a supervised learning fit to the integrand. In principle, such a fit can then be used to improve the overall estimate of the integral. In ‘fit-based’ MC and MCO one uses all the data at hand to fit the integrand and then uses that fit to improve the algorithm. In this paper, we concentrate on situations where the data at hand consist only of sample locations and the associated values of the integrand, but in other situations the data at hand may also include information like the gradient of the integrand at the sample points. In their most general form, fit-based MC and MCO include techniques to exploit such information. One natural Bayesian approach to fit-based MC uses Gaussian processes. Work adopting this approach, for the case where the data only contain sample locations and associated integrand values, is reviewed in Rasmussen and Gharamani (2003). In Sec. 6 we generalize that work on fit-based MC, e.g., to allow non-Bayesian approaches. In that section, we also consider fit-based MCO in general, and fit-based immediate sampling in particular. One of the ways cross-validation is used in these experiments is to set a regularization parameter. In immediate sampling, the regularization parameter plays the same role as the temperature does in simulated annealing. So intuitively speaking, our results show how to use cross-validation to set an annealing schedule adaptively for blackbox optimiza- tion, without extra calls to the oracle. We show in particular that such auto-annealing outperforms the best-fit exponential annealing schedule. There are more topics involving the connection between MCO, immediate sampling and PL, than can be explored in this single paper. One such topic is how to incorporate constraints on x in immediate sampling. Another important topic involves a derivation from first principles of the objective function used in immediate sampling. These two topics are briefly discussed in the appendices. Some other topics are mentioned, albeit even more briefly, in the conclusion. 1.3 Notation As a point of notation, we will use the term ‘distribution’ to refer either to a probability distribution or a density function, with the associated Borel field implicitly fixing the meaning. Similarly, we will write integrals even when we mean sums; the measure of the 2. See Wolpert and Macready (1997); Droste et al. (2002); Wolpert and Macready (2005); Corne and Knowles (2003); Igel and Toussaint (2004); Schumacher et al. (2001) for a discussion of the mathematics relating algorithms under such performance measures. integral is implicitly taken to be the one appropriate for the the argument. We will use Θ to indicate the Heaviside or indicator function, which is 1 if its argument is positive, and 0 otherwise. We will use P to mean the set of all distributions over X. We are primarily interested in X’s that are too large to permit computations involving all members of P. Accordingly, we will will work with parametrized subsets Q ⊂ P. We generically write that (possibly vector-valued)parameter as θ, and write the element of Q specified by θ as qθ. We use E to indicate the expectation of a random variable. Subscripts on E are sometimes used to indicate the distribution(s) defining the expectation. We take any oracle G to be an x-indexed set of independent stochastic processes, and use the symbol g to indicate the generic output of the oracle in response to any query. With some abuse of notation, we denote the output of the oracle for query x as P (g | x,G ). For a noise-free, or single-valued oracle, we write P (g | x,G ) = δ(g −G(x)) for some function G implicitly specified by G , where δ(.) is the Dirac delta function. When there is both a factual version of a random variable and a posterior distribution over counter-factual values of that variable, they must be distinguished. In general this requires extending the conventional Bayesian formalism (see Wolpert, 1997, 1996). Here, though, it suffices for us to indicate counter-factual values by a subscript c. Say there is a factual oracle G , and we are provided a data set D formed by sampling G . We use superscripts to denote different samples in that data set. Then D in turn induces a posterior over oracles, and we write that posterior as P (Gc | D). 2. MCO and PL In this section we review PL and MCO show that they are mathematically identical. 2.1 Overview of PL A broad class of parametric machine learning problems try to find (P1) : argminξ dx P (x)Rx(ξ). For subsequent purposes, it will be useful to write x as a subscript and ξ as an argument of R, even though x is the integration variable and ξ is the parameter being optimized. To perform this minimization, we have a set of function values D ≡ {Rxi(ξ)}, where we typically assume that the samples xi, i = 1, . . . , N were formed by IID sampling of P (x). The maximum likelihood approach to this minimization first makes the approximation∫ dx P (x)Rx(ξ) ≈ Rxi(ξ), Ri(ξ). (1) One then solves for the ξ minimizing the sum, and uses this as an approximation to the solution to P1. In practice, though, this procedure is seldom used directly: although the approximation in Eq. 1 is unbiased for any fixed ξ, minξ i(ξ) is not an unbiased estimate of minξ dx P (x)Rx(ξ). Therefore, when this approximation is exploited, it is modified to incorporate bias-reduction techniques. Example: Parametric Supervised Learning: Let X,Y be input and output spaces, respectively. Let L(y1, y2) : Y × Y → R be a loss function, and zξ : X → Y be a ξ-parametrized set of functions. In parametric machine learning with IID error our goal is to solve argminξ dx P (x) dy P (y | x)L(y, zξ(x)), argminξ dx P (x)Rx(ξ). (2) Intuitively, Rx(ξ) is the expected loss at x for the ‘fit’ zξ(x) to the x-indexed set of distri- butions P (y | x). To perform this minimization we have a training set of pairs D ≡ {xi, yi}, i = 1, . . . , N , that we assume were formed by IID sampling of P (x)P (y | x). The maximum likelihood approach to this minimization first makes the approximation∫ dx P (x) dy P (y | x)L(y, zξ(x)) ≈ L(yi, zξ(x Ri(ξ). One then solves for the ξ minimizing the sum, and uses this as an approximation to the solution to (P1). As discussed above, in practice, this minimization is rarely used directly, and is usually combined with a bias-reducing technique like cross-validation. 2.2 Overview of MCO Consider the problem (P2) : argminφ∈Φ dw U(w, φ). For now, we do not impose constraints on φ, nor restrict Φ. Monte Carlo Optimiza- tion (Ermoliev and Norkin, 1998) is a way to search for the solution of (P2). In MCO we use importance sampling to rewrite the integral in (P2) as∫ dw U(w, φ) = dw v(w) U(w, φ) dw v(w)rv,U,w(φ), (3) for some sampling distribution v. Following the usual importance sampling procedure, we IID sample v to form a sample set {U(wi, .) : i = 1, . . . N}, which specifies a set of N sample functions ri(φ) , rv,U,wi(φ). It is implicitly assumed that for any w, we can evaluate v(w) up to an overall normalization constant. In MCO, these N functions are used in combination with any prior information to estimate the solution to (P2). Conventionally, this is done by approximating the solution to (P2) with the solution to the problem (P3) : argminφ ri(φ). We define LU (φ) , dw U(w, φ), L̂v,U,{wi}(φ) , rv,U,wi(φ), φ̂v,U,{wi}(φ) , argmin[L̂v,U,{wi}(φ)]. For notational simplicity, the subscripts will usually be omitted in these expressions. We will use the term naive MCO to refer to solving (P3) by minimizing L̂ (φ). 2.3 Statistical Analysis of MCO The statistical analysis of MC estimation of integrals is a relatively mature field (see Robert and Casella, 2004; Fishman, 1996). We now show that when such MC estimation is com- bined with parameter optimization in MCO, the analysis becomes much more involved. 2.3.1 Review: MC Estimation First consider MC estimation, with no mention of MCO. We first need to to specify a loss function L(., .) that will couple our mathematics with real-world costs. The first argument of such an L is the output of the estimation algorithm under consideration. The second argument is the quantity statistically sampled by that algorithm. The associated value of L is the cost if the algorithm produces the output specified in that first argument, using the quantity specified in the second argument. As an example, consider importance-sampled MC estimation of an integral. Using the MCO notation just introduced, we use L̂ (φ) as an estimate of L (φ) for some fixed φ. The quantity being sampled is the function U(., φ), and the output of the algorithm is L̂ (φ). Accordingly, these are the arguments of the loss function. The most popular loss function in statistical analysis of MC integral estimation is quadratic loss, given below. L(L̂ (φ), U(., φ)) , [L̂ (φ)− dw U(w, φ)]2. Unless explicitly stated otherwise, we will henceforth use the term ‘expected loss’ to refer to the average of this loss function over sample sets. Since L̂ (φ) is an unbiased estimate of L (φ), the expected loss is the sample variance, Var(L̂ (φ)) = E([ U(wj , φ) Nv(wj) ]2) − [E( U(wj , φ) Nv(wj) dw v(w)[ U(w, φ) ]2 − [ dw v(w) U(w, φ) dw v(w)[ U(w, φ) ]2 − [L (φ)]2}. This expansion for the sample variance is quite useful. For example, one can solve for the v that minimizes this variance (and therefore minimizes expected loss) as a function of U(., φ). For nowhere-negative U , that optimal v is given by (see Robert and Casella, 2004) v(w) , U(w, φ)∫ dw′U(w′, φ) Given the formula for the optimal v, one can estimate it from a current sample set, and then use the estimated optimal v for future sampling. This is what is done in the VEGAS Algorithm (Lepage, 1978, 1980). Consideration of the sample variance has also led to algorithms that partition X and then run importance sampling on each partition element separately, for instance, stratified sampling (Fishman, 1996). MC estimators that do not use strict importance sampling may introduce bias. However, if the variance is sufficiently reduced, expected quadratic error is reduced. This can be exploited to tradeoff bias and variance. 2.3.2 From MC to MCO When we combine MC with parameter optimization in MCO, quantities like Var(L̂ (φ)) for one particular φ are not the main objects of interest. Instead, we are interested in expected loss of our iterated MCO algorithm, which involves multiple φ’s. So what is the appropriate loss function for analyzing MCO? From the very definition of (P2), it is clear that we want L(φ,U) to be minimized by the φ that minimizes dw U(w, φ). The simplest approach to doing this, which will be assumed from now on, stipulates that L(φ,U) = L (φ) = dw U(w, φ), (4) the same integral appearing in (P2). If we can solve (P2) exactly, then we will have produced the φ with minimal value of this loss function. Given this choice of loss function, expected loss in naive MCO is E(L | U, v) = dw1 . . . dwN v(wi)L (argminφ[L̂v,U,{wi}(φ)]) dw1 . . . dwN v(wi) dw′ U(w′, argminφ[ U(wi, φ) v(wi) ]). (5) The optimal v for naive MCO is the one that minimizes E(L | U, v). There is no direct relation between this v and the one that minimizes loss for some single φ. In stark contrast to the MC analysis in Sec. 2.3.1, in addition to the sample variance Var(L̂ (φ)) for any single φ, the expected loss E(L | U, v) now also depends on moments coupling the distributions of L̂ (φ) for different φ’s. Loosely speaking, the bias-variance tradeoff in Sec. 2.3.1 now becomes a more complicated bias-variance-covariance tradeoff. Now, setting w is more involved, but we can approach it as follows. Expressing the expected loss slightly differently gives us an important insight. Note that each sample set {wi} gives rise to an associated set of estimates for all φ ∈ Φ. Call this (possibly infinite dimensional) vector of estimates ~l, each of whose components is indexed by φ and is an estimate for that particular φ. Now, instead of computing expected loss by averaging over all possible sample sets, we average over all possible vectors ~l. In order to do this, we need to specify the probability of each vector ~l. Define πv,U,Φ(~l) , Pr({wi} : L̂v,U,{wi}(φ) = lφ ∀φ ∈ Φ). So, π v,U,Φ(~l) is the probability of a set of sample points {wi} such that for each φ ∈ Φ, the associated empirical estimate i rv,U,wi(φ) equals the corresponding component of ~l. For notational simplicity the subscripts of π v,U,Φ will sometimes be omitted. We can now write Eq. 5 succinctly as E(L | U, v) = d~l π(~l) L (argminφ[lφ]) (6) where ‘argminφ[lφ]’ means the index φ of the smallest component of ~l. The risk is the difference between this expected loss and the lowest possible loss. We can write that risk d~l πv,U,Φ(~l) [L (argminφ[lφ]) − minφ[L (φ)]]. (7) Our sample set constitutes a set of samples of πv,U,Φ occurring in Eq. 6,This fact can potentially be exploited to dynamically modify v and/or Φ to reduce E(L | U, v). Indeed, for the simpler case of MC estimation, this is essentially the kind of computation done in the VEGAS algorithm mentioned above. As a practical issue, it may be difficult to update v and/or Φ using the full formula Eq. 5. Instead, one could approximate that formula E(L | U, v) near a single φ of interest, e.g., about a current estimate for the optimal φ. Intuitively though, one would expect that for a fixed set of φ’s, everything else being equal, it would be advantageous to have small variances of unbiased estimators and large covariances between them. Such considerations based on the second moments may help one choose quantities like the sampling distribution v. Such considerations may also help one choose the set of candidate φ’s, Φ. For example, one way to have large covariances between the φ ∈ Φ is to have the associated functions over w, {U(., φ) : φ ∈ Φ}, all lie close to one another in an appropriate function space (e.g., according to an l∞ norm comparing such functions). However, choosing such a Φ will tend to mean there is a small ‘coverage’ of that set of functions, {U(., φ) : φ ∈ Φ}. More precisely, it will tend to prevent the best of those φ’s from being very good; minφ∈Φ[ dw U(w, φ)] will not be very low. This illustrates that, in choosing Φ, there will be a tradeoff between two quantities: The first quantity is the best possible performance with any of the φ ∈ Φ. The second quantity is the risk, that is, how close a given MCO algorithm operating on Φ is likely to come to that best possible performance of a member of Φ. Choosing Φ to have large covariances of the (MC estimators based on the) members of Φ, and in particular to have large covariances with the truly optimal φ, argminφ∈ΦL (φ), will tend to result in low risk. But it will also tend to result in poor best-possible performance over all φ ∈ Φ. Similarly, one would expect that as the size of Φ increases, there would be a greater chance that a sample set for one of the suboptimal φ ∈ Φ would have low expected loss ‘by luck’. This would then mislead one into choosing that suboptimal φ. So increasing the size of Φ may increase risk. However increasing Φ’s size should also improve best possible performance. So again, we get a tradeoff. It may be that such considerations involving the size of Φ and the covariances of its members can be encapsulated in a single number, giving an ‘effective size’ of Φ (somewhat analogously to the VC dimension of a set of functions). Such tradeoffs are specific to the use of MCO, and do not arise in plain (single-φ) MC. They are in addition to the usual bias-variance tradeoffs, which still apply to each of the separate MC estimators. An illustrative example of the foregoing is provided in App. C. A more complete sta- tistical analysis of risk in MCO, including Bayesian considerations, is in Sec. 6. MCO PL v(w) P (x) rv,w(φ) Rx(ξ) riv(φ) R Table 1: Correspondence between PL and MCO. 2.4 PL Equals MCO In MCO, we have to extrapolate from the sample set of w values to perform the integral minimization in Eq. 3. As discussed above, this can recast as having a set of sample functions φ→ ri(φ) that we want to use to estimate the φ that achieves that minimization. Similarly, in PL, we have to extrapolate from a training set of functions Ri(ξ) to minimize the integral dx P (x)Rx(ξ). Though not usually viewed this way, at the root of this extrapolation problem is the problem of using the sample functions ξ → Ri(ξ) to estimate the minimizer of Eq. 2. In addition, the analysis of Sec. 2.3 is closely related to the PL field of uniform conver- gence theory. That field can be cast in the terms of the current discussion as considering a broad class of U ’s, U . Its starting point is the establishment of bounds on how maxv,U∈U [ d~l πv,U,Φ(~l) Θ(L (argminφ[lφ]) − minφ[L (φ)] − κ) (8) depends3 on κ. Of particular interest is how the function taking κ to the associated bound varies with characteristics of U and Φ (see Vapnik, 1982, 1995). Eq. 8 should be compared with Eq. 7. All of this suggests that the general MCO problem of extrapolation from a sample set of empirical functions to minimize the integral of Eq. 3, is, in fact, identical to the general PL problem of extrapolation from a training set of empirical functions to minimize the integral of Eq. 2. This is indeed the case. As shown in Table 1, identify ξ ↔ φ, x ↔ w,P (x) ↔ v(w), Rx(ξ) ↔ rv,w(φ), riv(φ) ↔ Ri(ξ). Then the integrals in Eq. 3 and (P1) become identical. So the MCO expected loss function in Eq. 4 becomes identical to the PL expected loss. Similarly, the sample functions for MCO and PL become identical. In particular, in supervised learning, when there is no noise, P (y | x) becomes a single- valued function y(x), and the parametric supervised learning problem becomes argminξ dx P (x)[L(y(x), zξ(x))] This should be compared to the MCO problem as formulated in Eq. 3. For the same reasons that direct minimization of Eq. 1 is seldom used in PL, we now see that naive MCO will be biased, and should preferably not be used directly. Note that most sampling theory analysis of PL does not directly consider the biases and variances of the separate Monte Carlo estimators for each ξ, nor does it directly consider the moments that couple the distributions of those estimates. Rather, it considers a different 3. As an example, rewrite w → x, φ → α, v(x) → P (x). Also choose U to be all functions of the form U(w, φ) = U(x, α) , dy P (y | x)(y − F (x, α))2 for any function F and distribution P (y | x). Under this substitution, Eq. 8 becomes the archetypal uniform convergence theory problem for regression with quadratic loss. type of bias and variance — the bias and variance of an entire algorithm that chooses a ξ based on associated MC estimates of expected loss (Wolpert, 1997). In this sense, such PL analysis bypasses the issues considered in Sec. 2.3. The bias-variance-covariance approach described in this section might have important implications on PL analysis of learning algorithms, but for the moment, in our exploration of the identity between MCO and PL, we simply use PL-based techniques to reduce the bias or variance of our algorithms. 3. Review of PC This section cursorily reviews the previously investigated type of PC. It then briefly dis- cusses the advantages of PC for blackbox optimization and its relation to other blackbox optimization techniques. 3.1 Introduction to PC To introduce PC, consider the general (not necessarily blackbox) optimization problem (P4) : argminx∈XE(g | x,G ). For now, we ignore constraints on x. In PC we transform (P4) into the problem (P5) : argminqθ∈QFG (qθ), for some appropriate function FG . After solving (P5) we stochastically invert qθ to get an x (the ultimate object of interest), by sampling qθ. This type of “randomizing transform” contrasts with conventional transform techniques, where inversion is deterministic. Ideally, FG should be chosen in a first-principles manner, based on exactly how qθ will be sampled and how those samples used (see Sec. 6). In practice though, computational considerations might lead one to choose FG heuristically. Intuitively, such considerations might compel us to choose FG both so that (P5) is readily easy to solve, and so that any solution qθ to (P5) is concentrated about the solutions of (P4). Taking the parametrization to be implicit, we often abbreviate FG (qθ) as just FG (θ). In many variants of PC explored to date, FG (θ) is an integral transform4 over X, FG (θ) , dx dg P (g | x,G )F (g, qθ(x)). (9) dx rP (g|x,G )(x, θ) (10) As an example of such an integral transform, consider optimization with a noise-free (single- valued) oracle, P (g | x,G ) = δ(g −G(x)), where the transformed objective is the expected value of (g|x) under x ∼ qθ. In other words, FG = Eqθ [G(x)]. In addition, suppose that Q = P, that is, qθ can be any distribution. Under fairly weak assumptions, it can be shown that one solution to (P5) is given by the point-wise limit of Boltzmann distributions, p?(x) = lim pβ(x), where pβ(x) ∝ exp[−βG(x)]. In the case where Q ⊂P, we could choose FG (θ) to be a measure of the dissimilarity between such a Boltzmann (or other) ‘target’ distribution, and a given qθ. For instance, 4. An instance where this is not the case is with the elite objective function, described in App. B. we could use a Kullback-Leibler (KL) divergence between qθ and pβ , which we refer to as “pq” KL distance: FG (θ) = KL(p β || qθ) dx pβ(x)ln[ qθ(x) pβ(x) In terms of the quantities in Eq. 9, F (g, qθ(x)) ∝ e−βgln[qθ(x)], up to an overall additive constant. So rP (g|x,G )(x, θ) in Eq. 10 is the contribution to the KL distance between pβ and qθ given by the argument x. To see why this choice of FG (θ) is reasonable, first note that pβ(x) is large where G(x) is small. Indeed, as β →∞, pβ becomes a delta function about the x(s) minimizing G(x), that is, about the solution(s) to (P4). Now, suppose that Q is a broad enough class that it can approximate any sufficiently peaked distribution. That means that there is a qθ ∈ Q for which KL(pβ || qθ) is small for large β. In such a situation, the qθ solving (P5) will be highly peaked about the x(s) solving (P4). Accordingly, if we can solve (P5) for large β, sampling the resultant qθ will result in an x with a low E(g | x,G ). 3.2 Review of Delayed Sampling We now present a review of conventional, delayed-sampling PC. In this type of PC we exploit characteristics of the parametrization of qθ, and pursue the algebraic solution of (P5) as far as possible, in closed form. At some point, if there remain quantities in this algebraic expression that we cannot evaluate closed-form, we estimate them using Monte Carlo sampling. As an example, consider a noise-free oracle, and instead of pq Kullback-Leibler distance, choose FG (qθ) to be the expected value returned by the oracle under qθ, Eqθ,G (g) , dx dg gP (g | x,G )qθ(x) dx G(x)qθ(x) (11) where the second equality reflects the fact that we are assuming a noise-free oracle. To emphasize the fact that we’re considering noise-free oracles5, we will sometimes write Eqθ,G (g) = Eqθ,G(g). While Eqθ,G (g) is a linear function of qθ, in general it will not be a linear function of θ. Accordingly, finding the θ minimizing Eqθ,G (g) may be a non-trivial optimization problem. Since qθ must be a probability distribution, (P5) is actually a constrained optimization problem, involving |X| inequality constraints {qθ(x) ≥ 0 ∀x}, and one equality constraint,∫ dx qθ(x) = 1. As discussed by Wolpert et al. (2006), such a constrained optimization problem can be converted into one with no inequality constraints by the use of barrier function methods. These methods transform the original optimization problem into a sequence of new optimization problems, {(P5)i}, each of which is easier to solve than the original problem (P5). Solving those problems in sequence leads to a solution to the original problem (P5). Consider applying this method with an entropic barrier for the case where FG (qθ) = Eqθ,G(g). Then, it turns out that up to additive constants, each problem (P5) i is again 5. Even though it is noise-free, the oracle G may be a random variable — we may not know G, and may attempt to predict it probabilistically from data, in a Bayesian fashion. In such a situation, notation like ‘E(G )’ refers to the expected oracle under our prior distribution over oracles. So, we use Eqθ,G(g) rather than Eqθ (G), even though the latter is the notation we used in previous work on PC. of the form of (P5). However the FG (qθ) of each problem (P5)i is the ‘qp’ KL distance, KL(qθ || pβi), where βi is the value of the ‘barrier parameter’ specifying problem (P5)i. In other words, up to irrelevant additive constants, each (P5)i is the problem of finding the θ that minimizes FG,βi(qθ) = Eqθ,G(g)− βi −1S(qθ) where S(.) is conventional Shannon entropy6. In this case the barrier function method directs us to iterate the following process: Solve for the qθ that minimizes KL(qθ || pβi), and then update βi. At the end of this process we will have a local solution to (P5). In the case where X is a Cartesian product, we often use distributions parametrized as a product distribution, qθ = i qi(xi). Under this parametrization each problem (P5) i can be solved by gradient descent, where the gradient components of FG,βi(qθ) are given by ∂FG,βi(qθ) ∂qi(xi) = Eqθ,G(g | xi) + βi −1ln[qi(xi)] + λi where the Lagrange parameters λi enforce normalization of each qi. There are many better alternatives7 to simple gradient descent for minimizing each FG,βi(qθ), involving Newton’s method, block relaxation, and related techniques (Wolpert et al., 2006). In all such schemes investigated to date, we need to repeatedly evaluate terms like Eqθ,G(g | xi). Sometimes that evaluation can be done closed form (Macready and Wolpert, 2005, 2004). In blackbox optimization though, this is not possible. Typically, when we cannot evaluate the terms Eqθ,G(g | xi) in closed-form we use MC to estimate them. Since we have a product distribution, we can generate samples of the joint distribution qθ(x) by sampling each of the marginals qi(xi) separately. One can use those sample x’s as queries to the oracle. Then, by appropriately averaging the oracle’s responses to those queries, one can estimate each term Eqθ,G(g | xi) (Wolpert and Bieniawski, 2004). The product factorization implies that our iterative procedure can be performed in a completely decentralized manner, with a separate program controlling each component xi, and communicating only with the oracle8. In this scheme, once qθ is modified, samples of the oracle that were generated from preceding qθ’s can no longer be directly used to estimate the terms Eqθ,G(g | xi). However, there are several ‘data-aging’ heuristics one can employ to reuse such old data by down- weighting it. In all these schemes, while we ultimately use Monte Carlo in the PC, it is delayed as long as possible in the course of solving (P5). This is the basis for calling this variant of PC ‘delayed sampling’. 3.3 Advantages of PC The PC transformation can substantially alter the optimization landscape. For a noise- free oracle G(x), (P4) reduces to the problem of finding the x that minimizes G(x). In contrast, (P5) is the problem of finding the θ that minimizes FG (θ). The characteristics 6. This qp distance is just the free energy of qθ for Hamiltonian function G and inverse temperature βi. This gives a novel derivation of the physics injunction to minimize the free energy of a system. 7. One of these alternatives can be cast as a corrected version of the replicator dynamics of evolution- ary game theory (Wolpert, 2004b). This may have interesting implications for GAs, which presume evolutionary processes. 8. Each such program may be thought of as an ‘agent’ who updates his probability distribution, and this ‘collective’ of agents performs optimization in a decentralized manner. This led to the name ‘Probability Collective’. of the problems of minimizing G(x) and minimizing FG (θ) can be vastly different. For example, suppose qθ is log-concave in its parameters, and FG is pq KL distance. In this case, regardless of the function G(x), FG (θ) is a convex function of θ, over Q. So the PC transformation converts a problem with potentially many local minima into a problem with none. See Wolpert et al. (2006) for a discussion of the geometry of the surface FG (θ) : θ → R. Since it works directly on distributions, PC can handle arbitrary data types. X can be categorical, real-valued, integer-valued, or a mixture of all of these, but in each case, the distribution over X is parametrized by a vector of real numbers. This means that all such problems ‘look the same’ to much of the mathematics of PC. Moreover, PC can exploit extremely well-understood techniques (like gradient descent) for optimization of continuous functions of real-valued vectors, and apply them to problems in these arbitrary spaces. Optimizing over distributions can give sensitivity information: The distribution qθ pro- duced in PC will typically be tightly peaked along certain directions, while being relatively flat along other directions. This tells us the relative importance of getting the value of x along those different directions precisely correct. We can set the initial distribution for PC to be a sum of broad peaks, each centered on a solution produced by some other optimization algorithm. Then, as that initial distribution gets updated in the PC algorithm, the set of solutions provided by those other optimization algorithms are in essence combined, to produce a solution that should be superior to any of them individually. Yet another advantage to optimizing a distribution is that a distribution can easily provide multiple solutions to the optimization problem, potentially far apart in X. Those solutions can then be compared by the analyst in a qualitative fashion. As discussed later, there are other advantages that accrue specifically if one uses the immediate-sampling variant of PC. These include the ability to reuse all old data, the ability to exploit prior knowledge concerning the oracle, and the ability to leverage PL techniques. See Wolpert et al. (2006) for a discussion of other advantages of PC, in particular in the context of distributed control. 3.4 Relation to Other Work PC is related to several other optimization techniques. Consider, for instance, Response Surface Models (RSM)s Jones et al. (1998). In these techniques, one uses Design of Ex- periments (DOE) to evaluate the objective function at a set of points. Then, a low-order parametric function, often a quadratic, is fitted to these function values. Optimization of this ‘response surface’ or ‘surrogate model’ is considered trivial compared to the original optimization. The result of this surrogate optimization is then used to get more samples of the true objective at a different set of points. This procedure is then iterated using some heuristics, often in conjunction with trust-region methods to ensure validity of the low-order approximation. We note the similarities with PC in Table 2. As another example, some variants of PC exploit MC techniques as discussed above, and thus stochastically generate populations of samples. In their use of random populations these variants of PC are similar to simulated annealing (Kirkpatrick et al., 1983), and even more so to techniques like EDA’s Larraaga and Lozano (2001); De Bonet et al. (1997); Lozano et al. (2005) and the CE method (Rubinstein and Kroese, 2004). However, these other approaches do not explicitly pursue the optimization of the underlying distribution qθ, as in (P5). Accordingly, those approaches cannot exploit situations in which (P5) can be solved without using a stochastically generated population (Macready and Wolpert, 2005, 2004). See Macready and Wolpert (2005) for a more extensive discussion of the relation of PC to other techniques. RSM PC Fit parametric function to Fit parametric distribution to objective function values target distribution Heuristics to grow trust region Cross-validation for regularization DOE for sample points Random sampling for sample points Axis alignment of stencil matters Parametrization can address axis alignment Surrogate minimization not always easy Implicit, probabilistic ‘minimization’ of surrogate Table 2: Relation to RSM. 4. Immediate sampling This section introduces a new PC technique called immediate sampling, and cursorily compares it to delayed sampling. As we have just described, in delayed sampling, we use algebra for as long as possible in our solution of (P5). When closed-form expressions can no longer be evaluated, we resort to MC techniques. In immediate sampling, we form an MC sample immediately, rather than delaying it as long as possible. That sample gives us an approximation to our objective, FG (θ) for all θ ∈ Q. We then search for the θ that optimizes that sample-based approximate objective. 4.1 The General Immediate-sampling Algorithm We begin with an illustrative example. Consider an integral transform FG (qθ), and use importance sampling to rewrite it as∫ dx h1(x) dg P (g | x,G )F (g, qθ(x)) h1(x) dxdg h1(x)P (g | x,G ) F (g, qθ(x)) h1(x) , (12) dx h1(x)rP (g|x,G ),h1(θ). (13) where we call h1(x) the sampling distribution. Note that the r in Eq. 13 differs from the one defined in Eq. 10, as indicated by the extra subscript. This new r is used in the next section. We form a sample set of N pairs D1 ≡ {xi, gi} by IID sampling the distribution h1(x)P (g | x,G ) in the integrand of Eq. 12. That sampling is the ‘immediate’ Monte Carlo process. D1 is equivalent to a set of N sample functions rih1(x i, θ) , F (gi, qθ(xi)) h1(xi) : i = 1, . . . N. In the simplest version of immediate sampling, we would now use the functions ri (xi, θ), together with our prior knowledge (if any), to estimate the θ that minimizes FG (qθ). As an example, not using any prior knowledge, we could estimate FG (qθ) for any θ as∫ dx h1(x)rP (g|x,G ),h1(θ) ≈ (xi, θ) . (14) This estimate is both an unbiased estimate of FG (qθ) and the maximum likelihood estimate of FG (qθ). Moreover, it has these attributes for all θ. (This is the advantage of estimating FG (qθ) using importance sampling with a proposal distribution h that doesn’t vary with θ. ) Accordingly, to estimate the θ that minimizes FG (qθ) we could simply search for the θ that minimizes (xi, θ).9 Once again, even though the average in Eq. 14 is an unbiased estimate of FG (qθ) for any fixed θ, its minimizer is not an unbiased estimate of minθFG (qθ). This is because searching for the minimizing θ introduces bias. Therefore, one should use some other technique than directly minimizing the righthand side of Eq. 14 to estimate argminθFG (qθ). 4.2 Immediate Sampling with Multiple Sample Sets In general, we will not end the algorithm after forming a single sample set D1. Instead we will use a map η that takes D1 to a new sampling distribution, h2. We then generate new (x, g) pairs using h2, giving us a new sample set D2. We then iterate this process until we decide to end the algorithm, at which point we use all our samples sets together to estimate argminθFG (qθ). To illustrate this we first present an example of a θ-estimation procedure we could run at the end of the immediate sampling algorithm. This example is just the extension of the maximum likelihood θ-estimation procedure introduced above to accommodate multiple sample sets. Let N be the total number of samples, drawn from M sample sets, with Nj samples in the j’th sample set. Let hj be sample distribution for the j’th sample set, and ri,j the sample function value for the i’th element of the j’th sample set. Also define Dj ≡ {xi,j , gi,j : i = 1, . . . Nj}. Then i=1 r (xi,j , θ)/Nj is an unbiased estimate of FG (qθ) for any sample set j. Accordingly, any weighted average of these estimates is an unbiased estimate of FG (qθ): FG (qθ) ≈ (xi,j , θ) . (15) Modulo unbiasedness concerns, we could then use the minimizer of Eq.15 as our estimate of argminθFG (qθ). Say we have fixed on some such θ-estimation procedure to run at the end of the algo- rithm. The final step of each iteration of immediate sampling is to run η, the map taking the samples generated so far to a new h. Ideally, we want to use the η that, when repeatedly run during the algorithm, maximizes the expected accuracy of the final θ-estimation. However even for a simple θ-estimation procedure, determining this optimal η can be quite difficult. As discussed later, it is identical to the active learning problem in machine learning. In this paper we adopt a two-step heuristic for setting η. In the first step, at the end of each iteration, we estimate the optimal qθ based on all the sample sets generated so far, using Eq. 15. In the second step, we complete η by setting the new h to the current estimate of the optimal qθ. At that point, the new h is used to generate a new sample set, and the process repeats. 4.3 Immediate Sampling with MCMC For certain types of FG , it is possible to form samples using other sampling methods like Markov Chain Monte Carlo (MCMC) (see Mackay, 2003; Bernardo and Smith, 2000; 9. Since qθ is normalized, so is dx F (G(x), qθ(x)) = dx F (G(x),qθ(x))R dx qθ(x) . In Eq. 14 we fix the denominator integral to 1. In practice though, it may make sense to replace both of the integrals in this ratio with importance sample estimates of them. That means dividing the sum in Eq. 15 by q(xi)/h(xi) and then finding the θ that optimizes that ratio of sums, rather than the θ that just optimizes the numerator term (see Robert and Casella, 2004). For example, this can be helpful when one uses cross-validation to set β, as described below. Berger, 1985). For example, if FG (θ) is pq distance from the Boltzmann distribution pβ to qθ, then we can use MCMC to form a sample set of pβ (not of qθ). We can then use that sample set to form an unbiased estimate of FG (θ) for any θ. But if β were to change, these old samples cannot be used directly. One would have to resort to additional techniques like rejection sampling in order to reuse these samples. The advantage of using importance sampling is that all previous samples can be reused by the simple expedient of modifying their likelihood ratios. Therefore, in this paper, we only consider sampling distributions h that can be sampled directly, without any need for techniques like MCMC. 4.4 Advantages of Immediate-Sampling PC In contrast to delayed sampling, immediate sampling usually presents no difficulty with reusing old data, as shown above; all (xi,j , gi,j) pairs can be used directly. Note that we can also reuse data that was generated when F was different, for instance, data generated under a differerent βi during a KL distance minimization procedure. As long as we store hj(xi,j) in addition to gi,j and xi,j for every sample, we can always evaluate ri,j (xi,j , θ) for any F . Indeed, we can even comment on optimal ways of reusing this old data. Since each (xi,j , θ) is an unbiased estimate of the integral FG (θ), any weighted average of the (xi,j , θ)’s is also an unbiased estimate. This can be exploited in the θ-estimation pro- cedure. For instance, consider the estimator of Eq. 15. If we have good estimates of the variances of the individual ri,j (xi,j , θ), we can weight the terms ri,j (xi,j , θ) to minimize the variance of the associated weighted average estimator. Those weights are proportional to the inverses of the variances (see Macready and Wolpert, 2005; Lepage, 1978, 1980). As discussed in Sec. 2.3, the accuracy of the associated MCO algorithm could be expected to improve under such weighting. We can also shed light on how to go about gathering new data. As in the VEGAS Algorithm (Lepage, 1978, 1980), one could incorporate bias-variance considerations into the operator η that sets the next sampling distribution. To give an example, let Ξ be the range of η, and fix θ. Given Ξ and θ, one can ask what proposal distribution h ∈ Ξ would minimize the sample variance of the estimator in Eq. 14. Intuitively, this is akin to asking how best to do active learning. In general, the answer to this question, the optimal sampling distribution h(x), will be set by the function rP (g|x,G ),h(θ), viewed as mapping X → R. Accordingly, for any fixed θ, one can use the MC samples generated so far to estimate the x-dependence of rP (g|x,G ),h(θ), and thereby estimate the optimal h ∈ Ξ. One then uses that estimate as the next sampling distribution h. Another advantage of immediate sampling over delayed sampling is that the analysis in delayed sampling relies crucially on the parametrization of the q’s; some such parametriza- tions will permit the closed-form calculations of delayed sampling, and others will not. In immediate sampling, this problem disappears. 4.5 Implications of the Identity Between MCO and PL For the case where FG (θ) is an integral transform like Eq. 9, the PC optimization problem (P5) becomes a special case of minimizing a parametrized integral, the problem (P2). Formally, the equivalence is made by equating x with the parameter φ, g with w, and g × P (g | x,G ) with U(w, φ). In particular, immediate sampling is a special case of MCO. This identity means that we can exploit the extremely well-researched field of PL to improve many aspects of immediate sampling. In particular: • PL techniques like boosting (Schapire and Singer, 1999) and bagging (Breiman, 1994) can be used in (re)using old samples before forming new ones. • Variants of active learning10 can be used to set and update h. Some aspect of this are discussed in Sec. 6 below. • Cross-validation is directly applicable in many ways: In our context, the curse of dimen- sionality arises if Q is very large. We can address this the conventional PL way, by adding a regularization function of qθ to the objective function. The parameters controlling this regularization can be updated dynamically, as new data is generated, using cross-validation To use cross-validation this way, one forms multiple partitions of the current data. For each such partition, one calculates the optimal qθ for the training subset of that partition. One then examines error on the validation subset of that partition. More precisely, one calculates the unregularized objective value on the held-out data. • More generally, we can use cross-validation to dynamically update any parameters of the immediate sampling algorithm. For example, we can update the ‘temperature’ parameter β of the Boltzmann distribution, arising in both qp and pq KL distance, this way. Note that doing this does not involve making more calls to the oracle. • We can also use cross-validation to choose the best model class (parametrization) for qθ, among several candidates. • As an alternative to all these uses of cross-validation, one can use stacking to dynamically combine differrent temperatures, different parametrized density functions, and so on. • One may also be able to apply kernel methods to do the density estimation (see Macready, 2005). 5. Experiments In this section, we demonstrate the application of PL and immediate-sampling PC tech- niques to the unconstrained optimization of continuous functions, both deterministic and nondeterministic. We first describe our choice of FG , in this case pq KL distance. Next, as an illustrative example, we apply immediate sampling to the simplest of optimization problems, where the objective is a 2-D quadratic. Subsequently, we apply it to determinis- tic and stochastic versions of two well-known unconstrained optimization benchmarks, the Rosenbrock function and the Woods function. We highlight the use of PL techniques to enhance optimizer performance on these benchmark problems. In particular, we show that cross-validation for regularization yields a performance improvement of an order of magnitude. We then show that cross-validation for model-selection results in improved performance, especially in the early stages of the algorithm. We also show that bagging can yield significant improvements in performance. 5.1 Minimizing pq KL Distance Recall that the integral form of pq KL distance is KL(p‖q) = dx p(x) ln It is easy to show that when there are no restrictions on q being a parametrized density, pq KL distance is minimized if p = q. However, owing to sampling considerations, we 10. Active learning in the precise machine learning sense uses current data to decide on a new query x to feed to the oracle. We use the term more loosely here, to refer to any scheme for using current data to dynamically modify a process for generating for future queries. usually choose q to be some parametric distribution qθ. In this case, we want to find the parameter vector θ that minimizes KL(p‖qθ). Since the target distribution p is derived purely from G and is independent of qθ, minimizing pq KL distance is equivalent to the following cross-entropy minimization problem. minimize − dx p(x) ln (q(x)) , subject to dx q(x) = 1, q(x) ≥ 0 ∀x. 5.1.1 Gaussian Densities If q is log-concave in its parameters θ, the minimization problem (16) is a convex opti- mization problem. In particular, consider the case where X = Rn, and qθ is a multivariate Gaussian density, with mean µ and covariance Σ, parametrized as follows, qµ,Σ(x) = (2π)n/2|Σ|1/2 (x− µ)TΣ−1(x− µ) then the optimal parameters are given by matching first and second moments of p and qθ. dxx p(x), dx (x− µ?)(x− µ?)T p(x). It is easy to generalize this to the case where X ⊂ Rn, by making a suitable modification to the definition of p. This is described in Sec. 5.2.1. 5.1.2 Immediate Sampling with a Single Gaussian Using importance sampling, we can convert the cross-entropy integral in Eq. 16 to a sum over data points, as follows. p(xi) h(xi) where D is the data set {(xi, gi)}, i = 1, . . . , N . This sets up the minimization problem for immediate sampling for pq KL distance. minimize − p(xi) h(xi) . (17) Denote the likelihood ratios by si = p(xi)/h(xi). Differentiating Eq. 17 with respect to the parameters µ and Σ−1 and setting them to zero yields11 i(xi − µ?)(xi − µ?)T∑ 11. Remarks: 1. As expected, these formulæ, in the infinite-data limit, are identical to the moment-matching results for the full-blown integral case. 2. The formulæ resemble those for MAP density estimation, often used in supervised learning to find the MAP parameters of a distribution from a set of samples. The difference in this case is that each sample point is weighted by the likelihood ratio si, and is equivalent to ‘converting’ samples from h to samples from p. 5.1.3 Mixture Models The single Gaussian is a fairly restrictive class of models. Mixture models can significantly improve flexibility, but at the cost of convexity of the KL distance minimization problem. However, a plethora of techniques from supervized learning, in particular the Expectation Maximization (EM) algorithm, can be applied with minor modifications. Suppose qθ is a mixture of M Gaussians, that is, θ = (µ,Σ, φ) where φ is the mixing p.m.f, we can view the problem as one where a hidden variable z decides which mixture component each sample is drawn from. We then have the optimization problem minimize − p(xi) h(xi) i, zi) Following the standard EM procedure, we multiply and divide the quantity inside the logarithm by some Qi(zi), where Qi is a distribution over the possible values of zi. As before, let si be the likelihood ratio of the i’th sample. minimize − si ln qθ(xi, zi) Qi(zi) Then using Jensen’s inequality, we can take Qi outside the logarithm to get a lower bound. To make this lower bound tight, choose Qi(zi) to be the constant p(zi|xi). Finally, differ- entiating with respect to µj ,Σ j and φj gives us the EM-like algorithm: E-step: For each i, set Qi(zi) = p(zi|xi), that is, wij = qµ,Σ,φ(z i = j|xi), j = 1, . . . ,M. M-step: Set µj = i xi∑ i (xi − µj)(xi − µj)T∑ Since this is a nonconvex problem, one typically runs the algorithm multiple times with random initializations of the parameters. 5.2 Implementation Details In this section we describe the implementation details of an iterative immediate-sampling PC algorithm that uses the Gaussian mixture models described in the previous section to minimize pq KL distance to a Boltzmann target parametrized by β. We also describe the modification of a variety of techniques from parmetric learning that significantly improve performance of this algorithm. An overview of the procedure is presented in Algorithm 1. 5.2.1 Example: Quadratic G(x) Consider the 2-D box X = {x ∈ R2 | ‖x‖∞ < 1}. Consider a simple quadratic on X, GQ(x) = x 1 + x 2 + x1x2, x ∈ X. The surface and contours of this simple quadratic on X are shown in Fig. 1. Also shown are the corresponding Boltzmann target distributions pβ on X, for β = 2, 10 and 50. As Algorithm 1 Overview of pq minimization using Gaussian mixtures 1: Draw uniform random samples on X 2: Initialize regularization parameter β 3: Compute G(x) values for those samples 4: repeat 5: Find a mixture distribution qθ to minimize sampled pq KL distance 6: Sample from qθ 7: Compute G(x) for those samples 8: Update β 9: until Termination 10: Sample final qθ to get solution(s). can be seen, as β increases, pβ places increasing probability mass near the optimum of G(x), leading to progressively lower EpβG(x). Also note that since G(x) is a quadratic, pβ(x) ∝ exp(−βG(x)) is a Gaussian, restricted to X and renormalized. We now ‘fit’ a Gaussian density qθ to the Boltzmann pβ by minimizing KL(pβ‖qθ), for a sequence of increasing values of β. Note that qθ is a distribution over R2, and GQ is not defined everywhere in R2. Therefore, we extend the definition of GQ to all of R2 as follows. GQ(x) = x21 + x 2 + x1x2, x ∈ X. ∞ otherwise. Now pβ = 0 for all x /∈ X, and the integral for KL distance can be reduced to an integral over X. This means that samples outside X are not considered in our computations. 5.2.2 Constant β Figure 1: Quadratic G(x) and associated Gaussian targets First, we fix β = 5, and run a few iterations of the PC algorithm. To start with, we draw Nj = 30 samples from the uniform distribution on X. The best-fit Gaussian is computed using the immediate sampling procedure out- lined in the preceding section. At each successive itera- tion, Nj = 30 more samples are drawn from the current qθ and the algorithm proceeds. A total of 6 such iterations are performed. The 90% confidence ellipsoids correspond- ing to pβ (heavy line) and the iterates of qθ (thin line) are shown in Fig. 2. Also shown are the corresponding values of EqθG(x), computed using the sample mean of GQ(x) for 1000 samples of x drawn from each qθ, and KL(pβ‖qθ), computed as the sample mean of ln(pβ(x)/qθ(x)) for 1000 samples of x drawn according to pβ . 5.2.3 Varying β Next, we change β between iterations, in the ‘update β’ step shown in algorithm(1). With the same algorithm pa- rameters, we start with β = 10, and at each iteration, we use a multiplicative update rule β ← kββ, for some constant kβ > 1, in this case, 1.5. As the algorithm progresses, the increasing β causes the target density pβ to place increasing probability mass on regions with low G(x), as shown in Fig. 1. Since the distributions qθ (a) Constant β: Confidence ellipsoids (b) Constant β: KL distance and expected G (c) Varying β: Confidence ellipsoids (d) Varying β: KL distance and expected G Figure 2: PC iterations for quadratic G(x). are best-fits to p, successive iterations will generate lower EqθG(x). The 90% confidence ellipsoids and evolution of EqθG(x) and KL distance are shown in Fig. 2. 5.2.4 Cross-validation to Schedule β In more complex problems, it may be difficult to find a good value for the β update ratio kβ . However, we note that the objective KL(pβ‖qθ) can be viewed as a regularized version of the original objective, Eqθ [G(x)]. Therefore, we use the standard PL technique of cross- validation to pick the regularization parameter β from some set {β}. At each iteration, we partition the data set D into training and test data sets DT and DV . Then, for each β ∈ {β}, we find the optimal parameters θ?(β) using only the training data DT . Next, we test the associated qθ?(β) on the test data DV using the following performance measure. ĝ(θ) = qθ(xi)G(xi) h(xi)∑ qθ(xi) h(xi) , (18) The objective ĝ(θ) is an estimate12 of the unregularized objective Eqθ [G(x)]. Finally, we set β? = arg minβ∈{β} ĝ(θ?(β)), and compute θ?(β?) using all the data D. Note that the whole cross-validation procedure is carried out without any more calls to the oracle G . We demonstrate the functioning of cross-validation on the well-known Rosenbrock prob- lem in two dimensions, given by GR(x) = 100(x2 − x21) 2 + (1− x1)2, over the region X = {x ∈ R2 | ‖x‖∞ < 4}. The optimum value of 0 is achieved at x = (1, 1). The details of the cross-validation algorithm used are presented in Algorithm 2. For this Algorithm 2 Cross-validation for β. Initialize interval extension count extIter = 0, and maxExtIter and β0.. repeat At β = β0, consider the interval ∆β = [k1β0, k2β0]. Choose {β} be a set of nβ equally-spaced points in ∆β. Partition the data into K random disjoint subsets. for each fold k, do Training data is the union of all but the kth data partitions. Test data is the kth partition. for βi in {β}, do Use training data to compute optimal parameters θ?(βi, DTk). Use test data to compute held-out performance ĝ(θ?(βi, DVk)), from Eq. 18. end for end for Compute average held-out performance, g(βi), of ĝ(θ?(βi, DVk)). Fit a quadratic Q(β) in a least-squares sense to the data (βi, g(βi)). if Q is convex then Set optimum regularization parameter β? = arg minβ∈∆β Q(β). Fit a line L(β) in a least-squares sense to the data (βi, g(βi)). Choose β? = arg minβ∈∆β L(β). end if Increment extIter Update β0 ← β? until extIter > maxExtIter or Q is convex. experiment, we choose maxExtIter = 4, k1 = 0.5, k2 = 2, Nj = 10, nβ = 5, K = 10. The histories of EqG(x) and β are shown in Fig. 3. Also shown are plots of the fitted Q(β) at iterations 8 and 15. As can be seen, the value of β sometimes decreases from one 12. The reason for dividing by the sum of q(xi)/h(xi) is as follows. If the training data is such that no probability mass is placed on the test data, the numerator of bgqθ is 0, regardless of the parameters of qθ. In order to avoid this peculiar problem, we divide by the sum of q(x i)/h(xi), as desribed by Robert and Casella (2004). (a) EqG(x) history. (b) β history. (c) Fit Q(β), iteration 8. (d) Fit Q(β), iteration 15. Figure 3: Cross-validation for β: 2-D Rosenbrock G(x). iteration to the next, which can never happen in any fixed multiplicative update scheme. We now demonstrate the need for an automated regularization scheme, on another well-known test problem in R4, the Woods problem, given by Gwoods(x) = 100(x2 − x1)2 + (1− x1)2 + 90(x4 − x23) 2 + (1− x3)2 +10.1[(1− x2)2 + (1− x4)2] + 19.8(1− x2)(1− x4). The optimum value of 0 is achieved at x = (1, 1, 1, 1). We run the PC algorithm 50 times with cross-validation for regularization. For this experiment, we used a single Gaussian q, and set maxExtIter = 4, k1 = 0.5, k2 = 3, Nj = 20, nβ = 5, K = 10. From these results, we then attempt to find the best-fit multiplicative update rule for β, only to find that the average β variation is not at all well-approximated by any fixed update β ← kββ. This poor fit is shown in Fig. 4, where we show a least-squares fit to both β and log(β). In the fit to log(β) the final β is off by over 100%, and in the fit to (a) Least-squares fit to β: (b) Least-squares fit to log(β): Figure 4: Best-fit β update rule. β, the initial β is off by several orders of magnitude. We then compare the performance of cross-validation to that of PC algorithm using the fixed update rule derived from the best least-squares fit to log(β). From a comparison over 50 runs, we see that using this best-fit update rule performs extremely poorly - cross-validation yields an improvement in final EqθG(x) by over an order of magnitude, as shown in Fig. 5. (a) log(EqG) history. Figure 5: Cross-validation beats best-fit fixed β update: 4-D Woods G(x). 5.2.5 Bagging While regularization is a method to decrease bias, bagging is a well-known variance-reducing technique. Bagging is easily incorporated in our algorithm. Suppose, at some stage in the algorithm, we have N samples (xi, gi), we resample our existing data set exactly N times with replacement. This gives us a different set of data set D′, which also contains some duplicates. We compute optimal parameters θ?(D′). We repeat this resampling process kb times and uniformly average the resulting optimal densities qθ?(D′ ), k = 1, . . . , kb. We demonstrate this procedure, using the Rosenbrock function and a single Gaussian qθ. In this experiment, we also demonstrate the ability of PC to handle non-deterministic oracles by adding uniform random noise to every function evaluation, that is, (g | x,G) ∼ U [−0.25, 0.25]. For this experiment, Nj = 20, kb = 5. The β update is performed using the same cross-validation algorithm described above. Fig. 6 shows the results of 50 runs of the PC algorithm with and without bagging. We see that bagging finds better solutions, and Figure 6: Bagging improves performance: Noisy 2-D Rosenbrock. moreover, it reduces the variance between runs. Note that the way we use bagging, we are only assured of improved variance for a single MC estimation at a given θ, and not over the whole MCO process of searching over θ. 5.2.6 Cross-validation for Regularization and Model Selection In many problems like the Rosenbrock, a single Gaussian is a poor fit to pβ for many values of β. In these cases, we can use a mixture of Gaussians to obtain a better fit to pβ . We now describe the use of cross-validation to pick the number of components in the mixture model. We use an algorithm very similar to the one described for regularization. In these experiments, we use a greedy algorithm to search over the joint space of β and models: 1. We first pick the regularization parameter β, using Algorithm 2. 2. For that β, we use Algorithm 3 to pick the number of mixture components. For this experiment, the details are the same as the preceding section, but without bagging. The set of models {{φ}} is the set of Gaussian mixtures with one, two or three mixing components. Fig. 7 shows the variation of Eq(G) vs. iteration. The mixture model is much quicker to yield lower expected G, because the Boltzmann at many values of β is better approximated by a mixture of Gaussians. However, note that the mixture models performs poorly towards the end of the run. The reason for this is as follows: No shape regularization was used during the EM procedure. This means that the algorithm often samples from nearly degenerate Gaussians. These ‘strange’ sample sets hurt the subsequent performance of importance sampling, and hence of the associated MCO problem. This can be alleviated by using some form of shape regularization in the EM algorithm. Algorithm 3 Cross-validation for model selection. Initialize set {{φ}} of model classes {φ} to search over. Partition the data into K disjoint subsets. for each fold k, do Training data is all but the kth data partitions. Test data is the kth data partition. for {φi} in {{φ}} do Compute the optimal parameter set θ?(DTk) ∈ {φi} Compute held-out performance ĝ(θ?(DVk)) end for Compute the sample held-out performance, g({φi}), from Eq. 18. end for Choose best model class {φ?} = arg minφi g({φi}). Figure 7: Cross-validation for regularization and model-selection: 2-D penalty function G(x). 6. Fit-based Monte Carlo Thus far, we have not exploited the locations of the samples in constructing esimates. In this section, we discuss the incorporation of sample locations to improve both MC and 6.1 Fit-based MC Estimation of Integrals We first consider MC estimation of an integral, presented at the beginning of Sec. 2.3. Recall from that discussion, that to accord with MCO notation, we write the integral to be estimated as L (φ) = dw U(w, φ) for some fixed φ. In this notation the sampling of v provides a sample set {(wi, U(wi, φ)) : i = 1, . . . N). The associated sum L̂{(wi,U(wi,φ))} ≡ L̂ (φ) then serves as our estimate of the integral L ≡ L (φ). In forming the estimate L̂{(wi,U(wi,φ))} we do not exploit the relationships between the locations of the sample points and the associated values of the integrand. Indeed, since those locations {wi} do not appear directly in that estimate, L̂{(wi,U(wi,φ))} is unchanged even if those locations changed in such a way that the values {ri} stayed the same. The idea behind fit-based (FB) Monte Carlo is to leverage the location data to replace L̂{(wi,U(wi,φ))} with a more accurate estimate of L . The most straightforward FB MC treats the sample pairs {(wi, U(wi, φ)) : i = 1, . . . N)} as a training set for a supervised- learning algorithm. Running such an algorithm produces a fit Ũ(., φ) taking w’s into R. This fit is an estimate of the actual oracle U(, φ), and this fit defines an estimate for the full integral, L̃{wi,U(wi,φ)} ≡ dw Ũ(w, φ) (19) We will sometimes omit the subscript and just write L̃ (φ) or even just L̃ . In this most straightforward version of FB MC, we use L̃ as our estimate of L rather than L̂ . In some circumstances one can evaluate L̃ in closed form. A recent paper reviewing some work on how to do this with Gaussian processes is given by Rasmussen and Gharamani (2003). In other circumstances one can form low-order approximations to L , for example using Laplace approximations (see Robert and Casella, 2004). Alternatively, conventional deterministic grid-based approximation of the integral L can be cast as a degenerate version of closed-form fit-based estimation13 of L . More generally, one can form an approximation to the integral L̃ (φ) by MC sampling of Ũ(., φ). Generating these fictitious samples of Ũ(., φ) does not incur the expense of calling the actual oracle U(., φ). So, in this approach, we run MC twice. The first time, we generate the factual samples {(wi, U(wi, φ)) : i = 1, . . . N)}. Given those samples, we form the fit to them, Ũ(., φ). We then run a second MC process using the fictitious oracle Ũ(., φ). Note that in all of these approaches, the original sampling distribution v does not directly arise, that is, the values {v(wi)} do not arise. In particular, if one were to change those values without changing the factual sample locations {wi}, then the esti- mate L̃{(wi,U(wi,φ))} would not change. Of course, a different v would result in a different sample set, and thereby a different estimate, but given a sample set, the sampling distribu- tion is immaterial. This is typically the case with FB MC estimators, and it contrasts with the estimator L̂{(wi,U(wi,φ))}, which would change if v were changed without changing the factual sample locations. Note also that the factual samples underlying the fit Ũ(., φ) are exact samples of the factual oracle, U(., φ). In contrast, since in general Ũ(., φ) 6= U(., φ), the fictitious samples will be erroneous, if viewed as samples of U(., φ). Since we are ultimately concerned with an integral of U(., φ), this suggests that the fictitious samples should be weighted less than the factual samples. This might be the case even if one had infinitely many fictitious samples. In fact, even if one could evaluate L̃ in closed form, it might make sense not to use it directly as our final estimate of L . Instead, combining it with the importance-sampling estimate L̂ might improve the estimate. 13. To see this, modify the MC process to be sampling without replacement. Choose the proposal distribution v for this process to be a sum of delta functions. The centers of those delta functions give the points on a regular grid of points in the space of allowed x’s. Have the number of samples equal the number of such grid points. Finally, have our fit to the samples, Ũ(., φ), be a sum of step-wise constant functions, going through the sample points. The closed form integral of that fit given by Eq. 19 is just the Reimann approximation to the original integral, dw U(w, φ). 6.2 Bayesian Fit-based MCO We now extend the discussion to MCO by allowing φ to vary. As with the case of FB MC estimation of an integral, the most straightforward version of FB MCO uses L̃ (φ) rather than L̂ (φ) as our estimate of L (φ). This means that we use argminφ[L̃ (φ)] rather than argminφ[L̂ (φ)] as our estimate of the φ optimizing L (φ). L̂ (φ) is a sum, whereas L̃ (φ) is an integral. This means that different algorithms are required to find the φ optimizing them. Indeed, optimizing L̃ (φ) is formally the same type of problem as optimizing L (φ); both functions of φ are parametrized integrals over w. So if needed, we can use PLMCO techniques to optimize L̃ (φ). Again, as with MC, the integrand of L̃ (φ) is not the factual oracle. So, minimizing L̃ (φ) using PLMCO sampling techniques will not require making additional calls to the factual oracle. Consider a Bayesian approach to forming the fit. Our problem is to solve the MCO problem (P2), given that the factual oracle U is not known, and we only can generate samples of U . Adopting a fully Bayesian perspective, since U is not known, we must treat it as a random variable. So we have a posterior distribution over all possible oracles, reflecting all the data we have concerning the factual oracle. We then use that posterior to try to solve (P2). Say that in the usual way that our data contains the w’s and the associated functions U(w, .) of a sample set that was generated by importance sampling U (see Sec. 2). More generally, we may have additional data, for instance, the gradients of U at the sample points. For simplicity though, we restrict attention to the case where the provided information is only the sample set of functions, {wi, ri(.)}, together with v. We will use D to refer to a sample set for MC or MCO, and for immediate sampling in particular. So we write our posterior over oracles14 as P (Uc | D). Using this notation, the goal in Bayesian FB MCO is to exploit P (Uc | D) to improve our estimate of the φ that minimizes dw U(w, φ). How should we use P (Uc | D) to estimate the solution to (P2)? Bayesian decision theory tells us to minimize posterior expected loss, dUc P (Uc | D)L(φ,Uc). Given the loss function of Eq. 4, that means we wish to solve (P6): min dUc P (Uc | D)L(φ,Uc) = min dwcdUc P (Uc | D)Uc(wc, φ). To avoid confusion, the variable of integration is written as wc, to distinguish it from w’s in the integral dw U(w, φ). The solution to (P6) is our best possible guess of the φ solving problem (P2), given the sample set D. Finding that solution is a problem of minimizing a parametrized integral. Sometimes we may be able to solve (P6) in closed form, even when we cannot solve (P2) in closed form. Performing the integral over Uc may simplify the remaining integral over wc. More generally, we can address (P6) using MCO techniques, and in particular using PLMCO.15 To solve (P6) with PLMCO one generates fictitious samples by sampling one distri- bution over Uc’s and one over wc’s. This MC process does not involve calls to the actual oracle U , but samples a new distribution over Uc’s, to generate counter-factual Uc’s, and then samples those Uc’s. 14. Practically, when running a computer experiment, U is the actual oracle generating D according to a likelihood P (D | U). On the other hand, the posterior P (Uc | D) reflects both that likelihood and a prior P (Uc) assumed by the algorithm. So, Uc is a random variable, whereas U refers to the single true factual oracle. 15. To see that explicitly, rewrite the integral in (P2) as dz V (z, φ), and identify values of z with pairs (wc, Uc), while taking V (z, φ) = V (wc, Uc, φ) = P (Uc | D)Uc(wc, φ). 6.3 Example: Fit-based Immediate Sampling To illustrate the foregoing we consider the variant of MCO given by immediate sampling with a noise-free oracle. In the simple version of MCO considered just above, the estimate we make for φ has no effect on what points are chosen for any future calls we might make to the oracle. For simplicity, we restrict attention to the analogous formulation of immediate sampling. Using immediate sampling terminology, this means that we only consider the issue of how best to estimate θ after the immediate sampling algorithm has exhausted all its calls to the oracle. We do not consider the more general active learning issue, of how best to estimate θ when this estimate will be affect further calls to the oracle. However see Sec. 6.6 below. Recall that in immediate sampling, identifying wc with xc and φ with θ, Uc(wc, φ) becomes F (Gc(xc), θ). Given a sample set D of (x,G(x)) pairs generated from a noise-free factual oracle, our Bayesian optimization problem in immediate sampling is to find the θ that minimizes E(FGc(θ) | D) = EP (Gc|D)[ dxcF (Gc(xc), qθ(xc))] dxcdGc P (Gc | D)F (Gc(xc), qθ(xc)) , F̃D(θ). (20) Contrast this with Eq. 11. In particular, the inner integral in Eq. 20 runs over fictitious oracles Gc that are generated according to P (Gc | D), whereas in Eq. 11, G is the factual oracle. In some circumstances one can evaluate the integral in Eq. 20 algebraically, to give a closed form function of θ. In other cases, we can algebraically evaluate an accurate low-order approximation, to again give a closed form function of θ. For the rest of this subsection, however, we consider the situation where neither of these possibilities hold. To address this situation, we approximate the integral in Eq. 20 using importance sampling. However, to do this, we may have to importance-sample over two domains. The first sampling is over sample locations xc, using some sampling distribution hc. (As an example, we can simply choose hc = h.) The second sampling is over possible oracles Gc, using some sampling distribution H. More precisely, write E(FGc(θ) | D) = dxc dGc hc(xc)H(Gc) ] P (Gc | D) hc(xc)H(Gc) F (Gc(xc), qθ(xc)). (21) To approximate this integral generate NT locations, {xic}, by sampling hc. This gives us NT integrals TD,{xic}(θ) , hc(xic) dGc H(Gc) P (Gc | D) H(Gc) F (Gc(x c), qθ(x c)). (22) Sometimes, these integrals also can be evaluated algebraically, giving closed form sample functions of θ. As an example, suppose we sample oracles according to the posterior, that is, take H(Gc) = P (Gc | D), so that TD,{xic}(θ) = hc(xic) dGc P (Gc | D)F (Gc(xic), qθ(x c)). (23) Next, say that we have a Gaussian process prior over oracles, P (Gc) (Rasmussen and Williams, 2006), with a Gaussian covariance kernel. For this choice, and for some F ’s, we can compute TD,{xic}(θ) exactly for any θ, provided D is not too large. For example, this is the case for most F ’s whose dependence on their first argument lies in the expo- nential family.16 In other situations, while we cannot evaluate them exactly, the integrals TD,{xic}(θ) can be accurately approximated algebraically. This again reduces them to closed form sample functions of θ. In either of these cases, there is no need to sample H; samples of hc suffice. More generally, we may not be able to evaluate the NT functions TD,{xic}(θ) algebraically, and also cannot form accurate low-order algebraic approximations to them. In this situation, for each xic, we should generate one sample of Gc from H(Gc). 17 This provides us a total of NT sample functions TD,{xic,Gic}(θ) , P (Gic | D)F (Gic(xic), qθ(xic)) hc(xic)H(Gic) . (24) As an example, say we believe firmly that a particular posterior P (Gc | D) governs our problem, and can sample that posterior. Then we can take H(Gc) to equal P (Gc | D). Now Eq. 24 only requires that we have values of our sampled Gc’s at the {xic}, that is, we only need to have values Gic(x c), where G c ∼ H(Gc). So we only need to sample the NT separate one-dimensional distributions {P (Gc(xic) | D)}. In particular, we might be able to use Gaussian Process techniques to generate those values. Alternatively, as a rough approximation, one could simply fit a regression to the data in D, ω(x), and then add noise to the vector of values {ω(xic)} to get {Gic(xic)}. If we wanted to have multiple Gc’s for each xic, then we would simply generate more samples of each distribution {P (Gc(xic) | D)}.18 However we sample H, the resultant sample functions provide the estimate FD,{xic,Gic}(θ) , TD,{xic,Gic}(θ) ≈ E(F̃D,{xic,Gic}(θ) | D) (25) which we sometimes abbreviate as ˆ̃FD(θ). For the situation where we can evaluate the integral over Gc algebraically (or at least approximate it that way), we instead define FD(θ) , TD,{xic}(θ) (26) and rely on the context to decide which of the definitions of ˆ̃FD(θ) is meant. So for either case we can write F̃D(θ) ≈ FD(θ).19 16. Strictly speaking, since the oracle is noise-free, the likelihood P (D | Gc) is a delta function about having D lie exactly on the function Gc(x). In practice, this may make the computation be ill-behaved numerically. Typically such problems are addressed modeling the fictitious oracles as though the values they returned had a small amount of Gaussian noise added. 17. One could have the number of samples of H(Gc) not match the number of samples of hc(x). To avoid the associated notational overhead, here we just match up the two types of samples, one-to-one. 18. As an alternative, we could reverse the sampling order and sample P (Gc | D) first and hc second. Practically, this would mean generating NT samples of hc, and then sampling the single NT -dimensional distribution P (Gc(x c), . . . Gc(x c ) | D). (This contrasts to the case considered in the text in which H is sampled second, so one instead samples NT separate one-dimensional distributions {P (Gc(xic) | D)}.) If we do this using a ‘rough approximation’ based on a fit ω to D, the noise values added to the values of the fit, {ω(xic)}, would have to be correlated with each other, since they reflect the same Gc. 19. As a practical issue, we may want to divide the sum in Eq. 26 by the empirical average P (Gic|D) Similarly, if we cannot evaluate the integral over Gc algebraically, we may want to modify the estimate In both cases, under naive MCO, we search for the θ that minimizes ˆ̃FD(θ). We then use that θ as our estimate for the solution to (P6). More generally, rather than use naive MCO we can exploit our sample functions with PLMCO. For example, rather than minimizing ˆ̃FD(θ), we could minimize a sum of {F̃D(θ) and a regularization penalty term. However we arrive at our (estimated) optimal qθ, most simplistically, we can update h to equal that new qθ. In a more sophisticated approach, we could set h from the sample functions using active learning (see Sec. 6.6 below). Once we have that new h we can form samples of it to generate new factual sample locations x. These in turn are fed to the factual oracle G to augment our data set D. Then the process repeats. Note that unlike with non-FB immediate sampling, with FB immediate sampling we need to evaluate P (Gc | D) (or sample it, if we choose to have H(Gc) , P (Gc | D)). This may be non-trivial. On the other hand, that very same distribution P (Gc | D) that may cause difficulty also gives the major advantage of the fit-based approach; it allows any insights we have into how to fit a curve Gc to the data points D to be exploited. 6.4 Exploiting FB Immediate Sampling To illustrate how fitting might improve immediate sampling, consider the case where FG(qθ) is qp KL distance. Say that G(x) is a high-dimensional convex paraboloid in- side a hypercube, and zero outside of that hypercube. Suppose as well that we have a single factual sampling distribution h, which is concentrated on one side of the paraboloid. For example, if the peak of the paraboloid is at the origin, h might be a Gaussian (masked by the hypercube) whose mean lies several sigmas away from the origin. To start, consider importance sampling MC estimation of the integral FG(qθ) for one particular θ, without any concern about choosing among θ’s. Say that the factual sample D isn’t too large. Then it is likely that no elements of D are in regions where G reaches its lowest values. For such a D, the associated factual estimate F̂D(θ) , i, θ) is larger than the actual value, FG(qθ) (cf. Eq. 14). So straightforward importance- sampling integral estimation is likely to be badly off.20 Intuitively, the problem is that as far as the factual estimate F̂D(θ) is concerned, G could just as well be a sum of delta functions centered at the x’s in D, with low associated oracle sample values, as a paraboloid. If G were in fact such a sum, then F̂D would be correct. However by looking at the (x,G(x)) pairs in D, all of which lie on the same paraboloid, such an inference of G appears quite unreasonable. It makes sense to instead infer that G is a paraboloid. Fitting is a way to formalize (and exploit) such D-based insights. As an example, consider using a Bayesian PL algorithm to do the fitting. Typical choices for the prior P (Gc) used in PL would result in a posterior P (Gc | D) that would be far more tightly concentrated about the actual G’s paraboloid shape than about the sum of delta functions. Fitting would automatically reflect this, and thereby produce a better estimate of FG(θ) than F̂D(θ).21 in Eq. 25 by dividing by P (Gic|D) H(Gic)hc(x . Such divisions would accord with the analogous division we do in our non-FB immediate sampling experiments. 20. Since importance sampling is unbiased, this means its variance is likely to be large. 21. It might be objected that in a different problem G actually would be the sum of delta functions, not the paraboloid. In that case the FB estimate is the one that would be in error. However this possibility is Now we aren’t directly concerned with the accuracy of our estimate FG(qθ) for any single θ. We aren’t even concerned with the overall accuracy of that estimate for a set of θ’s. Rather we are concerned with the accuracy of the ranking of the θ’s given by those estimates. For example, consider naive MCO, under which we choose the θ minimizing F̃D(θ). Even if all of our estimates (one for each θ) were far from the associated actual values, if their signed errors were identical, the naive MCO would perform perfectly. In other words, ultimately we are interested in correlations between errors of our esti- mates of FG(qθ) for different θ’s. (See Sec. 2.3.) Nonetheless, we might expect that if we tend to have large error in our estimates of FG(qθ) for the θ’s, then everything else being equal, we would be likely to have large error in the associated estimate of an optimal θ. In Sec. 4.5 we exploited the equivalence between PL and MCO to improve upon naive MCO. However the parameter in MCO doesn’t specify a functional fit to a data set. Ac- cordingly, the incorporation of PL into MCO considered in Sec. 4.5 doesn’t involve fitting a function to D. This is why those PLMCO techniques don’t address the issue raised in this example; fitting does that. So in full FB MCO, we may use PL in two separate parts of the algorithm, both to form the fit to D, and then to use those fits to choose among the 6.5 Statistical Analysis of FB MC Before analyzing expected performance of FB MCO, we start with the simpler case of FB MC introduced at the beginning of this section. For simplicity we assume that the integral L̃D can be calculated exactly for any D, so that no fictitious samples arise. As discussed in Sec. 2.3, two important properties of an MC estimator of an integral L (φ) = dw U(w, φ) are the sample bias and the sample variance of that estimator. Together, these give the expected loss of the estimator under a quadratic loss function, conditioned on a fixed oracle U(., φ). This is just as true for a Bayesian fitting algorithm as for any other. For quadratic loss, for sample set D ≡ {wi, U(wi, φ)}, the Bayesian FB MC prediction for L is the posterior mean, L̃D = dw′ dUc(., φ) Uc(w ′, φ)P (Uc(., φ) | D). (28) Accordingly, the expected quadratic loss of Bayesian FBMC is∫ dD P (D | v, U(., φ))[L − L̃D]2 =∫ dw1 . . . dwN v(wi)[ dw′ U(w′, φ) − dw′ dUc(, φ) Uc(w ′, φ)P (Uc(., φ) | D)]2 (29) where v is the proposal distribution that is IID sampled N times to generate the sample In the usual way one can re-express this expected quadratic loss using a bias-variance decomposition. Whereas a conventional importance sample estimator of dw U(w, φ) is unbiased, the Bayesian estimator is biased in general; typically∫ dD P (D | v, U(., φ)) L̃D 6= L . (30) exactly what the prior P (Gc) addresses; if in fact you have reason to believe that a G that is a sum of delta functions is a priori just as likely as a paraboloid G, then that should be reflected in P (Gc). Doing so would in turn make the FB estimate more closely track the non-FB estimate. This bias is a general characteristic of Bayesian estimators. Furthermore, for some functions U(., φ), the Bayesian estimator will both be biased (unlike the factual sample estimator) and have higher variance than the factual sample estimator. So for those U(., φ), the Bayesian estimator has worse bias plus variance. In conventional importance sampling estimation of an integral, the sampling distribu- tion v is used twice. First it is used to form the sample set. Then, when the sample set has been formed, v is used again, to set the denominator values in the ratios giving the MC estimate of the integral (cf. Sec. 2.2). In contrast, Bayesian FB MC doesn’t care what v is. P (Uc(., φ) | D) is independent of the values v(wi). As mentioned at the beginning of this section, this is a typical feature of FB MC estimators. This feature does not mean that the sampling distribution is immaterial in FB MC however. Even though it does not arise in making the estimate, as Eq. 29 shows, v helps determine what the expected loss will be. Indeed, in principle at least, Eq. 29 can be used to guide the choice of the sampling distribution for Bayesian FB MC. It can even be used this way dynamically, at a midpoint of the sampling process, when one already has some samples of U(., φ). Such a procedure for using Eq. 29 to set v dynamically amounts to what is called ‘active learning’ in the PL literature (see Freund et al., 1997; Dasgupta and Kalai, 2005). We now generalize the foregoing to the case of a non-quadratic loss function L. The Bayesian estimator produces the estimate L̃D , argminρ∈R[ dUc(., φ) P (Uc(., φ) | D)L[L̃D, LUc ]] (31) Given that the factual oracle is U(., φ), the expected loss with that Bayesian estimator is dw1 . . . dwN v(wi)L[L̃{wi,U(wi,φ)}, dw′ U(w′, φ)]. (32) The expected loss in Eq. 32 is an average over data with the oracle held fixed. This contrasts with the analogous quantity typically considered in Bayesian analysis, which is an average over oracles with the data held fixed. That quantity is the posterior expected loss, ∫ dU(., φ)P (U(., φ) | D)L[L̃D,L U (φ)] (33) In general, different U(., φ)’s will give different risks for the same estimator. So we can adapt any measure concerning loss in which U(., φ) varies, to concern risk instead. In particular, the posterior expected risk is∫ dU(., φ)P (U(., φ) | D) {L[L̃D,LU (φ)] − minρ∈R[L[ρ,LU (φ)]]}. (34) Often the lower bound on loss is always 0, so that minρ∈R[L[ρ,LU (φ)]] = 0 ∀ U(., φ). In this case posterior expected risk just equals posterior expected loss. We can combine the non-Bayesian and Bayesian analyses, involving expected loss and posterior expected loss respectively. To do this we consider the prior-averaged expected loss, given by dU(., φ) P (U(., φ)) dw1 . . . dwN v(wi)L[L̃{(wi,U(wi,φ))}, dw′ U(w′, φ)]. (35) where P (U(., φ)) is a prior distribution over oracles. Note that the prior-averaged expected loss is an average over both oracles and sample sets. It reflects the following experimental test of our FB MCO algorithm: Multiple times a factual oracle U(., φ) is generated by sampling P (U(., φ)). For each such U(., φ), many times a factual sample set D is generated by sampling the likelihood P (D | U(., φ), v). That D is then used by the FMCO algorithm to calculate LD. In performing that calculation, the algorithm assumes the same likelihood as was used to generate D, but its prior P (Uc(., φ)) may not be the same function of Uc(., φ) as P (U(., φ)) is of U(., φ). Then the loss between LD and LU is calculated. The quantity in Eq. 35 is the average of that loss. Say that P (U(., φ)) is the same function of U(., φ) as P (Uc(., φ)) is of Uc(., φ). Then the Bayesian estimator is based on the actual prior. In this case, the Bayesian estimator L̃D will minimize the prior-averaged expected loss of Eq. 35.22 In general though, there is no reason to suppose that these two priors are the same. In the real world where those priors differ, expected loss for a Bayesian estimator is given by an inner product between the posterior used by that estimator, P (Uc(., φ) | D), and the true posterior, P (U(., φ) | D) (see Wolpert, 1997, 1996).23 As before, since U(.φ) varies in the integrand of prior-averaged expected loss, we can can adapt it to get a prior-averaged expected risk. This is given by dU(., φ) P (U(., φ)) dw1 . . . dwN v(wi) × {L[L̃{(wi,U(wi,φ))}, L (φ)] − minρ∈R[L[ρ, L (φ)]]}. (36) As before, if the minimal loss is always 0, then prior-averaged expected risk just equals prior-averaged expected loss. Broadly speaking, in Bayesian approaches to Monte Carlo problems, the sampling distribution that generated the samples is immaterial once one those samples have been generated (see Rasmussen and Gharamani, 2003) and references therein). So what differ- ence does the choice of a sampling distribution like v make to a Bayesian? The answer is that v determines how likely it is that we will generate a D with a high posterior variance of the quantity of interest. For example, say one wishes to form an importance sampling estimate of L = dx U(x) using sampling distribution v to generate sample set D. Then if one changes v, one changes the likelihoods of the possible D. Moreover, each D has its own posterior variance, Var(Lc | D). So what a good choice of v means is that a D with poor Var(Lc | D) is unlikely to be formed, that is, that dD P (D | v)Var(Lc | D) is low. 22. To see this, replace L̃D with some arbitrary function of D, f(D). Our task it to solve for the optimal f . First interchange the integrals over data and over oracles in Eq. 35. Next consider the integrand of the outer (data) integral, dU(., φ) P (U(., φ)) )L[f(D), , φ)]]. Since we are considering a noise-free oracle, we can write this asZ dU(., φ) P (U(., φ))P (D | v, U(., φ))L[f(D),LU (φ)]. Since P (U(., φ))P (D | v, U(., φ)) ∝ P (U(., φ) | v,D) = P (U(., φ) | D), this integral is minimized by setting f(D) = L̃D. QED. 23. It is in recognition of the fact that those functions might differ that we have been referring to ‘Bayesian’ rather than ‘Bayes-optimal’ estimators. 6.6 Statistical Analysis of FB MCO We can extend the statistical analysis of FB MC to the case of FB MCO by allowing φ to vary. The Bayesian choice of φ is the one that minimizes posterior expected loss, φ̃D , argminφ[ dUc P (Uc | D)L(φ,Uc)]. (37) Since P (Uc | v,D) = P (Uc | D), this estimator is independent of v. The same is true for the posterior expected loss of this Bayesian estimator,∫ dU P (U | D)L(φ̃D, U). (38) On the other hand, the expected loss associated with this estimator,∫ dw1 . . . dwN v(wi)L(φ̃{wi,U(wi)}, U), (39) explicitly depends on v. So does the prior-averaged expected loss,∫ dU P (U) dw1 . . . dwN v(wi)L(φ̃{wi,U(wi)}, U). (40) Next, the posterior expected risk is∫ dU P (U | D){L(φ̃D, U) − minφ′ [L(φ′, U)]} (41) where φ′ runs over the (implicit) set of all possible φ. In general minφ′ [L(φ′, U)] varies with U . (For example, this is the case with the loss function LU (φ) of Eq. 4.) Accordingly, unlike in Bayesian FB MC, typically in Bayesian FB MCO the posterior expected risk does not equal the posterior expected loss. Finally, the prior-averaged expected risk is∫ dUdw1 . . . dwN P (U) v(wi){L(φ̃{wi,U(wi)}, U) − minφ′ [L(φ′, U)]}. (42) Again, since minφ′ [L(φ′, U)] typically varies with U , in general this prior-averaged expected risk does not equal the prior-averaged expected loss. However the estimator that minimizes prior-averaged expected loss — φ̃D — is the same as the estimator that minimizes prior- averaged expected risk.24 For any particular fitting algorithm, our equations tell us how performance of the asso- ciated FB MCO depends on v and either P (U(., φ)) or the pair P (U(., φ)) and P (Uc(., φ)), depending on which equation we consider. So if we fix those prior(s), our equations tell us, formally, what the optimal v is. One can consider estimating that optimal v at a mid-way point of the algorithm, based on the algorithm’s behavior up to that point. One can then set v to that estimate for the remainder of the algorithm.25 Doing this essentially amounts to a type of active learning. 24. This follows from the fact that the prior-averaged lowest possible risk, the term subtracted in Eq. 42, is independent of the choice of the estimator. 25. Note though that if one intends to update v more than once, then strictly speaking the first update to v should take into account the fact that the future update will occur. That means the equations above for expected loss, prior-averaged expected loss, etc., no longer apply. As with Bayesian FBMC, we can analyze the effects of having P (U) not be the same function of U as P (Uc) is of Uc. Since PL and MCO are formally the same, such an analysis applies to parametric machine learning in addition to FB MCO. In particular, the analysis gives a Bayesian correction to the bias-variance decomposition of supervised learning. This correction holds even if the fitting algorithm in the supervised learning cannot be cast as Bayes-optimal for some assumed prior P (Uc). Intuitively speaking, the correction means that the bias-variance decomposition gets replaced by a bias-variance- covariance decomposition. That covariance is between the posterior distribution over target functions on the one hand, and the posterior distribution over fits produced by the fitting algorithm on the other (see Wolpert, 1997). 6.7 Combining FB and Non-FB Estimates in FB MCO Return now to the example in Sec. 6.6, where the factual sample is formed by importance sampling the factual oracle and we form a fictitious sample set using fictitious oracles.Then using only D, our estimate of FG(θ) would be the factual estimate, F̂D(θ) = k=1 r Using only our fictitious samples would instead give us the estimate ˆ̃FD(θ). On the one extreme, say we firmly believe that distribution we use for the posterior P (Gc | D) is correct. (So in particular we firmly believe that the factual oracle G was generated by sampling the prior P (Gc).) Then in the limit NT →∞, ∀θ our importance- sample estimate of F̃D will be exactly correct. So Bayesian decision theory would direct us to use the associated estimate ˆ̃FD(θ), and ignore F̂D(θ). At the other extreme, say that NT = 1, while N , the number of factual samples, is quite large. In such a situation, even if we believe our posterior is correct, it would clearly be wrong to use ˆ̃FD(θ) as our estimate, ignoring F̂D(θ). How should we combine the estimates in this latter situation? More generally, even when we believe our posterior is correct, unless the number of fictitious samples is far greater than the number of factual samples, we should combine the two associated estimates. How best to do that? Does the fact that ˆ̃FD(θ) is estimated via importance sampling over a much larger space than F̂D(θ) affect how we should combine them? More generally, say we don’t presume that our P (Gc | D) is exactly correct; how should we combine the estimates then? One is tempted to invoke Bayesian reasoning to determine how best to combine the two estimates. While that might be possible in certain situations, often determining the optimal Bayesian combination would necessitate yet more Monte Carlo sampling of some new integrals. It would be nice if some other approach could be used. One potential such approach is stacking (Wolpert, 1992; Breiman, 1996; Smyth and Wolpert, 1999). In this approach, one many times partitions the factual sample D into two parts, a ‘training set’ D1, and a ‘validation set’ D2. We write the values of w and U in D1 as {D1w(i)} and {D1U (i, φ)} respectively, and similarly for D 2. For each such partition one would run both the non-FB MCO algorithm and the FB MCO algorithm on D1. That generates the estimates φ̂v,U,D1 and φD1 , respectively. Those two φ’s give us two associated error values on the validation set, U (j, φ̂v,U,D1) U (j, φD1), respectively. More generally, we can evaluate the error on the the val- idation set of any φ, in addition to the errors of φ̂v,U,D1 and φD1 . Moreover, we can do this for the validation set of any of the partitions of D. Note, however, that only factual samples are used for cross-validation. This is what stacking exploits. In the most straightforward use of stacking, one searches for a function mapping the φ’s produced by our two algorithms to a composite φ. The goal is to find such a composite φ that will have as small validation set error (when averaged over all partitions) as possible. For example, if φ is a Euclidean vector, one could perform a regularized search for the weighted sum of φ’s that gives minimal partition-averaged validation set error. Let the weights produced by that search be bFB and bnon−FB . Then to find the final estimate for φ, one would use those weights to sum the outputs of the algorithms when run on all of D: bnon−FBφ̂v,U,D + bFBφ̃D. 7. Conclusion In this paper we explored the relationship between Monte Carlo Optimization of a parametrized integral, parametric machine learning, and ‘blackbox’ or ‘oracle’-based optimization. We made four contributions. First, we proved that MCO is identical to a broad class of parametric machine learning problems. This should open a new application domain for previously investigated para- metric machine learning techniques, to the problem of MCO. To test the use of PL in MCO one needs an MCO problem domain. The one we used was based on our second contribution, which was the introduction of immediate sampling. Immediate sampling is a way to transform an arbitrary blackbox optimization problem into an MCO problem. Accordingly, it provides us a way to test the use of PL to improve MCO, but testing whether it can improve blackbox optimization. In our third contribution we validated this way of improving blackbox optimization. In particular, we demonstratied that cross-validation and bagging improve immediate sam- pling. Conventional Monte Carlo and MCO procedures ignore some features of the sample data. In particular, they ignore the relationship between the sample point locations and the associated values of the integrand; only the values of the integrand at those locations are considered. We ended by presenting fit-based MCO, which is a way to exploit the information in the sample locations. There are many PL techniques that should be applicable to immediate sampling but that are not experimentally tested in this paper. These include density estimation active learning, stacking, kernel-based methods, boosting, etc. Current and future work involves experimental tests of the ability of such techniques to improve MCO in general and imme- diate sampling in particular. Other future work is to conduct experimental investigations of the three techniques that we presented in this paper but did not test. One of these is fit-based MCO (and fit-based immediate sampling in particular). The other two are the techniques described in the appendices: immediate sampling for constrained optimization problems, and immediate sampling with elite objective functions. There are also many potential application domains for immediate sampling PC for blackbox optimization that we intend to explore. Some of these exploit the ability of such PC to handle arbitrary (mixed) data types of x’s. In particular, one such data type is the full trajectory of a system through a space; for optimizing a problem over such a space, PC becomes a form of reinforcement learning. A. Constrained Optimization Under the PC transform we replace an optimization problem over X with one over Q. As discussed at the beginning of Sec. 3.3, the characteristics of the transformed objective can be very different from those of the original objective. Similarly, characteristics of any constraints on X in the original problem can also change significantly under this transformation. More precisely, say we add to (P4) equality and inequality constraints restricting x ∈ X to a feasible region. Then to satisfy those X- constraints we need to modify (P5) to ensure that the support of the solution qθ(.) is a subset of the feasible region in X. This appendix considers some ways of modifying PC to do this. For earlier work on this topic in the context of delayed sampling, see Wolpert et al. (2006); Bieniawski and Wolpert (2004); Bieniawski et al. (2004); Macready and Wolpert (2005). A.1 Guaranteeing Constraints Say we have a set of equality and inequality constraints over X. Indicate the feasible region by a feasibility indicator function Φ(x) = 1, x is feasible, 0, otherwise. For simplicity, we assume that for any x, we can evaluate Φ(x) essentially ‘for free’. The transformed version of this constrained optimization problem is (P5c) : minimize FG (qθ), subject to qθ(x)Φ(x) = 0. We now present a parametrization for q that ensures that it has zero support over infeasible x. First, let q̃ be any parametrized distribution overX, for instance, a mixture of Gaussians. Then using Φ(x ∈ X) as a ‘masking funtion’ we parametrize qθ(x) as qθ(x) , q̃θ(x)Φ(x)∫ dx′ q̃θ(x′)Φ(x′) , q̃Φ,θ(x). This qθ automatically meets the constraints; it places zero probability mass at infeasible x’s. It transforms the constrained problem (P5)c into the unconstrained problem (P5uc) : minimize FG (q̃Φ,θ). Now consider the case where FG is an integral over X. Typically in this case we are only concerned with the values of the associated integrand at feasible x’s. For example, when Eqθ (G) is of interest, it’s usually because our ultimate goal is to find a feasible x with as good a G(x) as possible. In this situation it makes no sense to choose between two candidate qθ’s based on differences in (the G values at) the regions of infeasible x that they emphasize. More formally, our choice between them should be independent of their respective values of dx [1 − Φ(x)]qθ(x)G(x). We can enforce this by replacing the objective Eqθ (G) = dx qθ(x)G(x) with dx Φ(x)qθ(x)G(x). If we then use the barrier function approach outlined above, our final objective becomes qp KL distance with the integral restricted to feasible x’s. Generalizing this, when we are not interested in behavior at infeasible x we can reduce the optimization problem further from (P5uc), by restricting the integral to only run over feasible x’s. More precisely, write the original problem (P5c) as the minimization of∫ dgP (g | x,G )]F (g, qθ(x)) , dx µ(x, qθ(x)), subject to the constraints on the support of qθ. By using the q̃ construction we can replace this constrained optimization problem with the unconstrained problem (A1): argminq dx Φ(x)µ[x, qθ(x)] = argminθ dx Φ(x)µ[x, q̃Φ,θ(x)], = argminθ dx Φ(x)µ[x, Φ(x)q̃(x)∫ dx′Φ(x′)q̃(x′) As an example, say our original objective function is pq KL distance. Define ZβΦ ≡∫ dx pβ(x)Φ(x). Then our new optimization problem is to minimize over θ KL(pβΦ || q̃Φ,θ) = KL( || q̃Φ) Φ(x)pβ(x) q̃θ(x)Φ(x)∫ dx′q̃θ(x′)Φ(x′) Φ(x)pβ(x) {ln[q̃θ(x)] + ln[Φ(x)]− ln[ dx′q̃θ(x ′)Φ(x′)]}. The q̃θ minimizing this is the same as the one that maximizes∫ Φ(x)pβ(x) ln[q̃θ(x)] − ln[ dx′ q̃θ(x ′)Φ(x′)]. (43) We can estimate ZβΦ using MC techniques. We can then apply MCO to estimate the θ that maximizes the integral difference26 in Eq. 43. To generate a sample of sample qθ(x) = q̃θ(x)Φ(x) we can subsample27 q̃θ according to Φ. In some cases though, this can be very inefficient (that is, one may get many rejections before getting a feasible x). To deal with such cases, we can first run a density estimator on the samples of feasible x’s we have so far, getting a distribution π. (Note that no extra calls to the feasibility oracle are needed to do this.) Next write qθ(x) = π(x)[qθ(x)/π(x)]. This identity justifies the generation of samples of qθ by first sampling π(x) and then subsampling according to qθ(x)/π(x) = q̃θ(x)Φ(x)/π(x). In an obvious modification to the foregoing, we can replace the hard restriction that supp(q) contain only feasible x’s, with a ‘soft’ constraint that q(x) ≤ κ ∀ infeasible x. A similar alternative is to ‘soften’ Φ(x) by replacing it with κ for all infeasible x, for some κ > 0. For either alternative we anneal κ down to 0, as usual, perhaps using cross- validation. 26. Note that in general this difference of integrals will not be convex in q̃θ for product distributions, unlike Φ || q̃θ). See the discussion at the end of Sec. 3.1 on product distributions and pq distance. 27. Say we want to sample a distribution A(x) ∝ B(x)C(x) where B is a distribution and C is non-negative definite, with c some upper bound on C. To generate such a sample by ‘subsampling B according to C’ we first generate a random sample of B(.), getting x′. We then toss a coin with bias C(x′)/c. If that coin comes up heads, we keep x′ as our sample of A. Otherwise we repeat the process (see Wolpert et al., 2006; Robert and Casella, 2004). A.2 Alternative FG Since we’re maximizing our expression over q̃θ, the second, correcting integral in Eq. 43 will tend to push q̃θ to have probability mass away from feasible regions. To understand this intuitively, say that q̃θ is a Gaussian and that the feasible region is ‘spiky’, resembling a multi-dimensional star-fish with a large central region and long, thin legs. For this situation, if we over-concentrate on keeping most of q̃θ’s mass restricted to feasible x, our Gaussian will be pushed away from any of the spikes of the feasible x’s, and concentrate on the center. If the solution to our original optimization problem is in one of those spikes, such over-concentration is a fatal flaw. The second integral in Eq. 43 corrects for this potential problem. More broadly, consider typical case behavior when one applies some particular con- strained optimization algorithm to any of the problems in a particular class of optimization problems. As a practical matter, there is a spectrum of such problem classes, indexed by how difficult it is just to find feasible solutions on typical problems of the class. On the one side of this spectrum are problem classes where it is exceedingly difficult to find such a solution, e.g., high-dimensional satisfiability problems with a performance measure G su- perimposed to compare potential solutions. On the other end are “simple” problem classes where it is reasonable to expect to find a feasible solution. The ‘starfish’ optimization problem is an example of a problem of the former type.28 For problems on the first side of the spectrum, where just getting a substantial amount of probability mass into the feasible region is very difficult, we may want to leave out the second integral in Eq. 43. In other words, we may want to minimize KL(pβΦ || q̃θ) rather than KL(pβΦ || q̃Φ,θ). The reason to make this change is so that q̃θ won’t get pushed away from the feasible region. (As an aside, another potential benefit of this change is that if we make it, then for product distribution q̃θ, FG (.) is convex.) Even if we do make this change, when we sample the resultant q̃θ we may not get a feasible x. If this happens, a natural approach is to repeatedly sample q̃θ until we do get a feasible x. However the resultant distribution of x’s is the same as that formed by sampling q̃Φ,θ for the same θ. So under this ’natural approach’ we work to optimize a distribution (q̃θ) different from the one we ultimately sample (q̃Φ,θ). This means that this approach may not properly balance our two conflicting needs for q̃θ: that it have most of its support in the feasible region, and that it be peaked about x’s with high pβ(x). To illustrate this issue differently, take q̃θ to be normalized, and to avoid multiplying and dividing by zero, modify Φ(x) to equal some very small non-zero value κ for infeasible x (as discussed above). Then under this ‘compound procedure’, we ultimately sample q̃Φ,θ(.). However we do not choose FG (q̃Φ,θ) = KL(p Φ || q̃Φ,θ) as the function of θ that we want to minimize. Instead we choose FG (q̃Φ,θ) = KL(p Φ || q̃Φ,θ) − ln[ q̃Φ,θ(x′) Φ(x′) ]. (44) A.3 Using Constraints for Unconstrained Optimization Return now to unconstrained optimization problems. Say that we have reason to expect that over a particular region R, the distribution pβ(x) has values approximately κ times as small as its value over X \R. It would be nice to reflect this insight in our parametrization of q, that is, to parametrize q in a way that makes it easy to match it to pβ(x) accurately. We can do this using a binary-valued function Φ and the approaches presented above. 28. Note that since we are discussing typical-case behavior, computational complexity considerations do not apply. To illustrate this, define q̃Φ,θ as above and choose the objective function FG (θ) = KL(pβ || q̃Φ,θ), where Φ(x) = κ over R, and equals 1 over X \R.29 Then working through the algebra, the qθ that minimizes this objective is given by the q̃θ that minimizes dxpβ(x)ln[q̃θ(x)] + ln[ dx′q̃θ(x ′)Φ(x′)] = KL(pβ || q̃θ) + ln[ dx′q̃θ(x ′)Φ(x′)] = KL(pβ || q̃θ) + dx′R κq̃θ(x dx′X\R q̃θ(x The logarithm on the right-hand side is a ‘correction’ to pq distance from pβ to q̃θ, a correc- tion that pushes q̃θ away from regions where Φ(x) = 1 (assuming κ < 1). To use immediate sampling with this parametrization scheme, once we find the q̃θ that minimizes the sum of pq distance plus that correction term, we would set h to the distribution (proportional to) q̃θ(x)Φ(x). So we would generate our new samples from q̃θ(x)Φ(x), for example by subsampling.30 B. The Elite Objective Function Not all PC objectives can be cast as an integral transform. Properly speaking, the choice of objective should be set by how the final qθ will be used. For instance, the concept of expected improvement suggested by Mockus et al. (1978), and used by Jones et al. (1998), considers an objective (to be maximized) given by max(Gbc − G(x), 0), where Gbc is the best of all the current samples, mini{G(xi)}. This means that at each step we will take a single sample, and want to maximize the improvement. This is a simplification; even though the next sample may yield any improvement, it may be informative, so that we get a good sample ten steps later. A less simplistic objective is the following: In blackbox optimization, no matter how many calls to the (factual) oracle we make, we will ultimately choose the best x (as far as the associated G value is concerned) out of all the ones that were fed to the oracle during the course of the entire run. Our true goal in BO is to have the G associated with that best x be as small as possible. For a discussion of distributions of extremal values, see Leadbetter et al. (1983); Resnick (1987). Given that qθ varies over the run in a way that we do not know beforehand, how can one approximate this goal as minimizing an objective function that is well-defined at all points during the run? One way to do this is to assume that there is some integer N such that, simultaneously, 1. It is likely that the best x will be one of the final N calls to the oracle during the run; 2. It is likely that qθ will not vary much during the generation of those final N samples. Under (2) we can approximate the qθ’s that are used to generate the final N calls as all being equal to some canonical qθ. Under (1), our goal then becomes finding the canonical 29. Note that we use pβ in this FG , not p Φ, which is what we used for constrained optimization. This is because our goal now is simply to find a qθ that matches p β(x). There are no additional aspects to the problem involving feasibility regions that have no a priori relation to G(x). 30. In practice, Φ(x) for this unconstrained case would not be provided by an oracle. Instead we would typically have to estimate it. We could do that for example by using a regression to form a fit to samples of pβ(x) and then use that regression to define the region R. qθ that, when sampled N times, produces a set of x’s whose best element is as good as possible. 31 In this appendix we make some cursory comments about this objective function, which we call the elite objective function. We focus on the use of Bayesian FB techniques with this objective. For a noise-free oracle the CDF for the elite objective is CDF(k) , 1− dx1 . . . dxN [qθ(x i)Θ(G(xi)− k)]. (46) So the associated density function is f(k) = d CDF(k) = Nqθ(k)[ dx qθ(x)Θ(G(x)− k)](N−1). (47) The associated expectation value, dk kf(k), is not linear in qθ. Writing it out, the posterior expected best-of-K value returned by the oracle when queries are generated by sampling qθ is dx1 . . . dxK dG P (G | D) dg1 . . . dgK P (gk | xk, G)mink{gk} (48) We want the θ minimizing this. Say we knew the exact posterior P (G | D) and could evaluate the associated integral in Eq. 48 closed-form. In this case there would be need for the parametric machine learning techniques used in the text. In particular there would be no need for regularization — an analogous role is played by the prior P (G) underlying P (G | D). When we cannot evaluate the integral in closed form we must approximate it. To illustrate this, as in Sec. 6, for simplicity consider a single-valued oracle G. This reduces Eq. 48 to dx1 . . . dxK dG P (G | D)mink{G(xk)}. (49) (The analogous FB MCO equation for objective functions involving a single integral Eq. 20; here the single ‘x’ in that equation is replaced with a set of K x’s sampled from qθ.) To approximate this integral we draw NT sample-vectors of K x’s each, using a sampling distribution hc(x) to do so. At the same time we draw NT fictitious oracles from some sampling distribution H over oracles. 31. An obvious variant of this reasoning is to have N vary across the run of the entire algorithm, at any iteration t being only the number of remaining calls to the oracle that we presume will be made. In this variant, one would modify the elite objective function to only involve the N(t) remaining samples whose G value is better than the best found by iteration t. For the case N = 1, this is analogous to the expected improvement idea in Jones et al. (1998). Note that this variant objective function will change during the run, which may cause stability problems. To simplify notation, let ~x indicate such a K-tuple of x’s. (So for multidimensional X, ~x is actually a matrix.) Also write hc(~x) , qθ(~x) , G(~x) , (G(x1), . . . , G(xK)). (50) With this notation, the estimate based on fictitious samples introduced in Sec. 6 becomes32 P (Gi | D) H(Gi) mink=1,...K{Gi(xki )}. (51) As discussed in Sec. 6, it is often good to set H(G) to be as close to P (G | D) as possible. So for example if we assume a Gaussian process model, typically we can set H(Gi) = P (Gi | D), and then directly sample H to get the values of one Gi at the K separate points xji . Alternatively, we can first form a fit φ(x) to the data in D. Next, for each of N samples ~xi, sample a colored (correlated) noise process over the K points {~xi} to get K real numbers. Finally, add those K numbers to the corresponding values {φ(~xji ) : j = 1, . . . ,K}. This gives our desired sample of {Gi(~xi)}. To illustrate the foregoing, suppose K = 1, and that we have no regularization on qθ. Then, in general, the sum in Eq. 51 is minimized by a qθ that is a delta function about that data point x1i with the best associated value Gi(x i )/hc(x i ). However for K > 1, even without regularization, the optimal qθ is not a delta function, in general.33 In addition to the regularization-based argument in the text, this gives a more formal reason why the optimal qθ should not be infinitely peaked. When K > 1, the peakedness of qθ parallels the peakedness of another non-negative function over x’s, namely P (G : G(x) is minimized at x | D). However, if we run a few iter- ations of FB MCO with the elite objective, thenD grows, and so P (G : G(x) is minimized at x | D) gets increasingly peaked over x’s. (Intuitively, the larger D is, the more confident we are about G, and consequently the more confident we are about what regions of x’s minimize G.) Accordingly, qθ gets increasingly peaked as the algorithm progresses. Note that this happens even though there is no external annealing schedule. This reflects the fact that the elite objective has no hyperparameter or regularization parameter like the β that appears in both the pq and qp objective functions. C. Gaussian Example for Risk Analysis The following example illustrates the foregoing for the case of Gaussian π, where only moments of π up to order 2 matter. To illustrate the foregoing, consider the simple case where there are only two φ’s, φ1 and φ2. Suppose that U and X are such that π is a two-dimensional Gaussian. Write π’s mean as µ. Say that one of π’s principal axes is parallel to the diagonal line, l1 = l2 (that 32. In practice there might be more efficient sampling procedures than Eq. 51. For example, one could form NK samples of hc(x) and N samples of H(G), to get two sets, which one then subsamples many times, to get pairs [~x,G(~x)]. 33. This suboptimality of a delta function qθ is similar to the suboptimality of having all K pulls in a multi-armed bandit problem be pulls of the same arm. is, one of the eigenvectors of π’s covariance matrix is parallel to the diagonal, and one is orthogonal to the diagonal). Write the standard deviation of π along that diagonal axis as σA, and write the standard deviation along the other, orthogonal axis as σB . Since π’s covariance matrix has identical diagonal entries, and since the trace of that matrix is preserved under rotations, those entries are both 1 [σ2A + σ B ]. Since the determi- nant is preserved, and since σA is the variance parallel to the diagonal, this in turn means that π’s (identical) off-diagonal entries are 1 [σ2A − σ B ]. The probability that MCO will choose φ1 is the integral of π over the half-plane where φ1 ≤ φ2: Pr(L̂ (φ2) > L̂ (φ1)) = erf( µ2 − µ1 ). (52) Next, define ∆L ≡ L (φ1)−L (φ2), ∆b ≡ [µ1 −L (φ1)] − [µ2 −L (φ2)] = [µ1 − µ2]−∆L. (53) So the difference in the value of the loss function between the two φ’s is ∆L, and the the difference in the biases of the two estimators L̂ (φ1) and L̂ (φ2) is ∆b. Note also that the variances of the two estimators are the same, Var[L (φ1)] = Var[L (φ2)] = σ2A + σ . (54) So if we shrink the variance of either of the estimators, then we shrink an upper bound on For this case of a fixed set of φ’s, it is illuminating to consider the difference between expected loss under a particular MCO algorithm and minimal expected loss over all φ’s, that is, the risk of the MCO algorithm. Assuming ∆L < 0, it is given by [Pr[L̂ (φ2) > L̂ (φ1)] − Θ[L (φ2)−L (φ1)]] × [L (φ1) − L (φ2)] [erf( µ2 − µ1 )−Θ(∆b+ µ2 − µ1)] × [µ1 − µ2 −∆b]. (55) Say that ∆b = 0. Then Eq. 55 shows that so long as µ1 6= µ2, as σB → 0 risk goes to its minimal possible value of zero. So everything else being equal, shrinking the variance of either estimator reduces risk, essentially minimizing it. 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Introduction Background on PL, MCO, Blackbox Optimization, and PC Roadmap of This Paper Notation MCO and PL Overview of PL Overview of MCO Statistical Analysis of MCO Review: MC Estimation From MC to MCO PL Equals MCO Review of PC Introduction to PC Review of Delayed Sampling Advantages of PC Relation to Other Work Immediate sampling The General Immediate-sampling Algorithm Immediate Sampling with Multiple Sample Sets Immediate Sampling with MCMC Advantages of Immediate-Sampling PC Implications of the Identity Between MCO and PL Experiments Minimizing pq KL Distance Gaussian Densities Immediate Sampling with a Single Gaussian Mixture Models Implementation Details Example: Quadratic G(x) Constant Varying Cross-validation to Schedule Bagging Cross-validation for Regularization and Model Selection Fit-based Monte Carlo Fit-based MC Estimation of Integrals Bayesian Fit-based MCO Example: Fit-based Immediate Sampling Exploiting FB Immediate Sampling Statistical Analysis of FB MC Statistical Analysis of FB MCO Combining FB and Non-FB Estimates in FB MCO Conclusion Constrained Optimization Guaranteeing Constraints Alternative FG Using Constraints for Unconstrained Optimization The Elite Objective Function Gaussian Example for Risk Analysis

0704.1275

On the Structure and Properties of Differentially Rotating Main-Sequence Stars in the 1-2 M_sun Range

ASTROPHYSICAL JOURNAL, ACCEPTED Preprint typeset using LATEX style emulateapj v. 10/10/03 ON THE STRUCTURE AND PROPERTIES OF DIFFERENTIALLY ROTATING, MAIN-SEQUENCE STARS IN THE 1 − 2 M⊙ RANGE K. B. MACGREGOR, STEPHEN JACKSON, ANDREW SKUMANICH, T. S. METCALFE High Altitude Observatory, NCAR, P. O. Box 3000, Boulder, CO 80307∗ Astrophysical Journal, Accepted ABSTRACT We conduct a systematic examination of the properties of models for chemically homogeneous, differentially rotating, main-sequence stars of mass 1 − 2 M⊙. The models were constructed using a code based on a re- formulation of the self-consistent field method of computing the equilibrium stellar structure for a specified conservative internal rotation law. The code has recently been upgraded with the addition of new opacity, equa- tion of state, and energy generation routines, and a mixing-length treatment of convection in the outer layers of the stellar interior. Relative to nonrotating stars of the same mass, these models all have reduced luminosities and effective temperatures, and flattened photospheric shapes (i.e., decreased polar radii) with equatorial radii that can be larger or smaller, depending on the degree of differential rotation. For a fixed ratio of the axial rotation rate to the surface equatorial rotation rate, increasingly rapid rotation generally deepens convective envelopes, shrinks convective cores, and can lead to the presence of a convective core (envelope) in a 1 M⊙ (2 M⊙) model, a feature that is absent in a nonrotating star of the same mass. The positions of differentially rotating models for a given mass M in the H-R diagram can be shifted in such a way as to approximate the non- rotating ZAMS over ranges in luminosity and effective temperature that correspond to a mass interval between M and about 0.7 M. We briefly note a few of the implications of these results, including (i) possible ambiguities arising from similarities between the properties of rotating and nonrotating models of different masses, (ii) a reduced radiative luminosity for a young, rapidly rotating Sun, (iii) the nuclear destruction of lithium and other light metallic species in the layers beneath an outer convective envelope, and (iv), the excitation of solar-like oscillations and the operation of a solar-like hydromagnetic dynamo in some 1.5 − 2 M⊙ stars. Subject headings: stars: interiors — stars: rotation 1. INTRODUCTION Rotation is a universal stellar physical attribute. This con- clusion is supported by an enormous body of observational data, accrued over a period of nearly 100 years, from which information about rotational speeds and periods, the depen- dence of these quantities on parameters such as mass and age (i.e., evolutionary state), and the effects of rotation on the shape, effective temperature, chemical homogeneity, and other basic properties of stars have been inferred. Despite in- creasing evidence that rotation can have a significant impact on a variety of stellar characteristics, it is generally not in- cluded as a component of the structural/evolutionary models which are the primary tools for the interpretation of observa- tions. This omission is, in part, a consequence of the increased complexity of the problem of determining the structure and evolution of a rapidly, differentially rotating star: a one- dimensional model necessarily becomes two-dimensional, the gravitational potential must be derived by solution of Pois- son’s equation, and the uncertain physics of convective and circulatory flows, along with other rotation-dependent mecha- nisms that contribute to angular momentum redistribution and chemical mixing, needs to be addressed. Some progress toward the development of a straightforward yet robust technique for computing the internal structure of a rotating star has recently been made with the implementa- tion of a new version of the self-consistent field (SCF) method (Jackson, MacGregor, & Skumanich 2005, hereafter Paper I). In its original form (Ostriker & Mark 1968; see also Jack- son 1970), the SCF approach to computing a model for an ∗The National Center for Atmospheric Research (NCAR) is sponsored by the National Science Foundation. assumed conservative law of rotation consisted of two sep- arate steps: (i) determination of the gravitational potential for a given distribution of the mass density ρ, followed by (ii) solution of the equations of stellar structure with the po- tential from (i) to update the equilibrium distributions of ρ and other quantities. The process was initiated with a trial distribution for ρ, and the two steps were executed sequen- tially and iterated upon until the densities from (i) and (ii) agreed to within a specified tolerance. While the SCF method was successfully used to construct detailed models of rotating upper-main-sequence stars (Bodenheimer 1971), it failed to converge for objects less massive than about 9 M⊙, behavior which precluded its application to intermediate- and low-mass stars (see, e.g., Chambers 1976; Clement 1978, 1979; Paper As described in Paper I (see also §2 below), the refor- mulated SCF method circumvents the difficulties responsi- ble for nonconvergence in lower-mass, more centrally con- densed stars with an approach that entails the specification of trial functions for the pressure P, the temperature T , and the shape of constant-density surfaces, together with iterative ad- justment of both the profiles and central values of P and T . With these modifications, the method is capable of produc- ing self-consistent models for rotating main-sequence stars of all masses. It has been validated through detailed compar- isons with stellar models (both rotating and nonrotating) for masses ≥ 2 M⊙ computed by other investigators using alter- native techniques (Paper I), and has been applied in an exam- ination of the photospheric shape of the Be star Achernar, an object revealed by interferometric observations as highly flat- tened by rapid rotation (Jackson, MacGregor, & Skumanich 2004, and references therein). More recently, the structure http://arxiv.org/abs/0704.1275v1 2 MACGREGOR ET AL. code based on the method has undergone considerable reno- vation, with the replacement of routines for the equation of state, nuclear energy generation rates, and opacities, and the addition of a mixing length treatment of convection. It is now equipped for use in an investigation of the effects of rotation on the structure and properties of stars with masses ≤ 2 M⊙. In the present paper, we use the formalism described above to conduct a detailed survey of the structural characteristics of differentially rotating stars having masses M in the range 1 ≤ M ≤ 2 M⊙. In the absence of rotation, this mass inter- val encompasses considerable variation in internal properties, with the gross structure of objects at the lower limit consist- ing of an inner radiative core and an outer convective enve- lope, changing to a convective core and radiative envelope for objects at the upper limit. The influence of rotation on the basic morphology of stellar interiors for these masses has not received much attention, with rotation-related effects usually treated as perturbations to the nonrotating structure, if at all (see, e.g., Thompson et al. 2003). However, observations in- dicate the occurrence of surface rotation speeds rapid enough to imply non-negligible modifications to many stellar proper- ties, particularly if the rotation is differential: the projected rotation speeds of 2 M⊙ main-sequence stars are typically in excess of 100 km s−1, and comparable values have been mea- sured for near zero-age main-sequence (ZAMS) 1 M⊙ stars in young clusters (Stauffer 1991; Wolff & Simon 1997; Tassoul 2000, and references therein). Motivated by these considerations, we have constructed an extensive set of self-consistent models for 1 − 2 M⊙ ZAMS stars, to systematically study the dependence of stellar char- acteristics on the rate and degree of differential rotation. Our SCF models extend previous computational results for stars in this mass range, some of which were obtained for uni- formly rotating configurations (Faulkner, Roxburgh, & Strit- matter 1968; Sackmann 1970; Kippenhahn & Thomas 1970; Papaloizou & Whelan 1973; Roxburgh 2004), while others were obtained using either approximate methods for noncon- servative differential rotation (Endal & Sofia 1981; Pinson- neault et al. 1989; Eggenberger, Maeder, & Meynet 2005) or a non-SCF, finite-difference technique in the case of conser- vative rotation (Clement 1979). In §2 we provide a synopsis of the new SCF method, briefly describing its implementa- tion in a code for computing the structure of chemically ho- mogeneous, differentially rotating main sequence stars, the improvements and extensions to the input physics that have been made since Paper I, and the results of tests of code reli- ability through comparisons with extant models for nonrotat- ing ZAMS stars with masses ≤ 2 M⊙. An examination of the properties of models spanning a wide range of internal rota- tion characteristics is presented in §3, with particular attention paid to the behavior of such quantitites as luminosity and ef- fective temperature, central thermodynamic properties, the lo- cation and extent of convective regions, and the size and shape of the stellar photosphere. Among the consequences of these rotation-induced changes in stellar properties is a shift in the position of objects in the classical H-R diagram (HRD); in- deed, for some adopted rotation laws, the resulting structural modifications can enable a differentially rotating star to have the same Te f f and L values as a non-rotating star of signifi- cantly lower mass and to thereby occupy the same position in the HRD. In the concluding section of the paper (§4), we use the model results to consider how the changes brought on by rapid differential rotation might affect other features of main- sequence stars in this mass range, such as the radiative lumi- nosity of the young Sun and associated effects on the planets, the abundance of lithium in the surface layers of mid-F dwarf stars, and the presence of strong, large-scale magnetic fields in some A stars. 2. THE NEW SCF METHOD 2.1. Implementation The new version of the SCF method (described in detail in Paper I) is an iterative scheme that is initialized by specify- ing (i) a pair of one-dimensional trial functions, one for the temperature distribution and one for the pressure distribution (each normalized by its central value and defined over a spa- tial range that is normalized by the equatorial radius of the star), and (ii) a two-dimensional normalized function describ- ing the shape of the equidensity surfaces. The normalized two-dimensional trial density distribution, which is used as the source term in Poisson’s equation for the gravitational po- tential, follows from the equation of state and the trial func- tions for P and T . For a conservative law of rotation, in which the angular velocity depends only on the perpendicular dis- tance from the axis of rotation, it is possible to define an effec- tive potential from which both the gravitational and centrifu- gal forces can be derived. The surfaces of constant effective potential (i.e., level surfaces) can then be identified and used to solve a set of ordinary differential equations analogous to the usual equations of stellar structure for nonrotating stars. This solution step yields updated temperature and pressure distributions, allowing the iterative cycle to be repeated and the process continued until convergence is achieved. When the input and output functions representing the normalized temperature and pressure profiles are in agreement, the two parameters corresponding to the central temperature (Tc) and central pressure (Pc) are adjusted by a Newton-Raphson tech- nique to bring them closer to the actual physical conditions at the center of the final equilibrium model. The entire pro- cedure (consisting of an SCF loop nested inside a Newton- Raphson loop) is repeated until an acceptable level of agree- ment between the input and output values of the two central parameters is attained. When used to construct models for differentially rotating main-sequence stars, this reformulation of the SCF method has been shown to converge for all masses in the range 0.6 ≤ M ≤ 30 M⊙, and for values of the dimen- sionless rotational kinetic energy t (the ratio of the rotational kinetic energy to the absolute value of the gravitational po- tential energy of the configuration) as high as 0.10–0.12 for intermediate- and high-mass models, and up to nearly 0.26 for some fully convective, highly flattened, disk-like, 1 M⊙ mod- els. By comparison, the largest t values among the non-SCF models computed by Clement (1978, 1979) were ≈ 0.18 for 30 M⊙ and ≤ 0.12 for models in the range 1.5 ≤ M ≤ 5M⊙, whereas Bodenheimer (1971) obtained a 60 M⊙ SCF model with t ≈ 0.24. As in Paper I (see also Jackson, MacGregor, & Skumanich 2004), the internal rotation of each of the stellar models dis- cussed in §3 is given by an angular velocity distribution of the Ω(ϖ) = αϖ/Re , (1) where ϖ = r sin θ, and Re is the equatorial radius of the star. For a given model, the constants Ω0 and α are prescribed parameters that characterize, respectively, the axial rotation rate and the ratio of the surface equatorial rotation rate to the axial rate (a measure of the degree of differential rotation), DIFFERENTIALLY ROTATING STARS 3 Ωe/Ω0 = 1/(1 +α2). In practice, the value of the first of these quantities is specified through the parameter η = Ω0/Ωcr, with Ωcr the equatorial angular velocity for which the magnitudes of the gravitational and centrifugal forces at Re are equal. Rigorous answers to questions about the existence and uniqueness of solutions to complicated systems of integro- differential equations are generally very difficult to obtain. Although specifying the two rotational parameters α and η for chemically homogeneous models of fixed mass and com- position does not always lead to a converged model, when it does, our experience with the code suggests that the model is unique. For a given set of (α,η) values, using the SCF code to converge models of the same mass and chemical composition from two or more different trial models always seems to lead to the same final converged model. On the other hand, it can be demonstrated that specifying the parametersα and Ω0 does not, in general, lead to a unique solution. A minor drawback to use of the parameters (α, η) in presenting the results is that several important global properties, including the luminosity L, the total angular momentum J, and the dimensionless rota- tional kinetic energy t, are not monotonic functions of η when α is held fixed (see also §3.1). Despite this shortcoming, we feel that the apparent uniqueness of the models corresponding to particular (α,η) values justifies the use of these quantities in presenting and discussing results. We emphasize that se- quences of models obtained by holding α fixed and varying η should not be interpreted as any sort of evolutionary sequence. The significance of models characterized by the same value of α is that they have the same degree of differential rotation, that is, the same Ωe/Ω0. The “half-width” of the rotation profile, ϖ1/2 = Re/α, does, however, change from model to model for constant α. 2.2. Input Physics Since the publication of Paper I, the input physics for the SCF code described therein has been updated considerably, with the installation of software components that at various times were parts of the stellar-evolution code developed by Don VandenBerg at the University of Victoria. All of the models presented in this paper were computed for the fol- lowing abundances by weight of H, He, and heavy elements: X = 0.7112, Y = 0.27, and Z = 0.0188. The opacities were ob- tained, as in VandenBerg et al. (2000), from tables of OPAL opacities calculated by Rogers & Iglesias (1992) and from ta- bles of low-temperature opacities calculated by Alexander & Ferguson (1994), using interpolation subroutines written by VandenBerg (1983). Other subroutines written by Vanden- Berg (1992) were utilized for the following: (i) the equation of state formulated by Eggleton, Faulkner, & Flannery (1973, EFF); (ii) nuclear energy generation rates for hydrogen burn- ing from Caughlan & Fowler (1988), including the effect of electron screening as treated by Graboske et al. (1973) for the case of equilibrium abundances of CNO isotopes; and (iii) a standard mixing-length treatment of surface convective zones (see, e.g., Baker & Temesvary 1966; Kippenhahn, Weigert, & Hofmeister 1967). We have made no attempt to incorporate any of the direct effects of rotation into the adopted convec- tion model; instead, we have simply modified the nonrotat- ing mixing-length description of convection by replacing the local gravitational acceleration, g, with the effective gravita- tional acceleration, ge f f = g −Ω(ϖ)2ϖ eϖ (i.e., g as reduced by the local centrifugal acceleration), averaged over equipo- tential surfaces. For all of the models, a value of 1.9 for the ratio of the mixing length to the pressure scale height has been adopted. 2.3. Validation In view of the revisions and updates that have been made to the SCF code of Paper I, it seems worthwhile to make a careful comparison of our nonrotating models for stars on the lower main sequence with models for the same mass ob- tained from a standard (nonrotating) stellar evolution code For this purpose, we have used the current version of the evolutionary code of Christensen-Dalsgaard (1982, hereafter referred to as the JCD code) to generate two evenly spaced sequences of seven models each, spanning the mass range 0.8 ≤ M ≤ 2.0 M⊙ along the ZAMS. These models have chemical composition X = 0.711, Z = 0.019, quite close to the abundances adopted for our SCF models, and were com- puted using the same value (1.9) of the mixing length pa- rameter. The basic EFF equation of state was used to con- struct one of the sequences, while the other was derived with the so-called CEFF equation of state, a modification of the EFF treatment that includes Coulomb corrections (see, e.g., Christensen-Dalsgaard & Dappen 1992). Aside from the in- clusion or omission of Coulomb effects in the equation of state, the package of input-physics routines used to generate the JCD models is very similar to that installed in the SCF code (see, e.g., Christensen-Dalsgaard, Proffitt, & Thompson 1993; Di Mauro & Christensen-Dalsgaard 2001). In addition to the intrinsic distinctions between the mathematical tech- niques used to obtain the two sets of models, the current ver- sion of the SCF code differs from the JCD code in the follow- ing ways: (i) the equation of state does not include treatment of Coulomb corrections; (ii) the photospheric pressure in the SCF code follows from the application of a different, sim- plified boundary condition (see Paper I); and (iii), there is a slight difference in the vintage of the nuclear energy genera- tion data. We have conducted a quantitative comparison of some of the important properties (R, L, Te f f , Pc, Tc, ρc) of SCF, JCD/EFF, and JCD/CEFF models for nonrotating ZAMS stars of the same mass. A theoretical HRD indicating the positions of the various models is displayed in Figure 1. The EFF and CEFF main sequences (the dashed and dotted lines, respec- tively) are very nearly coincident, with the former models shifted shifted along the locus relative to the latter models, toward somewhat lower luminosities and effective tempera- tures. On the basis of the preceding discussion, we expect the SCF models to agree better with EFF models than with CEFF models, and inspection of the results plotted in Figure 1 in- dicate that this is generally the case. For the luminosity, the most sensitive of the stellar properties, the largest discrepan- cies occur for the 0.8 M⊙ models, with the SCF L about 2% lower than that of the EFF model of the same mass; this latter value is, in turn, about 7% lower than L for the corresponding CEFF model. The luminosities of the SCF and EFF mod- els are essentially equal for 2 M⊙, while a comparison of the EFF and CEFF models for that mass reveals that the CEFF model is about 3% more luminous. The effective tempera- tures and radii of the SCF models deviate from those of the corresponding EFF models by less than 1%, except for those models between 1.2 and 1.4 M⊙ where the the magnitudes of the discrepancies are somewhat larger, ≈ 1% − 2%. The SCF values for Pc and ρc differ from the EFF values of those quan- tities by about 2%, with the relative difference in the Tc values . 1%. There is also good qualitative agreement between the SCF and JCD models with respect to the appearance or ab- 4 MACGREGOR ET AL. FIG. 1.— A theoretical HR diagram showing the positions of nonrotating, chemically homogeneous stellar models for the indicated masses, as com- puted using different codes and equations of state. The various symbols de- note models obtained using the SCF code of the present paper (∗), a current version of the code of Christensen-Dalsgaard (1982, JCD) with the simpli- fied equation of state of Eggleton, Faulkner, & Flannery (1973, EFF) (△), and the JCD code with a version of the EFF equation of state that includes Coulomb corrections (CEFF) (⊓⊔). The dashed and dotted lines are the com- puted ZAMS locations derived from the JCD/EFF and JCD/CEFF models, respectively. sence of convective cores and envelopes, and there is good quantitative agreement with respect to the radial extent and the enclosed mass of the principal radiative-convective inter- faces. 3. PROPERTIES OF DIFFERENTIALLY ROTATING STELLAR MODELS 3.1. Convergence Properties The properties of SCF models for rotating ZAMS stars with masses 1 ≤ M ≤ 2 M⊙ are summarized in Figures 2-8 and Table 1. To facilitate discussion of the computed stellar char- acteristics, we adopt the convention, established in Paper I, of identifying each model by its mass, M, and the two rotational parameters, α and η, for the reasons discussed in §2. Regions of the (α, η) parameter space in which converged 6 M⊙ SCF models can and cannot be obtained have been illustrated in Figure 3 of Paper I. While the (α, η) planes for the lower- mass SCF models considered here closely resemble that for the 6 M⊙ models, there are some important differences. Of particular relevance to the present paper are the regions cor- responding to Region II in Paper I, regions of relatively high angular momentum wherein the SCF method is incapable of producing converged models. These forbidden zones are con- siderably expanded for the 1 and 2 M⊙ models: for the for- mer models, the values (αt, ηt) corresponding to the lower tip of this region are αt = 1.39, ηt ≈ 2.4, while for the latter, αt = 2.83 and ηt ≈ 6.5. Along each constant-α sequence for 0 < α ≤ αt , quantities such as the axial angular velocity Ω0 and the total angular momentum J are non-monotonic func- tions of η, increasing to a maximum for η ≈ ηt and decreas- ing thereafter. As α→ αt , convergence in the vicinity of this maximum becomes significantly slower, and fails completely when α is large enough (i.e., > αt) to place the model within the forbidden zone. For αt <α≤ 7 (the largest value of α we have considered), converged models can be readily obtained on the low-η sides of these regions. On the high-η sides, converged models can be obtained only within the ranges FIG. 2.— Selected properties of differentially rotating, 1 M⊙ , ZAMS stellar models. The model characteristics are shown as functions of η for 0 ≤α≤ 5, where the parameters α and η specify the assumed internal angular velocity distribution given by equation (1). The quantities depicted in the various pan- els include: (a) the surface equatorial rotation speed Ve; (b) the luminosity L in units of L0, the luminosity of a nonrotating 1 M⊙ model; (c) the central temperature Tc relative to the corresponding value Tc0 for the nonrotating model; (d) the average effective temperature Te f f ; (e) the equatorial radius Re in units of R0, the radius of the nonrotating model; (f) the maximum per- pendicular distance Zmax from the equatorial plane to the photosphere (solid curves), and the polar radius Rp (dotted curves), as fractions of Re; (g) the radii of the base of the convective envelope rce and the convective core rcc, both measured in the equatorial plane relative to Re; and, (h) the temperature Tce at the base of the convective envelope. 1.39 ≤ α ≤ 3 for the 1 M⊙ models, and 2.83 ≤ α ≤ 3.47 for the 2 M⊙ models. The physical, mathematical, and computa- tional reasons for the lack of convergence of models on either side of the forbidden zone are discussed in considerable de- tail in Paper I. In this paper, we present results for complete constant-α sequences of models having α < αt , but confine our attention to just the low-η sides of forbidden zones for models with α > αt . Models on the high-η sides of these re- gions have highly rotationally flattened, disk-like structures that will be the focus of a subsequent paper. 3.2. 1 M⊙ Models In Figure 2, we show how some of the characteristics of the 1 M⊙ models depend on the dimensionless rotation parame- ter η for 0 ≤ α ≤ 5. The relation between η and the physical quantity Ve, the equatorial rotation speed at the stellar surface, is given in panel (a), from which it can be seen that for each DIFFERENTIALLY ROTATING STARS 5 TABLE 1. SELECTED MODELS Model I II III IV V VI VII VIII IX X XI M 0.8 1 1 1 1 1.2 1.2 1.6 2 2 2 α 0 0 1.5 3.75 5 0 4 0 0 3 4.75 η 0 0 1.55 3.58 4.15 0 3.74 0 0 5.64 5.9 t 0 0 0.044 0.079 0.052 0 0.065 0 0 0.047 0.078 J 0 0 6.44 7.27 5.49 0 8.93 0 0 23.71 23.73 Ve 0 0 219 123 81 0 112 0 0 205 127 L 0.224 0.653 0.332 0.224 0.336 1.591 0.653 6.075 15.47 7.560 6.083 Teff 4710 5540 4710 4700 5080 6110 5550 7490 9090 5890 7480 log gs 4.635 4.551 4.478 4.709 4.679 4.412 4.616 4.310 4.337 3.855 4.404 Re 0.714 0.878 0.934 0.744 0.767 1.129 0.904 1.466 1.590 2.927 1.512 Rp/Re 1 1 0.697 0.638 0.712 1 0.668 1 1 0.371 0.549 Zmax/Re 1 1 0.697 0.764 0.837 1 0.791 1 1 0.530 0.770 log Pc 17.078 17.178 17.100 16.967 17.019 17.259 17.101 17.326 17.287 17.317 17.282 log Tc 7.046 7.121 7.069 7.004 7.039 7.187 7.094 7.280 7.326 7.289 7.255 log ρc 1.892 1.918 1.893 1.823 1.841 1.935 1.869 1.911 1.827 1.893 1.892 ρc/ρ 25 40 35 17 20 73 29 114 96 446 83 rcc/Re . . . . . . . . . 0.138 0.138 0.046 0.090 0.094 0.122 0.054 0.110 mcc/M . . . . . . . . . 0.023 0.031 0.007 0.014 0.082 0.141 0.085 0.074 rce/Re 0.678 0.718 0.618 0.718 0.722 0.822 0.746 0.990 0.990 0.718 . . . mce/M 0.921 0.968 0.905 0.935 0.950 0.997 0.972 1.000 1.000 0.995 . . . log Tce 6.458 6.392 6.467 6.452 6.440 6.104 6.388 4.780 4.771 5.988 . . . Fig. . . . . . . 3a 3b 3c . . . 3d . . . . . . 3e 3 f NOTE. — Quantities listed (units in parentheses): total mass, M (M⊙); rotational parameters, α, η; dimensionless rotational kinetic energy, t; total angular momentum, J (1050 g cm2 s−1); equatorial velocity, Ve (km s −1); luminosity, L (L⊙); mean effective temperature, Teff (K); mean surface gravity, gs (cm s −2); equatorial radius, Re (R⊙); polar radius, Rp; maximum (normal) distance from the equatorial plane to the surface of star, Zmax; central pressure, Pc (dyn cm −2); central temperature, Tc (K); central density, ρc (g cm −3); mean density, ρ; distance in the equatorial plane from the center to the top of the convective core, rcc, and to the bottom of the convective envelope, rce; mass enclosed by the upper bounding surface of the convective core, mcc., and by the lower bounding surface of the convective envelope, mce; and temperature at the bottom of the convective envelope, Tce (K). of the α sequences depicted, Ve increases monotonically with η. Along the curves with α <αt in Figure 2, the plotted mod- els span the range from η = 0 (i.e., nonrotating) to the value η = 1 +α2 for which Ωe = Ωcr. Along the curves with α > αt , the last plotted model is located adjacent to the boundary of the forbidden region in the (α,η) plane; for these differen- tially rotating models, the centrifugal and gravitational forces have nearly equal magnitudes in the core of the star. We note that the degree of differential rotation increases with α, in the sense that the configurations corresponding to larger values of α have a greater difference between the axial and surface equatorial rates of rotation (see §2.1). As panel (b) of Figure 2 makes evident, the radiative lumi- nosities of these models are diminished relative to the lumi- nosity L0 of a nonrotating 1 M⊙ star. This is a well-known consequence of including rotation in the determination of the equilibrium stellar structure (see, e.g., Clement 1979; Boden- heimer 1971). In the results shown in Figure 2, the reduc- tion in L is larger for differentially rotating models than it is for models that are uniformly or nearly uniformly rotat- ing. A model with α = 0 rotating at the break-up rate (η = 1) has L/L0 = 0.78, while an α = 2 model with η = 2.42 has L/L0 = 0.15, a reduction of more than a factor of 6 from the nonrotating value. Much of the reason for this behavior lies in the effect of rotation on the thermodynamic conditions in the deep, energy-producing regions of the stellar interior. For these 1 M⊙ models, the contribution of the centrifugal force to supporting material against gravity enables the star to emulate an object of lower mass with correspondingly reduced values of Pc, Tc, and ρc (e.g., Sackmann 1970). The results presented in panel (c) illustrate the dependence of Tc on model rota- tional properties; similar variations are found for both Pc and ρc. For rigidly rotating configurations, this centrifugal sup- port is largest in the outermost layers of the interior, which contain only a small fraction of the stellar mass; in this case, Pc, Tc, and ρc are little changed from the values appropri- ate to a nonrotating star of the same mass. For the α = 0, η = 1 model noted previously, Pc/Pc0 = 0.94, Tc/Tc0 = 0.96, and ρc/ρc0 = 0.98, where the subscript 0 indicates the nonro- tating value. Alternatively, in models for higher values of α, the effects of rotation are increasingly concentrated toward the central regions of the star, with the result that the perturbations to the central thermodynamic quantities can be more substan- tial; for α = 2, η = 2.42, Pc/Pc0 = 0.56, Tc/Tc0 = 0.68, and ρc/ρc0 = 0.81. Panels (b) and (c) also indicate that the mag- nitudes of the changes in L, Tc, and other quantities depend on the assumed profile of internal differential rotation. The model for α = 5, η = 4 has L/L0 = 0.54, with Pc/Pc0 = 0.72, Tc/Tc0 = 0.84, and ρc/ρc0 = 0.85, smaller reductions relative to the nonrotating model than those for α = 2, η = 2.42. This behavior is an outgrowth of the structural modifications aris- ing from the centrifugal force distributions associated with the different rotation profiles. In the α = 5 model, the ratio of the centrifugal to gravitational force in the equatorial plane, 2r/g (r is the radial coordinate in the equatorial plane), is sharply peaked in the innermost portion of the stellar core, with maximum value 0.83 at the center, decreasing to ≈ 0.1 at r/Re = 0.3. For the shallower angular velocity profile of the α = 2 model, the force ratio decreases from a smaller central value of 0.42 to 0.23 at the stellar surface, 10 times the value found throughout the outer 50% of the interior of the α = 5 model. 6 MACGREGOR ET AL. In this connection, we note that Bodenheimer (1971) found that the luminosities of models for rotating 30 M⊙ stars, com- puted assuming a variety of internal rotation laws, depended primarily on the total angular momentum content of a given stellar model and not on the details of its distribution within the interior. Specifically, his results indicated that the lumi- nosities of models corresponding to four different prescribed distributions of the angular momentum per unit mass de- creased with increasing total angular momentum J, with the relation between L and J nearly the same for each model se- quence. If the luminosities of our 1 M⊙ models are plotted versus their respective J values, an analogous reduction in L for increasing J can be discerned. However, the relation be- tween L and J is roughly independent of α only for small J, and exhibits a clear dependence on α that becomes increas- ingly pronounced as J is made larger. That the luminosity can be even approximately expressed as a function of J necessar- ily reflects the modifications produced by rotation to the ther- modynamic conditions in the energy-producing core of the star, as described previously (see also Mark 1968). The na- ture and origin of the relation between L and J for stars with masses 1 − 2 M⊙ and higher, including its dependence on the structural characteristics of the models, will be addressed in detail in a subsequent paper in this series. As in Paper I, we define an average effective temperature through the relation Te f f = , where σ is the Stefan- Boltzmann constant and A the area of the stellar surface. The results for Te f f depicted in panel (d) exhibit dependences on α and η that are similar to those seen for the quantities plotted in the preceding panels. Reductions by more than 1500 K from the nonrotating value (Te f f = 5540 K) are possible as, for ex- ample, in the case α = 1.5, η = 2.18, for which Te f f = 3880 K. The rotation-induced variations in Te f f represent the com- bined effects of changes in both L and A. An indication as to the behavior of A can be gleaned from examination of the influence of rotation on the stellar size and shape. This in- formation is presented in panels (e) and (f), where we show, respectively, the equatorial radius Re (in units of the radius R0 of the nonrotating, spherical 1 M⊙ model), and the polar radius Rp together with Zmax, the maximum perpendicular dis- tance from the equatorial plane to the stellar surface. These latter two quantities are both given as fractions of Re; values of Rp/Re that are < 1 reflect a rotational flattening of the config- uration, while values of Zmax/Re that are 6= Rp/Re are indica- tive of a deviation from a convex spheroidal shape through the development of a concavity in each of the two polar regions of the star. Note that the response of the equatorial radius to increasing η differs greatly depending upon whetherα is . 1.5 or & 2. In the former case, the centrifugal force attains its largest value relative to the gravitational force at the stellar surface, conse- quently producing a distension of the outer, equatorial layers of the stellar interior and an overall increase in Re. Although Rp decreases somewhat relative to the radius of the corre- sponding nonrotating model, the net effect of the changes in Re and Rp is usually an increase in the volume of the rotating star. For some models with α ≈ αt , the decrease in Rp can more than compensate for the the increase in Re, and the vol- ume of the rotating star is reduced. In the case where α & 2, both Re and the stellar volume shrink with increasing η along a constant-α sequence. For these models, the centrifugal force is largest in comparison to the gravitational force in the central region of the stellar core, causing the central thermodynamic conditions, luminosity, and size (i.e, radius, surface area, vol- ume) to assume values that are characteristic of nonrotating stars of lower mass. This raises the possibility that a differen- tially rotating star can imitate a less massive nonrotating star in radius and effective temperature, as well as in luminosity. In panel (f), it can be seen that for α . 1.5, the curves for Zmax/Re (solid lines) and Rp/Re (dotted lines) are coincident and< 1, implying that the photospheric shape of these models is oblate spheroidal. Such a model (α = 1.5, η = 1.55) is de- picted in panel (a) of Figure 3. For α> 2, however, Zmax >Rp, symptomatic of the development of a “dimple” or indentation at either pole, as in the cases of the models shown in panels (b) (α = 3.75, η = 3.58) and (c) (α = 5, η = 4.15) of Figure 3. Panels (g) and (h) of Figure 2 contain results pertaining to the location, extent, and properties of convective regions in the models. In the absence of rotation, the internal structure of a 1 M⊙ star consists of an inner, radiative core that encom- passes ≈ 72% of the stellar radius and contains ≈ 97% of the stellar mass, surrounded by an outer, convective envelope. As is apparent in panel (g), the inclusion of rotational effects can modify this basic morphological picture in two ways: either by increasing the size of the convective envelope or by pro- moting the formation of a convective core. For lower values of α (i.e., α < 3 in panel [g]), the rota- tionally induced changes in internal structure cause the outer convection zone to deepen as η increases. For example, in the model with α = 1.5, η = 1.55 (see panel [a] in Figure 3), the radius in the equatorial plane of the base of the convective envelope is rce/Re = 0.618, significantly deeper than the the base radius rce/Re = 0.718 in the nonrotating model. Associ- ated with this reduction in rce is a decrease in the mass of the core (mc/M ≈ 0.90), and an increase in the temperature Tce at the bottom of the envelope, as can been seen in panel (h); for the α = 1.5, η = 1.55 model, Tce = 2.93× 10 6 K, as com- pared with Tce = 2.46×10 6 K for the nonrotating model. Since some chemical species (e.g., Li, Be, and B) can be destroyed by thermonuclear reactions at temperatures& 2.5×106 K, en- hancements of Tce of this magnitude are likely to have conse- quences for the surface abundances of these elements. In the case of the strongly differentially rotating models with α≥ 3, the largest centrifugal effects are concentrated in the inner- most portion of the core, so that the fractional thickness of the outer envelope is little affected. However, as the magni- tude of the centrifugal-to-gravitational force ratio in the core increases for larger η, the decreasing pressure gradient im- plied by the requirement of hydrostatic equilibrium forces the otherwise stably stratified central regions of the interior to be- come convective. Under these conditions, with the presence of a convective core, the structure of the deep interior resem- bles that of a higher-mass star. The models shown in panels (b) and (c) of Figure 3 are examples of 1 M⊙ stars with con- vective cores; in each of these models, the radial extent of this region in the equatorial plane is about 14% of Re and contains ≈ 2 − 3% of the stellar mass. For comparison, the convective core in a model for a nonrotating 2 M⊙ star has a radius that is about 12% of Re and contains ≈ 14% of the stellar mass. To illustrate the structural differences between differen- tially and near-uniformly rotating stars, in Figure 4 we show the profiles of several physical quantities for models with (α,η) = (3.00,3.26), and (1.00,1.20), along with the corre- sponding results for the nonrotating model. The luminosi- ties and total angular momenta of the two rotating models are L/L0 = 0.201 and J = 9.23 (α = 3), and L/L0 = 0.634 and J = 5.30 (α = 1), respectively, where J is meaasured in units DIFFERENTIALLY ROTATING STARS 7 FIG. 3.— Contours of level surfaces in the meridional plane for some of the nonspherical models listed in Table 1. The six rotating models shown are defined by the total mass and the two rotational parameters (M, α, η) as follows: (a) 1 M⊙ , 1.5, 1.55; (b) 1 M⊙ , 3.75, 3.58; (c) 1 M⊙, 5, 4.15; (d) 1.2 M⊙, 4, 3.74; (e) 2 M⊙, 3, 5.64; and, (f) 2 M⊙ , 4.75, 5.9. From the surface inward, the level surfaces depicted in each panel enclose a fraction of the total mass equal to 1.000, 0.995, 0.950, and 0.500, respectively. The fractional radii in the equatorial plane of these level surfaces for the various models are: (a) 1.00, 0.88, 0.71, 0.31; (b) 1.00, 0.90, 0.75, 0.40; (c) 1.00, 0.89, 0.72, 0.37; (d) 1.00, 0.87, 0.68, 0.34; (e) 1.00, 0.71, 0.37, 0.12; and, (f) 1.00, 0.66, 0.47, 0.24. Radiative portions of the interior are indicated in white, and convective regions are shaded gray. The fractional equatorial radii and enclosed masses for the interfaces between radiative and convective zones in the models are listed in Table 1. The numbers at the tops of the panels denote the total mass M and equatorial radius Re of each model. of 1050 g cm2 s−1. In panels (a) and (b), the temperature T and mass density ρ are depicted as functions of the radial po- sition r (measured in units of the present-day solar radius R⊙) in the equatorial plane of the star. The central values of both quantities exhibit rotation-induced reductions relative to the nonrotating case, the magnitudes of these modifications being larger for α = 3 (Tc/Tc0 = 0.68, ρc/ρc0 = 0.76) than for α = 1 (Tc/Tc0 = 0.92, ρc/ρc0 = 0.97). As noted previously, this be- havior is a consequence of differences in the magnitudes and distributions of the centrifugal force in the two rotating mod- els. These distinctions can be clearly seen in the profiles of the centrifugal-to-gravitational force ratio shown in panel (d). For α = 3, the force ratio is largest in the deep interior, attaining a maximum value of 0.96 at the stellar center and decreas- ing outwards to a magnitude ≈ 0.1 in the photosphere. For α = 1, the ratio increases monotonically throughout the inte- rior, rising from a central value of just 0.026 to a maximum of 0.36 at Re. The substantial contribution of the centrifu- gal force to the support of the innermost regions of the α = 3 model is responsible for the considerable enhancement of the density scale height there, evident in panel (b); ρ declines by only ≈ 10% over the inner 20% of the stellar interior. The resulting changes in the internal mass distribution (panel [c]) lead to a star with a smaller radius, Re = 0.72 R⊙ as opposed to Re = 0.88 R⊙ for the nonrotating model. Alternatively, for the α = 1 model, the lack of centrifugal support in the core of the star leads to temperature, density and mass distribu- tions therein that closely resemble those of the nonrotating model. Closer to the surface, however, the density distribu- tion becomes extended, a product of the increasing centrifu- gal reduction of gravity in the outer layers of the interior; as a result, the stellar radius is larger than that of the nonrotating model, Re = 1.00 R⊙. 3.3. Solar Look-Alike Models Figure 5 is a theoretical HRD for 1 M⊙ models that span a broad range of internal rotational states, from uniform rotation to extreme differential rotation (up to Ω0 = 50 Ωe for α = 7). The nonrotating ZAMS is delineated by a dotted line, with the positions of several specific models for 0.6≤ M ≤ 1.0 M⊙ in- dicated. Clearly, as already implied by panels (b) and (c) of Figure 2, the locations of rotating models in such a diagram are displaced to the right of and below the position they would occupy in the absence of rotation, toward lower values of both the luminosity and the effective temperature. Models with uniform or near-uniform internal rotation (i.e., α = 0, 1) lie 8 MACGREGOR ET AL. FIG. 4.— Profiles of selected physical quantities in the interiors of 1 M⊙ models with (α,η) = (3.00,3.26) (solid curves), (1.00,1.20) (dashed curves), and (0.00,0.00) (dotted curves). The profiles depict the dependence of each quantity on radial position r in the equatorial plane, from the center of the star to the surface. The various panels show (a) the temperature T , (b) the mass density ρ, (c) the fraction m/M of the total mass contained within a level surface that intersects the equatorial plane at r, and (d) the ratio (Ω2r/g) of the centrifugal force to the gravitational force. well above the nonrotating ZAMS, while those for which the degree of differential rotation is substantial (α ≥ 3) have po- sitions in the HRD that collectively approximate the nonrotat- ing ZAMS over virtually the entire mass range depicted. For example, the rotating 1 M⊙ model (α = 3.75, η = 3.58) shown in panel (b) of Figure 3 is characterized by Te f f = 4700 K and L/L⊙ = 0.224, which is indistinguishable from the val- ues Te f f = 4710 K, L/L⊙ = 0.224 for a nonrotating, 0.8 M⊙ model. Such a coincidence of the positions of stars of differ- ent mass in the classical HRD is made possible by the fact that the largest structural changes take place in the cores of strongly differentially rotating models. Specifically, the de- crease in L brought about by the smaller values of Pc, Tc, and ρc, together with the corresponding reduction in Re, enable such a model to effectively mimic a lower mass stellar model in which rotational effects are not included. Figure 6 illustrates some of the internal properties of three so-called solar look-alike models, that is, models for differ- FIG. 5.— A theoretical HR diagram showing the positions of models for rotating, ZAMS, 1 M⊙ stars. Luminosities are given in units of the luminos- ity L0 = 0.653 L⊙ of a non-rotating, 1 M⊙ star. The various symbols denote models constructed using the rotation law of equation (1) with the values of the parameters (α,η) listed in the Figure. The ZAMS for non-rotating stars is indicated by the dotted line, with the positions of models for masses 0.6, 0.7, 0.8, 0.9, and 1.0 along it marked by an ∗ symbol. entially rotating stars with masses > 1 M⊙ that have many physical attributes in common with the model for a nonrotat- ing, 1 M⊙ star. Inspection of panels (a) and (b) reveals that the profiles and the central values of the temperature and den- sity for the 1.1, 1.2, and 1.3 M⊙ models depicted therein are quite close to those of the solar-mass model with α = η = 0. A consequence of these structural similarities is that many of the general characteristics of the models are nearly iden- tical. For example, the 1.2 M⊙ model which, in the absence of rotation, would have L/L⊙ = 1.591, Re/R⊙ = 1.129, and Te f f = 6110 K instead has L/L⊙ = 0.653, Re/R⊙ = 0.904, and Te f f = 5546 K, compared with L/L⊙ = 0.653, Re/R⊙ = 0.878, and Te f f = 5545 K for the nonrotating, 1 M⊙ model. A cross- section in the meridional plane of the rotating 1.2 M⊙ inte- rior is shown in panel (d) of Figure 3, from which it can be seen that a solar-like convection zone having rce/Re = 0.746 is present in the layers beneath the photosphere. With a sur- face equatorial rotation speed Ve = 112 km s −1, such an object could be mistakenly identified as a rapidly rotating 1 M⊙ star, if observations were analyzed through comparison with non- rotating stellar models. That the values of measureable or inferrable stellar properties can be the same in rotating and nonrotating models for different masses represents a potential source of ambiguity in the interpretation of a variety of obser- vations. Panels (c) and (d) reveal subtle differences in the internal mass distributions and in the variation of the ratios of the centrifugal to gravitational force in the equatorial planes of these solar look-alike models. Future space-based astero- seismological observations may be capable of exploiting such differences in internal structure to distinguish slowly rotating stars from more rapidly, differentially rotating higher-mass stars that happen to have the same values of L and Te f f and thus occupy the same position in the HRD (see, e.g., Lochard et al. 2005). 3.4. 2 M⊙ Models In Figure 7, we present a summary of the rotational depen- dence of 2 M⊙ model characteristics, using the same format as adopted for Figure 2. The model for a non-rotating star of this DIFFERENTIALLY ROTATING STARS 9 FIG. 6.— Profiles in the equatorial plane of the temperature (a), den- sity (b), mass fraction (c), and the ratio of the centrifugal to gravitational acceleration (d) for a nonrotating, 1 M⊙ model (α = η = 0) and three so- lar look-alike models. The latter models were obtained for (M,α,η) = (1.1 M⊙,4.00,2.74), (1.2 M⊙,4.00,3.74) and (1.3 M⊙,4.00,4.39), and have values of Te f f and L that differ from those of the non-rotating, solar model by about 1% or less. mass has L0 = 15.47 L⊙, R0 = 1.59 R⊙, and Te f f = 9090 K, and a convective core with rcc/Re = 0.122, mcc/M = 0.141; such a model has a thin, subsurface convective layer, rce/Re = 0.99, that contains a negligible fraction of the stellar mass. The re- sults shown in Figure 7, which span the rotational parameter ranges 0 ≤α≤ 5, 0 ≤ η≤ 7, exhibit many of the same behav- iors as noted previously in connection with the 1 M⊙ models of Figure 2. In particular, along each constant-α sequence, the luminosity (panel [b]), central temperature, (panel [c]) and ef- fective temperature (panel [d]) all initially decrease as η is in- creased, with the largest reductions in these quantities occur- ring in the cases of differentially rotating models with α≥ 3. For α = 1, 2, both L/L0 and Tc/Tc0 pass through minima and then increase slightly with η beyond those points, whereas Te f f continues to decrease with η for all α sequences. The central pressure Pc (panel [e]), on the other hand, is an increas- ing function of η for 0 ≤ α≤ 3, and displays a non-montonic dependence on η for the model sequences corresponding to α = 4, 5. The origin of this variation is the mass dependence FIG. 7.— Selected properties of differentially rotating, 2 M⊙ , ZAMS stellar models, as in Figure 2. The quantities depicted in the various panels include: (a) Ve, (b) L, (c) Tc, (d) Te f f , (e) Pc, (f) Re, (g) Zmax and Rp, and (h) rce and of Pc for stars on the non-rotating ZAMS. In our SCF models without rotation, Pc has a maximum value for M near 1.6 M⊙, and is decreasing for both larger and smaller M values. The maximum occurs for a mass close to the value marking the in- ternal structural transition between stars with radiative cores and convective envelopes and stars with convective cores and radiative envelopes. Hence, insofar as the properties of a ro- tating star are like those of a non-rotating star of lower mass, we expect the Pc values of these 2 M⊙ models to increase as the effects of rotation become more pronounced. For cases in which the central thermodynamic conditions are significantly perturbed by rapid, differential rotation in the core of the star, this mass-lowering effect can be large enough to shift Pc to values characteristic of nonrotating stars with M < 1.6 M⊙ (i.e., on the other side of the central pressure peak), as in the 2 M⊙ models for α = 4, 5. Panels (f) and (g) of Figure 7 convey information pertain- ing to the photospheric sizes and shapes of the models. Those that rotate nearly rigidly are equatorially distended, and, in the direction perpendicular to the equatorial plane, have their largest dimension along the rotation axis (Zmax = Rp); since Rp < Re, these models (like their 1 M⊙ counterparts) have a flattened, spheroidal shape. Models with increasing degrees of differential rotation (i.e., with α & 3) develop polar con- cavities (as indicated by Zmax > Rp), and ultimately (i.e., for 10 MACGREGOR ET AL. α = 4, 5 and sufficiently large η) become more compact, with Re < R0. Panels (e) and (f) of Figure 3 give cross-sectional representations of the photospheric shapes of 2 M⊙ mod- els for (α, η) = (3.00, 5.64) and 4.75, 5.90), repectively; in the former model, Re = 2.93 R⊙ > R0, while for the latter, Re = 1.51 R⊙ < R0. The influence of rotation on the occurrence and extent of convective regions within the models is explored in panel (h) of Figure 7. The convective core, a salient feature of the non-rotating stellar interior, generally decreases in size as the value of η is increased. This contraction stems from a re- duction in the magnitude of the radiative gradient ∇rad in the core region of the star, a result of the way in which the cen- tral thermodynamic conditions are modified by rotation (see, e.g., MacGregor & Gilliland 1986); with ∇rad smaller, the size of the region wherein it exceeds the adiabatic gradient ∇ad shrinks accordingly. Only for the most rapidly, differen- tially rotating models does rcc show a modest increase com- pared to the non-rotating model; in panel [h], the model for (α, η) = (5.00,6.88) has rcc/Re = 0.124. In this case, as was seen for the 1 M⊙ models having convective cores in Fig- ure 2, the growth of rcc can be traced to a larger value of ∇rad , produced by a centrifugal-to-gravitational force ratio that is nearly unity at the stellar center. Note also that for some 2 M⊙ models, the convection zone underlying the stel- lar photosphere can extend into the stellar interior by more than the few 10−3 R0 that is the thickness of this region in the absence of rotation. As can be seen, for example, in panel (e) of Figure 3, the base of the convective envelope in the equatorial plane of the model with (α, η) = (3.00,5.64) is lo- cated at rce/Re = 0.718, a fractional depth which is the same as in the non-rotating 1 M⊙ model. This leads to the intrigu- ing possibility that solar-like oscillations, driven by turbulent convection, may be excited in stars that would normally be too massive to generate them. The results shown in panel (h) sug- gest that such a solar-like convective envelope is most likely to occur in 2 M⊙ models with intermediate differential rotation (say., α ≈ αt = 2.83). For a profile of this kind, the inner and outer portions of the interior can each rotate rapidly enough to both significantly perturb the thermodynamic conditions in the core and increase Re by extending the stellar envelope. In Figure 8, we show the positions of models for differen- tially rotating, 2 M⊙ stars in a theoretical HRD. As in Figure 5, the location of the non-rotating ZAMS is indicated by a dot- ted line, with the positions of models for stars with masses in the range 1.0 ≤M ≤ 2.0 M⊙ indicated along it. This HRD for rotating 2 M⊙ models has a number of features in common with the corresponding representation of 1 M⊙ properties. Without exception, models are shifted from the non-rotating location to new positions characterized by lower Te f f and L values. For α . 3, these new positions lie to the right-hand side of (i.e., above) the non-rotating ZAMS; for rotation that is increasingly differential, however, the model locations ap- proach the non-rotating ZAMS, moving just to the left-hand side of (i.e., below) it for α = 5. For α = 4.75, the model positions are distributed along the non-rotating ZAMS, de- lineating its track in the HRD for masses greater than about 1.5 M⊙. This coincidence again raises the possibility that a rapidly, differentially rotating 2 M⊙ star could effectively masquerade as a non-rotating star of lower mass. As an ex- ample, with L/L⊙ = 6.083, Re/R⊙ = 1.512, and Te f f = 7480 K the model for (α, η) = (4.75,5.90) shown in panel (f) of Fig- ure 3 closely resembles a non-rotating 1.6 M⊙ star, for which L/L⊙ = 6.075, Re/R⊙ = 1.466, and Te f f = 7490 K. FIG. 8.— A theoretical HR diagram showing the positions of models for rotating, 2 M⊙ ZAMS stars, as in Figure 5. Luminosities are given in units of the luminosity L0 = 15.468 L⊙ of a nonrotating, 2 M⊙ star. The ZAMS for nonrotating stars is indicated by the dotted line, with the positions of models for masses 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.8, and 2.0 M⊙ along it marked by an ∗ symbol. 4. SUMMARY AND DISCUSSION The SCF approach to computing the structure of rotating stars is robust and efficient, capable of yielding converged models of stars of all masses, with internal angular-velocity distributions covering the range from uniform to extreme dif- ferential rotation. The code described in Paper I has been up- graded throught the introduction of opacities, energy gener- ation rates, and an equation of state that are close to state- of-the-art, and with the introduction of a mixing-length treat- ment of convective energy transport. We have used this re- vised code to conduct a systematic study of the properties of models for rotating ZAMS stars with masses 1 ≤ M ≤ 2 M⊙, assuming that the rotation of the stellar interior can be ad- equately represented by the parameterized angular-velocity profile given in equation (1). Our results suggest that rotation affects virtually every char- acteristic of the intermediate- and low-mass stars considered herein, with the magnitude, and sometimes even the sense, of the changes in fundamental attributes depending on both the rate and degree of the differential rotation in the stellar interior. For models that are uniformly or nearly uniformly rotating, the centrifugal contribution to the hydrostatic sup- port of the star is largest in the outer layers of the interior; for models that are strongly differentially rotating, the cen- trifugal force is largest compared to gravity in the core of the star. Luminosities and effective temperatures are always di- minished relative to nonrotating stars of the same mass, while equatorial radii increase in uniformly rotating or moderately differentially rotating models, and decrease in models where the difference between the angular velocity at the center and that at the equator is sufficiently large. As the degree of dif- ferential rotation is increased, the photospheric shape of the star changes from a convex surface closely approximating a spheroid flattened along the rotation axis to a roughly oblate surface with deepening polar concavities; in such cases, the greatest perpendicular distance from the equatorial plane to the surface occurs away from the axis of rotation, and is al- ways less than the equatorial radius of the star. In 1 M⊙ mod- els, the effects of rotation can either increase the thickness of DIFFERENTIALLY ROTATING STARS 11 the outer convective envelope, or contribute to the formation of a convective core. In rotating 2 M⊙ models, the size of the convective core is diminished relative to that found in the ab- sence of rotation, and an extensive, solar-like convection zone can be present in the outer layers of what would otherwise be a stable radiative envelope. The extent to which the results of these computations accu- rately depict the stellar structural modifications arising from rotation depends on the validity of a number of simplifying assumptions and approximations made in the course of devel- oping the basic model, as well as on the inherent limitations of the SCF method itself. Here, we briefly address the most salient of the factors that might restrict the applicability of these results, noting where improvements and extensions can (or cannot) be made (see also Paper I). 1). The modified SCF iterative scheme utilized in the present investigation necessarily requires the angular velocity distribution within the star to be conservative, and thus ex- pressible as a function of just the perpendicular distance from the rotation axis, Ω = Ω(ϖ) (see eq. [1]). Although this speci- fication facilitates the construction of models through the con- siderable mathematical simplification it introduces, its effect on some computed stellar structural characteristics, such as the photospheric shapes and core physical properties of very rapidly, differentially rotating models, is likely to differ from that produced by a rotation law of (say) the form Ω = Ω(r). There is some evidence, from both simulations and observa- tions, for the occurrence of differential rotation of the type given by equation (1) (see, e.g., Dobler, Stix, & Brandenburg 2006, and references therein). Yet analyses of solar acous- tic oscillations yield a picture of the large-scale internal dy- namics of the Sun which is not in accord with the angular velocity being constant on cylindrical surfaces (see Thomp- son et al. 2003). We note that use of the SCF method pre- cludes consideration of non-conservative rotation laws (e.g., Ω = Ω(r)); however, detailed treatment of the hydrodynamical and magnetohydrodynamical processes affecting the internal rotational states of the Sun and stars is presently beyond the scope of any of the extant structural/evolutionary models. 2). The model treats only the prescribed rotational motion of the stellar interior, omitting any meridional circulatory flow and its consequent effects on the internal angular momentum distribution. For models with radiative envelopes, rotation- ally driven circulation will cause the rotation profile to devi- ate over time from the state given by equation (1), unless such evolution is mitigated by additional angular momentum trans- port mechansims (see, e.g., Maeder & Meynet 2000, and ref- erences therein). For models with convective envelopes, the rotation profile in the outer layers of the interior is the prod- uct of the complex interplay between meridional circulation and turbulent heat and angular momentum transport (Rempel 2005; Miesch, Brun, & Toomre 2006); whether Ω is solar- like or constant on cylindrical surfaces depends on the lati- tudinal entropy distribution in the subadiabatically stratified layers below the convection zone. 3). The model employs a simplified treatment of con- vection, locating unstable regions by application of the Schwarzschild criterion and utilizing an averaged, rotationally modified mixing-length description (see §2.2) to determine the structure of an outer convective envelope, if present. Use of the Solberg-Høiland condition (e.g., Ledoux 1965; Kip- penhahn & Weigert 1990) to ascertain the onset of convective instability would account for the direct influence of the cen- trifugal force. Since models computed for the rotation law of equation (1) have ∂ j/∂ϖ > 0, where j = Ωϖ2 is the specific angular momentum (see Paper I), this change would likely de- crease the equatorial-plane thickness of an outer convection zone, and produce a latitudinal variation in rce. However, for rapid rotation, there is considerable uncertainty regardless of the convection criterion/model adopted, because of both the rudimentary nature of extant treatments of convective energy transport and the universal assumption of axisymmetry among rotating stellar models. We reiterate that the present model describes a chemi- cally homogeneous, ZAMS star, and neglects effects associ- ated with the structural and compositional evolution of the stellar interior. We are presently developing a mean-field hydrodynamics-based treatment of turbulent chemical and an- gular momentum transport which maintains conservative ro- tation profiles, thus making it possible to investigate the main- sequence evolution of these SCF models. Despite the short- comings enumerated above, we believe that the basic model and method described herein compare favorably with alterna- tive approaches to determining the structure and evolution of rotating stars. The most widely used of these (e.g., Meynet & Maeder 1997) relies on the use of approximate, Roche- like equipotentials to represent the internal gravity of the star, thereby strictly limiting it to describing slowly rotating stars to ensure the accuracy of the computed models. Whatever its drawbacks, the SCF method yields two-dimensional, ax- isymmetric configurations that represent fully consistent so- lutions to both the set of stellar structure equations and Pois- son’s equation for the gravitational potential. The results presented herein have a number of implications for young stars with masses between 1 and 2 M⊙. Mea- surements of projected equatorial rotation speeds in excess of 100 km s−1 for some solar-type stars in young open clusters (see, e.g., Stauffer 1991) raise the possibility that the struc- ture and properties of these objects could be significantly al- tered from those of nonrotating stars of the same mass. Such stars, provided that their interior rotation is sufficiently differ- ential, could in actuality be somewhat more massive objects for which surface rotation as rapid as that indicated by obser- vations is the normally expected ZAMS state. The question of whether or not the kind of strong differential rotation required to produce such ambiguity is present within the interiors of some low- and intermediate-mass stars may ultimately be re- solved through space-based asteroseismological observations, which should allow low-resolution inversions of the rotation profiles in the inner ∼ 30% of the stellar radius (see Gough & Kosovichev 1993). Asteroseismology from space may also afford the means for identifying individual stars whose rota- tion enables them to pose as lower-mass objects, since, when effects associated with asphericity are not too large, the aver- age mode frequency spacing is sensitive to the mean density, a quantity which Table 1 reveals to be different for look-alike models. If rapid differential rotation is a possibility for objects in the mass range spanned by the models of §3, then the associated changes in structure and properties could have consequences for a variety of important astrophysical processes that take place within and around such stars. The reduced radiative luminosity of a rapidly rotating young Sun would likely influ- ence the evolution of the solar nebula, and further exacerbate the discrepancies between the properties of nonrotating mod- els and observational inferences indicating a higher ZAMS lu- minosity (see, e.g., Sackmann & Boothroyd 2003). The rota- tionally induced deepening of a sub-photospheric convection 12 MACGREGOR ET AL. zone, together with the increase in temperature of the mate- rial at the base, could contribute to the depletion of lithium in the stellar surface layers by reducing the thickness of the region through which chemical species must be transported in order to be destroyed by nuclear processes. The formation of a solar-like convective envelope in a young, differentially rotating, 2 M⊙ star could excite global oscillations, and be accompanied by the operation of a solar-like hydromagnetic dynamo. Dynamo-generated fields that diffuse into and are retained by the radiative interior (see, e.g., Dikpati, Gilman, & MacGregor 2006) could enable the star to remain magnetic long after spin-down and the elimination of nonuniform rota- tion have led to the disappearance of both the surface convec- tive layer and the dynamo. Each of these possibilities will be addressed in forthcoming papers. We wish to thank D. A. VandenBerg for providing soft- ware from his stellar-evolution code that was used to han- dle the input physics in our SCF code, and we wish to thank J. Christensen-Dalsgaard for the use of his stellar-evolution code to help us validate our models. This work was supported in part by an Astronomy & Astrophysics Postdoctoral Fel- lowship under award AST-0401441 (to T. S. 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0704.1276

SDSS J102146.44+234926.3: New WZ Sge-type dwarf nova

COMMISSIONS 27 AND 42 OF THE IAU INFORMATION BULLETIN ON VARIABLE STARS Number 5763 Konkoly Observatory Budapest 10 April 2007 HU ISSN 0374 – 0676 SDSS J102146.44+234926.3: NEW WZ SGE-TYPE DWARF NOVA GOLOVIN, ALEX1,2,3; AYANI, KAZUYA4; PAVLENKO, ELENA P.5; KRAJCI, TOM6; KUZNYETSOVA, YULIANA2,7; HENDEN, ARNE8; KRUSHEVSKA, VICTORIA2; DVORAK, SHAWN9; SOKOLOVSKY, KIRILL10,11; SERGEEVA, TATYANA P.2; JAMES, ROBERT12; CRAWFORD, TIM13; CORP, LAURENT14; Kyiv National Taras Shevchenko University, Kyiv, UKRAINE e-mail: astronom [email protected], [email protected] Main Astronomical Observatory of National Academy of Science of Ukraine, Kyiv, UKRAINE Visiting astronomer of the Crimean Astrophysical Observatory, Crimea, Nauchnyj, UKRAINE Bisei Astronomical Observatory, Ibara, Okayama, JAPAN Crimean Astrophysical Observatory, Crimea, Nauchnyj, UKRAINE AAVSO, Cloudcroft, New Mexico, USA 7 International Center of Astronomical and Medico-Ecological Researches, Kyiv, UKRAINE AAVSO, Clinton B. Ford Astronomical Data and Research Center, Cambridge, MA, USA Rolling Hills Observatory, Clermont, FL, USA Sternberg Astronomical Institute, Moscow State University, Moscow, RUSSIA 11 Astro Space Center of the Lebedev Physical Institute, Russian Academy of Sciences, Moscow, RUSSIA AAVSO, Las Cruses, NM, USA AAVSO, Arch Cape Observatory, Arch Cape, OR, USA AAVSO, Rodez, FRANCE The cataclysmic variable SDSS J102146.44+234926.3 (SDSS J1021 hereafter; α2000 = 10h21m46.s44; δ2000 = +23 ◦49′26.′′3) was discovered in outburst having a V magnitude of 13.m9 by Christensen on CCD images obtained in the course of the Catalina Sky Survey on October 28.503 UT 2006. In an archival image there is a star with V ∼ 21m at this position (Christensen, 2006) and there is an object in the database of the Sloan Digital Sky Survey Data Release 5 (Adelman-McCarthy et al., 2007; SDSS DR5 hereafter) with the following magnitudes, measured on January 17.455 UT, 2005: u = 20.83, g = 20.74, r = 20.63, i = 20.84, z = 20.45. In the USNO-B1.0 catalog this object is listed as USNO-B1.0 1138-0175054 with magnitudes B2mag = 20.79 and R2mag = 20.35. The large amplitude and the blue color imply that the object could be a dwarf nova of SU UMa or WZ Sge type (Waagen, 2006). Fig. 1 (left) shows the 8′ × 8′ image of the SDSS J1021 vicinity, generated from SDSS DR5 Finding Chart Tool (http://cas.sdss.org/astrodr5/en/tools/chart/chart.asp). Time resolved CCD photometry has been carried out from different sites by the authors since November 21, 2006 (the first night after the discovery was reported) until 2006 December 06 (Data available for download at http://www.aavso.org/data/download and from IBVS server; See Table 1 for log of observations). The photometry was done in the V and Rc bands as well as unfiltered; this did not affect the following period analysis. http://arxiv.org/abs/0704.1276v2 http://www.aavso.org/data/download 2 IBVS 5763 The error of a single measurement can be typically assumed to be ±0.m02. Fig. 1 (right) shows the overall light curve of the object. Here we assume mR = munfiltered. The light curve could be divided into three parts, denoting the plateau stage, dip and long-lasting echo-outburst (rebrightening). Before carrying out Fourier analysis for the presence of short-periodic signal in the light curve (superhumps), each observer’s data set was individually transformed to a uniform zero-point by subtracting a linear fit from each night’s observations. This was done to remove the overall trend of the outburst and to combine all observations into a single data From the periodogram analysis (Fig. 2, left) the value of the superhump period Psh = 0 .d05633 ± 0.00003 was determined. Such a value is typical for the WZ Sge-type systems and is just 58.7 seconds shorter than Psh of another WZ Sge-like system: ASAS 002511+1217.2 (Golovin et al., 2005). The superhump light curve (with 15-point binning used) folded with 0d.05633 period is shown on Fig. 2 (right). It is plotted for two cycles for clarity. Only JD 2454061.0- 2454063.6 data was included. Note the 0.m1 amplitude of variations and the double- humped profile of the light curve. There remain many questions concerning the nature of a double-humped superhumps in the WZ Sge-type stars. The explanation of a double- humped light curve could lie in a formation of a two-armed precessional spiral density wave in the accretion disk (Osaki, 2003) or a one-armed optically thick spiral wave, but with the occurrence of a self-eclipse of the energy emitting source in the wave (Bisikalo, 2006). Other theories concerning a double-peaked superhumps can be found in Lasota et al. (1995), Osaki & Meyer (2002), Kato (2002), Patterson et al. (2002), Osaki & Meyer (2003). Table 1. Log of observations JD Duration of (mid of observational Observatory Telescope CCD Filter obs. run) run [minutes] 2454060.9 214 Rolling Hills, FL, USA Meade LX200-10 SBIG ST-9 V 2454061.0 158 Cloudcroft, NM, USA C-11 SBIG ST-7 none 2454062.0 259 Cloudcroft, NM, USA C-11 SBIG ST-7 none 2454062.9 288 Cloudcroft, NM, USA C-11 SBIG ST-7 none 2454063.6 115 CrAO, UKRAINE K-380 SBIG ST-9 R 2454064.6 222 CrAO, UKRAINE K-380 SBIG ST-9 R 2454066.7 S.D.P. * Pic du Midi, FRANCE T-60 Mx516 None 2454067.6 90 CrAO, UKRAINE K-380 Apogee 47p R 2454067.9 S.D.P. Las Cruses, NM, USA Meade LX200 SBIG ST-7 V 2454069.0 S.D.P. Arch Cape, USA SCT-30 SBIG ST-9 V 2454069.0 S.D.P. Las Cruses, NM, USA Meade LX200 SBIG ST-7 V 2454069.6 63 CrAO, UKRAINE K-380 Apogee 47p R 2454071.9 S.D.P. Las Cruses, NM, USA Meade LX200 SBIG ST-7 V 2454072.9 S.D.P. Las Cruses, NM, USA Meade LX200 SBIG ST-7 V 2454073.9 S.D.P. Las Cruses, NM, USA Meade LX200 SBIG ST-7 V 2454074.9 S.D.P. Las Cruses, NM, USA Meade LX200 SBIG ST-7 V 2454075.9 S.D.P. Las Cruses, NM, USA Meade LX200 SBIG ST-7 V 2454166.8 S.D.P. Sonoita Observatory, USA 0.35 m telescope SBIG STL-1001XE V 2454167.7 S.D.P. Sonoita Observatory, USA 0.35 m telescope SBIG STL-1001XE V * S.D.P. - Single Data Point IBVS 5763 3 Figure 1. Left: SDSS image of the SDSS J1021 vicinity; Right: Light curve of SDSS J1021 during the outburst; Applying the method of ”sliding parabolas” (Marsakova & Andronov, 1996) we deter- mined, when it was possible (JD 2454061.0 - 2454063.6), the times of maxima of super- humps (with mean 1σ error of 0 .d0021) and calculated O-C residuals based on founded period. The moments of superhump maximua are given in Table 2. No period variations reaching the 3σ level were found during the time of observations. Another prominent feature of the SDSS J1021 light curve is the echo-outburst (or re- brightening - another term for this event) that occurs during the declining stage of the superoutburst. On Nov. 27/28 2006 (i.e. JD 2454067.61-2454067.68) a rapid brighten- ing with the rate of 0.m13 per hour was detected at Crimean Astrophysical Observatory (Ukraine; CrAO hereafter), that most probably was the early beginning of the echo- outburst. Judging from our light curve, we conclude that rebrightening phase lasted at least 8 days. Similar echo-outbursts are classified as ”type-A” echo-outburst according to classification system proposed by Imada et al. (2006) as observed in the 2005 superout- burst of TSS J022216.4+412259.9 and the 1995 superoutburst of AL Com (Imada et al., 2006; Patterson et al., 1996). Rebrightenings during the decline stage are observed in the WZ Sge-type dwarf novae (as well as in some of the WZ Sge-type candidate systems). However, their physical mechanism is still poorly understood. In most cases, just one rebrightening occurs (also observed sometimes in typical SU UMa systems), though a series of rebrightenings are also possible, as it was manifested by WZ Sge itself (12 rebrightenings), SDSS J0804 (11) and EG Cnc (6) (Pavlenko et al., 2007). There are several competing theories concerning what causes an echo-outburst(s) in such systems, though all of them predict that the disk must be heated over the thermal instability limit for a rebrightening to occur. See papers by Patterson et al. (1998), Buat-Menard & Hameury (2002), Schreiber & Gansicke (2001), Osaki, Meyer & Meyer-Hofmeister (2001) and Matthews et al. (2005) for a discussion of the physical reasons for echo-outbursts. Recent CCD-V photometry manifests that SDSS J1021 has a magnitude of 19.m72±0.07 and 19.m59 ± 0.07 as of 06 March and 07 March, 2007 (HJD = 2454165.80 and HJD = 2454167.74) respectively, at Sonoita Research Observatory (Sonoita, Arizona, USA) using a robotic 0.35 meter telescope equipped with an SBIG STL-1001XE CCD camera. Spectroscopic observations were carried out on November 21.8 UT with the CCD spec- 4 IBVS 5763 Figure 2. Left: Power spectrum, revealing the Psh of SDSS J1021; Right: Superhump profile of SDSS J1021 Table 2. Times of superhump maximums HJD E O-C σ(O−C) 2454061.03380 0 0 0.00120 2454061.88103 15 0.00228 0.00130 2454061.93507 16 -0.00001 0.00368 2454061.99121 17 -0.00020 0.00099 2454062.89325 33 0.00056 0.00179 2454062.94709 34 -0.00193 0.00214 2454063.00533 35 -0.00002 0.00156 2454063.62385 46 -0.00113 0.00464 trograph mounted on the 1.01-m telescope of Bisei Astronomical Observatory (Japan). The preliminary discussion of the spectra can be found in (Ayani & Kato, 2006). The spectral range is 400-800nm, and the resolution is 0.5 nm at Hα. HR 3454 (α2000 = 08h43m13.s475; δ2000 = +03 ◦23′55.′′18) was observed for flux calibration of the spectra. Standard IRAF routines were used for data reduction. Spectrum (Fig. 3) shows blue continuum and Balmer absorption lines (from Hǫ to Hα) together with K CaII 3934 in absorption. Very weak HeI 4471, Fe 5169, NII 5767 absorption lines may be present. Hǫ 3970 is probably blended by H Ca II 3968. The FeIII 5461 line resembles weak P-Cygni profile. Noteworthy, FeIII 5461 and NII 5767 may be artifacts caused by imperfect subtraction of city lights: HgI 5461 and 5770 (spectrum of the sky background which was subtracted, is available upon request). The HeI 5876 line (mentioned for this object in Rau et al., 2006) is not detectable on our spectrum. It is remarkable that Hα manifests a ”W-like” profile: an emission component embedded in the absorption component of the line. Table 3 represents EWs (equivalent widths) of detected spectral lines. EW was calcu- lated by direct numerical integration over the area under the line profile. The archive photographic plates from the Main Astronomical Observatory Wide Field Plate Archive (Kyiv, Ukraine; MAO hereafter) and Plate Archive of Sternberg Astro- nomical Institute of Moscow State University (Moscow, Russia; SAI hereafter) and plate from Crimean Astrophysical Observatory archive (Ukraine) were carefully scanned and inspected for previous outbursts on the plates dating from 1978 to 1992 from MAO, 1913 IBVS 5763 5 Table 3. Equivalent widths of spectral lines Line EW [Å] K CaII 3934 -5.8 Hǫ 3970 / H CaII 3968 -8.7 Hδ 4101 -6.4 Hγ 4340 -8.5 Hβ 4861 -6.4 Hα 6563 -7.7 Hα 6563 (emission) 2.3 HeI 4471 -0.95 FeII 5169 -0.65 NII 5767 -0.7 - 1973 from SAI and 1948 from CrAO archives. The number of plates from each archive is 22 for SAI, 6 for MAO and 1 for CrAO archives. For all plates the magnitude limit was determined (this data as well as scans of plates are available upon request). The selection of plates from MAO archive was done with the help of the database developed by L.K. Pakuliak, which is accessible at http://mao.kiev.ua/ardb/ (Sergeeva et al., 2004; Pakuliak, L.K. & Sergeeva, T.P., 2006;). No outbursts on the selected plates from the MAO, SAI and CrAO archives were detected. This implies that outbursts in SDDS J1021 are rather rare, which is typical for the WZ Sge-type stars. 4000 6000 8000 H / HCaII K CaII H H HFeIII NIIFeII Wavelength Figure 3. Spectra of SDSS J1021 obtained on November 21.8 UT on 1.01-m telescope of Bisei Astronomical Observatory (Japan) Table 4 (available only electronically from IBVS server or via AAVSO ftp-server at ftp://ftp.aavso.org/public/calib/varleo06.dat) represents BV RcIc photometric calibration of 52 stars in SDSS J1021 vicinity, which have a V-magnitude in the range of 11.m21-17.m23 and can serve as a comparison stars. Calibration (by AH8) was done at Sonoita Research Observatory (Arizona, USA). The large amplitude of the SDSS J1021 outburst of 7m, superhumps with a period 6 IBVS 5763 below the ”period gap”, rebrightening during the declining stage of superoutburst, rarity of outbursts and obtained spectrum allow to classify this object as a WZ Sge type dwarf nova. Acknowledgements: AG is grateful to Aaron Price (AAVSO, MA, USA) for his great help and useful discussions during the preparation of this manuscript. Authors are thankful to A. Zharova and L. Sat (both affiliated at SAI MSU, Moscow, RUSSIA) for the assistance on dealing with SAI Plate Archive and to V. Golovnya for the help concerning MAO Plate Archive (Kyiv, Ukraine). It is a great pleasure to express gratefulness to Dr. N. A. Katysheva, Dr. S. Yu. Shugarov (SAI MSU both) and Dr. D. Bisikalo (Institute of Astronomy RAS, Moscow, Russia) for useful discussions concerning the nature of SDSS J1021. IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. References: Adelman-McCarthy J. et al., 2007, submitted to ApJ Supplements Ayani, K. & Kato, T., 2006, CBET, 753, 1. Edited by Green, D.W.E. Bisikalo D.V. et al., 2006, Chinese Journal of Astronomy and Astrophysics, Supplement, 6, 159 Buat-Menard, V. & Hameury, J.-M., 2002, A&A, 386, 891 Christensen, E.J., 2006, CBET, 746, 1. Edited by Green, D.W.E. Golovin A. et al., 2005, IBVS No. 5611 Imada A. et al., 2006, PASJ, 58, L23 Kato, T., 2002, PASJ, 54, L11 Lasota, J.P., Hameury, J.M., Hure, J.M., 1995, A&A, 302, L29 Marsakova V., Andronov, I.L., 1996, Odessa Astronom. Publ., 9, 127 Matthews, O.M. et al., 2005, ASPC, 330, 171, in The Astrophysics of Cataclysmic Vari- ables and Related Objects, Eds. J.-M. Hameury and J.-P. Lasota. San Francisco: Astronomical Society of the Pacific Osaki, Y., Meyer, F. & Meyer-Hofmeister, E. 2001, A&A, 370, 488 Osaki, Y., & Meyer, F., 2002, A&A, 383, 574 Osaki, Y., & Meyer, F., 2003, A&A, 401, 325 Osaki, Y., 2003, PASJ, 55, 841 Pakuliak, L.K. & Sergeeva, T.P., 2006, in Virtual Observatory: Plate Content Digitiza- tion, Archive Mining and Image Sequence Processing, Eds.: Tsvetkov, M., et al., Sofia, p.129 Patterson, J., et al., 1996, PASP, 108, 748 Patterson, J., et al., 1998, PASP, 110, 1290 Patterson, J., et al., 2002, PASP, 114, 721 Pavlenko, E., et al., 2007, In Proc. of the 15th European White Dwarf Workshop ”EU- ROWD06”, in press. Rau, A., et al., 2006, The Astronomer’s Telegram, No. 951 Schreiber, M.R. & Gansicke, B.T., 2001, A&A, 375, 937 Sergeeva, T.P., et al., 2004, Baltic Astronomy, 13, 677 Templeton M. R. et al., 2006, PASP, 118, 236 Waagen, Elizabeth O., 2006, AAVSO Special Notice, #25

0704.1277

Discrete phase space and minimum-uncertainty states

DISCRETE PHASE SPACE AND MINIMUM-UNCERTAINTY STATES William K. Wootters and Daniel M. Sussman Department of Physics, Williams College Williamstown, MA 01267, USA The quantum state of a system of n qubits can be represented by a Wigner function on a discrete phase space, each axis of the phase space taking values in the finite field F2n . Within this framework, we show that one can make sense of the notion of a “rotationally invariant state” of any collection of qubits, and that any such state is, in a well defined sense, a state of minimum uncertainty. 1. INTRODUCTION A quantum state cannot be squeezed down to a point in phase space. But there are quantum states that closely approximate classical states, such as the coherent states of a harmonic oscillator. One characterization of the coherent states is based on the Wigner function: they are the only states for which the Wigner function is both strictly positive and rotationally symmetric around its center (here we assume a specific scaling of the axes appropriate for the given oscillator). One can also express the quantum mechanics of discrete systems in terms of phase space. In this paper we consider a system of n qubits described in the framework of Ref. [1], in which the discrete phase space can be pictured as a 2n × 2n array of points. In this framework, the discrete Wigner function preserves the tomographic feature of the usual Wigner function, but the points of the discrete phase space are defined abstractly and do not come with an immediate physical interpretation. As in the continuous case, a point in discrete phase space is illegal as a quantum state: it holds too much information. But one can ask whether there are quantum states that, like coherent states, approximate a phase-space point as closely as possible. We would like to identify such states and thereby to give more physical meaning to the discrete phase space. In this paper we focus primarily on the second of the two properties mentioned above: invariance under rotations. We will see that one can make sense of this notion in the discrete space and that rotationally invariant states exist for any number of qubits. The most interesting property of these states is that they minimize uncertainty in a well defined sense. The product ∆q∆p, where q and p are position and momentum, has no meaning in our setting because our variables have no natural ordering. We therefore express uncertainty in information-theoretic terms, specifically in terms of the Rényi en- tropy of order 2 (which we call simply “Rényi entropy” for short). Moreover we consider not just the “axis variables,” but also variables associated with all the other directions in the discrete phase space. (In the continuous case these other directions would be associ- ated with linear combinations of q and p.) We will find that each rotationally invariant state minimizes the Rényi entropy, averaged over all these variables. This will leave us with the question of picking out a “most pointlike” of the rotationally invariant states, if such a notion can be made meaningful; we address this question briefly in the conclusion. http://arxiv.org/abs/0704.1277v1 2. DISCRETE PHASE SPACE Over the years there have been many proposals for generalizing the Wigner function to discrete systems. (See, for example, Refs. [2, 3] and papers cited in Ref. [1].) Here we adopt the discrete Wigner function proposed by Gibbons et al. [1], which is well suited to a system of qubits. The basic idea is to use, instead of the field of real numbers in which position and momentum normally take their values, a finite field with a number of elements equal to the dimension d of the state space. There exists a field with d elements if and only if d is a power of a prime; so this approach applies directly only to quantum systems, such as a collection of qubits, whose state-space dimension is such a number. The two-element field F2 is simply the set {0, 1} with addition and multiplication mod 2, but the field of order 2n with n larger than 1 is different from arithmetic mod 2n. For example, F4 consists of the elements {0, 1, ω, ω + 1}, in which 0 and 1 act as in F2 and arithmetic involving the abstract symbol ω is determined by the equation ω2 = ω + 1. The discrete phase space for a system of n qubits is a two-dimensional vector space over F2n ; that is, a point in the phase space can be expressed as (q, p), where q and p, the discrete analogues of position and momentum, take values in F2n . In this phase space it makes perfect sense to speak of lines and parallel lines; a line, for example, is the solution to a linear equation. The key idea in constructing a Wigner function is to assign a pure quantum state, represented by a one-dimensional projection operator Q(λ), to each line λ in phase space. The only requirement imposed on the function Q(λ) is that it be “translationally covariant.” This means that if we translate the line λ in phase space by adding a fixed vector (q, p) to each point, the associated quantum state changes by a unitary operator T(q,p) associated with (q, p). The unitary translation operator T(q,p) is defined to be T(q,p) = X q1Zp1 ⊗ · · · ⊗XqnZpn, (1) whereX and Z are Pauli operators and qi and pi, which are elements of F2, are components of q and p when they are expanded in particular “bases” for the field: e.g., q = q1b1 + · · ·+ qnbn, where (b1, . . . , bn) is the basis chosen for the coordinate q. 1 One finds that the requirement of translational covariance severely constrains the construction: 1. States assigned to parallel lines must be orthogonal. A complete set of parallel lines, or “striation,” consists of exactly d lines; so the states associated with a given striation constitute a complete orthogonal basis for the state space. In other words, each striation is associated with a complete orthogonal measurement on the system. 2. The bases associated with different striations must be mutually unbiased. That is, each element of one basis is an equal-magnitude superposition of the elements of any of the other bases. There are exactly d+1 striations, so this construction generates a set of d+1 mutually unbiased bases. (Such a set is just sufficient for the complete tomographic reconstruction of an unknown quantum state.) Despite these constraints, there are many allowed functions Q(λ). This implies that there are many possible definitions of the Wigner function for a system of qubits, because 1The bases for q and p cannot be chosen independently: each must be proportional to the dual of the other [1]. once we have chosen a particular assignment of quantum states to phase-space lines, the Wigner function of any quantum state is uniquely fixed by the requirement that the sums over the lines of any striation be equal to the probabilities of the outcomes of the corresponding measurement. 3. ROTATIONALLY INVARIANT STATES In the finite field, consider a quadratic polynomial x2+ax+ b that has no roots. Then the equation q2 + aqp+ bp2 = c, (2) with c taking all nonzero values in F2n , defines what we will call a set of “circles” centered at the origin. Fixing the values of a and b—this is somewhat analogous to fixing the scales of the axes in the continuous case—we define a rotation to be any linear transformation of the phase space that leaves each circle invariant.2 (We consider only rotations around the origin. A state centered at the origin can always be translated to another point by T(q,p).) For example, in the two-qubit phase space, our circles can be defined by the equation q2 + qp+ ωp2 = c, (3) and an example of a rotation is the transformation R defined by ω + 1 ω . (4) One can check that this particular rotation has the property that if we apply it repeatedly, starting with any nonzero vector, it generates the entire circle on which that vector lies. In this sense R is a primitive rotation. With every unit-determinant linear transformation L on the phase space, one can associate (though not uniquely) a unitary transformation U on the state space whose action by conjugation on the translation operators T(q,p) mimics the action of L on the corresponding points of phase space [5, 1].3 One can show that every rotation has unit determinant and must therefore have an associated unitary transformation. For example, for the rotation R given above, if we expand both q and p in the field basis (b1, b2) = (ω, ω+1), the following unitary transformation acts in the desired way on the translation operators: 1 i i −1 i 1 −1 i 1 i −i 1 −i −1 −1 i . (5) Thus just as ω + 1 ω ω + 1 , (6) 2A different notion of rotation has been used in Ref. [4]. 3The argument in Appendix B.3 of Ref. [1] contains an error: Eqs. (B24) and (B25) implicitly assume that the chosen field basis is self-dual, which is not in fact the case. However, the proof can be repaired by starting with a self-dual basis to get those equations, and then changing to the actual basis via the argument of Appendix C.1. That there exists a self-dual basis for F2n is proved in Ref. [6]. we have that UT(1,0)U † = U(X ⊗X)U † = iX ⊗ (XZ) ∝ T(1,ω+1). (7) For any number n of qubits, let R be a primitive rotation, and let U be a unitary transformation associated with R in the above sense. (Techniques for finding U can be found in Refs. [1, 5].) Then from the action of U on the translation operators, it follows that U acts in a particularly simple way on the mutually unbiased bases associated with the striations of phase space: starting with any one of these bases, repeated applications of U generate all the other bases cyclically. That there always exists a unitary U generating a complete set of mutually unbiased bases for n qubits has been shown by Chau [5]. In our present context, we will reach the same conclusion by showing, in the following paragraph, that there always exists a primitive rotation. The existence of such a unitary matrix U leads naturally to a simple prescription for choosing the function Q(λ): (i) Use the translation operators to assign computational basis states to the vertical lines. (ii) Apply U repeatedly to these states, and R repeatedly to the lines, in order to complete the correspondence. This prescription results in a definition of the Wigner function that is “rotationally covariant,” in the sense that when one transforms the density matrix by U , the values of the Wigner function are permuted among the phase-space points according to R. How does one find a primitive rotation R? First, for any number of qubits, there always exists a primitive polynomial of the form x2+x+b [7], which one can use to define circles by the equation q2 + qp+ bp2 = c. Then the linear transformation is guaranteed to cycle through all the nonzero points of phase space [8], and it always takes circles to other circles. Raising L to the power d − 1 gives us a unit-determinant transformation that preserves circles and is indeed a primitive rotation. Moreover, one can write R explicitly in terms of b: R = Ld−1 = b−1 b−1 + 1 . (9) With Q(λ) chosen in the way we have prescribed, the eigenstates of U are our ro- tationally invariant states. When we apply U to any state, the Wigner function simply flows along the circles in accordance with the rotation R. But an eigenstate of U does not change under this action, so its Wigner function must be constant on each circle. 4. MINIMIZING ENTROPY Consider again our complete set of d+ 1 mutually unbiased bases, and let |ij〉 be the jth vector in the ith basis. These vectors together have the following remarkable property: for any pure state |ψ〉, the probabilities pij = |〈ψ|ij〉| 2 satisfy [9, 10] p2ij = 2. (10) Now consider the Rényi entropy HR = − log2 of the outcome-probabilities of the ith measurement when applied to the state |ψ〉. This entropy is a measure of our inability to predict the outcome of the measurement. The average of HR over all the mutually unbiased measurements can be bounded from below [11]: 〈HR〉 = − log2 ≥ − log2 = log2(d+ 1)− 1, with equality holding only if the Rényi entropy is constant over all the mutually unbiased measurements.4 Now, for any of the rotationally invariant states defined in the last section, the Rényi entropies associated with the d + 1 mutually unbiased measurements are indeed equal. By the inequality (11), such states therefore minimize the average Rényi entropy over all these measurements, that is, over all the directions in phase space. 5. EXAMPLES The one-qubit case is very simple. The three mutually unbiased bases generated in our construction are the eigenstates of the Pauli operators X , Y , and Z. It is not hard to find a unitary transformation that cycles through these three bases. Such a transformation rotates the Bloch sphere by 120◦ around the axis (x, y, z) = (1, 1, 1). The two eigenstates of this unitary transformation, which are the eigenstates of X + Y + Z, are rotationally invariant: each of their Wigner functions is constant on the only circle in the 2× 2 phase space. And each of these states minimizes the average Rényi entropy for the measurements X , Y , and Z. It is interesting to note that one of these two states has a positive Wigner function. Clearly there is nothing intrinsically special about these two states. They are special only in relation to the three measurements X , Y , and Z, which are associated with the three striations of the phase space. But in the context of quantum cryptography, the entropy-minimization property is quite relevant. In the six-state scheme (in which the signal states are the eigenstates of X , Y , and Z), if Eve chooses to eavesdrop by making a complete measurement on certain photons, her best choice is to make a measurement whose outcome-states are entropy-minimizing in our sense: it turns out that such a choice minimizes Eve’s own Rényi entropy about Alice’s bit. An interesting example comes from the 3-qubit case. The relevant field is F8, which can be constructed from F2 by introducing an element b that is defined to satisfy the equation b3 + b2 + 1 = 0. In our 8 × 8 discrete phase space, we can define circles via the equation q2 + qp+ p2 = c, (12) 4The analogous inequality in terms of Shannon entropy was proved in Refs. [12, 13]. where c can take any nonzero value. A primitive rotation preserving these circles is5 b3 b6 b6 b5 . (13) One finds that of the eight eigenvectors of any unitary U corresponding to R, all of which are rotationally invariant, exactly one has a positive Wigner function for a specific, fixed function Q(λ) associated with U . This state is also easy to describe physically. For a particular choice of U , it is of the form |ψ〉 = 1/3|+++〉+ 2/3| − −−〉, (14) where |+〉 and |−〉 are the two eigenstates (with a specific relative phase) of the oper- ator X + Y + Z. If we regard |ψ〉 as analogous to a coherent state at the origin, then the coherent-like states at the 63 other phase-space points can be obtained from |ψ〉 by applying Pauli rotations to the individual qubits. The Wigner function of each of these states has the value 0.319 at its center, the largest value possible for any three-qubit state. 6. CONCLUSION We have found that one can make sense of the notion of rotational invariance in a discrete phase space for a system of n qubits. The rotationally invariant states are in this respect analogous to the energy eigenstates of a harmonic oscillator, but the analogy is not perfect. Our rotationally invariant states are all states of minimum uncertainty with respect to the various directions in phase space, whereas except for the ground state, the harmonic oscillator eigenstates do not have this property (the uncertainty, even in our Rényi sense, increases with increasing energy). We have considered the further restriction to positive Wigner functions but so far have found examples of such states only for a single qubit and for three qubits. However, for any number of qubits, one can show that at least one of our rotationally invariant states takes a value at its center equal to the maximum value attainable by the Wigner function of any state. Perhaps this latter property, rather than positivity, should be taken as the defining feature of a “most pointlike” state. REFERENCES [1] K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, Phys. Rev. A 70, 062101 (2004). [2] J. H. Hannay and M. V. Berry, Physica D 1, 267 (1980). [3] S. Chaturvedi et al., Pramana 65, 981 (2005). [4] A. B. Klimov, C. Muñoz and J. L. Romero, quant-ph/0605113. [5] H. F. Chau, quant-ph/0212055 (2004); IEEE Trans. Inf. Theory 51, 1451 (2005). 5Even though Eq. (12) is not of the form we used in reaching Eq. (8), in that it is not based on a primitive polynomial, the matrix R is nevertheless a primitive rotation. http://arxiv.org/abs/quant-ph/0605113 http://arxiv.org/abs/quant-ph/0212055 [6] A. Lempel, SIAM J. Computing 4, 175 (1975). [7] O. Moreno, J. Comb. Theory A 51, 104 (1989). [8] R. Lidl and H. Niederreiter, Finite Fields, 2nd edition (Cambridge Univ. Press, 1997), Thm. 3.16 (p. 84) and Thm. 8.28 (pp. 408-409). [9] U. Larsen, J. Phys. A 23, 1041 (1990). [10] A. Klappenecker and M. Rötteler, Proceedings of the 2005 IEEE International Sym- posium on Information Theory (ISIT’05), p. 1740 (2005). [11] M. A. Ballester and S. Wehner, Phys. Rev. A 75, 022319 (2007). [12] I. D. Ivanovic, J. Phys. A 25, L363 (1992). [13] J. Sánchez, Phys. Lett. A 173, 233 (1993). Introduction DISCRETE PHASE SPACE ROTATIONALLY INVARIANT STATES MINIMIZING ENTROPY EXAMPLES CONCLUSION

0704.1278

Turbulent Mixing in the Surface Layers of Accreting Neutron Stars

ACCEPTED FOR PUBLICATION IN THE ASTROPHYSICAL JOURNAL Preprint typeset using LATEX style emulateapj v. 08/22/09 TURBULENT MIXING IN THE SURFACE LAYERS OF ACCRETING NEUTRON STARS ANTHONY L. PIRO1 AND LARS BILDSTEN Kavli Institute for Theoretical Physics, Kohn Hall, University of California, Santa Barbara, CA 93106; [email protected], [email protected] Accepted for publication in The Astrophysical Journal ABSTRACT During accretion a neutron star (NS) is spun up as angular momentum is transported through its surface layers. We study the resulting differentially rotating profile, focusing on the impact this has for type I X-ray bursts. The predominant viscosity is likely provided by the Tayler-Spruit dynamo, where toroidal magnetic field growth and Tayler instabilities balance to support a steady-state magnetic field. The radial and azimuthal components have strengths of ∼ 105 G and ∼ 1010 G, respectively. This field provides a Maxwell stress on the shearing surface layers, which leads to nearly uniform rotation at the depths of interest for X-ray bursts (near densities of ≈ 106 g cm−3). A remaining small shear transmits the accreted angular momentum inward to the NS interior. Though this shear gives little viscous heating, it can trigger turbulent mixing. Detailed simulations will be required to fully understand the consequences of mixing, but our models illustrate some general features. Mixing has the greatest impact when the buoyancy at the compositional discontinuity between accreted matter and ashes is overcome. This occurs at high accretion rates, at low spin frequencies (when the spin is small, the relative speed of the accreted material is larger), or may depend on the ashes from the previous burst. We then find two new regimes of burning. The first is ignition in a layer containing a mixture of heavier elements from the ashes. If ignition occurs at the base of the mixed layer, recurrence times as short as ∼ 5 − 30 minutes are possible. This may explain the short recurrence time of some bursts, but incomplete burning is still needed to explain these bursts’ energetics. When mixing is sufficiently strong, a second regime is found where accreted helium mixes deep enough to burn stably, quenching X-ray bursts. We speculate that the observed change in X-ray burst properties near one-tenth the Eddington accretion rate is from this mechanism. The carbon-rich material produced by stable helium burning would be important for triggering and fueling superbursts. Subject headings: accretion, accretion disks — stars: magnetic fields — stars: neutron — X-rays: bursts — X-rays: stars 1. INTRODUCTION As neutron stars (NSs) in low mass X-ray binaries accrete material from their companions they are expected to be spun up by this addition of angular momentum, possibly becoming millisecond pulsars (Bhattacharya & van den Heuvel 1991). This suspicion has received support by the discovery of ac- cretion driven millisecond pulsars (Wijnands & van der Klis 1998), as well as the millisecond oscillations seen during type I X-ray bursts (Chakrabarty et al. 2003), the unstable ignition of the accumulating fuel (for reviews, see Lewin et al. 1995; Strohmayer & Bildsten 2006; Galloway et al. 2006). The ma- jority of the rotational kinetic energy of the accreted material is dissipated at low densities (∼ 10−4 − 10−1 g cm−3) in the boundary layer (for example, Inogamov & Sunyaev 1999). Nevertheless, angular momentum must be transported into the NS interior if it is to be spun up, even if at times the angular momentum could be radiated as gravitational waves (Bildsten 1998b). This transport implies a non-zero, albeit small, shear throughout the outer liquid parts of the NS. Such shearing may lead to viscous heating as well as chemical mixing at depths far below the low density boundary layer where the majority of the shearing occurs. Such a picture was previously investigated by Fujimoto (1988, 1993) for both accreting NSs and white dwarfs. His studies demonstrated that the shearing profile of a NS could be large enough that mixing through baroclinic instabilities 1 Current address: Astronomy Department and Theoretical Astrophysics Center, 601 Campbell Hall, University of California, Berkeley, CA 94720; [email protected] may be important at the depths of type I X-ray bursts. Such a result is compelling because it may help to explain some of the remaining discrepancies between the theory and observa- tions of these bursts. Though a simple limit cycle picture has been successful in qualitatively explaining the bursts’ primary characteristics, including their energies (1039 − 1040 ergs), re- currence times (hours to days), and durations (∼ 10 − 100 s) (Fujimoto et al. 1981; Bildsten 1998a), outstanding problems still remain (Fujimoto et al. 1987; van Paradijs et al. 1988; Bildsten 2000). Chief among these is the critical accretion rate required for stable accumulation (which we discuss in more detail in §6). Another problem is the occurrence of suc- cessive X-ray bursts with recurrence times as short as ∼ 10 minutes (Galloway et al. 2006, and references therein), for which there has been speculation that this could be due to mixing during accretion (Fujimoto et al. 1987). In this present work we reassess the importance of angular momentum transport and the resulting mixing. We find that the hydrodynamic instabilities studied by Fujimoto (1993) are insufficient to prevent strong shearing of magnetic fields. This leads to generation of the Tayler-Spruit dynamo (Spruit 2002), where toroidal field growth is balanced by Tayler in- stabilities to create a steady-state magnetic field, which pro- vides a torque on the shearing layers that is larger than any purely hydrodynamic mechanism. The result is nearly uni- form rotation and little viscous heating, too little to affect either X-ray bursts or superbursts (thermonuclear ignition of ashes from previous X-ray bursts, Cumming & Bildsten 2001; Strohmayer & Brown 2002). Turbulent mixing is found to http://arxiv.org/abs/0704.1278v1 2 PIRO & BILDSTEN be non-negligible and in some cases may mix fresh mate- rial with the ashes of previous bursts. The key is whether the strong buoyancy due to the larger density of the ashes be- low can be overcome. When mixing occurs, we find two new regimes of burning for an accreting NS. The first is that fuel can be mixed down to depths necessary for premature unsta- ble ignition. The timescale for ignition of such bursts is short enough (∼ 5−30 minutes) to explain the short recurrence time bursts (Galloway et al. 2006). The second is that if material is mixed to sufficiently large depths (and therefore tempera- tures) it can burn stably, ceasing X-ray bursts altogether. Such a mechanism may be responsible for the quenching of X-ray bursts seen at surprisingly low accretion rates for many atoll class NSs (Cornelisse et al. 2003). We derive an analytic for- mula that estimates what spins and accretion rates necessary to transfer between these two regimes (eq. [77]). 1.1. The Basics of Angular Momentum Transport Before beginning more detailed analysis, it is useful to present some general equations for angular momentum trans- port. This demonstrates why we expect the NS interior to be shearing and explicitly relates the torque from the accreting matter to this shear. Figure 1 schematically shows the ex- pected rotation profile from the accretion disk, through the boundary layer, and into the NS interior. Material accreted at a rate Ṁ reaches the NS surface with a nearly Keplerian spin frequency ofΩK = (GM/R 3)1/2 = 1.4×104 s−1 M 1.4 R where M1.4 ≡ M/1.4M⊙ and R6 ≡ R/10 6 cm, which has a ki- netic energy per nucleon of ≈ 200 MeV nuc−1. The major- ity of this energy is dissipated in a boundary layer of thick- ness HBL ≪ R (as studied by Inogamov & Sunyaev 1999) and never reaches far into the NS surface. Nevertheless, angular momentum is being being added at a rate of Ṁ(GMR)1/2, so that a torque of this magnitude must be communicated into the NS. This implies a non-zero shear rate in the interior liq- uid layers, down to the solid crust. In the present work we are interested in the shear at the depths where X-ray bursts ignite, near ρ≈ 106 g cm−3, which is well below the boundary layer. Though we focus on a magnetic mechanism for angular mo- mentum transport, we expect the boundary layer and densities up to ≈ 5×103 g cm−3 to be dominated by hydrodynamic in- stabilities (which we discuss in more detail in §3). The pressure scale height at the depth of X-ray bursts is H ≈ 30 cm ≪ R, which allows us to use a Cartesian coordi- nate system with z as the radial coordinate. This is far below the depths considered by Inogamov & Sunyaev (1999) when they investigated the uneven covering of the NS surface by accreted fuel. Furthermore, all of the transport mechanisms we consider work most efficiently in directions perpendic- ular to gravity (because no work is performed), thus it is a good approximation to consider the surface layers as concen- tric spheres, each with constant Ω. Transfer of angular mo- mentum is reduced to a one-dimensional diffusion equation (Fujimoto 1993) (R2Ω) = , (1) where Ω is the NS spin frequency and ν is the viscosity. The total time derivative is given by d/dt = ∂/∂t +Vadv∂/∂z, where Vadv is the advecting velocity of the fluid in an Eulerian frame. In steady-state, we take ∂/∂t = 0 and set Vadv = −Ṁ/(4πR2ρ) FIG. 1.— A diagram demonstrating the rotation profile of material from the accretion disk, through the boundary layer, and into the NS. Material reaches the NS surface with a Keplerian spin frequency, ΩK, the majority of which is dissipated within the boundary layer (gray region) with thickness HBL ≪ R. Nevertheless, angular momentum is still being added to the NS, and a torque must be communicated through the NS. This implies a small, non-zero, amount of shear throughout the NS interior. so that (R2Ω) = . (2) Integrating from the surface where the spin is ΩK (due to the disk) down to a depth z where the local spin is Ω, and assum- ing dΩ/dz = 0 at the surface, − ṀR2ΩK + ṀR2Ω = −4πρνR4dΩ/dz. (3) Taking the limit Ω≪ ΩK we find, 4πR3ρνqΩ = ṀR2ΩK, (4) where q ≡ d logΩ/d logz is the shear rate. This equation shows that q > 0 when angular momentum is transported in- ward. In general, we will find that q is rather small (. 1) at the depths of interest. Nevertheless q is large enough to acti- vate instabilities that help to transport angular momentum, as well as mix material. 1.2. Outline of Paper We begin by comparing and contrasting some well-known hydrodynamic instabilities in §2 and estimate the resulting shear rates. In §3, we discuss the consequences that this shear- ing has on a magnetic field, which motivates implementation of the Tayler-Spruit dynamo. In §4, we calculate accumulat- ing NS envelopes without the effects of viscous angular mo- mentum transport. This enables us to judge when such effects must be incorporated. We calculate models including mixing in §5. We conclude in §6 with a summary of our results and a discussion of type I X-ray burst and superburst observations. 2. HYDRODYNAMIC VISCOSITY MECHANISMS In the following sections we discuss hydrodynamic insta- bilities. This is not an exhaustive survey (for further details, see Heger et al. 2000), but rather is meant to summarize those instabilities that are most crucial to our problem, so as to set the context for the magnetic transport mechanism we study later. TURBULENT MIXING IN ACCRETING NEUTRON STARS 3 2.1. Kelvin-Helmholtz Instability The Kelvin-Helmholtz instability (also referred to as the dy- namical shear instability) is governed by the Richardson num- , (5) where N is the Brunt-Väisälä frequency, which is composed of contributions from thermal and compositional buoyancy, N2 = N2T + N µ. (6) The thermal buoyancy is given by N2T = ∇ad − d lnT d lnP , (7) where g = GM/R2 = 1.87× 1014 cm s−2M1.4R 6 is the surface gravitational acceleration (ignoring the effects of general rela- tivity), χQ = ∂ lnP/∂ lnQ, with all other intensive variables set constant, ∇ad = (∂ lnT/∂ lnP)ad is the adiabatic temperature gradient, the asterisk refers to derivatives of the envelope’s profile, and the pressure scale height is, H = P/ρg = 33.1 cm µ−11.33T8, (8) where T8 ≡ T/10 8 K. For these analytic estimates we assume an ideal gas equation of state and use a pure helium composi- tion with a mean molecular weight of µ = 1.33µ1.33. We omit the scalings with mass and radius to simplify presentation. The compositional buoyancy is (Bildsten & Cumming 1998) N2µ = − d lnµe d lnP d lnµi d lnP , (9) where µe (µi) is the mean molecular weight per electron (ion). Linear analysis shows that Kelvin-Helmholtz instability oc- curs when Ri < 1/4, which develops into strong turbulence that readily transports angular momentum. This result as- sumes that thermal diffusion can be ignored for the unstable fluid perturbations, in other words, that the perturbations are adiabatic. Fluid perturbations with a characteristic size L and speed V become non-adiabatic when the timescale for thermal diffu- sion, L2/K, where K is the thermal diffusivity, is less than the timescale of the perturbation, L/V . The ratio of these two timescales is the Péclet number, Pe ≡ VL/K (Townsend 1958). The restoring force provided by thermal buoyancy is weakened when Pe < 1, which requires the substitution of N2T → PeN T and promotes instability for regions where NT > Nµ. Thermal diffusion is most efficient at small length- scales, which motivates setting LV/νk = Rec (Zahn 1992), where νk is the kinematic viscosity and Rec is the critical Reynolds number for turbulence, which is of order 1000. This gives the Péclet number approximately related to the Prandtl number, Pr, by Pe ≈ RecPr. The turbulent perturbations are thus non-adiabatic when (Zahn 1992) K > νkRec. (10) In the non-degenerate surface layers the kinematic viscosity is dominated by ions, and has a value of (Spitzer 1962) νk = 1.4× 10 −3 cm2 s−1ρ−16 T 8 , (11) where ρ6 ≡ ρ/10 6 g cm−3, and we assume a Coulomb loga- rithm of lnΛ = 20. Setting K = 16σSBT 3/(3cpκρ 2), where σSB the Stefan-Boltzmann constant, cp the specific heat, and κ the opacity, the thermal diffusivity is K = 48.8 cm2 s−1µ1.33κ 0.04ρ 8 , (12) where we approximate cp = 5kB/2µmp and scale the opac- ity to κ0.04 ≡ κ/0.04 cm 2 g−1 (the opacity is largely given by electron scattering, but is decreased by degeneracy effects, see Paczyński 1983; Bildsten 1998a). Substituting equations (11) and (12) into equation (10), we find that the perturbations are non-adiabatic at depths of ρ . 3× 107 g cm−3 T 8 . The new “secular” Richardson number associated with this limit is, Ris ≡ νkRec . (13) When Ris < 1/4, the so-called “secular shear instability” arises. The competing effects of accretion increasing q versus tur- bulence developing when Ris < 1/4 (and decreasing q) drive the surface layers toward marginally satisfying Ris = 1/4 (as- suming at this moment that the sole viscous mechanism is the Kelvin-Helmholtz instability). This expectation is borne out in the white dwarf studies of Yoon & Langer (2004). Thus we can trivially estimate the q due to this mechanism. The thermal buoyancy is (Bildsten 1998a) = 9.2× 105 s−1µ 1.33T 8 . (14) We substitute Ris = 1/4 into equation (13), and assuming Rec = 1000, solve for a shear rate of qKH = 223 κ 0.04ρ 0.1, (15) where Ω0.1 = Ω/0.1ΩK. A shear rate this large would promote prodigious viscous heating, as well as ample mixing, but as we soon show, such a large shear is prohibited by other insta- bilities. 2.2. Baroclinic Instability Another important hydrodynamic instability that has been studied extensively for accreting degenerate stars is the baro- clinic instability (Fujimoto 1993), which we quickly sum- marize here. The interested reader should consult Fujimoto (1987, 1988) for further details (also see the discussion in Cumming & Bildsten 2000). The baroclinic instability arises because surfaces of con- stant pressure and density no longer coincide if hydrostatic balance is to be maintained when differential rotation is present. In such a configuration, fluid perturbations along nearly horizontal directions are unstable, though with a suffi- cient radial component to allow mixing of angular momentum and material. The instability can roughly be broken into two limits, depending on a critical baroclinic Richardson number (Fujimoto 1987), RiBC ≡ 4 = 8.5× 103 µ1.33T 0.1. (16) When Ri > RiBC, Coriolis effects limit the horizontal scale of perturbations. This results in two parametrizations for viscos- 4 PIRO & BILDSTEN ity estimated from linear theory (Fujimoto 1993), νBC = Ri1/2 H2Ω, Ri ≤ RiBC, Ri3/2 H2Ω, Ri > RiBC, where a factor of order unity is usually included in these pre- scriptions, called αBC, to account for uncertainty in how lin- ear theory relates to the saturated amplitudes of the instability. For simplicity, we set αBC = 1 in our analysis below. By substituting νBC into the angular momentum equation (eq. [4]), we solve for the shearing profile. Since we are interested in how the shear relates to the bursting properties, it useful to write these results in terms of the local accretion rate ṁ = Ṁ/(4πR2), which is typically parametrized in terms of the local Eddington rate ṁEdd = (1 + X)RσTh 1.5× 105 g cm−2 s−1 (1 + X)R6 , (18) where X is the hydrogen mass fraction and σTh is the Thomson cross-section. The Richardson number in each case is then 2.2× 103 µ 1.33 ρ6T 0.1Ω0.1, Ri ≤ RiBC 4.4× 103 µ 1.33 ρ 0.1 Ω 0.1 , Ri > RiBC where ṁ0.1 = ṁ/0.1ṁEdd and we have used the hydrogen deficient (X = 0) value for ṁEdd (eq. [18]). The transi- tion to the case Ri & RiBC occurs roughly at depths ρ & 4× 106 g cm−3 T 8 . The shear rate for each case is qBC = 1.33ρ 0.1 Ω 0.1 , Ri ≤ RiBC 9.9 µ 1.33ρ 0.1 Ω 0.1 , Ri > RiBC This demonstrates that generally qBC ≪ qKH, so that the baro- clinic instability triggers before the Kelvin-Helmholtz insta- bility. This prevents the shear rate from ever becoming large enough for the Kelvin-Helmholtz instability to operate at depths of ρ& 6× 104 g cm−3. 2.3. Other Hydrodynamic Instabilities In addition to the two instabilities described above, there are a number of other possibilities including, but not limited to, Eddington-Sweet circulation (von Zeipel 1924a,b; Baker & Kippenhahn 1959), Solberg-Høiland insta- bility (Wasiutyński 1946; Tassoul 1978; Endal & Sofia 1978), Goldreich-Schubert-Fricke instability (Goldreich & Schubert 1967; Fricke 1968) and Ekman pumping (Pedlosky 1987). At this time we avoid assessing each of these individually. As we show below, magnetic fields are likely to dominate and are in- teresting since they have received the least attention in past works. 3. THE IMPORTANCE OF MAGNETIC EFFECTS Given the larger than order unity shear rates derived above, we estimate the consequences this has for a magnetic field. The point we want to emphasize is that even a reasonably small field will be wrapped by the shear flow until it becomes dynamically important. Assuming shellular rotation, a component of radial field is stretched to have a toroidal component, Bφ = nBr, where n is the number of differential turns, given as n = qΩt, and t is the duration of the shearing. The toroidal field growth is very fast. For the Kelvin-Helmholtz case, Bφ ∼ Br in merely ∼ 10−6 s (∼ 10−4 s for the baroclinic case). As the toroidal field becomes larger it exerts an azimuthal stress on the shearing layer equal to BrBφ/4π, which can be written as an effective viscosity, νe, = ρνeqΩ. (21) In a timescale t ≈ H2 /νe this torquing significantly decreases the spin of the layer, where HΩ = dz/d lnΩ = R/q is the shear- ing scale height. Setting Bφ = qΩtBr, we solve for the critical initial radial field needed to affect the shearing, Br,crit = (4πρ) 1/2 HΩ = (4πρ)1/2 , (22) so that t is basically the Alfvén travel time through a shearing scale height. Using ρ≈ 106 g cm−3 as burning density and t ≈ 1 hr, a fiducial timescale for accumulation, our estimates for qKH and qBC imply Br,crit ∼ 10 4 G and ∼ 105 G, respectively Any initial field larger than this gets so wound up that Lorentz forces alter the shearing profile. It is possible that the intrinsic magnetic field of the NS may be large enough that magnetic stresses never allow the shear rates to become so large. Spruit (1999) argued that this de- pends on the “rotational smoothing time,” the timescale for non-axisymmetric components of the magnetic field to be ex- pelled as differential rotational brings field with opposite po- larities together. This timescale is estimated as 3R2π2 ηΩ2q2 , (23) where η is the magnetic diffusivity, which in the non- degenerate limit is given by (Spitzer 1962) η ≈ 0.7 cm2 s−1T 8 . (24) If tΩ is larger than the Alfvén travel time through the layer (given by t in eq. [22]), then there is sufficient time for mag- netic torques to act and the shearing is merely a perturbation on the magnetic field that is quickly damped away. On the other hand, if tΩ is sufficiently small then the shear dominates and the magnetic field is made axisymmetric on a timescale tΩ. Substituting tΩ into equation (22), we estimate the critical field, below which shear dominates, BΩ = (4πρ) q−1/3 = 1.3× 107 G ρ 0.1 q −1/3. (25) The lack of persistent pulsations from accreting NSs implies a dipole field strength . 5× 107 G (Piro & Bildsten 2005). We therefore consider it plausible that they may have intrin- sic magnetic fields < BΩ. In this case, we expect a very axisymmetric magnetic field to be created in a timescale of tΩ ∼ 200 s, which is then wrapped until it becomes dynami- cally important. TURBULENT MIXING IN ACCRETING NEUTRON STARS 5 3.1. The Tayler-Spruit Dynamo As the toroidal magnetic field continues to wrap, it becomes increasingly important to the dynamics of the shearing and is also subject to magnetohydrodynamic instabilities. The com- bination of these effects have been shown to give rise to the Tayler-Spruit dynamo (Spruit 2002). In this picture, shear- ing grows the toroidal field, which then initiates Tayler in- stabilities (non-axisymmetric, pinch-like instabilities includ- ing stratification, Tayler 1973; Spruit 1999). This turbulently creates poloidal field components that once again shear to be toroidal. This cycle continues, creating a steady-state field. The minimum shear needed for this process to operate can be argued simply. (See Spruit 2006 for a mathematical deriva- tion that uses the dispersion relation from Acheson 1978.) We note that Denissenkov & Pinsonneault (2007) have recently given an alternate prescription for this same mechanism using solely heuristic arguments. Since their results have not been shown consistent with a more rigorous mathematical analysis we consider Spruit’s conclusions more reliable at this time. A vertical perturbation, lz, is limited by buoyancy forces to be (eq. [6] from Spruit 2002) lz < RωA/N, (26) where ωA = B/[(4πρ) 1/2R] is the Alfvén frequency. At small lengthscales, magnetic diffusion damps out perturbations. In the limit of Ω ≫ ωA, which we are considering, the Tayler instability growth rate is σB = ω A/Ω, so that l2z > η/σB = ηΩ/ω A. (27) Combining these two relations gives the minimum ωA needed for the dynamo to act, . (28) During the timescale for Tayler instability, σ−1B , Br is stretched into Bφ by an amount Bφ = σ B qΩBr. (29) The largest amplification is achieved for magnetic fields that extend the largest radial lengthscale available, so that assum- ing equation (26) is marginally satisfied along with the induc- tion equation we find Br/Bφ = lz/R = ωA/N. Combining this with equation (29) we obtain q = (N/Ω)(ωA/Ω). Substituting this into equation (28), we find (Spruit 2002) qmin = as the minimum shear needed for the Tayler instability to op- erate. Though this result is consistent with more rigorous analysis (Spruit 2006), it should be viewed with some caution as we apply it to accreting NSs. The thin shell geometry we consider is quite different than the spherical geometries typi- cally used when invoking the Tayler-Spruit dynamo. Simula- tions by Braithwaite (2006) demonstrate that the Tayler insta- bility is strongest along the rotation axis, which is only real- ized at the poles in the NS case. Since we find the dynamo to be so much stronger than any hydrodynamic transport mech- anisms, we consider it to be a reasonable approximation for these magnetic effects, even if its strength is decreased due to geometry. The value of N used to evaluate equation (30) depends on what is supplying the buoyancy. We follow Spruit (2002) and separately consider cases of Nµ ≫ NT and Nµ ≪ NT , denoted as case 0 and 1, respectively. In case 1 non-adiabatic effects become important when η/K < 1, and we must take N2T → (η/K)N2T (analogous to the above analysis of the secular shear instability). For this case, which dominates for most of the envelope, we find qmin = 0.10 κ 0.04ρ 0.1 . (31) Both qKH and qBC are considerably above this, thus the Tayler- Spruit dynamo activates long before the onset of purely hy- drodynamic instabilities. By assuming that equation (28) is marginally satisfied Spruit (2002) derived the steady-state field strengths. We summarize the relevant prescriptions needed for our study. The steady-state azimuthal and radial field components are Bφ0 = (4πρ) 1/2RqΩ2/Nµ, (32) Br0 = q Bφ0, (33) for case 0. For case 1, Bφ1 = (4πρ) 1/2RΩq1/2 )1/8( , (34) Br1 = )1/4( Bφ1. (35) The effective viscosities, as defined by equation (21), are ν0 = R , (36) ν1 = R )1/2( . (37) Although these viscosities are appropriate for angular mo- mentum transport, mixing of material is less efficient since it requires expending work to exchange fluid elements (versus just exerting shear stresses). The mixing diffusivities are D0 = R , (38) D1 = R )3/4( . (39) which is just equal to the effective turbulent magnetic diffu- sivity. In the Appendix we show that these prescriptions are consistent with energy conservation considerations. 3.2. Shearing Profile The Tayler-Spruit dynamo transports angular momentum, causes viscous heating, and mixes material. We follow the procedure we used for the baroclinic case and assume steady- state angular momentum transport. This is a good approxima- tion when the viscous timescale, tvisc = H 2/ν, is less than the timescale of accretion. For these initial estimates we focus on the NT ≫ Nµ limit (case 1) since this dominates except at compositional boundaries (which we revisit in §4). Substitut- ing ν1 into equation (4), the shear rate for the Tayler-Spruit 6 PIRO & BILDSTEN dynamo is qTS = )1/2( . (40) Scaling to values appropriate for accreting NSs, qTS = 0.38 κ 0.04T 8 ṁ0.1Ω 0.1 . (41) Note the scalings with ṁ and Ω. At high ṁ angular momen- tum is fed into the star faster, creating more shear. At low Ω the shear is greater because of the larger relative angular speed between the accreted material and the NS. These are generic feature we expect for any viscous mechanism (com- pare the scaling of qTS with qKH and qBC from eqs. [15] and [20], respectively). The viscosity that gives the small- est shear rate will likely be the most important at a given depth. Using this criterion, we find that the Tayler-Spruit dy- namo is dominant for densities ρ & 3 g cm−3 T or ρ & 5 × 103 g cm−3 for T ≈ 5 × 106 K. At shallower depths, Kelvin-Helmholtz instabilities damp the shear, which is consistent with the use of hydrodynamic instabilities by Inogamov & Sunyaev (1999) for understanding the initial spreading of accreted material in the boundary layer. The steady-state magnetic field components are Bφ = 1.3× 10 10 G κ 0.04 ρ 0.1 Ω 0.1 , (42) Br = 2.1× 10 5 G κ 0.04 ρ 0.1 Ω 0.1 . (43) Cumming & Bildsten (2000) argue that if Br & 10 6 G it would become dynamically important to determining the drift of X- ray burst oscillations (Muno et al. 2002). The interaction of such fields with the shearing of an X-ray burst from an accret- ing millisecond pulsar has been explored by Lovelace et al. (2007). At high accretion rates, Br increases and comes close to this limit, suggesting that these magnetic fields may be important for understanding the dynamics of X-ray burst os- cillations, even from NSs that do not show persistent pulsa- tions (which are preferentially seen at high accretion rates, Muno et al. 2004). 3.3. Viscous Heating Viscous shearing heats the surface layer, which can also be thought of as the rate of magnetic energy destruc- tion as the dynamo builds and destroys magnetic field (Maeder & Meynet 2004). The heating rate per unit mass is ν (qΩ)2 , (44) which for the Tayler-Spruit dynamo becomes ǫTS = 5.6× 10 10 ergs g−1 s−1κ 0.04ρ 0.1 . To put this number into perspective we express it in terms of the energy released per accreted nucleon. This is found by multiplying the above result by the total mass per unit area down to depth of interest, y ≈ Hρ (the column depth), and dividing by ṁ. We then find d lny = 0.13 keV nuc−1µ−11.33κ 0.04T 8 ṁ0.1Ω 0.1 . (46) In comparison, the thermal energy per nucleon at 108 K is ≈ 10 keV nuc−1 and burning of helium into carbon releases ≈ 0.6 MeV nuc−1. We conclude that viscous processes do not heat the layer sufficiently to alter X-ray bursts. 3.4. Turbulent Mixing It is not immediately clear how much of an observational impact is provided by the magnetic fields and viscous heating estimated above. In contrast, as we shall show, shear mixing produced by the dynamo could have important ramifications for the structure and composition of the surface layers. For this reason, we devote most of the remainder of our study to investigating the consequences of mixing. We first consider some estimates that highlight mixing’s im- portance. For this, we parametrize the mixing diffusivity as αTSD, where D is given by equations (38) or (39) and αTS is a factor of order unity that accounts for uncertainties in the Tayler-Spruit prescription. The features we find are general enough that other viscosities can be incorporated by increas- ing or decreasing αTS. The material becomes fully mixed over a scale height, H, in a time tmix = H 2/(αTSD), which gives for case 1 (NT ≫ Nµ), tmix = 4.3× 10 2 s α−1TSµ 1.33κ 0.04ρ 6 T8ṁ 0.1 , (47) Accretion is also advecting material downward, which hap- pens in a timescale tacc = y/ṁ ≈ ρH/ṁ, tacc = 2.2× 10 3 s µ−11.33ρ6T8ṁ 0.1. (48) The ratio of these timescales is = 0.19 α−1TSµ 1.33κ 0.04ρ 0.1 . (49) The scaling with ρ shows that mixing is generally more im- portant than advection at shallow depths (ρ. 3×107 g cm−3), while the scaling with Ω shows that mixing is more important for slower spinning NSs (as expected from our discussion of shear rates in §3.2). 4. ACCUMULATING, NON-MIXED MODELS We now use numerical calculations to consider the angu- lar momentum transport through the surface layers. In this section we calculate accumulating models without directly in- corporating the effects from mixing. This verifies many of the analytic estimates derived above and motivates when mixing must be included (which is done for models in §5). 4.1. Shear Profile Calculations We calculate the envelope profile for helium accumulating on an iron ocean (which represents the iron-peak ashes from previous X-ray bursts). As discussed in §2.2, we approxi- mate the surface as having a constant gravitational accelera- tion, and plane-parallel geometry. In addition to using z as our radial coordinate, we find it useful to use the column depth, y, defined as dy = −ρdz, giving a pressure P = gy from hy- drostatic balance. We solve for ρ using the analytic equa- tion of state from Paczyński (1983). For the liquid phase, when 1 ≤ Γ≤ 173 where Γ = (4πni/3) 1/3Z2e2/(kBT ), Z is the charge per ion, and ni is the ion density, we include the ionic free energy of Chabrier & Potekhin (1998). During the accumulating phase, a negligible amount of helium burning takes place, so we assume that the flux is constant and set by heating from the crust. Previ- ous studies of accreting NSs have shown that the in- terior thermal balance is set by electron captures, neu- TURBULENT MIXING IN ACCRETING NEUTRON STARS 7 tron emissions, and pycnonuclear reaction in the in- ner crust (Miralda-Escudé et al. 1990; Zdunik et al. 1992; Bildsten & Brown 1997; Brown & Bildsten 1998; Brown 2000, 2004) which release ≈ 1 MeV/mp ≈ 10 18 ergs g−1 (Haensel & Zdunik 1990, 2003). Depending on the accre- tion rate and thermal structure of the crust, this energy is ei- ther conducted into the core or released into the ocean such that for an Eddington accretion rate up to ≈ 92% of the en- ergy is lost to the core and exits as neutrinos (Brown 2000). We therefore set the heating to 150 keV nuc−1, giving a flux 2.2× 1021 ergs cm−2 s−1〈ṁ〉0.1, where 〈ṁ〉0.1 is the time aver- age accretion rate in units of 0.1ṁEdd and the average is over timescales of order the thermal time of the crust (many years). For simplicity we assume 〈ṁ〉 = ṁ. Recent calculations by Gupta et al. (2006) suggest that heating is stronger than previ- ously thought, but not sufficiently high to qualitatively change our results. We ignore the additional flux from compressional heating because it only contributes ∼ cpT ∼ 10 keV nuc (Bildsten 1998a). In a one-zone estimate ignition occurs at the base of the helium layer when dǫcool , (50) (Fujimoto et al. 1981) where ǫ3α is the heating rate from triple-α reactions (for which we use the rate from Fushiki & Lamb 1987 for numerical calculations), ǫcool = 4σSBT , (51) and the derivatives are both taken at constant pressure. We consider models where the column of accreted helium yacc that is just at this ignition point. This allows the maximum amount of time for angular momentum redistribution during the accu- mulating phase. The He/Fe boundary is assumed sharp, since the timescale for diffusion between the layers is much longer than the timescale of accumulation (Brown et al. 2002). We solve for the temperature profile by integrating the ra- diative diffusion equation, 16σSBT . (52) The opacity is set using electron-scattering (Paczyński 1983), free-free, and conductive opacities (Schatz et al. 1999). There is a sharp change in opacity at the He/Fe boundary (because the high Z of the iron makes it more opaque), which means that the top of the iron layer is convectively unstable. The characteristic convective velocity estimated from F ≈ ρV 3conv is Vconv ∼ 10 5 cm s−1, which is much less than the local sound speed of ∼ 108 cm s−1. This means that the convection is very efficient, and we therefore simply set (d lnT/d lnP)∗ = ∇ad in the convective region. This convective region has not been noted in previous studies, though a super-adiabatic tem- perature gradient is apparent in the figures shown in both Brown et al. (2002) and Cumming (2003). It is not yet clear how this impacts the bursting properties of NSs, and we delay exploring this in detail for a future study. Nevertheless, we must include convection so as to have accurate differentially rotating profiles. In Figure 2 we plot the temperature profile, K and η, and N for three accumulating models. The magnetic diffusivity is set using the conductivity from Schatz et al. (1999). The base of FIG. 2.— Accumulating models for accretion rates of 0.1ṁEdd (dotted line), 0.3ṁEdd (short-dashed line), and 1.0ṁEdd (solid line). For each model the base of the helium layer is taken to be at the unstable triple-α ignition depth where dǫ3α/dT = dǫcool/dT (thick long-dashed line in top panel). The top panel plots the temperature, and the solid circles mark the top and bottom of the convective zone. The middle panel plots the thermal diffusivity, K, and magnetic diffusivity, η. The bottom panel shows the Brunt-Väisälä fre- quency N for both thermal (NT , lines) and compositional (Nµ , solid squares) contributions. the accumulating helium is set where unstable helium ignition occurs, as designated by the thick long-dashed line in the top panel (eq. [50]). The convective zone begins just below this and is bracketed by solid circles at its top and bottom. The convective zone is also seen in the plot of N since NT is effec- tively zero in this region. The change in composition at the He/Fe boundary gives a large buoyancy contribution, which we estimate as ∆ lnµ , (53) where ∆ lnµ ≈ 0.44 is the logarithmic change in the mean molecular weight at the boundary. We denote this by a solid square for each model in the bottom panel. For the majority of the profile the Tayler-Spruit dynamo is given by case 1 (NT ≫ Nµ). Since K ≫ η, the perturbations creating the dynamo are non-adiabatic for case 1, as we described in §3.1. The only place case 0 (Nµ ≫ NT ) is important is at the He/Fe boundary. We next solve for the shear rates by solving equation (4) with either ν0 or ν1, depending on which case is appropriate. We assume angular momentum transport does not affect the thermal or compositional structure, in other words, that the transport is just happening “in the background.” This allows us to assess which effects are crucial for subsequent iterations that include angular momentum transport in the actual struc- ture calculation. 8 PIRO & BILDSTEN FIG. 3.— Angular momentum transport in the NS surface layers, for a spin of Ω = 0.1ΩK , and accretion rates of 0.1ṁEdd (dotted line), 0.3ṁEdd (dashed line), and 1.0ṁEdd (solid line). The panels display (from top to bottom) the viscous timescale for angular momentum transport across a scale height, tvisc, the shear rate, q, and the viscous energy deposition per logarithm column, dE/d lny. The squares indicate the corresponding values due to the compo- sitional discontinuity at the base of the accumulating layer. In Figure 3 lines denoted the profiles calculated assuming case 1 of the Tayler-Spruit dynamo. We assume that the con- vection is effectively instantaneous in transporting material and angular momentum, thus all the quantities become very small in the convective region. This is appropriate since the convective overturn timescale, H/Vconv ∼ 10 −4 s, is much less than the timescales for mixing or accretion. The squares in- dicate the corresponding values due to the compositional dis- continuity found by using case 0. Since the viscous timescale, tvisc = H 2/ν, is much less than the timescale it took to accrete to this base column tacc = yacc/ṁ ∼ 10 3 − 105 s (for the range of ṁ considered) our steady-state assumption is valid. These calculations highlight the importance of the composi- tional jump since tvisc, q, and dE/d lny are all amplified here. This is because the large buoyancy reduces angular momen- tum transport across this boundary. The viscous time is nearly independent of accretion rate in regions where NT ≫ Nµ be- cause the viscosity in case 1 is independent of q. The energy deposition is always much smaller than the heat coming the crust, and it falls off somewhat faster with depth than expected from the estimate presented in §3.3 due to electron degener- acy effects decreasing NT . New heat sources at a depth of ≈ 1012 g cm−2 could ease the difficulty calculations have in recreating the low ignition columns needed to explain super- burst recurrence times (Cumming et al. 2006), but this viscous heating is not nearly enough to correct this problem. In Figure 4 we plot the magnetic fields found within the radiative zones for the models from Figure 3. We assume FIG. 4.— Azimuthal and radial field components for a spin of Ω = 0.1ΩK . Lines have the same meaning as in Fig. 3. that within the convective zone the dynamo is not able to operate, and do not calculate a magnetic field here. It is also interesting to compare these fields to those derived by Cumming et al. (2001), who calculated the steady-state fields expected when Ohmic diffusion balances advection through accretion. Their principal result was that the steady-state hori- zontal field drops by ≈ ṁ/0.02ṁEdd orders of magnitude from the crust up through the ocean. In contrast, the magnetic fields we find are nearly constant with depth. We therefore do not expect these fields to persist if the accretion ceases for a time and instead to be expelled on an Ohmic diffusion time (∼ days near the top of the ocean). When the dynamo is ac- tive, the steady-state is reached quickly enough that Ohmic diffusion can be ignored. In Figure 5 we compare what happens as the spin is changed by plotting Ω = 0.03, 0.1, and 0.3ΩK (67, 220, and 670 Hz), all for ṁ = 0.1ṁEdd. Note that the shearing is most dramatic at smaller Ω. In fact, the shearing and heating profiles are very sensitive to the value of Ω, as was demonstrated by the ana- lytic estimates. We do not plot the associated magnetic field for these models since we have already plotted some exam- ples in Figure 4 and the analysis of §3.2 provides adequate estimates. These plots of q show that very little shearing is present. To emphasize this fact, in Figure 6 we plot the actual spin frequency as a function of depth found by integrating q for a range of accretion rates. The discontinuity at the He/Fe boundary complicates this estimate. Noting that qΩd lnr = − d lny, (54) we approximate the spin change at this boundary as ∆Ω≈ qΩH/R. (55) The change of spin is generally . 0.5 Hz across the accumu- lating layer, with the majority of the spin change occurring at the compositional boundary. The layer is very nearly in uniform rotation. TURBULENT MIXING IN ACCRETING NEUTRON STARS 9 FIG. 5.— Same as Fig. 3, but for spins of 0.03ΩK (solid line), 0.1ΩK (dotted line), and 0.3ΩK (dashed line), all for ṁ = 0.1ṁEdd . FIG. 6.— The spin frequency as a function of column depth for Ω = 0.1ΩK , with ṁ = 0.1 (dashed line), 0.3 (dotted line), and 1.0ṁEdd (solid line). The dot-dashed line shows a constant spin frequency for comparison. 4.2. Mixing and the Compositional Barrier The above calculations confirm that the viscosity is too large for either appreciable shearing or viscous heating. As demonstrated in §3.4, mixing should be important, but Fig- ures 3 and 5 argue that we must take into account the large Nµ at the He/Fe boundary. Assuming that Nµ scales like equation (53), we estimate Nµ = 1.6× 10 6 s−1 µ 1.33T 8 (∆ lnµ/0.44) 1/2. (56) We substitute this into ν0 to solve for q using equation (4), which is used to estimate a mixing timescale at the boundary, tmix = H 2/(αTSD0) = 1.7× 103s α−1TSµ 1.33 ρ 0.1 Ω0.1 ∆ lnµ Using equation (48) we find a ratio of ≈ 0.77 α−1TSµ 1.33 ρ 0.1 Ω0.1 ∆ lnµ Unlike in equation (49), this new ratio depends on ṁ. This is because the viscosity in case 0 has a different dependence on q. Now as ṁ is increased tmix decreases faster than tacc. Above a critical accretion rate of ṁcrit,1 = 4.6× 10 −2 ṁEdd α 1.33ρ6T ∆ lnµ mixing can no longer be ignored. For densities and tempera- tures expected at the base of the accumulating layer, this crit- ical accretion rate lies in the range of 0.1 − 1.0ṁEdd. Note that this depends very strongly on the spin rate, ṁcrit,1 ∝ Ω 3, thus we expect the slower spinning NSs to be considerably more affected by mixing. To test these analytic estimates we compare the mixing and accretion timescales in Figure 7. From the top panel to the bottom panel we increase ṁ (fixing Ω = 0.1ΩK). At low ṁ, the He/Fe boundary at yacc (shown by the filled squares) acts as a barrier to mixing since tmix > tacc at this depth. When this happens it is a good approximation to ignore mixing and assume two separate layers during the accumulation phase. As ṁ increases, tmix at yacc becomes less and less until fi- nally tmix < tacc, so that material should be mixed past yacc. When this occurs, our accumulating model can no longer ig- nore mixing. The mixing between helium and iron occurs down to a depth where tmix is equal to the length of time ac- cretion has been taking place, tacc = yacc/ṁ. The key point we want to emphasize is that because of the buoyancy barrier, the effect of mixing turns on abruptly, and when it does, mixing will occur well past yacc. 5. THE EFFECTS OF TURBULENT MIXING Once tmix < tacc at the buoyancy barrier, the compositional profile of the NS is very different, which we now explore. We treat the mixing as complete, which we diagram in Fig- ure 8 and summarize here. Material accretes at a rate ṁ for a time tacc with a helium mass fraction Y0, supplying a col- umn of material yacc = ṁtacc. The total column of helium that accretes during this time is therefore Y0yacc. Mixing causes this newly accreted material to mix past yacc to the mixing depth ymix defined as where tmix = tacc. The helium is fully mixed down to this depth, resulting in a diluted mass fraction Ymix = Y0yacc/ymix within the mixed layer. In the following sections we consider two scenarios that can result from the mixing: (1) unstable ignition of the mixed fuel, and (2) stable burning when the material mixes to sufficient depths. 5.1. Numerical Calculations of Mixed Ignition 10 PIRO & BILDSTEN FIG. 7.— Comparison of the mixing timescale, tmix (solid line) versus the accretion timescale tacc (dashed line) for ṁ = 0.1,0.3, and 1.0ṁEdd (top to bottom panel; all with Ω = 0.1ΩK and αTS = 1). The solid square denotes tmix due to the compositional discontinuity. At low ṁ, the squares are above the dashed line, demonstrating that tmix > tacc at the base of the accumulating layer, which prevents mixing to larger depths. At sufficiently high ṁ, tmix < tacc at depths below the accumulation depth, so that mixing between helium and iron can occur. FIG. 8.— Diagram demonstrating the main features of turbulent mixing. Material mixes down to where tmix = tacc, which defines the depth ymix. The total amount of material that accretes is yacc = ṁtacc , giving a total accreted column of helium of Y0yacc. Within the mixed layer the helium mass fraction is diluted to new mass fraction of Ymix = Y0yacc/ymix . As the column of accreted material grows, it can still reach the correct conditions for unstable ignition, but mixing causes two changes: (1) the ignited layer has a diluted helium frac- tion, Ymix < Y0, and (2) the ignition takes place at the base of the mixed layer, at a depth ymix > yacc, resulting in a re- currence time for ignition much less than when mixing is not included. Both of these effects are easiest to explore using a semi- analytic model. In this section all calculations use Y0 = 1. We consider other values of Y0 for our analytic estimates in the next section. We solve for the mixed accumulating structure by first assuming an amount of accretion yacc, which for a given ṁ implies an accretion time yacc/ṁ. We integrate the radiative diffusion equation (eq. [52]) down to a depth where tmix = tacc giving ymix, where we assume a constant flux pro- file as was discussed in §4 since little helium burning takes place during accumulation. We then estimate the mixed he- lium fraction as Ymix = yacc/ymix. This estimate is improved by substituting Ymix back into our envelope integration, and iter- ating until Ymix converges. The non-helium component in the layer is taken to be iron. In Figure 9 we plot the resulting profiles for a NS accreting at ṁ = 0.1ṁEdd and Ω = 0.1ΩK. In the four panels we consider values of yacc of 10 6, 3× 106, 107, and 4× 107 g cm−2 (from left to right and then up to down, denoted by the filled circles), which is meant to mimic the accumulation of fuel on the NS surface. For each integration the envelope profile continues down to a depth ymix, which also gives Ymix as displayed in each panel. The last model reaches the conditions necessary for ignition at the base of the mixed layer. The recurrence time for these mixed-ignition models is much less than for those without mixing. For the plotted model, the recurrence time is trec = 4× 10 7 g cm−2/1.5× 103 g cm−2 s−1 ≈ 44 minutes. In contrast the model shown in Figure 2 with the same accretion rate of ṁ = 0.1ṁEdd has trec ≈ 1.5 days! The shorter recur- rence time is not only due to mixing carrying helium down to deeper depth, but also a change in the thermal profile. Since significant iron is mixed up into the accumulating material, the free-free opacity, which scales κff ∝ Z 2/A, where Z and A are the charge and mass per nucleon, respectively, is now the dominate opacity mechanism. The accumulating layer is more opaque and therefore hotter for a given flux in compari- son to the pure-helium models considered before, which con- tributes to the shallow ignition depths. We calculate trec for a grid of models with various ṁ and Ω in Figure 10. The recurrence time is shorter for stronger mixing, which occurs at high ṁ or low Ω, and can be as short as ≈ 5 − 30 minutes. We warn though that all of these cal- culations assume that complete mixing can occur, and as we already showed in §4, buoyancy may prevent this (eq. [59]). Nevertheless, it is interesting to calculate the mixed-ignition conditions for a wide range of parameter space because the conditions left from previous bursts may vary, and quantities such as ∆ lnµ may be smaller at times if, for example, there is incomplete burning in a previous burst. At sufficiently high ṁ or low Ω the ignition takes place at high enough temperatures that the envelope does not ignite unstably. We consider this case in more depth in the following section. The short recurrence times that we find are similar to some seen for multiple bursts (Galloway et al. 2006, and references therein). However, our model cannot explain the energetics of these bursts. The energetics are typically quantified in a distance independent measure, the so-called α-value, which TURBULENT MIXING IN ACCRETING NEUTRON STARS 11 FIG. 9.— The four panels show how the fully mixed accumulating layer evolves in time until it reaches conditions necessary for unstable ignition. The parameters of the NS are ṁ = 0.1ṁEdd and Ω = 0.1ΩK . In each panel, the column of helium that has been accreted is denoted by a filled circle, which is from left to right, and up to down yacc = 106 , 3 × 106, 107, and 4× 107 g cm−2. Mixing takes place down to the column reached by the thin solid line. The mixed helium fraction, Ymix is displayed in the upper left-hand corner of each panel. The ignition curve associated with each Ymix is shown as a thick dashed line. FIG. 10.— The recurrence time for mixed-ignition models as a function of ṁ. The symbols denoted different spins, as shown in the key. Models that are at sufficiently high ṁ or low Ω do not ignite unstably, and thus are not plotted. is the ratio of energy released in the persistent emission be- tween bursts to the energy of the burst itself. For pure helium ignition α∼ 100. In contrast, the α-values for the short recur- rence time bursts are typically ∼ 10 (Boirin et al. 2007). Since the only helium that will burn in our mixed ignition models is that accreted since the last outburst, we always find α ∼ 100. Therefore we still require an additional nuclear energy source, such as incomplete burning from the previous X-ray burst, to explain such a low α-value. This in fact may not be a problem because incomplete burning may naturally explain the short FIG. 11.— The mixed helium mass fraction, Ymix as a function of ṁ and Ω. The crosses are mixed ignition models. The filled circles are for stably burn- ing models and give the amount of helium present in the steady-state mixing and burning layer. Steady-state burning therefore require either high ṁ or low Ω. The scaling of Ymix ∝ ṁ −0.21 is derived in equation (69) and is consis- tent with these numerical results. Without mixing all of the these considered models ignite unstably since stable accretion requires ṁ & 10ṁEdd . recurrence times, since this would lead to smaller composi- tional gradients and therefore stronger mixing. We plot the Ymix as a function of ṁ and Ω in Figure 11. The crosses correspond to models that ignite unstably. The filled circles are stable accreting models that are discussed on §5.3 and §5.4. The implications of this mixed ignition for the un- stable burning during the X-ray burst can be easily tested by more sophisticated numerical simulations by just considering a mixed accumulating column, for example, by artificially set- ting Y ≈ 0.1 − 0.6 as shown in this Figure 11. Since viscous heating is negligible, these initial tests do not need to resolve the shearing profiles. 5.2. Analytic Estimates of Mixed Ignition We now estimate the properties of the mixed ignition mod- els. These solutions directly show how mixing depends on the properties of the accumulating layer, in particular the pref- actor αTS and Y0, without having to consider a multitude of models. The effects of a free-free opacity are important in deriving the correct atmospheric conditions. To include this analytically, we use the free-free opacity from Schatz et al. (1999), simplified to a one-component plasma and with the dimensionless Gaunt factor set to unity, κff ≈ 3.77 cm 2 g−1 , (60) where we have estimated µe ≈ 2 as is correct within 10% for the any of the elements of interest. Integrating the radiative diffusion equation (eq. [52]), assuming a constant flux of F = 1021 ergs cm−2 s−1F21 and an ideal gas equation of state, the temperature as a function of column y is T (y) = 1.8× 108 K µ1.33F21Z )2/17 8 , (61) 12 PIRO & BILDSTEN where y8 ≡ y/10 8 g cm−2. Using equation (47), we find the mixing timescale as a function of y, tmix = 3.5× 10 3 s α−1TSµ −0.44 )−0.19 ṁ−10.1Ω 0.1 y Note that tmix ∝ y 1.37 is a higher power than tacc = y/ṁ ∝ y. This explicitly shows that mixing dominates at lower y, but accretion always wins at some depth. Setting tmix equal to tacc = yacc/ṁ gives the depth where mixing can extend to for a given column of accreted material yacc, ymix(yacc) = 1.6× 10 8 g cm−2α0.73TS µ )0.14 −0.55 0.1 y acc,8, (63) where yacc,8 = yacc/10 8 g cm−2. The mixed helium fraction down to this depth is Ymix(yacc) = Y0yacc ymix(yacc) = 0.63 α−0.73TS Y0µ −0.32 1.33 (F21Z 2/A)−0.14Ω0.550.1 y acc,8. Since Ymix ∝ y acc the strength of mixing decreases (Ymix gets larger) as more material accreted, which was demonstrated by the four panels in Figure 9. We next estimate what ignition depth is expected for this fully mixed accumulating layer. The energy generation rate for triple-α burning is approximated as ǫ3α = 5.3× 10 23 ergs g−1 s−1 f , (65) where f is factor that accounts for screening effects. To make progress analytically we expand the exponential as −44/T8 ≈ 7.95× 10−10(T8/2.1) 21. Using our tempera- ture profile (eq. [61]), the condition that dǫ3α/dT = dǫcool/dT (eq. [50]) implies an ignition depth of yign = 9.4× 10 7 g cm−2 f −0.15µ−0.571.33 F −0.13 ×(Z2/A)−0.28Y −0.44mix . (66) Setting yign = ymix from equation (63), we solve for yacc, the critical column of material that must be accreted to cause ig- nition. This is then substituted back into equations (63) and (64) to find that ignition occurs at a depth ymix,ign = 1.2× 10 8 g cm−2 α0.38TS f −0.13µ−0.331.33 Y −0.38 −0.039 ×(Z2/A)−0.17Ω−0.290.1 . (67) with a composition of Ymix,ign = 0.57 α −0.86 −0.047µ−0.561.33 Y −0.21 ×(Z2/A)−0.25Ω0.650.1 . (68) Comparing this with the numerical calculation is easiest if we assume F is set by ṁ (F21 = 2.2ṁ0.1) as well as scaling Z 12 and µ ≈ 2.1 as appropriate for the iron-rich composition. This gives Ymix,ign ≈ 0.20 α −0.86 TS ṁ −0.21 0.1 Ω 0.1 . (69) The recurrence time is trec = Ymix,ignymix,ign/ṁ, resulting in trec ≈ 950 s α −0.48ṁ−1.250.1 Ω 0.1 . (70) FIG. 12.— Diagram demonstrating the main features of steady-state mix- ing and burning. Material can mix further down to where tmix = t3α, which defines the depth ymix. During a timescale tmix(ymix) the amount of material which has accreted is yacc = ṁtmix(ymix). Within the mixed layer the helium mass fraction is diluted to new mass fraction of Ymix = Y0yacc/ymix. Equations (69) and (70) confirm the scalings found for the numerical calculations in Figures 10 and 11. We have plotted the Ymix ∝ ṁ −0.21 scaling in Figure 11 to emphasize this. These analytic results also show how strongly these results depend on the parameterαTS, for which Ymix,ign is especially sensitive. 5.3. Steady-State Mixing and Burning Another possibility is that the helium is mixed and burned by triple-α reactions in steady-state, leading to stable burning. The basic idea is similar to that described above for mixed ignition, except now the depth of the accumulating layer is set by the helium burning timescale, t3α = E3α/(Yǫ3α), where E3α = 5.84× 10 17 ergs g−1 is the energy per mass released from this burning. As shown in Figure 12, material is mixed to sufficient depths where t3α (which decreases with depth) is equal to tmix (which increases with depth), which defines a mixing (or burning) depth ymix = y(tmix = t3α). During a mixing timescale, the amount of material that is accreted is yacc = ṁtmix(ymix), so that the total column of helium that has been accreted is Y0ṁtmix(ymix). This is diluted over a depth ymix, so that the mixed helium fraction is Ymix = Y0yacc/ymix. This is all occurring in steady-state, material moves through the mixed layer at a rate ṁ, but this layer does not move up or down in pressure (column) coordinates, as the burning is stable. The basic results of steady-state mixing and burning are best shown using a simple numerical model. The equation that describes helium continuity, including depletion by triple- α burning, becomes, in the plane-parallel limit (Fujimoto 1993), . (71) We assume changes in ρ and D with depth are small in com- parison to changes in Y , and, following our derivation of the angular momentum equation in §2.2, we take the steady state TURBULENT MIXING IN ACCRETING NEUTRON STARS 13 limit to derive . (72) In the limit where tacc ≫ tmix that we are interested in, the term on the left-hand side can be dropped. Finally, making the approximation that ρDd/dz ≈ y/tmix, we find . (73) This equation mimics the properties we expect from mixing. When mixing is strong tmix/t3α ≪ 1, and dY/dy≈ 0, the com- position does not change with depth. At the depth where tmix ≈ t3α, dY/dy ≈ −Y/y and the helium is depleted expo- nentially. All the helium burns into carbon, so that carbon has a mass fraction X12 = 1 −Y . The envelope profiles are found from simultaneously in- tegrating three differential equations: (1) radiative transfer, equation (52), (2) the entropy equation, dF/dy = −ǫ3α, and (3) continuity of helium, equation (73). We integrate using a shooting method, but first we must set three boundary con- ditions for the flux, temperature, and helium mass fraction. Since all of the accreted helium must burn if the envelope is in steady-state, we set the surface flux to F = Y0E3αṁ + Fc, where Fc = 150 keV nuc −1〈ṁ〉 (as discussed in §4.1). The surface temperature is set from the radiative zero solution (Schwarzschild 1958). The helium abundance in the mixed region, Ymix, is an eigenvalue. It is varied until shooting gives the correct base flux of Fc. This is easily found through itera- tion since when Ymix is set too large, too much burning occurs, and the base flux is too small (and vice versa for small Ymix). In Figure 13 we plot a steady-state envelope using Y0 = 1, ṁ = 0.3ṁEdd, Ω = 0.1ΩK, and αTS = 1. The shooting method demonstrates that the initial helium abundance within the mixed layer is Ymix = 0.097. The top panel shows the tem- perature profile (thin solid line), and the critical curve for sta- bility where dǫ3α/dT = dǫcool/dT (thick dashed line). The bottom panel shows that the majority of triple-α burning oc- curs at ≈ 3×108 g cm−2 (thick solid line), which is where the helium is depleted. Comparing the middle and bottom panel shows that the majority of the burning takes place near where tmix = t3α (as required by construction), which is deeper than where tacc = t3α (the normal condition for steady burning). To make sure the steady-state models we find are physically realizable, we must check the thermal stability of the helium burning. In Figure 14 we compare the quantities dǫ3α/d lnT and dǫcool/d lnT for three different accretion rates. If the cooling derivative is always larger, then the model is thermally stable. We find stable accretion at ṁ’s considerably less than the stable accretion rate of ṁ ≈ 10ṁEdd ≈ 2× 10 6 g cm−2 s−1 expected for pure helium accretion estimated without mixing (Bildsten 1995, 1998a). Models that are found to be stable in this way are plotted as filled circles in Figure 11 (from §5.1). Only at large ṁ and small Ω are the models found to be stable. If we where to increase αTS, a wider range of the models in Figure 11 become stable. 5.4. The Critical ṁ for Stability Since we have found a set of models that can stably accrete, mix, and then burn helium, it is interesting to ask what accre- tion rates and spins are required for this to occur, and how does it depend on parameters such as αTS. FIG. 13.— An example steady-state mixing and burning envelope model using Y0 = 1, ṁ = 1.0ṁEdd , Ω = 0.03ΩK , and αTS = 1. The material within the mixed layer has Ymix = 0.097. The top panel shows the temperature profile (solid line), as well as the critical curve ignition using a helium mass fraction of 0.052 (thick dashed line, which is the helium mass fraction at the burning depth). The middle panel compares the key timescales. The bottom panel shows the helium (dashed line) and carbon abundances (dotted line), as well as the energy generation rate for helium burning (thick solid line). First we must derive the correct condition for stability in- cluding the fact that free-free opacity is important in setting the radiative profile. The one-zone condition for stability at the base of the mixed layer is d lnT dǫcool d lnT , (74) where the derivatives are taken at constant pressure. For ǫ3α ∝ T ζρχ, where ζ = 44/T8 −3, stability requires (Bildsten 1998a) ζ − 4 + ∂ lnκ ∂ lnT ∂ lnρ ∂ lnT ∂ lnκ ∂ lnρ < 0. (75) Substituting the scalings for a free-free opacity and ideal gas equation of state we find T8 > 3.26 is required for stability. By substituting the analytic form we found for the mixed ignition depth (eq. [67]) into the temperature profile (eq. [61]) we can find the temperature at the base of the mixed layer, Tign = 2.54× 10 8 K α0.09TS ṁ 0.1 Ω −0.067 0.1 . (76) By simply asking when Tign > 3.26× 10 8 K, we derive a sta- bilizing ṁ of ṁcrit,2 = 1.0ṁEddα −0.83 0.1 . (77) This is in reasonable agreement with Figure 11, which shows stability occurs for 0.9ṁEdd . ṁ . 1.0ṁEdd for Ω = 0.1ΩK. It is interesting that ṁcrit,2 is near (within an order of magnitude) a value where bursts are observed to change. It is conceivable 14 PIRO & BILDSTEN FIG. 14.— A comparison of dǫ3α/d lnT (solid lines) and dǫcool/d ln T (dashed lines) as a function of depth for ṁ = 0.1, 0.3, and 1.0ṁEdd (top to bottom panel). All the models use Ω = 0.03ΩK , αTS = 1, and Y0 = 1. This demonstrates that only the ṁ = 1.0ṁEdd model is stable out of these three. that this mechanism may act to stabilize X-ray bursts for the hydrogen-rich accreting systems, and given the strong scaling ṁcrit,2 has with αTS it is possible that more detailed calcula- tions could give results that agree even better with the critical accretion rates that are observed. We discuss this idea in more detail in the following section. 6. DISCUSSION AND CONCLUSION We have revisited the problem of angular momentum trans- port in the surface layers of accreting NSs. We found that the hydrodynamic instabilities used by Fujimoto (1993) in a pre- vious study are dwarfed by the magnetic effects of the Tayler- Spruit dynamo. The large viscosity provided by this process results in a very small shear rate and negligible viscous heat- ing. The turbulent mixing is sufficiently large to have impor- tant consequences for X-ray bursts. We constructed simple models, both analytic and numerical, to explore mixing for pure helium accretion. From these models we can make a few conclusions that are likely general enough to apply to most viscous mechanisms. As a guide, we show the different burn- ing regimes we find in Figure 15. These can be summarized as follows: • Mixing is strongest at large ṁ (when angular momen- tum is being added at greater rates) and small Ω (which gives a larger relative angular momentum between the NS and accreted material). • Mixing has trouble overcoming the buoyancy barrier at chemical discontinuities. But once mixing breaks FIG. 15.— Summary of the three regimes of burning found for models in- cluding mixing. The boundaries between each regime are shown as shaded regions to emphasize possible uncertainty in the strength of mixing (we con- sider αTS = 0.7 − 1.5). The heavy shaded region divides where the buoyancy barrier is overcome (ṁcrit,1, eq. [59]), and the light shaded region divides between stable and unstable mixed burning (ṁcrit,2, eq. [77]). An additional uncertainty in ṁcrit,1 is the value of ∆ lnµ, which could vary depending on the results of previous bursts. through this, it extends down to a depth where tmix = tacc, which is generally much deeper than the accreted col- umn. This means that mixing should turn on abruptly (as a function of either ṁ or Ω). It also means that the importance of mixing depends on the particular ashes left over from previous bursts. If, for example, incom- plete burning results in small compositional gradients, mixing would be important in subsequent bursts. • Mixing of freshly accreted material with the ashes from previous X-ray bursts can lead to two new effects. First, the layer may ignite, but now in a mixed environment with a short recurrence time of ∼ 5 − 30 minutes. Sec- ond, if the mixing is strong enough, accreted helium can mix and burn in steady-state, quenching X-ray bursts. Both of these regimes have observed analogs, namely the short recurrence time bursts (for example in, Boirin et al. 2007) and the stabilization of bursting seen at ≈ 0.1ṁEdd (Cornelisse et al. 2003). The mixed ignition case can be studied easily using the cur- rent numerical experiments (e.g., Woosley et al. 2004) by just artificially accreting fuel with a mixed composition of Ymix ≈ 0.1 − 0.6. These calculations are simplified by our conclusion that shearing and heating can be ignored, at least for initial studies. We next conclude by speculating about some of the other ramifications of turbulent mixing. 6.1. Superbursts An ongoing mystery in the study of bursting NSs is the recurrence times for superbursts, thermonuclear ig- nition of carbon in the X-ray burst ashes at columns of ≈ 1011 − 1012 g cm−2 (Cumming & Bildsten 2001; Strohmayer & Brown 2002). This problem could be alle- viated by enhanced heating from the core on the order of TURBULENT MIXING IN ACCRETING NEUTRON STARS 15 1 MeV nuc−1 (Cumming et al. 2006), but this is more than is expected theoretically, even in the newest calculations (Gupta et al. 2006). Shear heating is not large enough to solve this problem, as demonstrated in Figures 3 and 5. The regime of stable helium burning we have found may, however, create a carbon rich ocean that would assist in the ignition of superbursts. Carbon fractions of greater than 10% are needed to reproduce the lightcurves and recurrence times of superbursts (Keek et al. 2006). Calculations of rp- process burning show that unstable burning cannot give car- bon fractions this high (Schatz et al. 2003). Observationally, in’t Zand et al. (2003) showed that the α-value (see §5.1) is preferentially large for superbursting systems, indicating that some stable burning is occurring, perhaps due to the turbulent mixing we have studied. 6.2. Hydrogen-rich Accretion One of the main deficiencies of our calculations are the sim- plified compositions, since most accreting NSs are expected to be accreting a fuel abundant in hydrogen. If the only cru- cial burning is triple-α, such envelopes can be considered within the framework of our models by using a solar value of Y0 = 0.3. Our models fail, though, when sufficient carbon is produced by triple-α to feedback into hydrogen burning (which burns via the hot-CNO cycle, Hoyle & Fowler 1965). Such a scenario is interesting thought because it could poten- tially produce very hydrogen poor bursts. In the standard theoretical framework for X-ray bursts, flashes should be hydrogen poor at low ṁ (when there is suf- ficient time for the hot-CNO cycle to act), and then become mixed hydrogen-helium fuel at higher ṁ. Paradoxically, ob- servations show a transition at a seemingly universal luminos- ity of ≈ 2× 1037 ergs s−1 (approximately 0.1ṁEdd), but in the opposite sense (Cornelisse et al. 2003). Cooper & Narayan (2006b) argue that their models in fact show this transi- tion (also see Narayan & Heyl 2003) because carbon cre- ated during helium simmering increases the the hot-CNO rate and the temperature, which decreases the temperature sensitivity of triple-α reactions. To explain the discrep- ancies between their models and more detailed numerical simulations (Woosley et al. 2004; Heger et al. 2005) requires a decrease in the breakout reactions rate of 15O(α,γ)19Ne (Cooper & Narayan 2006a; Fisker et al. 2006). Unfortu- nately, the most recent experimental results do not support a decreased rate (Tan et al. 2007; Fisker et al. 2007). Further- more, helium accreting systems show this same transition in bursting properties, which is not explained within their frame- work. Another idea that may recreate this trend is that the frac- tion of the star covered by the fuel increases with the global accretion fast enough that the local accretion rate actually de- creases (Bildsten 2000). This interpretation is supported by the recent work of Heger et al. (2005), who claim that the mHz oscillations observed at around this same universal lu- minosity of 2× 1037 ergs s−1 (Revnivtsev et al. 2001) are a probe of the local accretion rate where bursting is occurring. The main problem with this suggestion is that it is difficult to understand how the covering of the surface can remain so anisotropic all the way down to the depths of where ignition occurs. Inogamov & Sunyaev (1999) calculated the spread- ing of accreted material from the equator, where the viscosity is due to a turbulent boundary layer, and find that spreading occurs orders of magnitude shallower in depth than where ig- nition takes place. Since mixing gets stronger with ṁ, it may be that the ob- served transition is in fact hydrogen being turbulently mixed and burned analogous to what we have found for pure- helium accretion. A possible implication of such an expla- nation is that slowly spinning NSs are more likely to have their hydrogen depleted resulting in helium-rich bursts. As- suming that the burst oscillation frequencies are indicative of the NS spin frequency (which is close to true or ex- actly true for all current explanations Heyl 2004; Lee 2004; Piro & Bildsten 2005; Payne & Melatos 2006), the systems can be broken into slow spinning (∼ 300 Hz) and fast spin- ning (∼ 600 Hz) classes. This slowly spinning class in- cludes 4U 1916 − 053 (270 Hz, Galloway et al. 2001), 4U 1702 − 429 (330 Hz, Markwardt et al. 1999) and 4U 1728 − 34 (363 Hz, Strohmayer & Markwardt 1999). Do these systems show helium-rich looking bursts, and if so, is this due to turbu- lent mixing destroying the hydrogen they are accreting? This can be answered by looking at the recent summary of RXTE burst observations by Galloway et al. (2006). 4U 1916 − 053 has bursts consistent with helium-rich fuel, but this system also has an orbital period of ≈ 50 min (Grindlay et al. 1988). This is an “ultracompact” system in which the donor is too small to be a H-rich star, so the accretion is probably helium- rich to begin with. The other two NSs both have bursts with decay times and α-values that suggest helium-rich fuel, which is also supported by model fits to radius expansion bursts from 4U 1728 − 34 (Galloway et al. 2006). Furthermore, the bursts of 4U 1728 − 34 have look very similar to those of 4U 1820 − 30 (Cumming 2003), a known ultracompact (see dis- cussion in Podsiadlowski et al. 2002, and references therein). While it is possible that 4U 1702 − 429 and 4U 1728 − 34 have helium-rich donors (and are thus ultracompacts), it may also be that they are accreting hydrogen-rich fuel and show helium-rich bursts because their low spins lead to mixing. As binary parameters of these systems become better known, it will be more clear whether turbulent mixing is indeed needed to explain their burst properties. We thank Henk Spruit for helpful discussions. This work was supported by the National Science Foundation under grants PHY 99-07949 and AST 02-05956. APPENDIX MIXING AND ENERGY CONSERVATION FOR THE TAYLER-SPRUIT DYNAMO In §3.1 we summarized the prescriptions that Spruit (2002) provides for the Tayler-Spruit dynamo. The mixing diffusivity is assumed to be equal to the turbulent magnetic diffusivity. Although this is plausible, it is not shown rigorously. Below we argue that such a scaling is consistent with energy conservation. Consider a layer differentially rotating with a speed ∆V . 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0704.1279

Wide-bandwidth, tunable, multiple-pulse-width optical delays using slow light in cesium vapor

Wide-bandwidth, tunable, multiple-pulse-width optical delays using slow light in cesium vapor Ryan M. Camacho, Michael V. Pack, John C. Howell Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA Aaron Schweinsberg, Robert W. Boyd Institute of Optics, University of Rochester, Rochester, NY 14627, USA We demonstrate an all-optical delay line in hot cesium vapor that tunably delays 275 ps input pulses up to 6.8 ns and 740 input ps pulses up to 59 ns (group index of approximately 200) with little pulse distortion. The delay is made tunable with a fast reconfiguration time (100’s of ns) by optically pumping out of the atomic ground states. There is considerable practical interest in developing all-optical delay lines that can tunably delay short pulses by much longer than the pulse duration. Slow light (i.e. the passage of light pulses through media with a small group velocity) has long been considered a possible mech- anism for constructing such a delay line. Most commonly, the steep linear dispersion associated with a single gain or transparency resonance provides the group delay. Most early work used the dispersion associated with electro- magnetically induced transparency [1, 2, 3, 4, 5, 6, 7], but recently other resonances have been explored, includ- ing coherent population oscillations [8, 9, 10], stimulated Brillouin scattering [11, 12, 13, 14], stimulated Raman scattering [15, 16], and spectral hole-burning [17]. In addition to single-resonance systems, double gain resonances have been used for pulse advancement [18, 19, 20, 21, 22, 23] and delay [24]. Widely spaced gain peaks create a region of anomalous dispersion, result- ing in pulse advancement. When the spacing between the gain peaks is small, a region of normal dispersion is created, resulting in pulse delay. Pulse advancement is also possible by the proper spacing of two absorbing resonances [22]. The possibility of pulse delay between two absorbing resonances has also received some atten- tion [25, 26, 27, 28, 29, 30]. Ideally, an optical delay line would delay high band- width pulses by many pulse lengths in a short propaga- tion distance without introducing appreciable pulse dis- tortion and be able to tune the delay continuously with a fast reconfiguration rate. Minimal pulse absorption is also desirable, but not necessary because absorption can be compensated through amplification. Relatively few experiments [1, 4, 7, 11, 17, 28] have directly mea- sured pulse delays longer than the incident pulse dura- tion, and of these, none has used pulses shorter than 2 ns or reported reconfiguration rates approaching the inverse pulse delay time. In this Letter, we demonstrate the tunable delay of a 1.6-GHz-bandwidth pulse by up to 25 pulse widths and the tunable delay of a 600-MHz-bandwidth pulse by up to 80 pulse widths by making use of a double absorption resonance in cesium. Furthermore, we show that the de- lay can be tuned with a reconfiguration time of 100’s of nanoseconds. In a medium with two Lorentzian absorption reso- nances, as illustrated in Figure 1, the complex index of refraction can be approximated as n(δ) = 1− A δ +∆+ + iγ δ −∆− + iγ where 2γ is the homogeneous linewidth (full width at half maximum, FWHM), g1 and g2 account for the possibility of different strengths for the two resonances, δ = ω−ω0− ∆ is the detuning from peak transmission, ω0 = (ω1 + ω2)/2, ω1 (ω2) is the resonance frequency for transition 1 (transition 2), ∆± = ω21 ±∆, ω21 = (ω2 − ω1)/2, and 1 − g 1 + g ω21. (2) For example, alkali atoms have two hyperfine levels as- sociated with their electronic ground state, leading to two closely spaced absorption resonances. We note that any other system with two similar absorbing resonances may also be used (e.g. quantum dots, microresonators, photonic crystals, etc). For a vapor of alkali atoms, the detunings satisfy ∆+ ≈ ∆− ≫ γ, and the strength of the resonance is given in SI units byA = N |µ|2/[ǫ0h̄(g1+g2)], where µ is the effective far-detuned dipole moment [31], and g1 and g2 are proportional to the degeneracies of the hyperfine levels. Eq. (1) is also applicable for inhomogeneously broad- ened lines, such as Doppler broadened atomic vapors, if the detunings ∆− and ∆+ are greater than the inhomo- geneous linewidth by an order of magnitude or more. This result holds because the homogenous Lorentzian lineshape has long wings while the inhomogeneous line- shape decreases exponentially. By expanding Eq. (1) about the point δ = 0, we find that the real part n′ and imaginary part n′′ of the index of refraction are given by n′(δ) ≈ 1 +K0 +K1 δ +K3 δ3 (3a) http://arxiv.org/abs/0704.1279v1 -5 0 5 Signal Detuning (GHz) FIG. 1: (a) CW signal transmission (asterisks–measured, solid–fit ) overlayed with the spectrum (dashed) of a 275 ps pulse and (b) index of refraction (solid) and group velocity (dashed), all versus signal detuning for cesium at approxi- mately 114 ◦C. All theory curves taken from Eq. (1) with A = 4 × 105 rad/s, g1 = 7/16 and g2 = 9/16. High-fidelity optical delay is observed for light pulses passing through the nearly transparent window between the two resonances. n′′(δ) ≈ K1 + 3K3 δ2, (3b) where 1 + g (2−i)/3 1 + (−1)i+1g (2−i)/3 and where we have assumed that n − 1 ≪ 1 in keeping only the first few terms in the expansion. Note that for the special case in which the two resonances are of equal strength (i.e. g1 = g2 = g), the coefficients are given by Ki = 2g for i odd and Ki = 0 for i even. For cesium, which has g1 = 7/16 and g2 = 9/16, the error introduced by assuming g1 = g2 is approximately 0.5%. For this reason, we make the simplifying assumption g1 = g2 = 1/2 throughout the remainder of the paper. Pulse propagation can be described in terms of various orders of dispersion, which can be determined through use of Eq. (3a) as djωn′(ω) ω=ω0+∆ , (5) Thus the group velocity is given by vg = 1/β1, and the group-velocity dispersion (GVD) and third-order- dispersion are given respectively by β2 and β3. The ab- sence of second-order (first-order) frequency dependence in Eq. (3a) (Eq. (3b)) means that near δ = 0 the GVD (absorption) is minimized regardless of possible differ- ences between g1 and g2. Thus, between two absorp- tion resonances, which can be described by Eq. (1), the maximum transparency is accompanied by a minimum in GVD. We next develop a simple model to provide an un- derstanding of the role of dispersion and absorption on pulse broadening. We provisionally define the pulse width as the square root of the variance of the tempo- ral pulse shape. For an unchirped Gaussian pulse, i.e. E(0, t) = E0 exp −t2/2T 20 , the pulse width defined in this way is simply T0. The pulse width after propagating through a distance L of dispersive medium is then given to third-order in δ by [32] T 2d = T 2T 20 where T0 is the initial pulse width. In the case of cesium, where ω0 ≈ 2π×3.5×1014 rad/s and ω21 ≈ π×9.2×109 rad/s, β2 can be neglected and Eq. (6) simplifies to T 2d ≈ T 20 + ω221T , (7) where β3 has been calculated using Eqs. (5) and (3) and where τd ≈ α0L/2γ is the pulse delay and α0L = ′′L/c is the optical depth at the pulse carrier fre- quency ω0. We further note that the change in pulse width due to absorption only can be approximated as [28, 33] T 2a = T , (8) so long as (Ta/T0 − 1) < 1. The fractional broadening due to dispersion, defined as Td/T0 − 1, scales as 1/T 30 , while the broadening due to absorption scales as 1/T0. In the present study, τd ≈ 10−8 s, ω21 ≈ 1011 rad/s, T0 ≈ 10−10 s, and γ ≈ 107 rad/s, indicating that dispersion is the dominant form of broadening by about three orders of magnitude, and the absorptive contribution to broadening can be ignored. Experimentally it is much easier to quantify pulse widths in terms of their FWHM rather than in terms of their variance as we have done in Eqs. (6) - (8). In the remainder of this Letter, we will quote pulse widths in terms of their FWHM. Our experimental setup is shown in Fig. 2. The signal laser is a CW diode laser with a wavelength of 852 nm. The signal frequency is tuned to obtain maximum trans- mission between the two Cs D2 hyperfine resonances and is pulsed at a pulse repetition frequency of 100 kHz using a fast electro-optic modulator (EOM). The signal beam is collimated to a diameter of 3 mm, and two different pulse widths are used, 275 ps or 740 ps FWHM, with a peak intensity of less than 10 mW/cm2. The pulses then pass through a heated 10-cm-long glass cell containing atomic cesium vapor. The 275 ps pulses are measured using a 7.5 GHz silicon photodiode, and the 740 ps pulses are mea- sured with a 1 GHz avalanche photodiode. All electrical signals are recorded with a 30 GHz sampling oscilloscope FIG. 2: Experimental schematic. A signal pulse passes through a heated cesium vapor cell. Two pump beams com- bine on a beamsplitter and counter-propagate relative to the signal beam through the vapor, to provide tunable delay of the signal pulse. triggered by the pulse generator. The pump beams are turned off except for the experiments reported in Figs. 5 and 6. Figure 1(a) shows the transmission of a CW optical beam as a function of frequency near the two cesium hy- perfine resonances, overlayed with the spectrum of a 275 ps Gaussian pulse. The data points are measured values and the solid line fits these points to the imaginary part of Eq. (1). The entire pulse spectrum lies well within the relatively flat transmission window between the res- onances, resulting in little pulse distortion from absorp- tion. Figure 1(b) shows the index of refraction (real part of Eq. (1)) and frequency-dependent group velocity asso- ciated with the absorption shown in Fig. 1(a). We note that, in the region of the pulse spectrum, the curvature of the frequency-dependent group velocity is greater than that of the absorption, suggesting that dispersion is the dominant form of pulse distortion. This is not the case for single-Lorentzian systems, where the spectral varia- tion of absorption is the dominant form of distortion [34]. While most slow light experiments have worked by mak- ing highly dispersive regions transparent, we have worked where a highly transparent region is dispersive. As shown above, the delay of a pulse is proportional to the optical depth of the vapor. Figure 3 shows that we can control the delay by changing the temperature (and thus optical depth) of the Cs cell. Using a 10 cm cell, and varying the temperature between approximately 90 ◦C and 120 ◦C, we were able to tune the delay of a 275 ps pulse between 1.8 ns and 6.8 ns. The theory curves in Fig. 3 were obtained using I(x, t) = n′(0)cǫ0|E(z, t)|2/2 where the electric field is given by E(z, t) = E0T0 exp [−i(ω0 +∆)t]√ dδexp ωn(δ) z − δt 2T 20 , (9) FIG. 3: Pulse shapes of 275 ps input pulses transmitted through a cesium vapor cell. Delays a large as 25 pulse widths are observed. The temperature range from 90 ◦C to 120 ◦C and where we have used Eq. (1) for the index of refrac- tion. The atomic density N has been chosen separately to fit each measured pulse. We note that a pulse may be delayed by many pulse widths relative to free-space propagation with little broadening. Longer pulses lead to delay with reduced pulse broad- ening because pulse broadening is approximately propor- tional to 1/T 30 (see Eq. (7)). To study the larger frac- tional delays enabled by this effect, we used longer 740 ps input pulses for which the dispersive broadening is sig- nificantly reduced. Figures 4(a) and 4(b) show the delay and broadening of a 740 ps pulse after passing through a sequence of three 10 cm cesium vapor cells. The plots correspond to a temperature range of approximately 110 ◦C to 160 ◦C. Even though the pulse experiences strong absorption at large delays, the fractional broadening of the pulse FWHM remains relatively low. FIG. 4: (a) Output pulse shapes and (b) fractional broaden- ing as functions of fractional delay for a 740 ps input pulse. Fractional delay is defined as (τd/T0) and fractional broaden- ing is defined as (T − T0)/T0. FIG. 5: Pulse output waveforms with auxiliary pump beams on (dotted) and off (solid). Two 275 ps input pulses separated by 1 ns are delayed by approximately 5.3 ns without pumping, but only 4.3 ns with pumping (a change of one bit slot) with little change in pulse shape. In addition to temperature tuning, the optical depth can be changed much more rapidly by optically pumping the atoms into the excited state using two pump lasers. As shown in Fig. 2 each pump laser is resonant with one of the D2 transitions in order to saturate the atoms without optical pumping from one hyperfine level to the other. The power of each pump beam is approximately 30 mW, and both pump beams are focused at the cell center. The signal beam overlaps the pump beams and is also focused to a 100 µm beam diameter. The pump beams are turned on and off using an 80 MHz AOM with a 100 ns rise/fall time. Being on resonance with the D2 transitions, the pump fields experience significant absorption (αL ∼ 300), and are entirely absorbed despite having intensities well above the saturation intensity. With the pump beams on, the decreases in effective ground-state atomic density leads to smaller delay. Fig- ure 5 shows a delayed pulse waveform consisting of two 275 ps input pulses separated by 1 ns, with the pump on and off. We note that pump fields create no noticeable change in the waveform shape or amplitude. Also, we measured that the change in delay is essentially propor- tional to the pump power. In Fig. 6 the measured signal delay is shown as a func- tion of the difference between arrival time ts of the signal at the cell and the turn-on time tp of the pump. The rise and fall times lie in the range 300-600 ns and vary slightly depending on the relative detunings of the pumps. In summary, we have observed large tunable fractional time delays of high-bandwidth pulses with fast recon- figurations rates and low distortion by tuning the laser frequency between the two ground-state hyperfine reso- nances of a hot atomic cesium vapor cell. We have shown that in such a medium dispersion is the dominant form of broadening, and we have characterized the delay, broad- ening, and reconfiguration rates of the delayed pulses. This work was supported by the DARPA/DSO Slow Light program, the National Science Foundation, andthe Research Corporation. FIG. 6: Pulse delay versus time following pump turn-on and turn-off, showing the reconfiguration time for optically tuning the pulse delay. The two pump beams are tuned to separate cesium hyperfine resonances and are switched on at the time origin and switched off 24 µs later. [1] A. Kasapi et al., Phys. Rev. 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Wang et al., Nature 406, 277 (2000). [20] A. Dogariu et al., Phys. Rev. A 63, 053806 (2001). [21] M. D. Stenner et al., Nature 425, 695 (2003). [22] B. Macke et al., Eur. Phys. J. D 23, 125 (2003). [23] G. S. Agarwal et al., Phys. Rev. A 70, 023802 (2004). [24] M. D. Stenner et al., Optics Express 13, 9995 (2005). [25] D. Grischkowsky, Phys. Rev. A 7, 2096 (1973). [26] H. Tanaka et al., Phys. Rev. A 68, 053801 (2003). [27] B. Macke et al., Phys. Rev. A 73, 043802 (2006). [28] R. M. Camacho et al., Phys. Rev. A 73, 063812 (2006). [29] Z. Zhu et al., Opt. Exp. 14, 7238 (2006). [30] R. M. Camacho et al., Physical Review Letters 98, 043902 (2007). [31] D. A. Steck. Cesium D line data, Technical Report No. LA-UR-03-7943, Los Alamos National Laboratory (2003), http://steck.us/alkalidata/. [32] G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995), p. 79. [33] R. W. Boyd et al., Phys. Rev. A 71, 023801 (2005). [34] R. W. Boyd et al., Progress in Optics (Elsevier, 2002), p. 497.

0704.1280

Controllable Quantum Switchboard

Controllable Quantum Switchboard D. Kaszlikowski,1 L. C. Kwek,2 C. H. Lai,1 and V. Vedral3 Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542 Nanyang Technological University, National Institute of Education, 1, Nanyang Walk, Singapore 637616 The School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom (Dated: September 6, 2021) All quantum information processes inevitably requires the explicit state preparation of an en- tangled state. Here we present the construction of a quantum switchboard which can act both as an optimal quantum cloning machine and a quantum demultiplexer based on the preparation of a four-qubit state. Many quantum information processes require the ex- plicit preparation of specially entangled quantum states. Two-qubit maximally entangled state often called Bell state, for instance, form an essential quantum resource needed in quantum teleportation [1]. The preparation of three-qubit maximally entangled state (like GHZ) could be harnessed for secure secret sharing [2]. In one-way quantum computing, a four-qubit entangled state called cluster state provides an efficient implementation of a universal quantum gate: arbitrary single-qubit unitary operation and the CNOT gate [3]. It is interesting to note that entangled states which are used as a common resource in quantum information pro- cesses generally need not even be maximally entangled at all. As long as the state is genuinely entangled, quan- tum computation and communication will generally be better than the classical counterparts. In particular, the non-maximally entangled W state has been experimen- tally implemented and proposed for controlled quantum teleportation and secure communication [4]. An essential component of any quantum computation is the ability to spread quantum information over vari- ous parts of quantum computer. The parts then undergo separate evolutions depending on the type of the quan- tum information processing we wish to implement. Ulti- mately we need be capable of navigating the relevant part of the information into a designated output. In a classi- cal computer this flow of information is achieved through a controllable switch. Is it possible to design a quantum analogue for such a device? An added complexity in a quantum switch would be the requirement that the in- formation flows down many possible channels coherently as well as the possibility of channeling it in one selected direction. Ideally we would like to realize the simplest such a device with the least number of qubits needed for this purpose. In addition these qubits will in practice be im- plemented in a physical system which will determine the nature of the qubits and couplings between them. There- fore, when designing our switch we should also take into account realistic interaction between the qubits, which severely limits the number of possible Hamiltonians to execute such a quantum switch. Here we present a pos- sible implementation of the switch that fulfills of all the above requirements. Let us consider an interesting four-qubit state de- scribed by |ψ〉 = 1√ (|(11)12〉|(11)34〉 − |(11)14〉|(11)23〉) , (1) where |(11)ij〉 = 1√2 (|0〉i|1〉j−|1〉i|0〉j) is the singlet state. Throughout the paper we use the following notation for the Bell states |(ab)〉 = (−1)kb√ |k, k + a〉 with sum- mation modulo 2. It turns out that this state is ideally suited for a quantum switchboard, i.e., a circuit that can be used to direct the flow of quantum information in a controllable manner. An interesting property of the presented switchboard is that in the case of failure the information is not entirely lost. The first qubit of the state (1) belongs to Alice, the sec- ond one to Bob, the third one to Charlene and the last one to Dick. Suppose Alice attaches some auxiliary qubit to the first qubit and perform a joint Bell measurement. Immediately after getting one of the four possible out- comes, she broadcasts two (classical) bits of information to Bob and Charlene as it is in the usual teleportation scheme. At this point, it is not necessary for Dick to know these two bits of information. Bob and Charlene can recover the state of the auxiliary qubit with the fidelity 5 by applying appropriate unitary transformation based on the knowledge of the broadcast classical bits. The given state at the beginning does not provide a universal cloning machine for three copies of the cloned state [5]. Thus, the qubit belonging to Dick is related to the Alice’s auxiliary qubit with the ”classical” fidelity 1 , i.e., the fidelity that can be achieved without prior entanglement. It is interesting to note that Bob and Charlene possess the optimum fidelity achievable under a symmetric cloning machine. Dick’s fidelity is allowed since there is no limitation on the production of clones with the fidelity below 2 Thus the presented protocol behaves like an optimal telecloner [6]. However, there is still an unused qubit held by Dick. Depending on Alice’s decision regarding to whom she wishes ultimately to send her auxiliary qubit, say Bob (Charlene) for instance, she can direct Dick to send his qubit to Charlene (Bob). As soon as Charlene receives Dick’s qubit, he can perform a Bell measurement on his qubit with Dick’s qubit and send the results of his http://arxiv.org/abs/0704.1280v1 Alice Charlene FIG. 1: Suppose Alice wishes to send her auxiliary qubit to Bob. She can direct Dick to send his qubit to Charlene. Charlene then performs a Bell measurement on his qubit with Dick’s qubit and send the results of his measurement to Bob. Using the information from Charlene, Bob can perfectly re- cover the state of the Alice’s auxiliary qubit. measurement to Bob. Using the information from Char- lene, Bob can perfectly recover the state of the Alice’s auxiliary qubit. The situation is entirely symmetric, i.e., Dick can send his qubit to Bob instead of Charlene with the result that now Charlene can obtain Alice’s auxiliary qubit with perfect fidelity. In short the state acts as a quantum switchboard in which Alice can direct optimal clones to Bob and Charlene or perform perfect quantum telepor- tation to Bob or Charlene by utilizing Dick’s qubit as in a quantum demultiplexer. A schematic diagram of this quantum switchboard protocol is shown in Fig. 1. By directing Dick’s qubit to either Bob (or Charlene), Alice can effectively transfer the unknown auxiliary qubit to Charlene (or Bob). Moreover, she can delay the transfer process to a later time as long as she has effective control over Dick’s qubit. Incidentally, other shared states may be able to achieve some aspect of our quantum switch but it is difficult to find a state with all desired properties. For instance with the GHZ state, shared among Alice, Bob and Charlene, one could in principle provide perfect quantum telepor- tation to both Bob and Charlene, but without the addi- tional benefit of an optimal quantum cloner. In this case, Alice teleclones to both Bob and Charlene with a classi- cal fidelity of 2/3. The eventual quantum teleportation to Bob (or Charlene) is performed with a measurement in the basis 1/ 2(|0〉 ± |1〉). It is also interesting to note that the sheer presence of singlets or dimer-like bonds in the four-qubit state may make it more robust to certain types of noise. One ex- ample would be fluctuating magnetic field or polarization drift, depending on how we implement our qubits. This kind of fault tolerance is absent in the GHZ state. Let us now prove the above statements. It is conve- nient to write the state |ψ〉 in the following way |ψ〉 = 1 (3|(11)12〉|(11)34〉+ |(10)12〉|(10)34〉+ |(01)12〉|(01)34〉 − |(00)12〉|(00)34〉). (2) We can immediately see that the state shared by Alice and Bob is the Werner state with 1 of noise. Taking into account that the fidelity of teleportation for the Werner state with the noise fraction 1 − p is given by p+1 we see that the fidelity of Bob’s qubit is 5 . It can be checked that the state between Alice and Dick is the Werner state that is an equal mixture of the three Bell states |(10)〉, |(01)〉, |(00)〉. Thus Dick’s clone of Alice’s auxiliary qubit has the fidelity 1 , which is the fidelity achievable classically. The state |ψ〉 is symmetric with respect to Bob and Charlene |ψ〉 = 1 (3|(11)13〉|(11)24〉+ |(10)13〉|(10)24〉+ |(01)13〉|(01)24〉 − |(00)13〉|(00)24〉). (3) therefore Charlene’s clone has the same fidelity as Bob’s Let us now write the state |ψ〉 together with the Al- ice’s auxiliary qubit |α〉 (particle with 0 index) in the form suitable for further analysis. To focus our attention we consider the scenario where Dick sends his qubit to Charlene. We have |α〉|ψ〉 = 1 k,l,m,n=0 λkl|(mn)01〉 ⊗ ⊗ Umn,kl|α〉|(kl)34〉, (4) where λ11 = 3, λ01 = λ10 = 1, λ00 = −1 and Umn,kl is a usual unitary transformation that appears in the pro- cess of teleportation with the Bell state |(kl)〉 and with the outcome of Bell measurement (mn). For instance, U01,11 = σx. Suppose now that the outcome of Alice’s measurement is (mn). The collapsed state |χmn〉 shared by Bob, Char- lene and Dick is |χmn〉 = k,l=0 Umn,kl|α〉|(kl)34〉. (5) Therefore, Bob’s state ρmn reads ρmn = k,l=0 |λkl|2Umn,kl|α〉〈α|U †mn,kl. (6) After receiving two bits (mn) of classical information from Alice, Bob can recover Alice’s state with the fi- delity 5 as mentioned before. However, when Charlene performs the Bell measurement on his and Dick’s qubit, obtains the result (kl) and sends it to Bob, Bob receives the state |λkl| Umn,kl|α〉, (7) which he can transform back to the state |α〉 by applying the inverse unitary transformation U mn,kl The symmetry of the state |ψ〉 allows us to repeat the same argument for the case in which Alice decides to send her qubit to Bob so that now Charlene can obtain the state |α〉 with perfect fidelity. It is interesting to note that the relative phase between the components of the state |ψ〉 is crucial for desired functionality. Other phase choices or, for that matter, the complete lack of coherence, will not give us the same quantum switch. Finally we emphasise that our quantum switch state is a ground state, albeit degenerate, of the Majundar- Ghosh (MG) model [10]. This spin chain belongs to a class of many- body Hamiltonians that provide a good qualitative account of materials like Cu2(C5H12N2)2Cl4, CuGeO3 and YCuO2.5[11]. Therefore our quantum switch is very realistic since it may already exist in some solid state systems. MG is essentially a one- dimensional quantum spin chain with nearest- and next- nearest-neighbor exchange interactions described by the the Hamiltonian HMG = J 2~Si~Si+1 + α~Si ~Si+2 , (8) where J > 0 and N is the number of sites in the one- dimensional lattice with periodic boundary condition. The Hamiltonian is exactly solvable for α = 1 and has a quantum phase transition from an ordered phase to a disordered spin-liquid-like phase as α varies from zero to some critical value αcrit = 0.482[12]. At α = 1 and for an even N , there is a two-fold de- generate ground state subspace spanned by two dimer configurations |(11)12〉|(11)34〉 . . . |(11)(N−1)N〉 |(11)23〉|(11)45〉 . . . |(11)N1〉, (9) superposition of which, for N = 4, gives us the state |ψ〉. In conclusion, we have provided a quantum switch- board which could act both as an optimal quantum cloning machine or a quantum demultiplexer. Moreover, we also note that it is possible to extend the switchboard to higher spins and higher dimensional spaces as long as we have a configuration of dimer-like neighboring bonds. We also note that apart from spin chains, it is possible that the four-qubit state considered in this paper could also be created from multi-photon entangled states gen- erated with spontaneous parametric down conversion and linear optics apparatus. I. ACKNOWLEDGMENT D.K. would like to thank Alastair Kay and Ravis- hankar Ramanathan for useful discussions. [1] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. Wootters,Phys. Rev. Lett. 70,1895 (1993); For experimental work, see D. Bouwmeester et al. Nature 390, 575 (1997); D. Boschi et al. Phys. Rev. Lett. 80, 1121 (1998); A. Furusawa et al. Science 282, 706 (1998); M.A. Nielsen et al. Nature 396, 52 (1998); I. Marcikic et al. Nature 421, 509 (2003); M. Riebe et al. Nature 429, 734 (2004); M.D. Barret et al. Nature 429, 737 (2004); R. Ursin et al. Nature 430, 849 (2004). [2] M. Hillery, V. Buzek, and A. Berthiaume, Phys. Rev. A 59, 1829 (1999). [3] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001). [4] M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer,and H. Weinfurter, Phys. Rev. Lett. 92, 077901 (2004); J. Joo, Y-J. Park, S. Oh and J. Kim, New J. Phys. 5, 136 (2003) [5] V. Scarani, S. Iblisdir, N. Gisin, and A. Aćın, Rev. Mod. Phys. 77, 1225-1256 (2005). [6] M. Murao, D. Jonathan, M.B. Plenio, and V. Vedral, Phys. Rev. A 59, 156 (1999). [7] P. Horodecki, M. Horodecki, and R. Horodecki, Phys. Rev. A, 60, 1888 (1999). [8] S. Chen, H. Büttner and J. Voit, Phys. Rev. Lett. 87, 087205 (2001); Phys. Rev. B 67, 054412 (2003). [9] E. Dagotto and T.M. Rice, Science, 271, 618 (1996). [10] C.K. Majumdar and D.P. Ghosh, J. Math. Phys., 10, 1388 (1969). [11] G. Castilla, S. Chakravarty, and V. J. Emery, Phys. Rev. Lett., 75, 1823 (1995); M. Azuma, Z. Hiroi, M. Takano, K. Ishida and Y. Kitaoka, Phys. Rev. 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0704.1281

The Galactic Center

Black Holes: from Stars to Galaxies – across the Range of Masses Proceedings IAU Symposium No. 238, 2006 V. Karas & G. Matt, eds. c© 2006 International Astronomical Union DOI: 10.1017/S1743921307004851 The Galactic Center Reinhard Genzel1,2 and Vladimı́r Karas3 1 Max-Planck Institut für Extraterrestrische Physik, Garching, Germany 2 Department of Physics, University of California, Berkeley, USA 3 Astronomical Institute, Academy of Sciences, Prague, Czech Republic Abstract. In the past decade high resolution measurements in the infrared employing adaptive optics imaging on 10m telescopes have allowed determining the three dimensional orbits stars within ten light hours of the compact radio source at the center of the Milky Way. These observations show the presence of a three million solar mass black hole in Sagittarius A* beyond any reasonable doubt. The Galactic Center thus constitutes the best astrophysical evidence for the existence of black holes which have long been postulated, and is also an ideal ‘lab’ for studying the physics in the vicinity of such an object. Remarkably, young massive stars are present there and probably have formed in the innermost stellar cusp. Variable infrared and X-ray emission from Sagittarius A* are a new probe of the physical processes and space-time curvature just outside the event horizon. Keywords. Galaxy: center – black hole physics 1. Introduction – Sagittarius A* The central light years of our Galaxy contain a dense and luminous star cluster, as well as several components of neutral, ionized and extremely hot gas (Genzel, Hollenbach & Townes 1994). The Galactic Center also contains a very compact radio source, Sagit- tarius A* (Sgr A*; Balick & Brown 1974) which is located at the center of the nuclear star cluster and ionized gas environment. Short-wavelength centimeter and millimeter VLBI observations have established that its intrinsic radio size is a mere 10 light min- utes (Bower et al 2004; Shen et al 2005). Sgr A* is also an X-ray emission source, albeit of only modest luminosity (Baganoff et al 2001). Most recently, Aharonian et al (2004) have discovered a source of TeV γ-ray emission within 10 arcsec of Sgr A*. It is not yet clear whether these most energetic γ-rays come from Sgr A* itself or whether they are associated with the nearby supernova remnant, Sgr A East. Sgr A* thus may be a supermassive black hole analogous to QSOs, albeit of much lower mass and luminosity. Because of its proximity – the distance to the Galactic Center is about 105 times closer than the nearest quasars – high resolution observations of the Milky Way nucleus offer the unique opportunity of stringently testing the black hole paradigm and of studying stars and gas in the immediate vicinity of a black hole, at a level of detail that will not be accessible in any other galactic nucleus in the foreseeable future. Since the center of the Milky Way is highly obscured by interstellar dust particles in the plane of the Galactic disk, observations in the visible light are not possible. Investi- gations require measurements at longer wavelengths – the infrared and microwave bands, or at shorter wavelengths – hard X-rays and γ-rays, where the veil of dust is transparent. The dramatic progress in our knowledge of the Galactic Center over the past two decades is a direct consequence of the development of novel facilities, instruments and techniques across the whole range of the electromagnetic spectrum. http://arxiv.org/abs/0704.1281v1 174 R. Genzel & V. Karas Figure 1. Left: VLA radio continuum map of the central parsec (Roberts & Goss 1993). The radio emission delineates ionized gaseous streams orbiting the compact radio source Sgr A*. Spectroscopic measurements in the radio band (Wollman et al 1977) provided the first dynamical evidence from large gas velocities that there might be a hidden mass of 3–4 million solar masses located near Sgr A*. Right: A diffraction limited image of Sgr A* (∼ 0.05 arcsec resolution) from the 8m ESO VLT, taken with the NACO AO-camera and an infrared wavefront sensor at 1.6/2.2/3.7 µm (Genzel et al 2003b). The central black hole is located in the centre of the box. NACO is a collaboration between ONERA (Paris), Observatoire de Paris, Observatoire Grenoble, MPE (Garching), and MPIA (Heidelberg) (Lenzen et al 1998; Rousset et al 1998). 2. High angular resolution astronomy The key to the nature of Sgr A* obviously lies in very high angular resolution measurements. The Schwarzschild radius of a 3.6 million solar mass black hole at the Galactic Center subtends a mere 10−5 arcsec. For the high-resolution imaging from the ground it is necessary to correct for the distortions of an incoming electromagnetic wave by the refractive and dynamic Earth atmosphere. VLBI overcomes this hurdle by phase- referencing to nearby QSOs; sub-milliarcsecond resolution can now be routinely achieved. In the optical/near-infrared wavebands the atmosphere smears out long-exposure images to a diameter at least ten times greater than the diffraction limited resolution of large ground-based telescopes (Fig. 1). From the early 1990s onward initially speckle imaging (recording short exposure images, which are subsequently processed and co- added to retrieve the diffraction limited resolution) and then later adaptive optics (AO, correcting the wave distortions on-line) became available. With these techniques it is possible to achieve diffraction limited resolution on large ground-based telescopes. The diffraction limited images are much sharper and also much deeper than the seeing limited images. In the case of AO (Beckers 1993) the incoming wavefront of a bright star near the source of interest is analyzed, the necessary corrections for undoing the aberrations of the atmosphere are computed (on time scales shorter than the atmospheric coherence time of a few milli-seconds) and these corrections are then applied to a deformable optical element (e.g. a mirror) in the light path. The requirements on the brightness of the AO star and on the maximum allowable separation between star and source are quite stringent, resulting in a very small sky coverage of natural star AO. Fortunately, in the Galactic Center there is a bright infrared star only 6 arcsec away from Sgr A*, such that good AO correction can be achieved The Galactic Center 175 with an infrared wavefront sensor system. Artificial laser beacons can overcome the sky coverage problem to a considerable extent. For this purpose, a laser beam is projected from the telescope into the upper atmosphere and the backscattered laser light can then be used for AO correction. The Keck telescope team has already begun successfully exploiting the new laser guide star technique for Galactic Center research (Ghez et al 2005a). After AO correction, the images are an order of magnitude sharper and also much deeper than in conventional seeing limited measurements. The combination of AO techniques with advanced imaging and spectroscopic instruments (e.g. integral field imaging spectroscopy) have resulted in a major breakthrough in high resolution studies of the Galactic Center. 3. Nuclear star cluster and the paradox of youth One of the big surprises is a fairly large number of bright stars in Sgr A*, a number of which were already apparent on the discovery infrared images of Becklin & Neugebauer (1975, 1978). High-resolution infrared spectroscopy reveals that many of these bright stars are actually somewhat older, late-type supergiants and AGB stars. Starting with the discovery of the AF-star (Allen et al 1990; Forrest et al 1987), however, an ever increasing number of the bright stars have been identified as being young, massive and early type. The most recent counts from the deep SINFONI integral-field spectroscopy yields about one hundred OB stars, including various luminous blue supergiants and Wolf-Rayet stars, but also normal main-sequence OB stars (Paumard et al 2006a). The nuclear star cluster is one of the richest concentrations of young massive stars in the Milky Way. The deep adaptive optics images also trace the surface density distribution of the fainter stars, to about K 17–18 mag, corresponding to late B or early A stars (masses of 3–6 solar masses), which are a better probe of the density distribution of the overall mass density of the star cluster. While the surface brightness distribution of the star cluster is not centered on Sgr A*, the surface density distribution is. There is clearly a cusp of stars centered on the compact radio source (Genzel et al 2003b; Schödel et al 2006). The inferred volume density of the cusp is a power-law ∝R−1.4±0.1, consistent with the expectation for a stellar cusp around a massive black hole (Alexander 2005). If there is indeed a central black hole associated with Sgr A* the presence of so many young stars in its immediate vicinity constitutes a significant puzzle (Allen & Sanders 1986; Morris 1993; Alexander 2005). For gravitational collapse to occur in the presence of the tidal shear from the central mass, gas clouds have to be denser than∼ 109(R/(10′′))−3 hydrogen atoms per cm−3. This ‘Roche’ limit exceeds the density of any gas currently observed in the central region. Recent near-diffraction limited AO spectroscopy with both the Keck and VLT shows that almost all of the cusp stars brighter than K ∼ 16 mag appear to be normal, main sequence B stars (Ghez et al 2003; Eisenhauer et al 2005a). If these stars formed in situ, the required cloud densities approach the conditions in outer stellar atmospheres. Several scenarios have been proposed to account for this paradox of youth. In spite of this effort the origin of central stars (S-stars) is not well understood: models have difficulties in reconciling different aspects of the Galaxy Centre – on one side it is a low level of present activity, indicating a very small accretion rate, and on the other side it is the spectral classification that suggests these stars have been formed relatively recently; see Alexander (2005) for a detailed discussion and references. The most prominent ideas to resolve the apparent problem are in situ formation in a dense gas accretion disk that can overcome the tidal limits, re-juvenation of older stars by collisions or stripping, and 176 R. Genzel & V. Karas Figure 2. Positions on the sky as a function of time for the central stars orbiting the compact radio source Sgr A*. Left: the data from the UCLA group working with the Keck telescope (Ghez et al 2005b). Right: the data from the MPE–Cologne group at the ESO-VLT (Schödel et al 2003; Eisenhauer et al 2005a; Gillessen et al, in preparation). rapid in-spiral of a compact, massive star cluster that formed outside the central region and various scattering a three body interaction mechanisms, including resonant relaxation (Alexander 2005). Several other mechanisms have been proposed that could set stars on highly eccentric orbits and bring them to the neighbourhood of the central black hole (e.g., Hansen & Milosavljević 2003; McMillan & Portegies Zwart 2003; Alexander & Livio 2004; Šubr & Karas 2005), but the problem of the S-stars remains open. 4. Compelling evidence for a central massive black hole With diffraction limited imagery starting in 1991 on the 3.5m ESO New Technology Telescope and continuing since 2002 on the VLT, a group at MPE was able to determine proper motions of stars as close as∼ 0.1 arcsec from Sgr A* (Eckart & Genzel 1996, 1997). In 1995 a group at the University of California, Los Angeles started a similar program with the 10m diameter Keck telescope (Ghez et al 1998). Both groups independently found that the stellar velocities follow Kepler laws and exceed 103 km/s within the central light month. Only a few years later both groups achieved the next and crucial step: they were able to determine individual stellar orbits for several stars very close to the compact radio source (Fig. 2; Schödel et al. 2002, 2003; Ghez et al 2003, 2005b; Eisenhauer et al 2005a). In addition to the astrometric imaging they obtained near-diffraction limited Doppler spectroscopy of the same stars (Ghez et al 2003; Eisenhauer et al 2003a,b), yielding precision measurements of the three dimensional structure of several orbits, as well as the distance to the Galactic Center. At the time of writing, the orbits have been determined for about a dozen stars in the central light month. The central mass and stellar orbital parameters derived by the two teams agree mostly very well. The orbits show that the gravitational potential indeed is that of a point mass centered on Sgr A* within the relative astrometric uncertainties of ∼ 10 milliarcsec. Most of the mass must be concentrated well within the peri-approaches of the innermost stars, ∼ 10–20 light The Galactic Center 177 hours, or 70 times the Earth orbit radius and about 1000 times the event horizon of a 3.6 million solar mass black hole. There is presently no indication for an extended mass greater than about 5% of the point mass. Simulations indicate that current measurement accuracies are sufficient to reveal the first and second order effects of Special and General Relativity in a few years time (Zucker et al 2006). Observations with future 30m+ diameter telescopes will be able to measure the mass and distance to the Galactic Center to ∼ 0.1% precision. They should detect radial precession of stellar orbits due to General Relativity and constrain the extended mass to < 10−3 of the massive black hole (Weinberg, Milosavljevic & Ghez 2005). At that level a positive detection of a halo of stellar remnants (stellar black holes and neutron stars) and perhaps dark matter would appear to be likely. Future interferometric techniques will push capabilities yet further. Long-term VLBA observations have set 2σ upper limits of about 20 km/s and 2 km/s (or 50 micro-arcsec per year) to the motion of Sgr A* itself, along and perpendicular to the plane of the Milky Way, respectively (Reid & Brunthaler 2004; see also Backer & Sramek 1999). This precision measurement demonstrates very clearly that the radio source itself must indeed be massive, with simulations indicating a lower limit to the mass of Sgr A* of ∼ 105 solar masses. The intrinsic size of the radio source at millimeter wavelengths is less than 5 to 20 times the event horizon diameter (Bower et al 2004; Shen et al 2005). Combining the radio size and proper motion limit of Sgr A* with the dynamical measurements of the nearby orbiting stars leads to the conclusion that Sgr A* can only be a massive black hole, beyond any reasonable doubt. An astrophysical dark cluster fulfilling the observational constraints would have a life-time less than a few 104 years and thus can be safely rejected, as can be a possible fermion ball of hypothetical heavy neutrinos. In fact all non-black hole configurations can be excluded by the available measurements (Schödel et al 2003; Ghez et al 2005b) – except for a hypothetical boson star and the gravastar hypothesis, but it appears that the two mentioned alternatives have difficulties of their own, and they are less likely and certainly much less understood than black holes (e.g. Maoz 1998; Miller et al 1998). We thus conclude that, under the assumption of the validity of General Relativity, the Galactic Center provides the best quantitative evidence for the actual existence of (massive) black holes that contemporary astrophysics can offer. 5. Zooming in on the accretion zone and event horizon Recent millimeter, infrared and X-ray observations have detected irregular, and sometimes intense outbursts of emission from Sgr A* lasting anywhere between 30 min- utes and a number of hours and occurring at least once per day (Baganoff et al 2001; Genzel et al 2003a; Marrone et al 2006). These flares originate from within a few milli- arcseconds of the radio position of Sgr A*. They probably occur when relativistic elec- trons in the innermost accretion zone of the black hole are significantly accelerated, so that they are able to produce infrared synchrotron emission and X-ray synchrotron or inverse Compton radiation (Markoff et al 2001; Yuan et al 2003; Liu et al 2005). This interpretation is also supported by the detection of significant polarization of the infrared flares (Eckart et al 2006b), by the simultaneous occurrence of X- and IR-flaring activity (Eckart et al 2006a; Yusef-Zadeh et al. 2006) and by variability in the infrared spectral properties (Ghez et al 2005b; Gillessen et al 2006a; Krabbe et al. 2006). There are in- dications for quasi-periodicities in the light curves of some of these flares, perhaps due to orbital motion of hot gas spots near the last circular orbit around the event horizon (Genzel et al 2003a; Aschenbach et al 2004; Bélanger et al 2006). 178 R. Genzel & V. Karas Figure 3. Photo-center wobbling (left) and light curve (right) of a hot spot on the innermost stable orbit around Schwarzschild black hole (inclination of 80 deg), as derived from ray-tracing computations. Dotted curve: ‘true’ path of the hot spot; dashed curves: apparent path and a pre- dicted light curve of the primary image; dash-dotted curves: the same for secondary image; solid curves: path of centroid and integrated light curve. Axes on the left panel are in Schwarzschild radii of a 3 million solar-mass black hole, roughly equal to the astrometric accuracy of 10 arcsec; the abscissa axis of the right panel is in cycles. The loop in the centroids track is due to the secondary image, which is strongly sensitive to the space-time curvature. The overall motion can be detected at good significance at the anticipated accuracy of GRAVITY. Details can be obtained by analyzing several flares simultaneously (Gillessen et al 2006b; Paumard et al 2005). The infrared flares as well as the steady microwave emission from Sgr A* may be important probes of the gas dynamics and space-time metric around the black hole (Broderick & Loeb 2006; Meyer et al 2006a,b; Paumard et al 2006b). Future long-baseline interferometry at short millimeter or sub-millimeter wavelengths may be able to map out the strong light-bending effects around the photon orbit of the black hole. It is interesting to realize that the angular size of the “shadow” of black hole (Bardeen 1973) is not very far from the anticipated resolution of interferometric techniques and it may thus be accessible to observations in near future (Falcke, Melia & Agol 2000). Polarization measurements will help us to set further constraints on the emission processes responsible for the flares. Especially the time-resolved lightcurves of the polar- ized signal carry specific information about the interplay between the gravitational and magnetic fields near Sgr A* horizon, because the propagation of the polarization vec- tor is sensitive to the presence and properties of these fields along the light trajectories (Bromley, Melia & Liu 2001; Horák & Karas 2006; Paumard et al 2006b). Polarization is also very sensitive also to intrinsic properties of the source – its geometry and details of radiation mechanisms responsible for the emission. Synthesis of different techniques will be a promising way for the future: the as- trometry of central stars gives very robust results because the stellar motion is almost unaffected by poorly known processes of non-gravitational origin, while the flaring gas occurs much closer to the black hole horizon and hence it directly probes the innermost regions of Sgr A*. Eventually the two components – gas and stars of the Galaxy Center – are interconnected and form the unique environment in which the flaring gas is influenced by intense stellar winds whereas the long-term motion and the ‘non-standard’ evolution of the central stars bears imprints of the gaseous medium though which the stars pass. Eisenhauer et al (2005b) are developing GRAVITY (an instrument for ‘General Relativity Analysis via VLT Interferometry’), which will provide dual-beam, 10 micro- arcsecond precision infrared astrometric imaging of faint sources. GRAVITY may be able to map out the motion on the sky of hot spots during flares with a high enough resolu- The Galactic Center 179 tion and precision to determine the size of the emission region and possibly detect the imprint of multiple gravitational images (see Fig. 3). In addition to studies of the flares, it will also be able to image the orbits of stars very close to the black hole, which should then exhibit the orbital radial oscillations and Lense-Thirring precession due to General Relativity. Both the microwave shadows as well as the infrared hot spots are sensitive to the space-time metric in the strong gravity regime. 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0704.1282

A geometric proof that $e$ is irrational and a new measure of its irrationality

Microsoft Word - eIrrationalAddendum.doc A Geometric Proof that e is Irrational and a New Measure of its Irrationality Jonathan Sondow 1. INTRODUCTION. While there exist geometric proofs of irrationality for √2 [2], [27], no such proof for e, π , or ln 2 seems to be known. In section 2 we use a geometric construction to prove that e is irrational. (For other proofs, see [1, pp. 27-28], [3, p. 352], [6], [10, pp. 78-79], [15, p. 301], [16], [17, p. 11], [19], [20], and [21, p. 302].) The proof leads in section 3 to a new measure of irrationality for e, that is, a lower bound on the distance from e to a given rational number, as a function of its denominator. A connection with the greatest prime factor of a number is discussed in section 4. In section 5 we compare the new irrationality measure for e with a known one, and state a number- theoretic conjecture that implies the known measure is almost always stronger. The new measure is applied in section 6 to prove a special case of a result from [24], leading to another conjecture. Finally, in section 7 we recall a theorem of G. Cantor that can be proved by a similar construction. 2. PROOF. The irrationality of e is a consequence of the following construction of a nested sequence of closed intervals In . Let I1 = [2, 3]. Proceeding inductively, divide the interval In−1 into n (≥ 2) equal subintervals, and let the second one be In (see Figure 1). For example, I2 = 2![ ] , I3 = 163!,173![ ] , and I4 = 654!,664![ ] . Figure 1. The intervals I1, I2, I3 , I4 . The intersection In = {e} (1) is then the geometric equivalent of the summation (see the Addendum) n!n=0 ∑ = e . (2) When n > 1 the interval In+1 lies strictly between the endpoints of In , which are and for some integer a = a(n) . It follows that the point of intersection (1) is not a fraction with denominator n! for any n ≥ 1. Since a rational number p q with q > 0 can be written p ⋅(q −1)! , (3) we conclude that e is irrational. • Question. The nested intervals In intersect in a number—let's call it b. It is seen by the Taylor series (2) for e that b = e . Using only standard facts about the natural logarithm (including its definition as an integral), but not using any series representation for log, can one see directly from the given construction that log b = 1? 3. A NEW IRRATIONALITY MEASURE FOR e. As a bonus, the proof leads to the following measure of irrationality for e. Theorem 1. For all integers p and q with q > 1 (S(q) +1)! , (4) where S(q) is the smallest positive integer such that S(q)! is a multiple of q. For instance, S(q) = q if 1 ≤ q ≤ 5 , while S(6) = 3 . In 19l8 A. J. Kempner [13] used the prime factorization of q to give the first algorithm for computing S(q) = min{k > 0 : q k!} (5) (the so-called Smarandache function [28]). We do not use the algorithm in this note. Proof of Theorem 1. For n > 1 the left endpoint of In is the closest fraction to e with denominator not exceeding n!. Since e lies in the interior of the second subinterval of In , (n +1)! (6) for any integer m. Now given integers p and q with q > 1, let m = p ⋅S(q)! q and n = S(q) . In view of (5), m and n are integers. Moreover, p ⋅S(q)! q S(q)! . (7) Therefore, (6) implies (4). • As an example, take q to be a prime. Clearly, S(q) = q . In this case, (4) is the (very weak) inequality (q +1)! . (8) In fact, (4) implies that (8) holds for any integer q larger than 1, because S(q) ≤ q always holds. But (4) is an improvement of (8), just as (7) is a refinement of (3). Theorem 1 would be false if we replaced the denominator on the right side of (4) with a smaller factorial. To see this, let p q be an endpoint of In , which has length . If we take q = n! , then since evidently S(n!) = n (9) and e lies in the interior of In , S(q)! . (10) (If q < n! , then (10) still holds, since n > 2 , so p q is not an endpoint of In−1, hence S(q) = n .) 4. THE LARGEST PRIME FACTOR OF q. For q ≥ 2 let P(q) denote the largest prime factor of q. Note that S(q) ≥ P(q) . Also, S(q) = P (q) if and only if S(q) is prime. (If S(q) were prime but greater than P(q) , then since q divides S(q)!, it would also divide (S(q) −1)! , contradicting the minimality of S(q) .) P. Er ʹ′ ʹ′ d os and I. Kastanas [9] observed that S(q) = P (q) (almost all q). (11) (Recall that a claim Cq is true for almost all q if the counting function N(x) = #{q ≤ x :Cq is false} satisfies the asymptotic condition N(x) x→ 0 as x→ ∞ .) It follows that Theorem 1 implies an irrationality measure for e involving the simpler function P(q) . Corollary 1. For almost all q, the following inequality holds with any integer p: (P(q) +1)! . (12) When q is a factorial, the statement is more definite. Corollary 2. Fix q = n!>1. Then (12) holds for all p if and only if n is prime. Proof. If n is prime, then P(q) = n , so (4) and (9) imply (12) for all p. Conversely, if n is composite, then P(q) < n , and (10) shows that (12) fails for certain p. • Thus when q > 1 is a factorial, (12) is true for all p if and only if S(q) = P (q) . To illustrate this, take p to be the left endpoint of I4 . Then P(q) = 3 < 4 = S(q) , and (12) does not hold, although of course (4) does: 0.00833 . . . = = 0.00994 . . . < = 0.04166 . . . . 5. A KNOWN IRRATIONALITY MEASURE FOR e. The following measure of irrationality for e is well known: given any ε > 0 there exists a positive constant q (ε) such that q2+ ε (13) for all p and q with q ≥ q(ε) . This follows easily from the continued fraction expansion of e. (See, for example, [23]. For sharper inequalities than (13), see [3, Corollary 11.1], [4], [7], [10, pp. 112-113], and especially the elegant [26].) Presumably, (13) is usually stronger than (4). We state this more precisely, and in a number-theoretic way that does not involve e. Conjecture 1. The inequality q2 < S(q)! holds for almost all q. Equivalently, q2 < P(q)! for almost all q. (The equivalence follows from (11).) This is no doubt true; the only thing lacking is a proof. (Compare [12], where A. Iv ʹ′ i c proves an asymptotic formula for the counting function N(x) = #{q ≤ x : P(q) < S(q)} and surveys earlier work, including [9].) Conjecture 1 implies that (13) is almost always a better measure of irrationality for e than those in Theorem 1 and Corollary 1. On the other hand, Theorem 1 applies to all q > 1. Moreover, (4) is stronger than (13) for certain q. For example, let q = n! once more. Then (4) and (9) give (6), which is stronger than (13) if n > 2, since (n +1)! < (n!)2 (n ≥ 3). (14) 6. PARTIAL SUMS VS. CONVERGENTS. Theorem 1 yields other results on rational approximations to e [24]. One is that for almost all n, the n-th partial sum sn of series (2) for e is not a convergent to the simple continued fraction for e. Here s 0 = 1 and sn is the left endpoint of In for n ≥ 1. (In 1840 J. Liouville [14] used the partial sums of the Taylor series for e2 and e−2 to prove that the equation ae2 + be−2 = c is impossible if a, b, and c are integers with a ≠ 0 . In particular, e4 is irrational.) Let qn be the denominator of sn in lowest terms. When qn = n! (see [22, sequence A102470]), the result is more definite, and the proof is easy. Corollary 3. If qn = n! with n ≥ 3 , then sn cannot be a convergent to e. Proof. Use (4), (9), (14), and the fact that every convergent satisfies the reverse of inequality (13) with ε = 0 [10, p. 24], [17, p. 61]. • When qn < n! (for example, q19 =19! 4000—see [22, sequence A093101]), another argument is required, and we can only prove the assertion for almost all n. However, numerical evidence suggests that much more is true. Conjecture 2. Only two partial sums of series (2) for e are convergents to e, namely, s1 = 2 and s 3 = 8 3 . 7. CANTOR'S THEOREM. A generalization of the construction in section 2 can be used to prove the following result of Cantor [5]. Theorem 2. Let a0, a1, . . . and b1, b2, . . . be integers satisfying the inequalities bn ≥ 2 and 0 ≤ an ≤ bn −1 for all n ≥ 1. Assume that each prime divides infinitely many of the bn . Then the sum of the convergent series b1b2b3 + ⋅ ⋅ ⋅ is irrational if and only if both an > 0 and an < bn −1 hold infinitely often. For example, series (2) for e and all subseries (such as Σn≥0 (2n)! = cosh1 and (2n+1)! = sinh1) are irrational, but the sum Σn≥1 =1 is rational. An exposition of the "if" part of Cantor's theorem is given in [17, pp. 7-11]. For extensions of the theorem, see [8], [11], [18], and [25]. ADDENDUM. Here are some details on why the nested closed intervals In constructed in section 2 have intersection e. Recall that I1 = [2, 3], and that for n ≥ 2 we get In from In−1 by cutting it into n equal subintervals and taking the second one. 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Monthly 61 (1954) 235-241. 19. A. E. Parks, π , e, and other irrational numbers, Amer. Math. Monthly 93 (1986) 722- 723. 20. L. L. Pennisi, Elementary proof that e is irrational, Amer. Math. Monthly 60 (1953) 474. 21. P. Ribenboim, My Numbers, My Friends, Springer-Verlag, New York, 2000. 22. N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (2005), published electronically at http://www.research.att.com/~njas/sequences/. 23. J. Sondow, Irrationality measures, irrationality bases, and a theorem of Jarník (2004, preprint); available at http://arXiv.org/abs/math/0406300. 24. _____, Which partial sums of the Taylor series for e are convergents to e?, with an Appendix by K. Schalm (2006, preprint); available at http://arXiv.org/find/math/1/au:+sondow/0/1/0/all. 25. M. R. Spiegel, On a class of irrational numbers, Amer. Math. Monthly 60 (1953) 27- 26. B. G. Tasoev, Rational approximations to certain numbers, Math. Notes 67 (2000) 786-791. 27. B. Turner, A geometric proof that √2 is irrational, Math. Mag. 50 (1977) 263. 28. E. W. Weisstein et al, Smarandache function, MathWorld—A Wolfram Web Resource, published electronically at http://mathworld.wolfram.com/SmarandacheFunction.html. 209 West 97th Street, New York, NY 10025 [email protected]

0704.1283

Scanning tunnelling microscopy for ultracold atoms

Scanning tunnelling microscopy for ultracold atoms Corinna Kollath,1 Michael Köhl,2, 3 and Thierry Giamarchi1 DPMC-MaNEP, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom Institute of Quantum Electronics, ETH Zürich, CH-8093 Zürich, Switzerland (Dated: October 19, 2021) We propose a novel experimental probe for cold atomic gases analogous to the scanning tunnelling microscope (STM) in condensed matter. This probe uses the coherent coupling of a single particle to the system. Depending on the measurement sequence, our probe allows to either obtain the local density, with a resolution on the nanometer scale, or the single particle correlation function in real time. We discuss applications of this scheme to the various possible phases for a two dimensional Hubbard system of fermions in an optical lattice. PACS numbers: 73.43.Nq 03.75.Ss 71.10.Pm Recent advances in the field of ultracold atoms have led to a close connection between quantum gases and condensed matter physics. The achievement of strongly correlated systems and their remarkable tunability open the possibility to realize ‘quantum simulators’ for quan- tum many-body phenomena. To name one example, ul- tracold fermionic systems clarified the crossover between a BCS-state of paired fermions to a Bose-Einstein con- densate of ultracold bosonic molecules [1, 2, 3, 4, 5]. Fur- ther investigations of strongly correlated systems were initialized by the successful loading of ultracold bosonic [6] and fermionic [7] atoms into three-dimensional opti- cal lattices. In these periodic lattice potentials created by counter-propagating laser beams the physics of different lattice models can be mimicked [8, 9]. In particular the fermionic Hubbard model, which plays an important role on the way of understanding high-temperature supercon- ductivity, can be realized naturally given the short range nature of the interactions between the neutral atoms. Whereas achieving the exotic quantum phases exper- imentally appears feasible with today’s technology their clear identification remains an obstacle. Compared to condensed matter the neutrality of the cold atoms is both an advantage and a drawback since they cannot be per- turbed as easily as electrons in a solid. Possible probes are thus more sophisticated than their condensed mat- ter counterparts. In addition, for ultracold atomic gases an inhomogeneous confining potential causes the coex- istence of different spatially separated quantum phases [8, 10, 11]. This makes their realization and observation in the presence of a trapping potential very involved and creates a need for a method of probing the systems lo- cally. However, the existing probes [10, 11] still involve an averaging over regions of various densities and new techniques which allow for a local detection need to be developed. In condensed matter physics, a remarkable local probe was provided by the scanning tunneling microscope (STM) [12]. It allowed to explore and image the sur- face topography with atomic resolution, paving the way to control and analyze quantum phenomena on solid sur- faces [13]. In addition to the density analysis with un- precedented resolution, the STM has also become a spec- troscopic tool probing the local density of states. This spectroscopic method had a major impact on the un- derstanding of the physical properties of strongly corre- lated systems for which the local density of states pro- vides unique information on the physics of the system. In particular the STM has made significant contributions to the field of high temperature superconductors [14]. In this work we propose a novel experimental setup to locally probe cold atomic systems in an approach similar to and as versatile as the STM. The probe relies on the coupling of a single particle to the system. Different ‘op- erating modes’ yield either a measurement of the local density or of the single particle Green’s function in time. The realization of such a probe will open the possibili- ties to investigate exotic quantum phases in great detail as we show on the example of the Hubbard model. In extension to the conventional STM in condensed matter physics our scheme would allow for measurements in a three-dimensional sample. The key idea for the realization of an STM-like scheme with cold atoms, the ‘cold atom tunneling microscope’ is sketched in Fig. 1. A single trapped particle is used as a probe of local quantities by inducing a controlled interac- tion between the probe particle and the quantum many- body state. To allow for a precise control over the motion of the probe and to facilitate a convenient readout mech- anism, we suggest to employ a single atomic ion trapped in the vibrational ground state of a radio-frequency Paul trap [15]. In this case the spatial resolution of the micro- scope relies on the excellent control over the position and motion of trapped ions on the sub-micron scale. However, the working principle of the microscope does not depend on the charge of the probe particle and it also applies to a neutral atomic quantum dot [16] provided that the trap- ping potential of the dot has only a negligible influence http://arxiv.org/abs/0704.1283v1 FIG. 1: Sketch of the cold atom tunneling microscope. As an example the application to an anti-ferromagnetic state with alternating spin states labeled by different colors is shown. on the quantum many-body system [17]. The controlled interaction between the probe particle (the ion) and the quantum many-body system could be provided by a two- photon Raman coupling. As we show below the cold atom tunnelling microscope facilitates a local detection of the density on individual lattice sites and, quite remarkably, it even allows to per- form a spin-resolved detection of the density. The real- ization of a spin-resolved STM is a long-sought goal in condensed matter systems but has not yet been achieved. The cold atom tunnelling microscope also allows to per- form spectroscopy by observing the local single particle Green’s function 〈c σ,j(t0)cσ,j(0)〉F in time. Here cσ,j is the annihilation operator for the neutral atom on a site j with spin σ = {↑, ↓} and 〈·〉F stands for taking the expec- tation value with respect to the atomic system only. The temporal decay of this function directly reflects the na- ture of the excitations and gives thus direct information on the quantum phases present in the system. We first show taking the example of fermionic atoms in two different spin states ↑ and ↓ in an optical lattice how a measurement of the local density, the ’scanning mode’, can be achieved. It is facilitated by a two-photon Raman coupling between the ion |i〉 and an atom |aj〉 in a lattice well j by which a weakly bound molecular ion |i+aj〉 can be created (see Fig. 2). This coupling can be described by the expression σ Ωσ(t)M σIcσ,j + h.c. . Here Mσ and I are the annihilation operators for the molecular ion and the atomic ion, respectively. The coupling strength Ωσ(t) can be controlled experimentally. By choosing the correct frequency and polarization of the laser fields, the coupling is dependent on the atomic ‘spin’ state paving the way for the spin-resolved microscopy. The experimental sequence to detect the local density is as follows: At time t = 0 the atomic many-body system is prepared in its ground state |Ψ0〉. The ion is introduced into the lattice well j in state |i〉 and the Raman cou- pling is switched on for a duration δt, i.e. Ωσ(t) = Ωσ,0 if t ∈ [0, δt] and vanishes otherwise. The time δt has to be short compared to the internal time-scales of the internuclear separation Ii> x Iaj> FIG. 2: Two-photon Raman coupling of the ion |i〉 and an atom |aj〉 in a harmonic potential well to a molecular ion bound state |i + aj〉. The single photon coupling is detuned by ∆ from a resonant transition to suppress spontaneous emis- sion from the intermediate excited state. The effective two- photon Rabi frequency Ω0 is proportional to the coefficients of the single photon transitions and inversely proportional to the detuning ∆, i.e. Ω0 ∝ Ω1Ω2/∆. probed system (i.e. time-scales set by the atom-atom in- teraction U , the kinetic energy of the atoms J , and the atom-ion contact interaction Uai) to avoid a change of the many-body state during the probe sequence. The Raman coupling generates a superposition of the initial state |Ψ0〉 ⊗ |i〉 and the state cσ,j|Ψ0〉 ⊗ |i+ aj〉 in which one atom is removed from the system and a molecular ion is formed. The ratio between the amplitudes of the two states depends on the density of atoms in the well j. Hence detecting the probability (i.e. the average of the outcome of several quantum measurements) for molecule formation after the application of the Raman pulse mea- sures the local density of atoms in the lattice well j by the relation 〈 σMσ〉 = σ sin 2 (Ωσ,0δt)〈nσ,j〉F with 〈nσ,j〉 the local atomic density. The outcome of the photo-association process can be detected by measuring the changed oscillation frequency of the heavier molec- ular ion in the Paul trap or by observing the absence of resonant light scattering of the molecular ion and its reappearance after photo-dissociation [18]. The proce- dure can be repeated scanning different lattice sites as sketched in Fig. 1 with a spatial resolution on the order of 20 nm [19]. To facilitate the measurement of the den- sity with a good signal to noise ratio, the lattice potential could be increased such that the density profile on dif- ferent lattice sites is frozen and sequential measurements on single lattice site are feasible. Using the cold atom tunnelling microscope with a dif- ferent sequence, the ’tunneling mode’, allows to perform spectroscopy and to measure time dependent correlations locally. The experimental sequence is sketched in Fig. 3. As in the scanning mode we start at t = 0 in the state |Ψ0〉 ⊗ |i〉, i.e. the ground state |Ψ0〉 of the atomic sys- tem and a single atomic ion. A two-photon Raman pro- cess is applied over a short time interval δt1 to couple the ion with an atom present in the lattice well. Sub- sequently, the superposition state of the atomic and the molecular ion |i〉+ α|i+ aj〉 is removed from the system such that they are non-interacting with the remaining quantum many-body system, for example their center- of-mass position can be shifted by applying a small dc voltage. After a variable time of free evolution t0 in this isolated position they return into the addressed lattice well and the application of the two-photon Raman pro- cess is repeated for a time interval δt2. The outcome of the molecule formation is detected afterwards [28]: 〈M †M〉 = A(δt1, δt2) + sin 2(δt2Ω0) [cos(δt1Ω0)− 1] {[cos(δt1Ω0)− 1]〈nj(0)nj(t0)nj(0)〉+ 〈nj(t0)nj(0)〉} sin2(δt2Ω0) + sin 2(δt1Ω0) 〈nj(t0)〉+ sin 2(δt1Ω0) [cos(δt2Ω0)− 1] 〈c j(0)cj(t0)c j(t0)cj(0)〉 (1) A(δt1, δt2) = 2 sin(δt1Ω0) sin(δt2Ω0) cos(δt2Ω0)ℜ e−i(εM−εI )t0/h̄ j(t0)cj(0)〉 ︸ ︷︷ ︸ +(cos(δt1Ω0)− 1) 〈nj(0)c j(t0)cj(0)〉 ︸ ︷︷ ︸ Ii > Ii >+α Ii+aj> IΨ0>+β.cj IΨ0>IΨ0> detection Raman interaction Raman interaction FIG. 3: Schematics of the experimental sequence for the tun- nelling mode. The atomic ion |i〉 is introduced at the lat- tice site j into the many-body system in state |Ψ0〉. The two-photon Raman process (red) couples an atom at this lat- tice site |aj〉 to the ion with a certain amplitude. Subse- quently, the ion and the many-body system are separated for the probe time t0 during which they evolve individually. Af- ter recombination, the Raman interaction is applied again and the molecule formation is detected. We supressed the spin index and on the right hand side additionally the index F . Using appropriate measurement sequences different correlation functions can be extracted. To obtain the temporal correlation function 〈c σ,j(t0)cσ,j(0)〉F the de- scribed measurement procedure is applied sequentially: first, using δt1 = δt2 = δt for both pulses, second using δt1 = δt for the first pulse and δt2 = 2π/Ωσ,0− δt for the second pulse. Subtracting the outcome for the molecule formation of the two measurements gives ∆〈M †σMσ〉 = 2A(δt, δt). For small values of (δtΩσ,0) the pre-factor of the first summand A1 in A(δt, δt) is quadratic in (δtΩσ,0), whereas the prefactor of the second term A2 is quar- tic. Since additionally in many systems the decay of the correlation function 〈nσ,j(0)c σ,j(t0)cσ,j(0)〉 is faster or comparable to the decay of the single particle correla- tion function the second term can safely be neglected. The expression (εI −εM )t0/h̄ represents the phase dif- ference the atomic ion and the molecular ion collect dur- ing the time t0. In principle this quantity could be zeroed by choosing a suitable combination of the optical lattice field and the ion trapping fields. However, this cancella- tion is not necessary if εI − εM ≫ U, J because then the temporal evolution of the correlation function is encoded simply in the envelope of the detection signal. One direct application of the cold atom tunnelling microscope would be the identification of the quan- tum many-body phases of the two dimensional Hubbard model. In addition to the normal (Fermi liquid) quantum fluid of fermions, this model can lead to broken symmetry phases such as an anti-ferromagnet, and a strongly corre- lated (Mott) insulator. An important and yet open ques- tion is whether other more exotic phases can exist in this model, such as inhomogeneous distribution of the den- sity (stripes and checkerboards) or even superconducting phases with d-wave symmetry for the pairing. Our local probe directly detects symmetry broken phases such as the anti-ferromagnet in which the spin density is modu- lated (cf. Fig. 1) and even more inhomogeneous phases with a modulation of the density (stripes and checker- boards [20]). Additionally, even for phases with homogeneous den- sity and spin density, such as a quantum fluid or a su- perconductor, the ’tunnelling mode’ reveals the nature of the excitations by probing the single particle density of states. This, for example, allows to characterize di- rectly an s-wave or a d-wave superconductor. In Fig. 4 we plot the Fourier transform of the correlation function σ,j(t0)cσ,j(0)〉F for both an s-wave superconducting and a d-wave superconducting phase on a two-dimensional lattice. Both are obtained using the phenomenological BCS-approach using the energy dispersion on the lattice −2J(cos(kxa) + cos(kya)). In the s-wave superconduct- 0 1 2 3 ω /∆0 d-wave superconductor s-wave superconductor FIG. 4: The Fourier transform of the temporal correlation function in an s-wave and a d-wave superconducting state is shown for a gap value of ∆0 = 0.3J . ing phase, the gap ∆s(k) ≡ ∆0, clearly leads to a strong divergence of the correlation signal and a zero response in the gap below ∆0. For the d-wave superconducting phase with ∆d(k) = (cos(kxa) − cos(kya)) [21], one observes a quite different signal having a spectral weight below the gap energy. Thereby the structure of the su- perconducting order parameter and the size of the gap can be extracted from the proposed measurement. The independent control over the single particle and the neutral atomic quantum gas lies at the heart of the cold atom tunnelling microscope. To a good approxi- mation the ion experiences only the ion trapping poten- tial, the atom only the optical lattice potential and the weakly bound molecular ion both potentials. This makes the single ion a particularly attractive choice for shut- tling the atomic and the molecular ion in and out of the lattice without influencing the neutral atomic quantum many-body state. For example a displacement of 1.2mm within 50µs has been achieved without exciting vibra- tional quanta [22]. The binding energies of the weakly bound states of the atom-ion interaction potential (see Fig. 2) are determined by its asymptotic behavior scaling as −C4/r 4. Here C4 is proportional to the electric dipole polarizability of the neutral atom and r is the internuclear separation. The binding energy of the most weakly bound molecular state is two orders of magnitude less than for typical neutral atom interactions [23, 24]. Several more deeply bound states with binding energies in the 10-100MHz range are available for Raman photo-association [25]. The gener- ation of weakly bound molecules using two-photon Ra- man coupling in optical lattices has already been demon- strated for pairs of neutral atoms [26] and even the coher- ent coupling of free atomic and bound molecular states has been observed [27] which is the prerequisite for the tunneling mode. In order to probe the quantum many-body state with- out perturbations the time scales set by the parameters of the atomic system should be larger than the time inter- vals of the Raman pulses. To realize a strongly correlated phase in the lattice, the atom-atom scattering length aaa needs to be enhanced by a Feshbach resonance. Assum- ing aaa ≈ aai ≈ 10 3 a0 results in U ≃ Uai ≃ 20 kHz for the fermionic isotope 40K, whereas J is typically one order of magnitude smaller. Therefore the condition for the proposed ‘scanning’ mode J , U , Uai ≪ 1/δt can for example be fulfilled using an effective Raman coupling Ωσ,0 = 2π× 10 kHz applied over a time interval δt = 5µs resulting in a molecule formation probability of ≈ 0.1. For the ‘tunnelling’ mode the Raman coupling needs to be one order of magnitude stronger since the above condi- tion has to be fulfilled for both Raman pulses of duration δt and 2π/Ωσ,0 − δt, respectively. The shortness of the photo-association pulse has other direct benefits: first, the level shift due to the interaction Uai is not resolved and thus the measurement is independent of the occupa- tion of the lattice well by an atom in the second hyperfine state which is not probed in the spin-resolved mode. Sec- ondly, the short pulse and the subsequent removal of the molecular ion from the quantum many-body system en- sures also the stability of the microscope scheme against three-body recombination in a lattice well. In conclusion, we have proposed a novel experimental setup, the cold atom tunnelling microscope, to observe locally the (spin-resolved) density and the single parti- cle Green’s function. In contrast to previous work this measurement procedure does not average over spatially different regions of the system with coexisting quantum phases, but can resolve single lattice wells. A modifica- tion of the proposed scheme would give also access to nonlocal single particle correlation functions. The re- quired modification consists of moving the probe particle during the tunnelling mode scheme to a different lattice well, say m, before the second Raman pulse is applied. The outcome of the molecule formation then will be re- lated to the correlation function 〈c†σ,m(t0)cσ,j(0)〉F of the atomic system. Additionally the proposed setup opens the possibility to create single particle excitations in a controlled way. We would like to thank C. Berthod and H. Häffner for fruitful discussions. This work was partly supported by the SNF under MaNEP and Division II. [1] C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92, 040403 (2004). [2] M. Bartenstein et al., Phys. Rev. Lett. 92, 120401 (2004). [3] M. Zwierlein et al., Phys. Rev. Lett. 92, 120403 (2004). [4] J. Kinast et al., Phys. Rev. Lett. 92, 150402 (2004). [5] T. Bourdel et al., Phys. Rev. Lett. 93, 050401 (2004). [6] M. Greiner et al., Nature 415, 39 (2002). [7] M. Köhl et al., Phys. Rev. Lett. 94, 080403 (2005). [8] D. Jaksch et al., Phys. Rev. Lett. 81, 3108 (1998). [9] W. Hofstetter et al., Phys. Rev. Lett. 89, 220407 (2002). [10] S. Fölling, Phys. Rev. Lett. 97, 060403 (2006). [11] G. K. Campbell et al., Science 313, 469 (2006). [12] G. Binnig and H. Rohrer, Helv. Phys. Acta 55, 726 (1982). [13] M. Crommie, C. Lutz, and D. Eigler, Science 262, 218 (1993). [14] Ø. Fischer, M. Kugler, I. Maggio-Aprile, and C. Berthod, submitted to Rev. Mod. Phys. (2006). [15] D. Wineland et al., J. Res. Natl. Inst. Stand. Tech. 103, 259 (1998). [16] A. Recati et al., Phys. Rev. Lett. 94, 040404 (2005). [17] M. Bruderer and D. Jaksch, New J. Phys. 8, 87 (2006). [18] K. Sugiyama and J. Yoda, Phys. Rev. A 55, 10 (1997). [19] J. Eschner et al., Nature 413, 495 (2001). [20] J. Hoffman et al., Science 295, 466 (2002). [21] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991). [22] M.A. Rowe et al., Quantum Inf. Comp. 2, 257 (2002). [23] G. F. Gribakin and V. V. Flambaum, Phys. Rev. A 48, 546 (1993). [24] R. Cote, V. Kharchenko, and M. Lukin, Phys. Rev. Lett. 89, 093001 (2002). [25] R. J. LeRoy and R. B. Bernstein, The Journal of Chem- ical Physics 52, 3869 (1970). [26] T. Rom et al., Phys. Rev. Lett. 93, 073002 (2004). [27] C. Ryu et al., cond-mat/0508201 (2005). [28] Note that most of the terms do only appear if the system under consideration is a quantum many body system.

0704.1284

Center Manifold and Lie Symmetry Calculations on a Quasi-chemical Model for Growth-death Kinetics in Food

Center Manifold and Lie Symmetry Calculations on a Quasi-chemical Model for Growth-death Kinetics in Food Rachelle C. DeCoste Department of Mathematical Sciences The United States Military Academy West Point, NY 10996 Tel.: 845-938-2530 [email protected] Louis Piscitelle U.S. Army RDECOM, Natick Soldier Center Kansas Street, Natick, MA 01760 Tel.: 508-233-4294 [email protected] November 28, 2018 Abstract Food scientists at the U.S. Army’s Natick Solider Center have devel- oped a model for the lifecyle of the bacteria Staphylococcus aureus in intermediate moisture bread. In this article, we study this model us- ing dynamical systems and Lie symmetry methods. We calculate center manifolds and Lie symmetries for different cases of parameter values and compare our results to those of the food scientists. 1 Introduction 1.1 The model To ensure the food that U.S. soldiers receive is as safe as possible, the growth of bacteria such as Staphylococcus aureus (S. aureus) needs to be addressed. The system of equations considered in this paper arises from a “quasi-chemical” kinetics model for the phases of the microbial life cycle of S. aureus in intermedi- ate moisture bread. The food scientists who developed the model confirmed its usefulness by fitting to it the data from observations on bread crumbs with vary- ing conditions of water activity, pH and temperature. The model differs from http://arxiv.org/abs/0704.1284v1 CENTER MANIFOLDS AND LIE SYMMETRIES 2 previous models in its attempt to model continuously the growth and death of the microorganism rather than focusing solely on either growth or inactivation. The model was developed by food scientists at the Natick Soldier Center, see Taub, et al [4] for more on their techniques. The model arose from the observation of four phases in the life cycle of S. aureus. The cells pass through the various stages of metabolizing (M), mul- tiplying (M∗), sensitization to death (M∗∗), and dead (D). Additionally, the scientists hypothesized that there was an antagonist (A) present that would af- fect the cells. They found that without this added element their original model did not fit the observed data with any accuracy. The first step in the process describes cells moving from lag phase to growth phase (M →M∗). In the next step, cells multiply via binary division and then the newly multiplied cells in- teract with an antagonist (M∗ → 2M∗ + A). The last two steps represent two different pathways to death: the first with cells interacting with an antagonist, then passing to sensitization before death (A+M∗ →M∗∗ → D) and lastly the cells experiencing natural death (M∗ → D). The following equations represent the velocities of each of the above steps (v) as they relate to the concentrations of cells in various the phases. Each equation has a rate constant (k) associated to it. v1 = k1M (1) v2 = k2M ∗ (2) v3 = (10 −9)k3M ∗A (3) v4 = k4M ∗ (4) Finally these velocities are represented by the following system of ordinary dif- ferential equations: Ṁ = −v1 = −k1M (5) Ṁ∗ = v1 + v2 − v3 − v4 = k1M +M ∗(G− εA) (6) Ȧ = v2 − v3 =M ∗(k2 − εA) (7) Ḋ = v3 + v4 =M ∗(k4 + εA) (8) where G = k2−k4 is the net natural growth rate and ε = 10 −9k3. It is assumed that all the rate constants have non-negative values. The initial conditions at time zero areM(0) = I, the inoculum level I ≈ 103− 104, and M∗(0) = A(0) = D(0) = 0. 1.2 A simplification We notice that the fourth equation is uncoupled since there are no terms involv- ing the variable D in any of the other equations and Ḋ depends on M∗ and A. Therefore to investigate the dynamics of our system, we reduce to a system of three equations. Renaming our variables (y1 =M, y2 = M ∗, y3 = A) we have CENTER MANIFOLDS AND LIE SYMMETRIES 3 the following system equivalent to equations 5-8: −k1 0 0 k1 G 0 0 k2 0 −εy2y3 −εy2y3  (9) 2 Normalizing the system To consider the invariant manifold structure of a system, it is necessary to write the system in normal form as follows: ẋ = Ax+ g(x, y) (10) ẏ = By + j(x, y) (11) with (x, y) ∈ Rn × Rm, the n × n matrix A having eigenvalues with zero real part and the m×m matrix B having eigenvalues with nonzero real part. The functions g(x, y) and j(x, y) must be zero with zero first partial derivatives at the origin. The system (9) above is not in normal form since the y′3 equation corresponds to the zero eigenvalue piece and the nonlinear term of y′3 does not have all zero partial derivatives at the origin. Thus we must normalize by a change of coordinates using the eigenvectors of the matrix of the linear terms of the equation. We will investigate the invariant manifolds in a neighborhood of G = 0. Writing our system in normal form for nonzero G does not depend on the sign of G, so we treat the negative and positive case simultaneously. We let T be the matrix of eigenvectors of the eigenvalues of the linear terms of our system and let  = T 0 0 G+ k1 0 1 −k1  . (12) Using the inverse of the matrix T we can solve for u, v and w, find their derivatives and finally write our system in normal form as follows: u′ = 0 · u+ f(u, v, w) (13) 0 −k1 f(u, v, w) where f(u, v, w) = − ε (v−k1w)(k2v+G(u+k2w)). Since f(u, v, w) and its first partials with respect to u, v, and w are all zero at the origin (u, v, w) = (0, 0, 0), we have our system in normal form and we see immediately that we have a one-dimensional center manifold in the case that G 6= 0. For G > 0, we also have a one-dimensional stable and a one-dimensional unstable manifold. For G < 0, we have a two-dimensional stable manifold. The system reduced to the center manifold simply becomes u′ = 0. (15) CENTER MANIFOLDS AND LIE SYMMETRIES 4 For the case G = 0, we have a slightly simpler system of equations: −k1 0 0 k1 0 0 0 k2 0 −εy2y3 −εy2y3  (16) Note that we now have two zero eigenvalues and one negative eigenvalue for the matrix in the linear term. Since zero is a repeated eigenvalue, we must use generalized eigenvectors to find the normalization of this system. Three such eigenvectors are (0, 0, 1), (0, 1, 0), and (1,−1, k2 ). Then to transform our system we again let T be the matrix consisting of these eigenvectors and let  = T 0 0 1 0 1 −1 1 0 k2  . (17) As above, this allows us to write our system in normal form: g(u, v, w) g(u, v, w) w′ = −k1w + 0 (19) where g(u, v, w) = −ε(v−w)(u+ k2 w). Since g(u, v, w) is zero at the origin and all of its first partial derivatives are also zero at the origin, we can see that we have a two dimensional center manifold and a one dimensional stable manifold. 3 Center manifold calculations Recall that a center manifoldW c = {(x, y)|y = h(x)} is described by h(x) where h(0) = Dh(0) = 0. We consider a system written in normal form ẋ = Ax+ g(x, y) (20) ẏ = By + j(x, y) (21) with A having eigenvalues with zero real part and B eigenvalues with nonzero real part. Then we determine h(x) by finding the function that satisfies the following condition: (Mh)(x) = Dh(x)[Ax + g(x, h(x))] −Bh(x)− j(x, h(x)) = 0. The sign of G does not change the outcome of this calculation, thus we treat the case G 6= 0 at once. We have h : V → R2, V ⊂ R a neighborhood of the origin. Thus let h(x) = (h1(x), h2(x)) = (ax 2+ bx3+O(x4), cx2+dx3+O(x4)). Then f(x, h1(x), h2(x)) = ε(−a+ k1c)x 3 +O(x4) resulting in (Mh)(x) = −Gax2 + (−Gb− εa+ k1εc)x 3 +O(x4) 2 + k1dx 3 +O(x4) CENTER MANIFOLDS AND LIE SYMMETRIES 5 Solving for (Mh)(x) = 0, h1(x) = h2(x) = O(x 4). Thus up to third order, we have h1(x) = h2(x) = 0, so a center manifold is simply the u−axis. Next we consider the case G = 0. Here h : V → R, V ⊂ R2, a neighborhood of the origin. We let h(x) = h(x1, x2) = ax 2+cx1x2+dx 2. Then we calculate (Mh)(x) = hx1(x1, x2), hx2(x1, x2) k2x2 + g(x1, x2, h(x1, x2)) g(x1, x2, h(x1, x2)) + k1h(x1, x2) = (2ak2 + ck1)x1x2 + (ak1)x 1 + (ck2 + bk1)x 2 + (dk1)x 1 + (jk2 + ek1)x +(3dk2 − 2aε− cε+ fk1)x 1x2 + (2fk2 − cε− 2bε+ jk1)x1x resulting in h(x1, x2) = O(x 4), thus h(x1, x2) = 0 up to order three. Hence in this case the uv−plane is a center manifold. 4 Lie Symmetry Recall that a Lie symmetry is a map from the set of solutions of a system of differential equations to the set itself. For a system of first order ordinary differential equations k = ωk(t, y1, y2, . . . , yn), k = 1, . . . , n (22) the Lie symmetries that transform the variables t, y1, . . . , yn have infinitesimal generators of the form X = ξ∂t + η1∂y1 + η2∂y2 + · · ·+ ηn∂yn (23) where ξ = ξ(t, y1, y2, . . . , yn) and ηk = ηk(t, y1, y2, . . . , yn) for all k. The in- finitesimal generator must satisfy the Linearized Symmetry Condition: (1)(y′k − ωk) = 0, k = 1, . . . , n (24) when (22) holds. In this case the prolongation of X is as follows: X(1) = X + η 1 ∂y′1 + η 2 ∂y′2 + · · ·+ η n ∂y′n (25) where η is defined as η = Dtηk − y kDtξ. The total derivative Dt in this case is Dt = ∂t + y 1∂y1 + · · ·+ y n∂yn . Thus we have the following: = ∂tηk+y 1∂y1ηk+y 2∂y2ηk+· · ·+y n∂ynηk−y k(∂tξ+y 1∂y1ξ+y 2∂y2ξ+· · ·+y n∂ynξ). A system of first order ODEs has an infinite number of symmetries. We find symmetries by solving for the functions ξ, ηk that satisfy the Linearized Symmetry Condition (24). This condition reduces to a system of PDEs which are computationally difficult to solve. We use the “Intro to Symmetry” package in Mathematica and a script included in Cantwell [1] to calculate the symmetries CENTER MANIFOLDS AND LIE SYMMETRIES 6 for our system. We are limited in the symmetries we can calculate by our computing power. In the case G 6= 0 we calculate symmetries up to third order in our original coordinates y1, y2, and y3 and then use a change of coordinates on our symmetries to rewrite in the coordinates u, v, and w of our equations in normal form. Since the case G = 0 involves simpler equations, we are able to calculate these symmetries directly from the equations in normal form, however we followed the same method as in the G 6= 0 case since we want to be able to compare cases. 4.1 The case G 6= 0 The infinitesimals of the Lie symmetries (up to order 3) are listed in an array with {ξ, η1, η2, η3}, representing the infinitesimal generator X = ξ∂t + η1∂y1 + η2∂y2 + η3∂y3 . X1 = {1, 0, 0, 0} X2 = {y2,−k1y1y2, k1y1y2 +Gy 2 − εy 2y3, k2y 2 − εy X3 = {y3,−k1y1y3, k1y1y3 +Gy2y3 − εy2y 3 , k2y2y3 − εy2y X4 = {0,−y1, y1 + y2y3, y2y3} X5 = { t,−ty1, ty1 + ty2 − ty2y3, ty2 − ty2y3} X6 = { y1y2 + y1y2y3, y1y2 + y1y2y3} Then we transform the infinitesimal generators of the Lie symmetries found in the yi coordinates as follows. If X is an infinitesimal generator in yi, then X̃ = (Xt)∂t+(Xu)∂u+(Xv)∂v+(Xw)∂w is the corresponding infinitesimal generator for a Lie symmetry in the u, v, w coordinates The transformed symmetries in the form X̃ = {ξ̃, η̃1, η̃2, η̃3} where X̃ = ξ̃∂t + η̃1∂u + η̃2∂v + η̃3∂w: X̃1 = {1, 0, 0, 0} X̃2 = {j(u, v, w), (G− k2)j(u, v, w)f(u, v, w), j(u, v, w)(Gv + f(u, v, w)),−k1wj(u, v, w)} X̃3 = {l(u, v, w), (G− k2)l(u, v, w)f(u, v, w), l(u, v, w)(Gv + f(u, v, w)),−k1wl(u, v, w)} X̃4 = {0, (G− k2)f(u, v, w), (Gv + f(u, v, w)),−w} X̃5 = { (G− k2)f(u, v, w), (Gv + f(u, v, w)),−tw} X̃6 = {m(u, v, w), (G− k2)m(u, v, w)f(u, v, w),m(u, v, w)(Gv + f(u, v, w)),−k1wm(u, v, w) where f(u, v, w) is as above, j(u, v, w) = v − k1w, l(u, v, w) = u + v + k2w and m(u, v, w) = − 1 (G+ k1)w. CENTER MANIFOLDS AND LIE SYMMETRIES 7 4.2 The case G = 0 Again we calculate the infinitesimals of the Lie symmetries (up to order 3) of the original system with coordinates {y1, y2, y3} and list them as X = {ξ, η1, η2, η3}, representing the infinitesimal generator X = ξ∂t + η1∂y1 + η2∂y2 + η3∂y3 . X1 = {1, 0, 0, 0} X2 = {y2,−k1y1y2, k1y1y2 − εy 2y3, k2y 2 − εy X3 = {y3,−k1y1y3, k1y1y3 − εy2y 3 , k2y2y3 − εy2y X4 = {0, y1 + y2y3, y2 + y2y3} X5 = { ty1 + ty2y3, ty2 + ty2y3} X6 = { 1 + y1y2y3, y1y2 + y1y2y3} Then we transform these to the u, v, w coordinate system as above with X̃ = {ξ̃, η̃1, η̃2, η̃3} where X̃ = ξ̃∂t + η̃1∂u + η̃2∂v + η̃3∂w: X̃1 = {1, 0, 0, 0} X̃2 = {v − w, (v − w)n(u, v, w),−ε(v − w)p(u, v, w),−k1(v − w)w} X̃3 = {u+ w)n(u, v, w),−ε(u + w)p(u, v, w),−k1(u+ X̃4 = {0, n(u, v, w), p(u, v, w), X̃5 = { tn(u, v, w), tp(u, v, w), X̃6 = { wn(u, v, w), wp(u, v, w), where n(u, v, w) = εk2w(−v + w) + k1(k2v + εu(−v + w)) and p(u, v, w) = (v − w)(u + k2 5 The connections between the center manifold and the Lie symmetry Recently Cicogna and Gaeta [2] have written about the connections between dynamical systems and Lie symmetries. We are interested in particular in their results on invariant manifolds. They have commented that any Lie symmetry of the system will leave invariant both the stable and unstable manifolds. Due to the non-uniqueness of center manifolds, a Lie symmetry will map a center manifold to another (possibly the same) center manifold. The following result indicates when a center manifold given by ω(u) will be invariant under a given Lie symmetry, in their notation X = φ∂u + ψ∂v. CENTER MANIFOLDS AND LIE SYMMETRIES 8 Lemma 5.1 (Lemma 4 of [2] Chapter 7). A center manifold w(u) is invariant under a Lie symmetry X = φ∂u + ψ∂v if and only if ψ(u, ω(u)) = (∂u(ω(u))) · φ(u, ω(u)). For the case G 6= 0, ω(u) = {0, 0} giving zero on the right side of this equality. Thus the left side of this equation evaluated on the center manifold must always be zero if our center manifold is to be invariant under the action of the symmetry. This is the case with all of our Lie symmetries as given above. For example consider X2 with φ(u, v, w) = (G − k2)j(u, v, w)f(u, v, w) and ψ(u, v, w) = {j(u, v, w)(Gv + f(u, v, w)),−k1wj(u, v, w)}. Since j(u, 0, 0) ≡ 0, ψ(u, ω(u)) = ψ(u, 0, 0) = {0, 0}, thus satisfying the necessary and sufficient condition of the lemma. It is easy to determine that the remainder of the symmetries in this case also leave the center manifold invariant. Thus the center manifolds inherit these Lie symmetries. However, in this case, since v = w = 0, all of our symmetries become trivial. Recall that in the case G = 0 we found a center manifold to be the uv−plane. Now, in the notation of our lemma, ω(u) = 0, and again the right side of our equation is zero. Thus we must have ψ(u, v, 0) = 0 for any symmetry that leaves invariant this center manifold. It can be easily checked to see that all of the symmetries listed above do indeed satisfy this necessary and sufficient condition. In this case the center manifold again inherits the Lie symmetries which are now nontrivial. The restriction of the system to our center manifold, the uv−plane, is u′ = k2v − εuv (27) v′ = −εuv. (28) The nontrivial symmetries inherited by this system are X̂2 = {v, vn(u, v, 0),−εvp(u, v, 0), 0} X̂3 = {u, un(u, v, 0),−εup(u, v, w), 0} X̂4 = {0, n(u, v, 0), p(u, v, 0), 0} X̂5 = { tn(u, v, 0), tp(u, v, 0), 0} If we transform back to our original variables, we see that on the center manifold u = y3 and v = y2, resulting in the system: y′2 = −εy2y3 (29) y′3 = k2y2 − εy2y3 (30) CENTER MANIFOLDS AND LIE SYMMETRIES 9 and the symmetries: X̂2 = y2∂t + 2 − εy ∂y2 − εy 2y3∂y3 (31) X̂3 = y3∂t + k2y2y3 − εy2y ∂y2 − εy2y 3∂y3 (32) X̂4 = y2 + y2y3 ∂y2 + y2y3∂y3 (33) X̂5 = t∂t + ty2 + ty2y3 ∂y2 + ty2y3∂y3 (34) While we have calculated the infinitesimal generators, it would be interesting to determine the actual Lie symmetries on the center manifolds. We would like to say precisely what these maps do to various trajectories on the center manifold and to the flow in general. This is however, a very difficult question. There is no known method that allows us to take the infinitesimal generators of any Lie symmetry and integrate them to find the actual symmetries. The difficulty of this question is analogous to the solving of a system of differential equations analytically. For example, if we consider X̂4 with η2(t, y2, y3) = y2 + y2y3 and η3 = y2y3, this means that, letting γ be the parameter of the one-parameter Lie group, we need to solve the following for ŷ2 and ŷ3, giving us the map (ŷ2, ŷ3) as our symmetry: ŷ2 + ŷ2ŷ3 (35) = ŷ2ŷ3 (36) This is equivalent to the system above. Attempting to solve this system we find it equivalent to solving the following: ŷ2 = e +ŷ3)dγ (37) ŷ3 = e ŷ2dγ (38) with the initial conditions ŷ2(γ, y2, y3)|γ=0 = y2 and ŷ3(γ, y2, y3)|γ=0 = y3. This is something we continue to work on for this particular system as well as in general. 5.1 Comparison to previous results Based on numerical solutions of the original system of equations Ross et. al [3] predicted trajectories for M, M∗, A and D with particular emphasis on the concentrations of M∗ (cells undergoing multiplication) and A (the antagonist). They found that the behavior depended on the values of the various constants ki. In particular, with k3 = 0 and G > 0, they found unrestrained growth of both M∗ and A. For the values k3 = 0 and a negative G, A increases toward an upper limit and M∗ increases slightly but then begins to decrease toward zero. CENTER MANIFOLDS AND LIE SYMMETRIES 10 For k3 > 0 and G > 0, both M ∗ and A increase initially, but then M∗ reaches a maximum and begins to decline while A approaches an upper bound. All of these analyses combined to indicate to the food scientists that the necessary constraints for growth-death kinetics are non-zero values for k3 and positive values of G. In our consideration of the system, we also found thatM∗(= y3) and A(= y2) were the two variables that determined the behavior of the system. In the G 6= 0 case, the center manifold is the u-axis, which corresponds to A when all other variables are zero, as on the center manifold. When G = 0, the reduced system on the center manifold is given by equations 27 and 28. An inspection of this system, noting that u = A and v =M∗, shows that the behavior is qualitatively identical to that found numerically in [4] for the case k = [1 4 100 4], i.e. k2 = k4 = 4 resulting in G = 0. In both the results are that M ∗ goes to zero and A approaches a constant value. Acknowledgements This research was performed while the first author held a National Research Council Research Associateship Award jointly at the U.S. Army Natick Soldier Center, Natick, Massachusetts and the United States Military Academy, West Point, New York. References [1] B. J. Cantwell, Introduction to Symmetry Analysis, Cambridge Univer- sity Press, Cambridge, United Kingdom, 2002. [2] G. Cicogna and G. Gaeta, Symmetry and Perturbation Theory in Non- linear Dynamics, Springer-Verlag, 1999. [3] E. Ross, I. Taub, C. Doona, F. Feeherry, K. Kustin, The math- ematical properties of the quasi-chemical model for microorganism growth – death kinetics in food, International Journal of Food Microbiology, 99 (2005), pp. 157–171. [4] I. A. Taub, F. E. Feeherry, E. W. Ross, K. Kustin, and C. J.Doona, A Quasi-Chemical Kinetics Model for the Growth and Death of Staphylococcus aureus in Intermediate Moisture Bread, Journal of Food Science, 68, No. 8 (2003), pp. 2530–2537. Introduction The model A simplification Normalizing the system Center manifold calculations Lie Symmetry The case G=0 The case G=0 The connections between the center manifold and the Lie symmetry Comparison to previous results

0704.1285

Absence of commensurate ordering at the polarization flop transition in multiferroic DyMnO3

Absence of commensurate ordering at the polarization flop transition in multiferroic DyMnO3 J. Strempfer,1 B. Bohnenbuck,2 M. Mostovoy,3 N. Aliouane,4 D.N. Argyriou,4 F. Schrettle,5 J. Hemberger,5 A. Krimmel,5 and M. v. Zimmermann1 Hamburger Synchrotronstrahlungslabor HASYLAB at Deutsches Elektronen-Synchrotron DESY, 22605 Hamburg, Germany Max-Planck-Institut für Festkörperforschung, 70569 Stuttgart, Germany Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, Netherlands Hahn-Meitner-Institut, 14109 Berlin, Germany Experimentalphysik V, Universität Augsburg, 86159 Augsburg, Germany (Dated: October 26, 2018) Ferroelectric spiral magnets DyMnO3 and TbMnO3 show similar behavior of electric polarization in applied magnetic fields. Studies of the field dependence of lattice modulations on the contrary show a completely different picture. Whereas in TbMnO3 the polarization flop from P‖c to P‖a is accompanied by a sudden change from incommensurate to commensurate wave vector modulation, in DyMnO3 the wave vector varies continuously through the flop transition. This smooth behavior may be related to the giant magnetocapacitive effect observed in DyMnO3. The ability to control ferroelectric polarization (P ) with an applied magnetic field (H) in manganite per- ovskites and related materials has caused a resurgence of interest in magneto-electric phenomena. One of the most striking effects is the five-fold increase of the dielec- tric constant of DyMnO3 in magnetic field, named gi- ant magnetocapacitance.1 The key to this unprecedented sensitivity is the spiral magnetic ordering stabilized by competing exchange interactions, which forces positive and negative ions to shift in opposite directions and which can be rather easily influenced by applied mag- netic fields. Unlike conventional ferroelectrics, in the RMnO3 man- ganite perovskites such as R=Tb and Dy the emergence of ferroelectricity2,3 arises from the peculiar coupling of the lattice to a spiral ordering of Mn-spins.4,5,6 Spiral ordering is defined by the wave vector κ and the axis e around which the spins rotate. For R=Tb and Dy these two vectors are perpendicular to each other.4 The coupling of a uniform electric polarization P to an inho- mogeneous magnetization M is phenomenologically de- scribed by a term linear in the gradient ∇M , the so- called Lifshitz invariant. Such a coupling breaks the in- version symmetry of the crystal lattice in the spiral mag- netic state and induces the direction of the polarization P= γχM1M2[e×κ] , where γ and χ are a coupling con- stant and the dielectric susceptibility, M1 and M2 are amplitudes of the magnetic moments in directions per- pendicular to e.6 For R =Tb and Dy, κ is parallel to the b-axis and e parallel to the a-axis, so that the fer- roelectric polarization induced below the spiral ordering temperature TC , is parallel to the c-axis (P‖c). The application of magnetic field either parallel to the a- (H‖a) or b-axis (H‖b) leads to a flop of the polariza- tion from P‖c to P‖a. The flop in the polarization is interpreted as the flop of the vector e from the a- (spins within the bc-plane) to the c-direction (spins within the ab-plane). For R=Tb these field-induced flops of P oc- cur at critical field of HaC∼8 T and H C∼4.5 T at 4 K for field parallel to the a- and b-axis respectively and are as- sociated with a first order transition from an incommen- surate (IC) low field magnetic ordering to a commensu- rate (CM) high field phase. The H-T -phase diagram of DyMnO3 is very similar to the one of TbMnO3, with the same characteristic flops of P but at lower critical fields, HaC∼6.5 T and H C∼1 T at 2 K. In this paper we show, that although the phase di- agrams of R=Dy and Tb may be qualitatively similar, their structural behavior at HC is completely different. We find that for DyMnO3 the polarization flop is not as- sociated with a transition to a CM phase as in the case of R =Tb, but rather the magnitude of the wave vec- tor changes very little across HC for both H‖a and H‖b configurations. We argue that the magnitude of the in- commensurability for R=Dy does not lie sufficiently close to a CM value, as opposed to R=Tb, making the IC high field phase energetically more favorable. For R=Dy, Mn-spins order to form a longitudinal spin density wave (SDW) below TN∼39 K with wave vector κ = δmb ∗ with δMnm ∼ 0.36...0.385 3 determined on the basis of lattice reflections with δl=2δm 3,7,8 that arise from a coupling of the IC magnetic ordering to the lattice via a quadratic magneto-elastic coupling.9 With further cool- ing δMnm decreases down to TC=19 K, where a second transition into the spiral phase occurs. Coincident with the transition to a spiral phase a spontaneous electric po- larization parallel to the c-axis is found.2 Below T Dy magnetic moments order commensurately with prop- agation vector 1 ∗.8 In the study presented here, struc- tural first and second harmonic reflections related to the magnetic first harmonic reflections are investigated. Single crystalline samples were prepared using the floating zone technique at the HMI. Details of sample preparation and characterization are given elsewhere.8 Due to the high neutron absorption cross section of Dy, in-field neutron diffraction is not ideal to investigate field induced magneto-structural transitions in DyMnO3. Rather to investigate the structural response to the field http://arxiv.org/abs/0704.1285v1 induced polarization flop we have utilized in-field syn- chrotron x-ray diffraction with the two field configura- tions H‖a and H‖b. Measurements with H‖a were per- formed at beamline X21 at the National Synchrotron Light Source at Brookhaven National Laboratory with a photon energy of 9.5 keV, using a 13 T Oxford cry- omagnet with vertical field. Measurements in the H‖b configuration were carried out at beamline BW5 at HA- SYLAB with a photon energy of 100 keV using a 10 T Cryogenics cryomagnet with horizontal field. Magne- tization measurements were performed with a Physical Properties Measurement System (PPMS) on a 14.5 mg DyMnO3 sample. The spontaneous electric polarization as a function of field and temperature was determined from the pyro-current recorded in a PPMS-system using an electrometer. Measurements of the spontaneous electric polarization performed on the same samples used in our diffraction ex- periments in magnetic field configurations H‖a and H‖b (insets in Fig. 1c and Fig. 2 a) confirm that spontaneous ferroelectric polarization is present below TC=19 K as already reported in Ref. 3. The application of H‖b be- low TC(Dy) results in the suppression of P‖c (inset of Fig. 1c). Kimura et al. show in addition that this de- crease is accompanied by an increase in P‖a indicative of the flop in the ferroelectric polarization.3 In Fig. 1a-b we show the temperature dependence of the wave vector and integrated intensity of the second harmonic reflection (0, 4-2δ, 0), both for decreasing and increasing temperature. The dependences are compared for zero field (P‖c), and µ0H‖b = 10 T (P‖a). In both data sets, we find a significant hysteresis in the intensity as well as in the magnitude of δ around TC=18 K, which is associated with the onset of ferroelectricity. However, despite the fact that P lies along different axes for 0 and 10 T, we observe no significant change between these two measurements. In Fig. 1c-d, we show the field depen- dence of δ and of the integrated intensity of the same reflection at T=2 K. Here we find a small initial increase of δ up to µ0H C=1 T where the flop in polarization is found while the intensity of the same reflection shows a steady increase with increasing field. Above HbC , we find a small decrease of δ with increasing field. This behavior is in sharp contrast to TbMnO3 where δ varies slowly with increasing field and locks in above HbC into a CM value of κ= 1 ∗ at HbC . In Fig. 1e-f, the temperature dependence of δ and the intensities of first and second harmonic reflections is shown for data measured in field cooling with µ0H‖b = 2.5 T. Here we find a different behavior in the intensities of IC reflections. For wave vector along the b∗-direction (0, 4-2δ, 0), a strong hysteresis is observed in its inten- sity as a function of temperature, whereas this hystere- sis is absent for wave vectors that are mainly along the c∗-direction ( (0, δ, 5) and (0, 2δ, 5)) (Fig. 1f). Neverthe- less, δ and its hysteresis are the same for all reflections (Fig. 1e). The described intensity behavior is similar to what we have recently observed in zero field temperature µ0H=10T 0 10 20 30 40 T (K) (0 4−2δ 0) (a) H||bTC 0 10 20 30 40 T (K) t. In sity (a µ0H=0T 0 2 4 6 8 10 µ0H (T) 0 2 4 6 8 10 µ0H (T) t. In sity (a (0 4−2δ 0) 0 10 20 30 40 T (K) µ0H||b = 2.5 T TC TNTPF 0 10 20 30 40 T (K) t. In sity (a (0 δ 5)(f) TNTPF (0 2δ 5)*10 (0 4−2δ 0) 0 10 20 30 40 T (K) ) H||b FIG. 1: (color online) Temperature dependence of (a) the in- commensurability δ and (b) the respective intensities of the (0, 4-2δ, 0) structural reflection. Data are shown for µ0H = 0 T and µ0H‖b = 10 T for decreasing (open symbols) and in- creasing (closed symbols) temperature. In panel (c) and (d), δ and intensity variation as function of magnetic field, respec- tively, are shown for a sample temperature of T = 2 K, in the same axes ranges as in (a) and (b). In the inset, spontaneous electric polarization P‖c is shown as function of tempera- ture for magnetic field orientation H‖b. Temperature depen- dence of (e) wave vector and (f) the respective intensities for µ0H‖b = 2.5 T and wave vectors (0, δ, 5), (0, 2δ, 5) and (0, 4-2δ, 0) for decreasing (open symbols) and increasing (closed symbols) temperature. The (0, 2δ, 5) intensities in (f) are multiplied by a factor of 10. dependent measurements using resonant x-ray scatter- ing from a single crystal of DyMnO3. 11 In these mea- surements a similar hysteresis was observed to be associ- ated with the induced ordering of Dy-spins with the same propagation vector as that for the Mn spin ordering. Fi- nally for this field configuration we note that at 2.5 T, the polarization flop from P‖a to P‖c is expected with increasing temperature at TPF ∼ 12 K (inset in Fig. 1c). However our diffraction measurements find no anomaly either on the temperature dependence of the wave vector or in the intensities at this temperature as it was found in TbMnO3. We now turn our attention to measurements conducted 0 2 4 6 8 10 µ0H (T) t. In sity (a its)(0 2δ 5) Hc(a) (0 1−δ 5) 0 2 4 6 8 10 µ0H (T) Hc(a) 0 10 20 30 40 T (K) ) H||a FIG. 2: (color online) Field dependences with field H‖a of (a) the incommensurability and (b) the integrated intensity mea- sured at T = 8 K. The error bars in (a) represent the HWHM of the superlattice reflection. In the inset, spontaneous elec- tric polarization P‖c is shown as function of temperature for magnetic field H‖a. in the H‖a configuration. In Fig. 2a-b the field depen- dence of the intensity and wave vector of the first and second harmonic reflections (0, 1-δ, 5) and (0, 2δ, 5) is shown, measured at T = 8 K above the ordering tempera- ture of Dy of T ∼ 5 K. The second harmonic reflection decreases in intensity with increasing field and vanishes at 5 T, while the first harmonic appears only at a field of 4 T and saturates in intensity at 7 T. This behavior is not directly related to the polarization flop but reflects a variation of the modulation direction of the strain wave leading to the superlattice reflections. However the most remarkable behavior is found for the field dependence of the wavevector. Here, the wave vector of the first har- monic reflection does not change significantly through the polarization flop transition at µ0H C∼ 6.5 T. 3 Again this is in sharp contrast to the behavior of TbMnO3, where a discontinuous transition to a CM phase is found. Below 6K, Dy spins order commensurately with δDym = . Previously we have argued that the lattice distor- tion associated with this magnetic ordering is not CM but rather IC with δl ∼ 0.1b ∗.8 In our measurements we found the half-integer magnetic reflection to be extremely weak and we focused our attention to the lattice δl ∼ 0.1 satellite measured at (0, 0.9, 5). In Fig. 3a we show a series of scans at different fields applied along the a-axis at 3 K which show the rapid suppression of the δl ∼ 0.1 satellite which vanishes for fields µ0H‖a > 1 T. The CM magnetic reflection (0, 0.5, 5) is only measurable at zero field and can not longer be observed at 1 T. The tem- perature dependence of the intensity and wavevector of the (0, 0.9, 5) reflection is shown in Fig. 3b-c, revealing the rapid suppression of the intensity of this reflection at low temperature in accordance with the appearance of ferromagnetic order (inset Fig. 3c), indicating an easy axis along the b-direction. The suppression of this re- flection with magnetic field is analogous to a similar be- havior found for R =Tb which coincided with the ferro- magnetic ordering of Tb-spins for the same field size and configuration.10 The polarization flop is driven by the field-induced 0.4 0.6 0.8 k (r.l.u.) s) (a) µ0H=0T µ0H=1T µ0H=1.5T (0 0.9 5), µ0H = 0T (0 0.9 5), µ0H = 0.5T (0 2δ 5), µ0H = 0T (0 0.9 5), µ0H = 1T 0 5 10 15 20 T (K) (0 0.9 5) 0 2 4 6 8 10 B (T) ) H||b (T=2K) H||a (T=8K) FIG. 3: (color online) (a) Scans along (0, k, 5) at T = 3 K for different applied magnetic fields with intensity in logarithmic scale. (b) Temperature dependence as function of field H‖a of the integrated intensity of the superlattice reflections (0, 0.9, 5) in the Dy ordered phase and the second harmonic reflection (0, 2δ, 5) due to Mn order. Open and closed symbols repre- sent increasing and decreasing temperature, respectively. In (c) the wave vector of the (0, 0.9, 5) reflection is shown as function of temperature for different fields. The inset shows magnetization data of DyMnO3 for H‖a and H‖b. flop of the axis of rotation of the magnetic spiral e. Its direction is determined by magnetic anisotropy terms, which to lowest order of the free energy expansion in powers of magnetization of Mn spins have the form,∑ α=a,b,c α. Phenomenologically the spin flop tran- sition results from the field dependence of the coefficients aα. Below the critical field HC , ab < ac < aa, which fa- vors spins rotating in the bc plane (e‖a), while aboveHC , ab < aa < ac, favoring the rotation in the ab-plane (e‖c). From the perspective of symmetry there is no restric- tion that the high field spiral phase must be CM. In this view the fact that there is a magneto-elastic phase tran- sition to a CM phase associated with the flops in the polarization for R =Tb would appear to be a special case, especially when compared to R =Dy where we find no such transition at HC . The difference in behavior between these two multiferroics is not of fundamental nature but rather simply lies in the magnitude of the incommensurability. For R=Tb, δm=0.28 r.l.u. is close to the CM value of 1 . For this CM value of the wave vector the amplitude of the Mn magnetic moment is not modulated10 and may thus be energetically more favor- able than an IC amplitude modulated phase. In the phe- nomenological approach the CM state with δ = 0.25 r.l.u. is stabilized by M4 terms in the Landau expansion. For R=Dy the value of δ=0.38 r.l.u. is further away from a CM value (1 ) and thus a transition to it is not favorable energetically. Therefore the Mn spin spiral may indeed flop as expected but without a change in δ.13 Clearly here the modulation period in real space is much shorter, and the amplitude of the Mn magnetic moment is modulated. A significant difference between the two multiferroics is found in the behavior of the complex magnetic or- dering of R-ions. For R=Tb, Tb spins are induced to order along the a-axis below TN with the same period- icity as Mn-spins and below T TbN =7 K they order sepa- rately with δTbm =0.42 r.l.u. At low temperatures, when H‖a∼1 T, Tb-spins are aligned ferromagnetically, while for field along the b-axis, δTbm jumps discontinuously to a CM value of 1 at 1.75 T.10 The behavior for R=Dy is much simpler, as below T =6 K the Dy spins order with δDym = . Here, the suppression of the IC-reflections together with the magnetization data indicates a melting of the antiferromagnetic Dy ordering, a behavior different to TbMnO3. On the basis of the CM ordering of Mn-spins for the P‖a phase for R=Tb it has been suggested that ferro- electricity may arise in the absence of a spiral magnetic ordering. Here an exchange striction mechanism pro- posed from competing ferromagnetic and antiferromag- netic super-exchange interactions predicts a P‖a phase for a CM ordering with δ = 1 . This model holds strictly for a CM ordering and suggests that for R=Dy the spi- ral phase must be maintained at high fields to support a ferroelectric state. The smooth magnetic field dependence of the spiral wave vector in DyMnO3 at the spin flop transition may explain the large increase of the dielectric constant εa, observed in this material.1,3 If higher-order terms in the Landau expansion of free energy could be neglected, then at the spin flop transition the magnetic excitation spec- trum of the spiral would acquire a zero mode, since for aa (HC) = ac (HC) there is a freedom to rotate the spi- ral plane around the b axis. This mode can be excited by electric field E‖a normal to the spiral bc-plane and is the electromagnon studied in Ref. 12. Its softening at H = HC would result in divergence of static dielectric susceptibility εa. In reality, due to higher-order terms in the Landau expansion the spin flop transition is of first order, the softening of the magnetic mode is not complete and the peak value of the dielectric constant is finite. Still, in DyMnO3 this transition is close to a second-order one in the sense that the spirals above and below the critical field are essentially the same except for their orientation. The softness of the spiral magnetic or- dering at the critical field may be the reason behind the large magnetocapacitance observed in DyMnO3 which becomes truly gigantic close to the tricritical point at the crossing of the collinear and two spiral phases with P‖c and P‖a, where higher-order terms are small. In TbMnO3 the spin flop transition is strongly discontin- uous due to the concomitant IC-CM transition, which limits the growth of the dielectric constant. In summary, in-field synchrotron X-ray diffraction measurements from a DyMnO3 single crystal have shown that there is no change of the wave vector δ associated with the flop of the ferroelectric polarization P at HC for both H‖a and H‖c. This is in sharp contrast to similar measurements reported for TbMnO3 were a transition to a CM phase is found at HC for the same field configu- rations. We argue that the magnitude of the incommen- surability for R =Dy does not lie sufficiently close to a CM value, as opposed to R=Tb, making the IC high field phase energetically more favorable. Acknowledgments We would like to thank C.S. Nelson for the assistance at the experiment at NSLS. Work at Brookhaven was supported by the U.S. Department of Energy, Division of Materials Science, under Contract No. DE-AC02- 98CH10886. 1 T. Goto, T. Kimura, G. Lawes, A. P. Ramirez, and Y. Tokura, Phys. Rev. Lett. 92, 257201 (2004). 2 T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, Nature 426, 55 (2003). 3 T. Kimura, G. Lawes, T. Goto, Y. Tokura, and A. P. Ramirez, Phys. Rev. B 71, 224425 (2005). 4 M. Kenzelmann, A. B. Harris, S. Jonas, C. Broholm, J. Schefer, S. B. Kim, C. L. Zhang, S.-W. Cheong, O. P. Vajk, and J. W. Lynn, Phys. Rev. Letters 95, 087206 (2005). 5 H. Katsura, N. Nagaosa, and A.V. Balatsky, Phys. Rev. Lett. 95, 057205 (2005). 6 M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006). 7 T. Kimura, S. Ishihara, H. Shintani, T. Arima, K. T. Taka- hashi, K. Ishizaka, , and Y. Tokura, Phys. Rev. B 68, 060403(R) (2003). 8 R. Feyerherm, E. Dudzik, N. Aliouane, and D. N. Argyriou, Phys. Rev. B 73, 180401(R) (2006). 9 M. B. Walker, Phys. Rev. B 22, 1338 (1980). 10 N. Aliouane, D. N. Argyriou, J. Strempfer, I. Zegkinoglou, S. Landsgesell, and M. v. Zimmermann, Phys. Rev. B 73, 020102(R) (2006). 11 O. Prokhnenko, R. Feyerherm, E. Dudzik, S. Landsgesell, N. Aliouane, L.C. Chapon, and D.N. Argyriou, Phys. Rev. Letters 98, 057206 (2007). 12 A. Pimenov, A. A. Mukhin, V. Yu Ivanov, V. D. Travkin, A. M. Balbashov, and A. Loidl, Nature Physics 2, 97 (2006). 13 We note that a value of δ = 1 is observed for Tb1−xDyxMnO3, see Arima et al. Phys. Rev. Lett. 96, 097202 (2006).

0704.1286

Study of a finite volume - finite element scheme for a nuclear transport model

arXiv:0704.1286v1 [math.NA] 10 Apr 2007 IMA Journal of Numerical Analysis (2007) Page 1 of 26 doi: 10.1093/imanum/dri017 Study of a finite volume- finite element scheme for a nuclear transport model CATHERINE CHOQUET LATP, Université Aix-Marseille 3, 13013 Marseille Cedex 20, France SÉBASTIEN ZIMMERMANN Laboratoire de mathématiques, École Centrale de Lyon, 36 avenue Guy de Collongue, 69134 Ecully, France. [Received on 24 July 2007] We consider a problem of nuclear waste contamination. It takes into account the thermal effects. The temperature and the contaminant’s concentration fulfill convection-diffusion-reaction equations. The ve- locity and the pressure in the flow satisfy the Darcy equation, with a viscosity depending on both con- centration and temperature. The equations are nonlinear and strongly coupled. Using both finite volume and nonconforming finite element methods, we introduce a scheme adapted to this problem. We prove the stability and convergence of this scheme and give some error estimates. Keywords: porous media, miscible flow, nonconforming finite element, finite volume. 1. Introduction A part of the high-level nuclear waste is now stored in environmentally safe locations. One has to consider the eventuality of a leakage through the engineering and geological barriers. It may cause the contamination of underground water sources far away from the original repository’s location. In the present paper, we consider such a problem of nuclear waste contamination in the basement. We take into account the thermal effects. The evolution in time of the temperature and of the contaminants concen- tration is then governed by convection-diffusion-reaction equations. The velocity and the pressure in the flow satisfy the Darcy equation, with a viscosity depending on both concentrations and temperature in a nonlinear way. The velocity satisfies an incompressibility constraint. We introduce a scheme adapted to this problem. We use both finite volume and nonconforming finite element methods. It ensures that a maximum principle holds and that the associated linear systems have good-conditioned matrices. We prove the stability and convergence of the scheme and give some error estimates. Let us briefly point out some previous works. A complete model coupling concentrations and pres- sure equations is very seldom studied, since the system is strongly coupled. Instead, each equation is considered separately. In the general context of convection-diffusion-reaction equations, numerous schemes are available (see [15] or [19] and the references therein). Finite difference schemes are some- times used for the convective term (in [20] for instance). But they are not adapted for the complex geometry of a reservoir. More recently, finite volume methods were developed and analysed. Let us just cite the book [2] or [13] and the references therein. Finite elements (for the diffusive term) and finite volumes (for the convective term) are coupled for instance in [3, 8] . In convection dominant problems, the equations are of degenerate parabolic type. This setting is considered in [9, 18] . The reaction terms are specifically studied through operator splitting methods in [14] . Now, in the specific context of porous media flow, we mention [17] who consider only the evolution of the pressure. In [11, 12] a more complete set of equations is used, and a mixed finite element approximation is developed. We stress that IMA Journal of Numerical Analysis c© Institute of Mathematics and its Applications 2007; all rights reserved. http://arxiv.org/abs/0704.1286v1 2 of 26 C. CHOQUET AND S. ZIMMERMANN in all theses works, as in most, the thermic effects are not taken into account. The present paper is organized as follows. Section 2 is devoted to the derivation of the model. In section 3, we introduce the discrete tools used in this paper. It allows us to define the numerical scheme of section 5. The analysis of the scheme uses the properties of section 4. We then prove the stability and convergence of the scheme, in sections 6 and 7 respectively. We conclude with some error estimates in section 8. 2. Model of contamination The thickness of the medium is significantly smaller than its length and width. Hence it is reasonable to average the medium properties vertically and to describe the far-field repository by a polyhedral domain Ω of R2 with a smooth boundary ∂Ω . It is characterized by a porosity φ and a permeability tensor K. The time interval of interest is [0,T ]. We denote by p the pressure, by (ai) i=1 the concentrations of the Nr radionuclides involved in the flow and by θ the temperature. The Darcy velocity is represented by u. We assume a miscible and incompressible displacement. Due to the mass and energy conservation principles, the flow is governed by the following system satisfied in Ω × [0,T ], with i = 1, ..,Nr −1 (see [10]). φ Ri ∂tai + div(ai u)− div(φ Dc ∇ai) = si − sai −λi Ri φ ai + j=1, j 6=i ki j λ j R j φ a j , (2.1) φ Cp ∂tθ + div(θ u)− div(φ Dθ ∇θ ) =−sθ − s(θ −θ∗) , (2.2) divu = s, u+ (a j) j=1 ,θ ) ∇p = f. (2.3) In (2.1) the retardation factors Ri > 0 are due to the sorption mechanism. The real λ−1i > 0 denotes the half life time of radionuclide i. The term −λi Ri φ ai describes the radioactive decay of the i–th specy. Meanwhile, the quantity ∑ j 6=i ki j λ j R j φ a j is created by radioactive filiation. The molecular diffusion effects are given by the coefficient Dc > 0. The contamination is represented by the source term si and s=∑Nri=1 si. In (2.2) the coefficient Cp > 0 is the relative specific heat of the porous medium. The thermic diffusion coefficient is denoted by Dθ > 0. The real θ∗ > 0 is a reference temperature. The constitutive relation (2.3) is the Darcy law and f is a density of body forces. For a large range of temperatures µ has the form µ(a,θ ) = µR(a)exp where µR is a nonlinear function. For instance, in the Koval model for a two-species mixture [16], we µR(a) = µ(0)(1+(M1/4 − 1)a1)−4 where M = µ(0)/µ(1) is the mobility ratio. We notice that the equations (2.1)-(2.3) are strongly coupled. Moreover, every concentration equation (2.1) involves a different time scale. Therefore, it is difficult to build a numerical scheme that captures all the physical phenomena. We have to transform these equations. We first assume that only serial or parallel first-order reactions occur, so that ki j = yi with y1 = 0. Next, following [4], we assume that no two isotopes have identical decay rates and we set c1 = a1, ci = ai + yl+1λl λl −λi a j for i = 2, ..,Nr. (2.4) Study of a finite volume-finite element scheme for a nuclear transport model 3 of 26 Lastly, without losing any mathematical difficulty (see remark 2.2 below), we set Ri = 1 for i= 1, ..,Nr− 1 and φ = 1, Cp = 1. We also set sci = si +∑ ∏i−1l= j yl+1λl λl−λi s j and κ(c,θ ) = K/µ i=1 ,θ . The contamination problem is now modelized by the following parabolic-elliptic system ∂tci + div(ci u)−Dc ∆ci = sci − sci −λi ci , (2.5) ∂tθ + div(θ u)−Dθ ∆θ =−sθ − s(θ −θ∗) , (2.6) divu = s, u+κ(c,θ )∇p = f , (2.7) with i = 1, ..,Nr − 1. These equations are completed with the boundary and initial conditions ∇ci ·ν = 0 , ∇θ ·ν = 0 , u ·n = 0 , (2.8) ci|t=0 = ci0, θ |t=0 = θ0. (2.9) The pressure p is normalized by Ω pdx = 0. Equations (2.5) are all similar. Thus, for the sake of simplicity, we will assume that there is only one. We set Nr = 2 and c = c1, c0 = c 0, sc = sc1 , λ = λ1. The results of this paper readily extend to the general case. We conclude with some notations and hypothesis. Let D be a bounded open set of Rk with k > 1. We denote by C ∞0 (D) the set of functions that are continuous on D together with all their derivatives, and have a compact support in D. For p ∈ {2,∞}, we use the Lebesgue spaces Lp(D),‖.‖Lp(D) Lp(D),‖.‖Lp(D) with Lp = (Lp)2. We also use the Sobolev spaces W p,q(D) for p ∈ [1,∞[ and q ∈ [1,∞[. In the case D = Ω we use the following conventions. We drop the domain dependancy. We denote by |.| (resp. ‖.‖∞) the norms associated to L2 = L2(Ω) and L2 = L2(Ω) (resp. L∞ and L∞). We set L20 = {v ∈ L2; Ω v(x)dx = 0}. For p ∈ [1,∞[ we define (H p,‖.‖p) and (Hp,‖.‖p) with H p = W p,2 and Hp = (H p)2. Now let (X , |.|) be a Banach space. The functions g : [0,T ] → X such that t → ‖g(t)‖X is continuous (resp. bounded and square integrable) form the set C (0,T ;X) (resp. L∞(0,T ;X) and L2(0,T ;X)). The associated norm for the space L∞(0,T ;X) (resp. L2(0,T ;X)) is defined by ‖g‖L∞(0,T ;X) = supt∈[0,T ] ‖g(t)‖X (resp. ‖g‖L2(0,T ;X) = 0 ‖ f (t)‖2X dt ). Finally in all computations we use C > 0 as a generic constant. It depends only on the data of the problem. We assume the following regularities for the data in (2.5)–(2.7) κ ∈W 1,∞((0,1)× (0,∞)) , s, sc, sθ ∈ L2 , f ∈ C (0,T ;L2). (2.10) We also assume that κ > κin f with κin f > 0. For the initial data, we assume that c0 ∈ H1, θ0 ∈ H1, and that we have a.e. in Ω 0 6 c0(x)6 1 , θ− 6 θ0(x)6 θ+ (2.11) with θ− > 0. Finally we assume that we have a.e. in Ω 2s(x)+λ > sc(x)> 0 , 2(θ−−θ ∗)s(x)+ sθ (x)6 0 , 2(θ+−θ ∗)s(x)+ sθ (x)> 0. (2.12) These conditions ensure a maximum principle (proposition 6.1 below). REMARK 2.1 We have assumed that first-order reactions occur, so that the coefficients ki j in (2.1) depend only on i. If ki j depends on i and j, one can still uncouple the equations by iterating the transfor- mation (2.4), provided that k1 j = 0 for j = 2, ..,Nr −1. This assumption means that the first long-lasting isotope disappears and is not created anymore. It is satisfied by many radionuclides. 4 of 26 C. CHOQUET AND S. ZIMMERMANN REMARK 2.2 We have assumed that the retardation factors R j are identical. If it is not the case, the difficulty and the approach remain the same. Indeed, let us consider the Fourier transform of (2.1). For a Fourier mode â j(k, t) we obtain (φ R j â j) =−φ (λ j R j + k2 Dc − ik u) â j + yi R j−1 φ â j−1 =−λ ′j R j φ â j + yi R j−1φ â j−1 with λ ′j = λ j +(k 2 Dc − ik u)/R j for j = 1, ..,Nr − 1. A transform analogous to (2.4) uncouples the problem. By taking the partial differential equation counterpart, we obtain an equation similar to (2.5). 3. Discrete tools 3.1 Mesh and discrete spaces Let Th be a triangular mesh of Ω . The circumscribed circle of a triangle K ∈ Th is centered at xK and has the diameter hK . We set h = maxK∈Th hK . We assume that all the interior angles of the triangles of the mesh are less than π2 , so that xK ∈ K. The set of the edges of the triangle K ∈ Th is EK . The symbol nK,σ denotes the unit normal vector to an edge σ ∈ EK and pointing outward K. We denote by Eh the set of the edges of the mesh. We distinguish the subset E inth ⊂ Eh (resp. E h ) of the edges located inside Ω (resp. on ∂Ω ). The middle of an edge σ ∈ Eh is xσ and its length is |σ |. For each edge σ ∈ E inth let Kσ and Lσ be the two triangles having σ in common; we set dσ = d(xKσ ,xLσ ). For all σ ∈ E exth only the triangle Kσ located inside Ω is defined and we set dσ = d(xKσ ,xσ ). Then for all σ ∈ Eh we set τσ = We assume the following on the mesh (see [13]): there exists C > 0 such that ∀σ ∈ Eh , dσ >C |σ | and |σ |>C h. It implies that there exists C > 0 such that ∀σ ∈ E inth , τσ = |σ |/dσ >C. (3.1) We define on the mesh the following spaces. The usual space for finite volume schemes is P0 = {q ∈ L2 ; ∀K ∈ Th, q|K is a constant}. For any function qh ∈ P0 and any K ∈ Th we set qK = qh|K . We also consider Pd1 = {q ∈ L2 ; ∀K ∈ Th, q|K is affine} , Pc1 = {qh ∈ Pd1 ; qh is continuous over Ω} , Pnc1 = {qh ∈ Pd1 ; ∀σ ∈ E inth , qh is continuous at the middle of σ}. We have Pc1 ⊂ H1. On the other hand Pnc1 6⊂ H1, but Pnc1 ⊂ H1d with H1d = {q ∈ L2 ; ∀K ∈ Th, q|K ∈ H1(K)}. Thus we define ∇̃h : H1d → L2 by setting ∀qh ∈ H1d , ∀K ∈ Th, ∇̃hqh|K = ∇qh|K (3.2) and the associated norm ‖.‖1,h is given by ∀qh ∈ H1d , ‖qh‖1,h = (|qh|2 + |∇̃hqh|2)1/2. We then have the following Poincaré-like inequality for the space Pnc1 ∩L20 (see [1]). Study of a finite volume-finite element scheme for a nuclear transport model 5 of 26 PROPOSITION 3.1 There exists C > 0 such that |qh|6C |∇̃hqh| for all qh ∈ Pnc1 ∩L20. We also define discrete analogues of the norms H1 and H−1 for the space P0 by setting ‖qh‖h = σ∈E inth τσ (qLσ − qKσ )2 and ‖qh‖−1,h = sup ψh∈P0 (qh,ψh) ‖ψh‖h for any function qh ∈ P0. Note that for any ph ∈P0 and qh ∈ P0, (ph,qh)6 ‖qh‖−1,h‖qh‖h. The following Poincaré-like inequality holds for the space P0 ∩L20 (see [13]). PROPOSITION 3.2 There exists C > 0 such that |qh|6C‖qh‖h for all qh ∈ P0 ∩L20. Finally we set P0 = (P0)2, Pd1 = (P 2 and use the Raviart-Thomas spaces [7] RTd0 = {vh ∈ Pd1 ; ∀K ∈ Th , ∀σ ∈ EK , vh|K ·nK,σ is a constant} , RT0 = {vh ∈ RTd0 ; ∀σ ∈ E inth , vh|Kσ ·nKσ ,σ = vh|Lσ ·nKσ ,σ and vh ·n|∂Ω = 0}. For all vh ∈ RT0, K ∈ Th and σ ∈ EK , we set (vh ·nK,σ )σ = vh|K ·nK,σ . 3.2 Projection operators We associate with the spaces of section 3.1 some projection operators. First, we define ΠPc1 : H d → Pc1 by setting ∀q ∈ H1d , ∀φh ∈ Pc1 , ∇(ΠPc1 q),∇φh = (∇q,∇φh). (3.3) Next, we consider the space P0. Let Cd = {qh ∈ L2 ; qh is equal a.e. to a continuous function}. We define ΠP0 : L 2 → P0 and Π̃P0 : Cd → P0 by setting (ΠP0 p)K = p(x)dx , (Π̃P0q)K = q(xK) , (3.4) for all p ∈ L2, q ∈ Cd and K ∈ Th. We also set ΠP0 = (ΠP0)2. For the space Pnc1 , we define Π̃Pnc1 : L Pnc1 and ΠPnc1 : H 1 → Pnc1 . For all p ∈ L2 and q ∈ H1, Π̃Pnc1 p and ΠPnc1 q satisfy ∀ψh ∈ Pnc1 , (Π̃Pnc1 p,ψh) = (p,ψh) ; ∀σ ∈ Eh , (ΠPnc1 q)dσ = qdσ . (3.5) For the space RT0, we define Π̃RT0 : L 2 → RT0 and ΠRT0 : H1 → RT0. For all v ∈ L2 and w ∈ H1, Π̃RT0v and ΠRT0w satisfy ∀wh ∈ RT0 , (Π̃RT0v,wh) = (v,wh) ; ∀σ ∈ E (w−ΠRT0w) ·nKσ ,σ dσ = 0. (3.6) The operators ΠP0 , Π̃Pnc1 (resp. ΠP0 , Π̃RT0) are L 2 (resp. L2) projection operators. They are stable for the L2 (resp. L2) norms. The operators Π̃P0 , ΠPnc1 and ΠRT0 are interpolation operators. The following estimates are classical ([6] p.109 and [7]). PROPOSITION 3.3 There exists C > 0 such that for all q ∈ H1 and v ∈ H1 |q−ΠP0q|6C h‖q‖1, |v−ΠRT0v|6C h‖v‖1. 6 of 26 C. CHOQUET AND S. ZIMMERMANN For all p ∈ H1 and q ∈ H2 we have |p−ΠPnc1 p|6C h‖p‖1 , |∇̃h(q−ΠPnc1 q)|6C h‖q‖2. For all q ∈ Hd1 we have |q−ΠPc1 q|6C h‖q‖1,h. Finally, using the Sobolev embedding theorem, one checks that |ΠP0q− Π̃P0q|6C h‖q‖W1,r (3.7) for all q ∈W 1,r with r > 1 (see [22]). 3.3 Discrete operators Equations (2.5)–(2.7) use the differential operators gradient, divergence and laplacian. We have to define analogous operators in the discrete setting. The discrete gradient operator ∇h : Pnc1 →P0 is the restriction to Pnc1 of the operator ∇̃h given by (3.2). The discrete divergence operator divh : P0 → Pnc1 is defined by ∀σ ∈ E inth , (divh vh)(xσ ) = 3 |σ | |Kσ |+ |Lσ | (vLσ − vKσ ) ·nK,σ , ∀σ ∈ E exth , (divh vh)(xσ ) =− 3 |σ | |Kσ | vKσ ·nK,σ , for all vh ∈ P0. It is adjoint to ∇h (proposition 4.1 below). The discrete laplacian operator ∆h : P0 → P0 is the usual one for finite volume schemes (see [13]). For all qh ∈ P0 and K ∈ Th we have |K| ∑ σ∈EK∩E inth qLσ − qKσ . (3.8) Let us now consider the convection terms in (2.5) and (2.6). We define b̃ : H1 ×H1 → L2 by b̃(v,q) = div(qv) (3.9) for all q ∈ H1 and v ∈ H1. In order to define a discrete counterpart to b̃ we use the classical upwind scheme (see [13]). The discrete operator b̃h : RT0 ×P0 → P0 is such that b̃h(vh,qh) |K| ∑ σ∈EK∩E inth (vh ·nK,σ )+σ qK +(vh ·nK,σ )−σ qLσ (3.10) for all vh ∈ RT0, qh ∈ P0 and K ∈ Th. We have set a+ = max(a,0) and a− = min(a,0) for all a ∈ R. Integrating by parts the convection terms also leads to consider b : L2 ×L2 ×L∞ →R defined by b(v, p,q) =− pv ·∇qdx (3.11) for all v ∈ L2, p ∈ L2 and q ∈ L∞. The discrete counterpart is bh : RT0 ×P0×P0 → R with bh(vh, ph,qh) = ∑ σ∈EK∩E inth (vh ·nK,σ )+σ pK +(vh ·nK,σ )−σ pLσ (3.12) for all vh ∈ RT0, ph ∈ P0 and qh ∈ P0. Study of a finite volume-finite element scheme for a nuclear transport model 7 of 26 4. Properties of the discrete operators The properties of the discrete operators are analogous to the ones satisfied by their continuous counter- part. The gradient and divergence operators are adjoint. For the operators ∇h and divh we state in [21] the following. PROPOSITION 4.1 For all vh ∈ P0 and qh ∈ Pnc1 we have (vh,∇hqh) =−(qh,divhvh). Let us now consider the convection terms. Let q ∈ L∞ ∩H1, v ∈ L2 with divv ∈ L2 and divv(x) > 0 a.e. in Ω . We obtain b(v,q,q) = 2/2)divvdx > 0 by integration by parts. For bh we state in [21] a similar result. PROPOSITION 4.2 Let vh ∈ RT0 with divvh > 0. We have bh(vh,qh,qh)> 0 for all qh ∈ P0 . The following stability properties are used to prove the error estimates in section 8. PROPOSITION 4.3 There exists C > 0 such that for all ph ∈ P0, qh ∈ P0 and vh ∈ RT0 with divvh = 0 |bh(vh, ph,qh)|6C |vh|‖ph‖h‖qh‖h. (4.1) There exists C > 0 such that for all ph ∈ P0, qh ∈ P0 ∩L20, vh ∈ RT0 |bh(vh, ph,qh)|6C (|vh|‖ph‖h + |divvh|‖ph‖∞ )‖qh‖h. (4.2) PROOF. For all K ∈ Th and σ ∈ EK ∩E inth we write (vh ·nK,σ )+σ pK +(vh ·nK,σ )−σ pLσ = (vh ·nK,σ )σ pK −|(vh ·nK,σ )σ |(pLσ − pK). Thus (3.12) reads bh(vh, ph,qh) = S1 + S2 with S1 =− ∑ σ∈EK∩E inth |σ | |(vh ·nK,σ )σ |(pLσ − pK) , S2 = ∑ pK qK ∑ σ∈EK∩E inth |σ |(vh ·nK,σ )σ . Using the Cauchy-Schwarz inequality we write |S1|= ∣∣∣ ∑ σ∈E inth |σ | |vh(xσ ) ·nK,σ |(pLσ − pK)(qLσ − qK) 6 h‖vh‖∞ σ∈E inth (pLσ − pKσ )2 )1/2( ∑ σ∈E inth (qLσ − qKσ )2 Since vh ∈ RT0 ⊂ (Pd1 )2 we have h‖vh‖∞ 6C |vh| ([6] p. 112). Moreover (3.1) implies ∑σ∈E inth (pLσ − pKσ ) 6C ∑σ∈E inth τσ (pLσ − pKσ ) 2 =C‖ph‖2h and ∑σ∈E inth (qLσ − qKσ ) 6C‖qh‖2h. Thus |S1|6C |vh|‖ph‖h ‖qh‖h. (4.3) We now consider S2. We have (vh ·nK,σ )σ = 0 for all K ∈ Th and σ ∈ EK ∩E exth . Thus we write σ∈EK∩E inth |σ |(vh ·nK,σ )σ = ∑ |σ |(vh ·nK,σ )σ = divvh dx. 8 of 26 C. CHOQUET AND S. ZIMMERMANN It gives the following relation. S2 = ∑ pK qK divvh dx = ph qh divvh dx. Thus if divvh = 0 then S2 = 0 and estimate (4.3) gives (4.1). Let us prove (4.2). Since qh ∈ P0 ∩L20 we can apply proposition 3.2. Using the Cauchy-Schwarz inequality we get |S2|6 ‖ph‖∞ |qh| |divvh|6C‖ph‖∞ ‖qh‖h |divvh|. This latter estimate together with (4.3) gives (4.2). Lastly, we claim that b̃h is a consistent approximation of b̃ [21]. PROPOSITION 4.4 Let r > 0. There exists C > 0 such that for all functions q ∈ H2 and v ∈ H1+r with v ·n|∂Ω = 0 ‖ΠP0 b̃(v,q)− b̃h(ΠRT0v,Π̃P0q)‖−1,h 6C h‖q‖1‖v‖1+r. Let us now consider the discrete laplacian. We have a coercivity and stability result. PROPOSITION 4.5 For all ph ∈ P0 and qh ∈ P0, we have −(∆h ph, ph) = ‖ph‖2h , |(∆h ph,qh)|6 ‖ph‖h‖qh‖h. PROOF. Definition (3.8) implies (∆h ph,qh) = ∑ σ∈EK∩E inth τσ (pLσ − pKσ ) =− ∑ σ∈E inth τσ (pLσ − pKσ )(qLσ − qKσ ). (4.4) Setting qh = ph gives the first part of the result. Using the Cauchy-Schwarz inequality, we get the second We also deduce from (4.4) the following property. PROPOSITION 4.6 For all ph ∈ P0 and qh ∈ P0 we have (∆h ph,qh) = (ph,∆hqh). Lastly, we state that ∆h is a consistent approximation of the laplacian. The proof follows the lines of the one of proposition 1.14 in [22]. PROPOSITION 4.7 There exists C > 0 such that for all q ∈ H2 with ∇q ·n|∂Ω = 0 we have ‖ΠP0(∆q)−∆h(Π̃P0q)‖−1,h 6C h‖q‖2. 5. The finite element-finite volume scheme We now introduce the scheme for (2.5)-(2.9). The interval [0,T ] is split with a constant time step k = T/N. We set [0,T ] = m=0[tm, tm+1] with tm = mk. The time derivatives are approximated using a first order Euler scheme. The convection terms are discretized semi-implicitly in time and the other ones in an implicit way. We set sh = ΠP0s, s h = ΠP0sc, s h = ΠP0sθ and f h = ΠP0f(tm) for all m ∈ {0, . . . ,N}. Since ΠP0 (resp. ΠP0) is stable for the L 2 (resp. L2) norm we have |sh|6 |s| , |sch|6 |sc| , |sθh |6 |sθ | , |fmh |6 |f(tm)|6 ‖f‖L∞(0,T ;L2). (5.1) Study of a finite volume-finite element scheme for a nuclear transport model 9 of 26 The initial values are c0h = ΠP0c0 and θ h = ΠP0θ0. Then for all n ∈ {0, . . . ,N − 1}, the quantities cn+1h ∈ P0, θ h ∈ P0, p h ∈ P 1 ∩L20, u h ∈ RT0 are the solutions of the following problem. cn+1h − c −Dc ∆hcn+1h = s h − (sh +λ )cn+1h − b̃h(u h ), (5.2) θ n+1h −θ −Dθ ∆hθ n+1h =−s h − sh (θ h −θ∗)− b̃h(u h ), (5.3) divh(κn+1h ∇h p h ) = divh f h − Π̃Pnc1 sh , (5.4) un+1h = Π̃RT0(f h ∇h p h ), (5.5) with κn+1h = κ(c h ) ∈ P0. This term is defined thanks to proposition 6.1 below. Note also that the boundary conditions are implicitly included in the definition of the discrete operators (section 3.3). The existence of a unique solution to (5.2) and (5.3) is classical (see [13]). Since κn+1h > κmin > 0 and pn+1h ∈ L 0 equation (5.4) also has a unique solution (see [6]). We have a discrete equivalent for the divergence condition (2.7). PROPOSITION 5.1 For all m ∈ {1, . . . ,N} we have divumh = sh . PROOF. Let m ∈ {1, . . . ,N} and n = m− 1. We compare the solution of (5.4)–(5.5) with the solution of the following mixed hybrid problem. Let E0 = µh : ∪σ∈Eh → R ; ∀σ ∈ Eh , µh|σ is constant . Then ũmh ∈ RT 0 , p h ∈ P0 and λ h ∈ E0 are the solution of (see [7]) ∀vh ∈ RTd0 , (ũmh ,vh)+ ∑ κmK ∑ σ ′∈EK |σ ′|λ mσ ′ (vh|K ·nK,σ ′)− ∑ |K|κmK pmK divvh|K = (fmh ,vh) , (5.6) ∀µh ∈ E0, ∑ µh (ũmh ·n)dσ = 0 , ∀K ∈ Th , div ũmh dx = sdx , (5.7) and p̃mh ∈ Pnc1 is defined by h dσ = λ σ for all σ ∈ Eh. Let σ ∈ Eh. We define φσ ∈ Pnc1 by setting φσ (xσ ) = 1 and φσ (xσ ′) = 0 for all σ ′ ∈ Eh\{σ}. We set vh = ∇hφσ ∈ P0 ⊂ RTd0 in (5.6). We have κmK ∑ σ ′∈EK |σ ′|λ mσ ′ ∇hφσ |K ·nK,σ ′ = ∑ κmK ∇hφσ |K · ∑ σ ′∈EK |σ ′|λ mσ ′ nK,σ ′ and according to the gradient formula σ ′∈EK |σ ′|λ mσ ′ nK,σ ′ = ∑ σ ′∈EK p̃mh nK,σ ′ dσ ∇h p̃ h dx. Thus we get from (5.6) (ũmh ,∇hφσ )+ (κ h ∇h p̃ h ,∇hφσ ) = (f h ,∇hφσ ). (5.8) The first term in (5.8) is treated as follows. Integrating by parts we get (ũmh ,∇hφσ ) =−(φσ ,div ũmh )+ ∑ σ ′∈EK φσ (ũmh |K ·nK,σ ′)dσ ′. Since (5.7) implies that ũmh ∈ RT0, we have σ ′∈EK φσ (ũmh ·nK,σ ′)dσ ′ = ∑ σ∈E inth |σ |φσ (xσ )(ũmh |Lσ ·nKσ ,σ − ũmh |Kσ ·nKσ ,σ ) = 0. 10 of 26 C. CHOQUET AND S. ZIMMERMANN Thus (ũmh ,∇hφσ ) =−(φσ ,div ũmh ). Then, using (5.7), we get (ũmh ,∇hφσ ) =−(φσ ,sh) =−(φσ ,Π̃Pnc1 sh). Furthermore, according to proposition 4.1, we have (κmh ∇h p̃ h ,∇hφσ ) = − φσ ,divh(κmh ∇h p̃ (fmh ,∇hφσ ) =−(φσ ,divhfmh ). Hence we deduce from (5.8) that ∀φσ ∈ Pnc1 , φσ ,divh(κmh ∇h p̃ h )− divhf h + Π̃Pnc1 s Since (φσ )σ∈Eh is a basis of P 1 , we get divh(κ h ∇h p̃ h ) = divhf h − Π̃Pnc1 s h . Thus, by (5.4), there exists a real C such that p̃mh = p h +C. We now compare ũ h with u h . Since for all vh ∈ RT0 we have κmK ∑ σ ′∈EK |σ ′|λ mσ ′ (vh|K ·nK,σ ′) = ∑ σ∈E inth |σ |φσ (xσ )(ũmh |Lσ ·nKσ ,σ − ũ h |Kσ ·nKσ ,σ ) = 0, it follows from (5.6) that (ũmh ,v) = (f h −κmh ∇h p̃mh ,v) for any v ∈ RT0. It means that ũmh = Π̃RT0(f h −κmh ∇h p̃mh ) = Π̃RT0(f h −κmh ∇h pmh ) = umh . Thus umh = ũ h satisfies (5.7) and divu h = sh. 6. Stability analysis We first check that a maximum principle holds. It ensures that the computed concentration and temper- ature are physically relevant. PROPOSITION 6.1 For any m ∈ {0, . . . ,N} we have 0 6 cmh 6 1 and θ− 6 θ h 6 θ+. PROOF. We prove the result by induction. Since c0h = ΠP0c0 and θ h = ΠP0θ0 the result holds for m = 0 thanks to (2.11) and (3.4). Let us assume that it is true for m = n ∈ {0, . . . ,N−1}. Let K ∈Th. Equation (5.2) implies (1+ k sK + k λ )cn+1K = c K + k s K + k Dc ∑ σ∈EK∩E inth τσ (cn+1Lσ − c K )− k b̃h(u We consider the last term of this relation. Since for any σ ∈ EK ∩E inth we have (unh ·nK,σ )+σ cn+1K +(u h ·nK,σ )−σ cn+1Lσ = (u h ·nK,σ )σ cn+1K +(−u h ·nK,σ )+σ (cn+1K − c We deduce from (3.10) −b̃h(unh,cn+1h ) −cn+1K ∑ σ∈EK∩E inth |σ |(unh ·nK,σ )σ + ∑ σ∈EK∩E inth (−unh ·nK,σ )+σ (cn+1Lσ − c Since unh ∈ RT0, (unh ·nK,σ )σ = 0 for any σ ∈ EK ∩E exth . It implies that ∑σ∈EK∩E inth |σ |(u h ·nK,σ )σ = ∑σ∈EK |σ |(u h ·nK,σ )σ . Thus using the divergence formula and proposition 5.1 we obtain |K| ∑ σ∈EK∩E inth |σ |(unh ·nK,σ )σ = divunh dx = sK . Study of a finite volume-finite element scheme for a nuclear transport model 11 of 26 Therefore we get (1+ 2k sK + k λ )cn+1K = c K + k s K + k Dc ∑ σ∈EK∩E inth τσ (cn+1Lσ − c |K| ∑ σ∈EK∩E inth (−unh ·nK,σ )+σ (cn+1Lσ − c K ). (6.1) We consider Ki ∈ Th such that cn+1Ki = minK∈Th c K . According to hypothesis (2.11) and definition (3.4) we have 2sKi +λ > 0 and s > 0. Thus, using the induction hypothesis, we deduce from (6.1) cn+1K = c cnKi + ks 1+ 2k sKi + k λ k scKi 1+ 2k sKi + k λ We now consider Ks ∈ Th such that cn+1Ks = maxK∈Th c K . Using again hypothesis (2.11) we have 2sKs +λ > s > 0. Thus, using the induction hypothesis, we deduce from (6.1) cn+1K = c cnKs + k s 1+ 2k sKs + k λ 1+ k scKs 1+ 2k sKs + k λ A similar work for equation (5.3) proves that θ− 6 minK∈Th θ K and maxK∈Th θ K 6 θ+. Thus the induction hypothesis still holds for m = n+ 1. We now state the stability of the scheme (5.2)-(5.5). PROPOSITION 6.2 For any 1 6 m 6 N we have ‖cnh‖2h + k ‖θ nh ‖2h 6C , (6.2) |umh |+ |∇h pmh |6C. (6.3) PROOF. Let 0 6 n 6 N − 1. Multiplying (5.2) by 2k cn+1h we get (cn+1h − c h )− 2k Dc (∆hc h )+ k (sh +λ )cn+1h ,c + k bh(u h ) = k (s We have (cn+1h − c h ) = |c 2 −|cnh| 2 + |cn+1h − c 2. Thanks to propositions 4.2 and 4.5 −2k (∆hcn+1h ,c h ) = 2k‖c h , bh(u h )> 0. Using the Cauchy-Schwarz and Young inequalities we write k (sch,c h )6 k |s h| |cn+1h |6C k |c h |6 k |cn+1h | 2 +C k. Finally thanks to (2.11) and (3.4) we have sh > 0. Thus we obtain |cn+1h | 2 −|cnh| 2 + 2k Dc‖cn+1h ‖ h + k |cn+1h | 6C k. 12 of 26 C. CHOQUET AND S. ZIMMERMANN Let m ∈ {1, . . . ,N}. Summing up the latter relation from n = 0 to m− 1 we get |cmh |2 + 2k Dc ‖cnh‖2h 6 |c0h| k 6C , thanks to proposition 6.1. With a similar work on equation (5.3), we get (6.2). We now prove (6.3). Let n = m− 1 ∈ {0, . . . ,N − 1}. Multiplying equation (5.4) by −pn+1h and using proposition 4.1, we get (κn+1h ∇h p h ,∇h p h ) = (f h ,∇h p h )+ (Π̃Pnc1 sh, p h ). (6.4) The left-hand side term satisfies (κn+1h ∇h p h ,∇h p h ) > κin f |∇h p 2. We now consider the right- hand side. Using (5.1), the Cauchy-Schwarz and Young inequalities we write |(fn+1h ,∇h p h )|6 |f h | |∇h p κin f |∇h pn+1h | 2 +C‖f‖2L∞(0,T ;L2). Also, the stability of Π̃Pnc1 for the L 2-norm, proposition 3.1 and the Young inequality lead to |(Π̃Pnc1 sh, p h )|6 |sh| |p h |6C |p h |6C |∇h p κin f |∇h pn+1h | 2 +C. Thus we deduce from (6.4) that |∇h pn+1h | 2 = |∇h pmh | 6C. Then (5.5) imply |umh |= |un+1h |6 |f h |+ |κ h ∇h p h |6 ‖f‖L∞(0,T ;L2)+ ‖κ‖W1,∞((0,1)×(0,∞)) |∇h p h |6C. Estimate (6.3) is proven. 7. Convergence analysis Let ε = max(h,k). In this section we study the behavior of the scheme (5.2)-(5.5) as ε → 0. We first define the applications cε : R→ P0, c̃ε : R→ P0, θε : R → P0, pε : R→ Pnc1 , sε : R→ P0, scε : R→ P0 and uε : R→ RT0, fε : R→ P0 by setting for all n ∈ {0, . . . ,N − 1} and t ∈ [tn, tn+1] cε(t) = c h , c̃ε(t) = c (t − tn)(cn+1h − c h), θε(t) = θ pε(t) = p h , sε(t) = sh, s ε (t) = s h, uε(t) = u h, fε(t) = f and for all t 6∈ (0,T ) cε(t) = c̃ε(t) = θε(t) = pε(t) = sε (t) = scε (t) = 0, uε(t) = fε(t) = 0. We recall that the Fourier transform f̂ of a function f ∈ L1(R) is defined for any τ ∈ R by f̂ (τ) = e−2iπτt f (t)dt. (7.1) We begin with the following estimate. PROPOSITION 7.1 Let 0 < γ < 14 . There exists C > 0 such that for all ε > 0 |τ|2γ (|ĉε (τ)|2 + |θ̂ε(τ)|2)dτ 6C. Study of a finite volume-finite element scheme for a nuclear transport model 13 of 26 PROOF. Since equations (5.2) and (5.3) are similar we only prove the estimate on ĉε . We first define gε : R→ P0 ∩L20 as the solution of ∆hgε = Dc ∆hcε + scε − (sε +λ )cε − b̃h(uε ,cε ). Multiplying this equation by −gε we obtain −(∆hgε ,gε) =−Dc ∆hcε ,gε scε − (sε +λ )cε ,gε + bh(uε ,cε ,gε). (7.2) Proposition 4.5 allows us to write −(∆hgε ,gε) = ‖gε‖2h, −(∆hcε ,gε)6 ‖cε‖h ‖gε‖h. Thanks to the Cauchy-Schwarz inequality, (5.1) and proposition 3.2 we have ∣∣(scε − (sε +λ )cε ,gε )∣∣6C (|sc|+ |s|+λ ) |gε |6C‖gε‖h. According to proposition 4.3, then proposition 6.1 and (6.3), we have uε ,cε ,gε )∣∣6C‖cε‖∞‖gε‖h |divuε |+C‖cε‖h ‖gε‖h |uε | 6C‖gε‖h |divuε |+C‖cε‖h ‖gε‖h. Let us plug these estimates into (7.2) and integrate from 0 to T . We get ‖gε‖h dt 6C |divuε |dt +C ‖cε‖h dt 6C , because of proposition 5.1 and (6.2). Definition (7.1) then leads to ∀τ ∈ R , ‖ĝε(τ)‖h 6C. (7.3) We now use this estimate to prove (7.1). Equation (5.2) reads c̃ε = ∆hgε +(c0hδ0 − cNh δT ) where δ0 and δT are Dirac distributions respectively localized in 0 and T . Let τ ∈ R. Applying the Fourier transform to the latter equation we obtain −2iπτ ̂̃cε (τ) = ∆hĝε(τ)+ (c0h − cNh e−2iπτT ). Let us take the scalar product of this relation with isign(τ)̂̃cε(τ). Applying propositions 3.2 and 4.5 leads to 2π |τ| |̂c̃ε(τ)|2 6C ‖ĝε(τ)‖h + |c0h|+ |c ‖̂̃cε(τ)‖h. We assume that τ 6= 0 and multiply this estimate by |τ|2γ−1. Using proposition 6.1 and (7.3) we get |τ|2γ |̂c̃ε(τ)|2 6C |τ|2γ−1 ‖̂̃cε(τ)‖h. Using the Young inequality and integrating over {τ ∈ R ; |τ|> 1}, we obtain |τ|>1 |τ|2γ |̂c̃ε(τ)|2 dτ 6 |τ|>1 |τ|4γ−2 dτ +C |τ|>1 ‖̂̃cε(τ)‖2h dτ. 14 of 26 C. CHOQUET AND S. ZIMMERMANN For |τ|6 1, we have |τ|2γ |̂c̃ε (τ)|2 6 |̂c̃ε(τ)|2 6C‖̂̃cε(τ)‖2h according to proposition 3.2. Thus |τ|61 |τ|2γ |̂c̃ε(τ)|2 dτ 6C |τ|61 ‖̂̃cε(τ)‖2h dτ. By combining the bounds for |τ|> 1 and |τ|6 1 we get |τ|2γ |̂c̃ε(τ)|2 dτ 6 |τ|>1 |τ|4γ−2 dτ +C ‖̂̃cε(τ)‖2h dτ. Since 4γ − 2 <−1, we have |τ|>1 |τ|4γ−2 dτ 6C. Thanks to the Parseval theorem and (6.2) ‖̂̃cε(τ)‖2h dτ 6 ‖c̃ε‖2h dt 6C k‖c0h‖ h + k ‖cnh‖2h because ‖c0h‖h = ‖ΠP0c0‖h 6C‖c0‖1 (see [13] p. 776). Hence the result. We can now prove the following convergence result. PROPOSITION 7.2 There exists a subsequence of (cε ,θε , pε ,uε)ε>0, not relabeled for convenience, such that the following convergences hold for ε → 0 cε → c in L2(0,T ;L2), θε → θ in L2(0,T ;L2), (7.4) pε ⇀ p weakly in L 2(0,T ;H1), uε ⇀ u weakly in L 2(0,T ;L2). (7.5) The limits (c,θ , p,u) satisfy the following properties. We have c ∈ L2(0,T ;H1), θ ∈ L2(0,T ;H1), p ∈ L∞(0,T ;H1) and u ∈ L∞(0,T ;L2). We also have 0 6 c(x, t) 6 1 and θ− 6 θ (x, t) 6 θ+ a.e. in Ω × [0,T ]. For all φ ∈ C ∞0 (Ω × (−1,T )), c and θ satisfy (c,∂tφ)+Dc (∇c,∇φ)− c(u ·∇φ)− sc − (s+λ )c c0,φ(·,0) , (7.6) (θ ,∂tφ)+Dθ (∇θ ,∇φ)−θ (u ·∇φ)+ sθ + s(θ −θ∗) θ0,φ(·,0) . (7.7) Lastly we have u = f−κ(c,θ )∇p in L∞(0,T ;L2) , divu = s in L2. (7.8) PROOF. In what follows, the convergence results hold for extracted subsequences. They are not rela- beled for convenience. We begin by proving (7.4). According to proposition 6.1, the sequence (cε)ε>0 is uniformly bounded in L∞(0,T ;L2). Thus there exists c ∈ L∞(0,T ;L2) such that cε ⇀ c weakly in L 2(0,T ;L2). Using the Fourier transform, we prove that this convergence is strong. Let dε = cε − c and M > 0. We use the following splitting |d̂ε(τ)|2 dτ = |τ|>M |d̂ε(τ)|2 dτ + |τ|6M |d̂ε(τ)|2 dτ = IMε + JMε . (7.9) Study of a finite volume-finite element scheme for a nuclear transport model 15 of 26 Since |d̂ε(τ)|2 6 2|ĉε(τ)|2 + 2|ĉ(τ)|2 we have IMε 6 2 |τ|>M |ĉε(τ)|2 dτ + 2 |τ|>M |ĉ(τ)|2 dτ. Using proposition 7.1 we write |τ|>M |ĉε(τ)|2 dτ 6 |τ|>M |τ|2γ |ĉε(τ)|2 dτ 6 Hence IMε 6 |τ|>M |ĉ(τ)|2 dτ. This implies that for all ε > 0, IMε → 0 when M → ∞. We now consider JMε . Let τ ∈ [−M,M]. Since cε(t) ∈ P0 for all t ∈ R, and cε ⇀ c weakly in L2(0,T ;L2), we deduce from (7.1) that ĉε(τ) ∈ P0 and ĉε(τ)⇀ ĉ(τ) weakly in L2. Extanding ĉε(τ) by 0 outside Ω , one checks ([13] p.811) that ∀η ∈ R2 , |ĉε(τ)(·+η)− ĉε(τ)|6C‖ĉε(τ)‖h |η |(|η |+ h). (7.9) Then, using estimate (6.2), we deduce from [13] (p.834) that ĉε(τ)→ ĉ(τ) strongly in L2. Thus d̂ε(τ) = ĉε(τ)− ĉ(τ) → 0 in L2, so that JεM → 0 when ε → 0. Now, let us report the limits for IεM and JεM into (7.9). Using the Parseval identity we get |d̂ε(τ)|2 dτ = |dε |2 dt = |cε − c|2 dt → 0. Thus we have proven that cε → c in L2(0,T ;L2). A similar work proves that θε → θ in L2(0,T ;L2) with θ ∈ L∞(0,T ;L2). Hence (7.4) is proven. Moreover using proposition 6.1 we obtain 0 6 c(x, t)6 1 and θ− 6 θ (x, t) 6 θ+ a.e. in Ω × [0,T ]. Lastly, using (6.2) and (7.9), we get as in [13] (p.811) that c ∈ L2(0,T ;H1) and θ ∈ L2(0,T ;H1). Let us now consider the sequences (pε)ε>0 and (uε )ε>0. According to (3.3) and (6.3) the sequence (ΠPc1 pε)ε>0 is bounded in L ∞(0,T ;H1). It implies that there exists p ∈ L∞(0,T ;H1) such that ΠPc1 pε ⇀ p weakly in L2(0,T ;H1). Using proposition 3.3 we get pε ⇀ p weakly in L 2(0,T ;H1). Moreover, according to (6.3), the sequence (uε)ε>0 is bounded in L∞(0,T ;L2) . Thus we have uε ⇀ u weakly in L2(0,T ;L2) with u ∈ L∞(0,T ;L2). We check the properties of u. Using a Taylor expansion, the Cauchy-Schwarz inequality, and a density argument, we have ‖κ(c,θ )−κ(cε,θε )‖L2(0,T ;L2) 6 ‖κ‖W1,∞((0,1)×(0,∞)) (‖c− cε‖L2(0,T ;L2)+ ‖θ −θε‖L2(0,T ;L2)). Thus, using the strong convergence of the sequences (cε)ε>0 and (θε )ε>0, we have κ(cε ,θε )→ κ(c,θ ) in L2(0,T ;L2). Since ∇h pε ⇀ ∇p weakly in L2(0,T ;L2), we deduce from this κ(cε ,θε )∇h pε ⇀ κ(c,θ )∇p weakly in L2(0,T ;L2). (7.10) Now let v ∈ L2(0,T ;(C ∞0 )2). According to (5.5) we have (uε ,ΠRT0v) = (fε −κ(cε ,θε)∇h pε ,v). Using proposition 3.3 one checks easily that (fε ,ΠRT0v)→ (f,v) and (uε ,ΠRT0v)→ (u,v) in L1(0,T ). Using moreover convergence (7.10) and a density argument, we deduce from this that u= f−κ(c,θ )∇p. And since divuε = sε by proposition 5.1, we also have divu = s. 16 of 26 C. CHOQUET AND S. ZIMMERMANN We finally prove that c satisfies (7.6). For all t ∈ (0,T ) equation (5.2) reads c̃ε −Dc ∆hcε + b̃h(uε ,cε) = scε − (sε +λ )cε . Let ψ ∈ C ∞0 (Ω × (−1,T )) and ψh = Π̃P0ψ . Multiplying the latter equation by ψh and integrating over [0,T ], we obtain c̃ε ,ψh dt −Dc (∆hcε ,ψh)dt + bh(uε ,cε ,ψh)dt = (scε − (sε +λ )cε ,ψh) dt. (7.11) We now pass to the limit ε → 0 in this equation. We begin with the term 0 bh(uε ,cε ,ψh)dt. We use the splitting b(u,c,ψ)− bh(uε ,cε ,ψh) = Aε1 +Aε2 +Aε3 with Aε1 = b(u,c,ψ)− b(uε ,c,ψ), Aε2 = b(uε ,c,ψ)− div(cuε)ψh dx , Aε3 = div(cuε)ψh dx− bh(uε ,cε ,ψh). According to definition (3.11) Aε1 = b(u,c,ψ)− b(uε ,c,ψ) =− c(u−uε) ·∇ψ dx. We know that c∇ψ ∈ L2(0,T ;L2). Since uε ⇀ u in L2(0,T ;L2) we get 1 dt → 0. We now consider Aε2. We have Aε2 = (ψ −ψh)div(cuε)dx = (ψ −ψh)(uε ·∇c+ cdivuε)dx. Using the Cauchy-Schwarz inequality we get |Aε2|dt 6 ‖ψ −ψh‖L∞(Ω×(0,T)) (‖uε‖L2(0,T ;L2)+ ‖divuε‖L2(0,T ;L2))‖c‖L2(0,T ;H1). Using a Taylor expansion, one checks that ‖ψ −ψh‖L∞(Ω×(0,T)) 6 h‖∇ψ‖L∞(Ω×(0,T )). Thus 2 dt → 0. Finally we estimate Aε3. For all triangles K ∈ Th and L ∈ Th sharing an edge σ , we set cK,L = cK if uε ·nK,σ > 0 and cK,L = cL otherwise. Using the divergence formula, we deduce from definition (3.12) Aε3 = ∑ σ∈EK∩E inth (c− cK,Lσ )(uε ·nK,σ )dσ σ∈E inth (ψKσ −ψLσ ) (c− cKσ ,Lσ )(uε ·nKσ ,σ )dσ . Using definition (3.5) this reads Aε3 = ∑ σ∈E inth (ψKσ −ψLσ ) (ΠPnc1 c− cKσ ,Lσ )(uε ·nKσ ,σ )dσ σ∈E inth (ψKσ −ψLσ ) |σ | (ΠPnc1 c)(xσ )− cKσ ,Lσ (uε ·nKσ ,σ )σ . Study of a finite volume-finite element scheme for a nuclear transport model 17 of 26 Using a Taylor expansion, one checks that |ψKσ −ψLσ |6 h‖∇ψ‖L∞(Ω×(0,T )). Moreover |σ |6 h. Thus, using the Cauchy-Schwarz inequality, we have |Aε3| 6 C h2 ∑ σ∈E inth |uε (xσ )| ∣∣∣(ΠPnc1 c)(xσ )− cKσ ,Lσ 6 C h2 σ∈E inth |uε (xσ )|2 )1/2( ∑ σ∈E inth |(ΠPnc1 c)(xσ )− cKσ ,Lσ | Using the assumption on the mesh, one checks that |K| > C h2 for all K ∈ Th. Thus, thanks to a quadrature formula, we have |Aε3| 6 C σ∈EK∩E inth |uε (xσ )|2 )1/2( σ∈EK∩E inth |(ΠPnc1 c)(xσ )− cKσ | 6 C |uε | |ΠPnc1 c− cε |. We write ΠPnc1 c− cε = (ΠPnc1 c− c)+ (c− cε) and we use proposition 3.3 . We obtain with (6.3) |Aε3|dt 6C‖uε‖L∞(0,T ;L2) (h‖c‖L2(0,T ;H1)+ ‖c− cε‖L2(0,T ;L2)). Since cε → c in L2(0,T ;L2) when ε = max(h,k) → 0, we conclude that 3 dt → 0. Gathering the limits for Aε1, A 3, we obtain bh(uε ,cε ,Π̃P0ψ)dt → b(u,c,ψ)dt. We now consider the other terms in (7.11). Proposition 4.6 leads to (∆hcε ,Π̃P0ψ) = cε ,∆h(Π̃P0ψ) cε ,∆h(Π̃P0 ψ)−∆ψ +(cε ,∆ψ). (7.12) According to proposition 4.7 cε ,∆h(Π̃P0ψ)−∆ψ )∣∣∣ 6 ‖cε‖h ‖∆h(Π̃P0ψ)−∆ψ‖−1,h 6C h‖cε‖h‖ψ‖2. We then apply the Cauchy-Schwarz inequality and use (6.2). We obtain cε ,∆h(Π̃P0 ψ)−∆ψ )∣∣∣ dt 6C h ‖cε‖2h dt ‖cnh‖ 6C h. Moreover, since cε → c in L2(0,T ;L2), we have 0 (cε ,∆ψ)dt → 0 (c,∆ψ)dt. Thus we deduce from (7.12) ∫ T (∆hcε ,Π̃P0ψ)dt → (c,∆ψ)dt. We are left with two terms. First, using Taylor expansions, one checks that ψh → ψ , ∂tψh → ∂tψ in L2(Ω × (−1,T)) , ψh(·,0)→ ψ(·,0) in L2. (7.13) 18 of 26 C. CHOQUET AND S. ZIMMERMANN We know that cε → c in L2(0,T ;L2). Thus (sc − (s+λ )cε ,ψh) dt → (sc − (s+λ )c,ψ) dt. Finally, integrating by parts the first term of (7.11), we get c̃ε ,ψh dt = (c̃ε ,ψh)t=T − (c̃ε ,ψh)t=0 − (c̃ε ,∂tψh)dt. Since ψ ∈ C ∞0 (Ω × (−1, ,T )) we have (c̃ε ,ψh)t=T = 0. Using proposition 3.3 one checks that c0h = ΠP0c0 → c0 in L2; using moreover (7.13) we get (c̃ε ,ψh)t=0 = c0h,ψh(·,0) ΠP0c0,ψh(·,0) c0,ψ(·,0) For the last term, one easily checks that ‖c̃ε − cε‖L2(0,T ;L2) → 0. Thus, since cε → c in L2(0,T ;L2), we also have c̃ε → c in L2(0,T ;L2). Using moreover (7.13) we get 0 (c̃ε ,∂t ψh)dt → 0 (c,∂tψ)dt. Therefore ∫ T c̃ε ,ψh dt →− c0,ψ(·,0) (c,∂tψ)dt. By gathering the limits we have obtained in (7.11) we get (7.6). The relation (7.7) for θ is proven in a similar way. 8. Error estimates We have proven in section 7 that the problem (2.5)–(2.9) has a weak solution (c,θ , p,u). From now on, we assume the following regularity for this solution: c,θ ∈ C (0,T ;H2) , ct ,θt ∈ L2(0,T ;H1+r)∩C (0,T ;L2) , ctt ,θtt ∈ L2(0,T ;L2) , p ∈ C (0,T ;H2) , u ∈ C (0,T ;H1+s) , with r > 0 and s > 0. We also assume that f ∈ C (0,T ;H1). 8.1 Definitions For all m ∈ {0, . . . ,N}, we define the following errors emh,c = c(tm)− cmh , emh,θ = c(tm)−θ mh , emh,p = p(tm)− pmh , emh,u = u(tm)−umh . We have the following splittings emh,c = ε h,c +η h,c, e h,θ = ε h,θ +η h,θ , emh,p = ε h,p +η h,p, e h,u = ε h,u+η with the discrete errors εmh,c = Π̃P0c(tm)− c h , ε h,θ = Π̃P0θ (tm)−θ εmh,p = ΠPnc1 p(tm)− p h , ε h,u= ΠRT0u(tm)−u Study of a finite volume-finite element scheme for a nuclear transport model 19 of 26 and the interpolation errors ηmh,c = c(tm)− Π̃P0c(tm), η h,θ = θ (tm)− Π̃P0θ (tm), ηmh,p = p(tm)−ΠPnc1 p(tm), η h,u= u(tm)−ΠRT0u(tm). The interpolation errors are estimated as follows. We write |ηmh,c| 6 |c(tm)−ΠP0c(tm)|+ |ΠP0c(tm)− Π̃P0c(tm)| and the same for ηmh,θ . Using proposition 3.3 and (3.7) we obtain |ηmh,c|6C h‖c(tm)‖1 6C h‖c‖L∞(0,T ;H1) , |ηmh,θ |6C h‖θ‖L∞(0,T ;H1). (8.1) According to proposition 3.3 we also have |ηmh,p|+ |∇̃hηmh,p|6C h‖p(tm)‖2 6C h‖p‖L∞(0,T ;H2), (8.2) |ηmh,u|6C h‖u(tm)‖1 6C h‖u‖L∞(0,T ;H1). (8.3) We now have to estimate the discrete errors. PROPOSITION 8.1 For all n ∈ {0, . . . ,N − 1} and ψh ∈ Pnc1 we have εn+1h,c − ε −Dc ∆hεn+1h,c + b̃h εnh,u,Π̃P0c(tn+1) + b̃h(u h,c )+ (s h +λ )ε h,c =C h,1 +C h,2 , (8.4) εn+1h,θ − ε −Dθ ∆hεn+1h,θ + b̃h εnh,u,Π̃P0θ (tn+1) + b̃h(u h,θ )+ sh ε h,θ =Θ h,1 +Θ h,2 , (8.5) κ(cn+1h ,θ h )∇hε h,p ,∇hψh (κn+1h,1 ε h,c +κ h,2 ε h,θ )∇p(tn+1),∇hψh Pn+1h ,∇hψh , (8.6) εn+1h,u =−Π̃RT0 (κn+1h,1 ε h,c +κ h,2 ε h,θ )∇p(tn+1)+κ(c h )∇hε −Un+1h . (8.7) For all m ∈ {0, . . . ,N}, the consistency errors Cmh,1, C h,2, Θ h,1, Θ h,2, P h and U h are defined in (8.9), (8.10), (8.14), (8.15) and the terms κmh,1 and κ h,2 are given by (8.13) below. PROOF. Let n ∈ {0, . . . ,N − 1}. Equation (2.5) for t = tn+1 reads ∂tc(tn+1)−Dc ∆c(tn+1)+ b̃ u(tn+1),c(tn+1) = sc − (s+λ )c(tn+1). We introduce the time discretization by setting Rn+1 = (c(tn+1)− c(tn) − ct(tn+1) u(tn)−u(tn+1),c(tn+1) We get c(tn+1)− c(tn) −Dc ∆c(tn+1)+ b̃ u(tn),c(tn+1) = sc − (s+λ )c(tn+1)+Rn+1. We apply ΠP0 to this equation. By subtracting the result from (5.2) we get (c(tn+1)− c(tn) cn+1h − c ΠP0∆c(tn+1)−∆hc +ΠP0 b̃ u(tn),c(tn+1) − b̃h(unh,cn+1h )+ΠP0 (s+λ )c(tn+1) − (sh +λ )cn+1h = ΠP0R n+1. (8.8) 20 of 26 C. CHOQUET AND S. ZIMMERMANN We now introduce the discrete errors as follows. Since c(tn+1)− c(tn) = ∫ tn+1 tn ct(s)ds one checks that (c(tn+1)− c(tn) cn+1h − c ∫ tn+1 ΠP0ct(s)− Π̃P0ct(s) (εn+1h,c − ε h,c). We also have ΠP0∆c(tn+1)−∆hc h = ΠP0∆c(tn+1)−∆h Π̃P0c(tn+1) +∆hεn+1h,c . Using the linearity of b̃h, one easily checks that ΠP0 b̃ u(tn),c(tn+1) − b̃h(unh,cn+1h ) = b̃h(u h,c )+ b̃h εnh,u,Π̃P0c(tn+1) +ΠP0 b̃ u(tn),c(tn+1) − b̃h ΠRT0u(tn),Π̃P0c(tn+1) Lastly (s+λ )c(tn+1) − (sh +λ )cn+1h = ΠP0 (s+λ )ηn+1h,c +(sh +λ )εn+1h,c . Using these relations in (8.8) we get (8.4). For any m ∈ {1, . . . ,N}, the consistency errors Cmh,1 ∈ P0 and Cmh,2 ∈ P0 are given by Cmh,1 = ΠP0 (c(tm)− c(tm−1) − ct(tm)+ b̃ u(tm−1)−u(tm),c(tm) + ΠP0 (s+λ )ηmh,c Π̃P0ct(s)−ΠP0ct(s) ds, (8.9) Cmh,2 = Dc ΠP0∆c(tm)−∆h Π̃P0c(tm) ΠP0 b̃ u(tm−1),c(tm) − b̃h(ΠRT0u(tm−1),Π̃P0c(tm) A similar proof leads to (8.5) where the consistence errors Θ mh,1 ∈ P0 and Θ mh,2 ∈ P0 are defined for any m ∈ {1, . . . ,N} by Θ mh,1 = ΠP0 (θ (tm)−θ (tm−1) −θt(tm)+ b̃ u(tm−1)−u(tm),θ (tm) + ΠP0(sη h,θ )− Π̃P0θt(s)−ΠP0θt(s) ds, (8.10) Θ mh,2 = Dθ ΠP0∆θ (tm)−∆h Π̃P0θ (tm) ΠP0 b̃ u(tm−1),θ (tm) − b̃h ΠRT0u(tm−1),Π̃P0θ (tm) We now consider the problem associated with the pressure. Let n ∈ {0, . . . ,N − 1} and ψh ∈ Pnc1 . Mul- tiplying equation (2.7) written for t = tn+1 by ψh and integrating by parts, we get κ(c(tn+1),θ (tn+1))∇p(tn+1),∇hψh = (f(tn+1),∇hψh)+ (s,ψh). (8.11) On the other hand, using (5.4) and proposition 4.1, we have κ(cn+1h ,θ h )∇h p h ,∇hψh = (fn+1h ,∇hψh)+ (Π̃Pnc1 sh,ψh). Since ∇hψh ∈ P0, one checks that (fn+1h ,∇hψh) = (ΠP0f(tn+1),∇hψh) = (f(tn+1),∇hψh). According to (3.5) we also have (Π̃Pnc1 sh,ψh) = (sh,ψh). Thus κ(cn+1h ,θ h ) ∇h p h ,∇hψh = (f(tn+1),∇hψh)− (sh,ψh). Study of a finite volume-finite element scheme for a nuclear transport model 21 of 26 Substracting (8.11) from the latter relation, we obtain κ(c(tn+1),θ (tn+1))∇p(tn+1)−κ(cn+1h ,θ h )∇h p h ,∇hψh =−(s− sh,ψh). (8.12) We split the left-hand side as κ(cn+1h ,θ h )(∇p(tn+1)−∇h p κ(c(tn+1),θ (tn+1))−κ(cn+1h ,θ ∇p(tn+1). Using a Taylor expansion, one can check that κ(c(tn+1),θ (tn+1))−κ(cn+1h ,θ h ) = (ε h,c +η h,c )κ h,1 +(ε h,θ +η h,θ )κ h,2 . We have set for any m ∈ {0, . . . ,N} and s ∈ [0,1] cmh (s) = c h +(c(tm)− c h )s, θ h (s) = θ h +(θ (tm)−θ h )s, κmh,1 = κx(cmh (s),θ h (s))ds, κ h,2 = κy(cmh (s),θ h (s))ds. (8.13) We also have ∇p(tn+1)−∇hpn+1h = ∇hε h,p + ∇̃hη h,p . Plugging these relations into (8.12) we get (8.6). For all m ∈ {0, . . . ,N} we have Pmh = (κ h,c +κ h,θ )∇p(tm)+κ(c h )∇hη h,p. (8.14) We end with the equation associated with u. Let n ∈ {0, . . . ,N − 1}. Applying the operator Π̃RT0 to (2.7) for t = tn+1 we obtain Π̃RT0u(tn+1) = Π̃RT0f(tn+1)− Π̃RT0 κ(c(tn+1),θ (tn+1))∇p(tn+1) Let us substract this equation from (5.5). Since fn+1h = ΠP0f(tn+1) we get Π̃RT0u(tn+1)−u h = Π̃RT0 f(tn+1)−ΠP0f(tn+1) − Π̃RT0 c(tn+1),θ (tn+1) ∇p(tn+1)−κn+1h ∇h p One easily checks that Π̃RT0u(tn+1)−u h = Π̃RT0(u(tn+1)−ΠRT0u(tn+1))+ ε h,u . Thus we get (8.7). For all m ∈ {0, . . . ,N}, we have Umh = Π̃RT0 (f(tm)−ΠP0f(tm))−ηmh,u−P . (8.15) This ends the proof of proposition 8.1. 22 of 26 C. CHOQUET AND S. ZIMMERMANN 8.2 Error estimates We first estimate the consistency errors. PROPOSITION 8.2 For all m ∈ {1, . . . ,N} the consistency errors satisfy |Cnh,1| 2 + k |Θ nh,1| 6C (h2 + k2), (8.16) ‖Cnh,2‖2−1,h + k ‖Θ nh,2‖2−1,h 6C h2 , (8.17) |Pmh |+ |Umh |6C h. (8.18) PROOF. Let n ∈ {1, . . . ,N}. Since the operator ΠP0 is stable for the L2-norm we have |ΠP0R n|6 |Rn|6 c(tn)− c(tn−1) − ct(tn) ∣∣∣b̃ u(tn−1)−u(tn),c(tn) )∣∣∣ . Using a Taylor expansion and the Cauchy-Schwarz inequality, we get c(tn)− c(tn−1) − ct(tn) ∫ tn−1 |tn−1 − s| |ctt(s)|ds 6 (∫ tn−1 |ctt(s)|2 ds On the other hand, since ∇c(tn)|∂Ω = 0, we deduce from (3.10) by integrating by parts b̃(u(tn−1)−u(tn),c(tn)) = (u(tn−1)−u(tn)) ·∇c(tn). Using a Taylor expansion and the Cauchy-Schwarz inequality, we get |(u(tn−1)−u(tn)) ·∇c(tn)|6 k‖c‖L∞(0,T ;H1) (∫ tm−1 |ut(s)|2 ds |ΠP0R (∫ tm−1 |ctt(s)|2 ds k‖c‖L∞(0,T ;H1) (∫ tm−1 |ut(s)|2 ds Thanks to the stability of ΠP0 for the L 2-norm and to (8.1) we have ∣∣ΠP0 (s+λ )ηnh,c )∣∣6C h(‖s‖L∞(0,T ;L2)+λ )‖c‖L∞(0,T ;H1). The Cauchy-Schwarz inequality and (3.7) allow us to write ∣∣ΠP0ct(s)− Π̃P0ct(s) ∣∣ds 6C h (∫ tm ‖ct(s)‖21+r ds By plugging these estimates into definition (8.9) we get k |Cnh,1|2 6 k2 ‖c‖2L∞(0,T ;H1) |ut(s)|2 ds+ k2 |ctt(s)|2 ds + C h2 ‖ct(s)‖21+r ds+C k h2‖c‖2L∞(0,T ;H1). Study of a finite volume-finite element scheme for a nuclear transport model 23 of 26 Summing up the latter relation for n = 1 to m ∈ {1, . . . ,N} and using a similar work on Θ mh,1 we get (8.16). Now let n ∈ {1, . . . ,N}. Using propositions 4.7 and 4.4 we have ‖ΠP0∆c(tn)−∆h Π̃P0c(tn) ‖−1,h 6C h‖c‖L∞(0,T ;H2) ‖ΠP0 b̃(u(tn−1),c(tn))− b̃h(ΠRT0u(tn−1),Π̃P0c(tn))‖−1,h 6C h‖c‖L∞(0,T ;H1)‖u‖L∞(0,T ;H1+s). Plugging these estimates into definition (8.9) and summing up from n = 1 to m, we obtain ‖Cnh,2‖2−1,h 6C h2 ‖c‖2L∞(0,T ;H1) ‖u‖ L∞(0,T ;H1+s)+ ‖c‖ L∞(0,T ;H2) A similar work on Θ mh,2 then leads to (8.17). We finally prove (8.18). Let m ∈ {1, . . . ,N}. On the one hand, we have by (8.14) |Pmh |6 (‖κmh,1‖∞ |ηmh,c|+ ‖κmh,2‖∞ |ηmh,θ |) |∇p(tm)|+ ‖κ(cmh ,θ mh )‖∞|∇hηmh,p|. Using estimates (8.1)–(8.3) we get |Pmh |6C h‖κ‖W1,∞((0,1)×(0,∞))(‖c‖L∞(0,T ;H1)+ ‖θ‖L∞(0,T ;H1)+ ‖p‖L∞(0,T ;H2)). On the other hand definition (8.15) leads to |Umh |6 ∣∣Π̃RT0 f(tm)−ΠP0f(tm) )∣∣+ |ηmh,u|+ |P Using the stability of Π̃RT0 for the L 2-norm and proposition 3.3 we have ∣∣∣Π̃RT0 f(tm)−ΠP0f(tm) )∣∣∣6 |f(tm)−ΠP0f(tm)|6C h‖f‖L∞(0,T ;H1). Using moreover (8.3) we obtain |Umh |6C h‖κ‖W1,∞((0,1)×(0,∞))(‖c‖L∞(0,T ;H1)+ ‖θ‖L∞(0,T ;H1)) +C (‖p‖L∞(0,T ;H2)+ ‖f‖L∞(0,T ;H1)+ ‖u‖L∞(0,T ;H1)). We have proven (8.18). Using the former proposition we are now able to estimate the discrete errors. PROPOSITION 8.3 There exists some real k0 > 0 such that for any k < k0 and m ∈ {1, . . . ,N} |εmh,c|2 + |εmh,θ |2 + k ‖εmh,c‖2h + ‖εmh,θ‖2h 6C (h2 + k2), (8.19) |∇hεmh,p|+ |εmh,u|6C (h+ k). (8.20) PROOF. Multiplying (8.4) by 2k εn+1h,c , we obtain (εn+1h,c − ε , 2k εn+1h,c − 2Dc k (∆hεn+1h,c ,ε h,c )+ 2k bh unh,ε h,c ,ε + 2k λ |εn+1h,c | = 2k (Cn+1h+1 +C h,2 ,ε h,c )− 2k (sh, |ε h,c | 2)− 2k bh εnh,u,Π̃P0c(tn+1),ε . (8.21) 24 of 26 C. CHOQUET AND S. ZIMMERMANN Using an algebraic identity we have (εn+1h,c − ε ,2k εn+1h,c = |εn+1h,c | 2 −|εnh,c|2 + |εn+1h,c − ε h,c|2. We know by propositions 4.2 and 4.5 that −2k (∆hεn+1h,c ,ε h,c ) = 2k‖ε h,c ‖ h , bh unh,ε h,c ,ε We have 2k (sh, |εn+1h,c | 2)6 2k‖sh‖∞ |εn+1h,c | 6C k |εn+1h,c | Using the Young inequality, we also write ∣∣∣2k (Cn+1h,1 ,ε h,c ) ∣∣∣6 2k |Cn+1h,1 | |ε h,c |6 k λ |ε h,c | 2 +C k |Cn+1h,1 | |2k (Cn+1h,2 ,ε h,c )|6 2k‖C h,2 ‖−1,h‖ε h,c ‖h 6 Dc ‖εn+1h,c ‖ h +C k‖Cn+1h,2 ‖ −1,h. We are left with the term bh εnh,u,Π̃P0c(tn+1),ε . We have εnh,u∈ RT0. Using the divergence formula, proposition 5.1 and (3.5), one easily checks that divεnh,u= 0. Thus we can apply proposition 4.3 to get ∣∣bh(εnh,u,Π̃P0c(tn+1),ε h,c ) ∣∣6C |εnh,u|‖Π̃P0c(tn+1)‖h‖ε h,c ‖h. (8.22) Let us first bound ‖Π̃P0c(tn+1)‖h. We have ‖Π̃P0c(tn+1)‖h 6 ‖Π̃P0c(tn+1)−ΠP0c(tn+1)‖h + ‖ΠP0c(tn+1)‖h. Using an inverse inequality (see proposition 1.2 in [22]) and (3.7) ‖Π̃P0c(tn+1)−ΠP0c(tn+1)‖h 6 |Π̃P0c(tn+1)−ΠP0c(tn+1)|6C. Moreover, according to [13] (p. 776), we have ‖ΠP0c(tn+1)‖h 6C‖c‖L∞(0,T ;H1). Thus ‖Π̃P0c(tn+1)‖h 6 C. We now estimate |εnh,u|. Using the stability of Π̃RT0 for the L 2-norm and the Cauchy-Schwarz inequality in (8.7) we get |εnh,u|6 ‖κ‖W1,∞((0,1)×(0,∞)) ‖∇p‖L∞(0,T ;L2) (|ε h,c |+ |ε h,θ |)+ |∇hε h,p | + |Un+1h |. (8.23) We bound εn+1h,p as follows. Setting ψh = ε h,p in (8.6) and using the Cauchy-Schwarz inequality, we get κ(cn+1h ,θ h )∇hε h,p ,∇hε 6 ‖κ‖W1,∞((0,1)×(0,∞))‖∇p‖L∞(0,T ;L2) (|εn+1h,c |+ |ε h,θ |) |∇hε h,p | + |Pn+1h | |∇hε h,p |+ |s− sh| |ε h,p |. The left-hand side is such that (κ(cn+1h ,θ h )∇hε h,p ,∇hε h,p )> κin f |∇hε h,p | 2. As for the right-hand side, we have sh = ΠP0s and ε h,p ∈ P 1 ∩L20. Thus, according to propositions 3.1 and 3.3 |s− sh| |εn+1h,p |6C h‖s‖L∞(0,T ;H1) |∇hε h,p |. Study of a finite volume-finite element scheme for a nuclear transport model 25 of 26 Finally |Pn+1h |6C h thanks to (8.18). Therefore we obtain |∇hεn+1h,p |6C (h+ |ε h,c |+ |ε h,θ |). (8.24) Let us plug this estimate into (8.23). Since |Un+1h |6C h thanks to (8.18), we get |εnh,u|6C (h+ |ε h,c |+ |ε h,θ |). (8.25) Now, plugging this bound into (8.23) and using the Young inequality, we obtain ∣∣bh(εnh,u,Π̃P0c(tn+1),ε h,c ) ∣∣ 6 C k (h+ |εn+1h,c |+ |ε h,θ |)‖ε h,c ‖h ‖εn+1h,c ‖ h +C k (h 2 + |εn+1h,c | 2 + |εn+1h,θ | Now we have treated all the terms in (8.21). This equation implies |εn+1h,c | 2 −|εnh,c|2 +Dc k‖εn+1h ‖ h 6C k h2 + |εn+1h,c | 2 + |εn+1h,θ | 2 + |Cn+1h,1 | 2 + ‖Cn+1h,2 ‖ Let m ∈ {1, . . . ,N}. Let us sum up the latter estimate from n = 0 to m− 1. Thanks to (3.3) |ε0h,c|= |Π̃P0c0 − c h|= |Π̃P0c0 −ΠP0c0|6C h‖c‖L∞(0,T ;H2). Using moreover estimates (8.16) and (8.17) we get |εmh,c|2 +Dc k ‖εnh,c‖2h 6C k (|εnh,c|2 + |εnh,θ |2)+C (h2 + k2). Summing this relation with the one obtained by a similar work on (8.5) we obtain |εmh,c|2 + |εmh,θ |2 + k (Dc ‖εnh,c‖2h +Dθ ‖εnh,θ‖2h)6C k (|εnh,c|2 + |εnh,θ |2)+C (h2 + k2). Using a discrete Gronwall lemma (see lemma 5.2 in [22]) we get (8.19). Then (8.24) and (8.25) imply (8.20). By combining proposition 8.3 with estimates (8.1)-(8.3), we obtain finally the following result. Theorem 8.1 There exists a real k0 > 0 such that for all k < k0 and m ∈ {1, . . . ,N} |emh,c|2 + |emh,θ |2 + k ‖Π̃P0e h,c‖2h + ‖Π̃P0e h,θ‖2h 6C (h2 + k2) , |∇̃hemh,p|+ |emh,u|6C (h+ k). REFERENCES [1] ACHDOU, Y., BERNARDI, C. & COQUEL, F. (2003) A priori and a posteriori analysis of finite volume discretiza- tions of Darcy’s equations, Numer. Math., 96(1) 17–42. [2] AFIF, M.,AMAZIANE, B. (2002) Convergence of finite volume schemes for a degenerate convection-diffusion equation arising in flow in porous media, Comput. Methods Appl., 191 5265–5286. 26 of 26 C. CHOQUET AND S. ZIMMERMANN [3] ANGOT, P., DOLJŠI, V., FEISTAUER, M. & FELCMAN, J. (1998) Analysis of a combined barycentric finite volume-nonconforming finite element method for nonlinear convection-diffusion problems, Appl. Math., 43(4) 263–310. [4] BATEMAN, H. (1910) The solution of a system of differential equations occuring in the theory of radioactive transformations, Proc. Camb. Philos. Soc., 15 423–427. [5] BEAR, J. (1972) Dynamics of Fluids in Porous Media, New York: American Elsevier. [6] BRENNER, S. C., SCOTT, L. R. & RIDGWAY, L. (2002) The mathematical theory of finite element methods, Texts in applied mathematics, 16, New-York: Springer-Verlag. [7] BREZZI, F. & FORTIN, M. (1991) Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, 15, New-York: Springer-Verlag. [8] CHAVENT, G., JAFFRÉ, J. & ROBERTS, J. E. (1995) Mixed hybrid finite elements and cell-centered finite volumes for two-phase flow in porous media. In: A. P. BOURGEAT AND AL, ed. Mathematical modeling of flow through porous media, London: World Scientific, 100–114. [9] CHEN, Z. (2000) Formulation and numerical methods of the black oil model in porous media, SIAM J. Numer. Anal., 38(2) 489–514. [10] CHOQUET C. (2004) Existence result for a radionuclide transport model with unbounded viscosity , J. Math. Fluid Mech., 6(4) 365–388. [11] DOUGLAS J., JR. & SPAGNUOLO, A. M. (2001) The approximation of nuclear contaminant transport in porous media, J. Korean Math. Soc., 38 723–761. [12] EWING, R. E., YUAN, Y. & LI, G. (1989) A time-discretization procedure for a mixed finite element approxi- mation of contamination by incompressible nuclear waste in porous media, In: Mathematics for large scale computing, Lecture Notes in Pure and Appl. Math., 120, 127–145. [13] EYMARD, R., GALLOUËT, T. & HERBIN, R. (2000) The finite volume method (chapter III), In P.G. CIARLET AND J. L. LIONS, eds Handbook of numerical analysis, 7, 762–835. [14] GEISER, J. (2002) Numerical simulation of a model for transport and reaction of radionuclides, In: Algoritmy 2002, Vysoké Tatry - Podbanské, Slovakia, Bratislava: Slovak university of technology. [15] JOHN, V., MAUBACH, J. M. & TOBISKA, L. (1997) Nonconforming streamline-diffusion finite-element methods for convection-diffusion problems, Numer. Math., 78(2) 165–188. [16] KOVAL, E. J. (1963) A method for predicting the performance of unstable miscible displacements in heterogeneous media, SPEJ trans. AIME, 228 145–154. [17] LIN P. & YANG, D. (1998) An iterative perturbation method for the pressure equation in the simulation of miscible displacement in porous media, SIAM J. Sci. Comput., 19(3) 893–911. [18] MICHEL, A. (1999) Convergence of a finite volume scheme for a nonlinear convection-diffusion problem, In: Proceedings of the Second international symposium on finite volumes for complex applications, Duisburg. [19] MORTON, K. W. (1996) Numerical solution of convection-diffusion problems, In: Applied Mathematics and math- ematical Computation, 12, London: Chapman and Hall. [20] VERWER, J. G., BLOM, J. G. & HUNDSDORFER, W. (1996) An implicit-explicit approach for atmospheric trans- port chemistry problems, Appl. Numer. Math., 20(1-2) 191–209. [21] ZIMMERMANN, S. (2006) Stability of a finite volume scheme for incompressible fluids, preprint. [22] ZIMMERMANN, S. (2006) Étude et implémentation de méthodes de volumes finis pour les fluides incompressibles. Thesis (PhD). Blaise Pascal university (France).

0704.1287

Realizable Hamiltonians for Universal Adiabatic Quantum Computers

Realizable Hamiltonians for universal adiabatic quantum computers Jacob D. Biamonte1, ∗ and Peter J. Love2, † Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX1 3QD, United Kingdom. Department of Physics, 370 Lancaster Ave., Haverford College, Haverford, PA USA 19041. It has been established that local lattice spin Hamiltonians can be used for universal adiabatic quantum computation. However, the 2-local model Hamiltonians used in these proofs are general and hence do not limit the types of interactions required between spins. To address this concern, the present paper provides two simple model Hamiltonians that are of practical interest to experi- mentalists working towards the realization of a universal adiabatic quantum computer. The model Hamiltonians presented are the simplest known quantum-Merlin-Arthur-complete (QMA-complete) 2-local Hamiltonians. The 2-local Ising model with 1-local transverse field which has been realized using an array of technologies, is perhaps the simplest quantum spin model but is unlikely to be universal for adiabatic quantum computation. We demonstrate that this model can be rendered universal and QMA-complete by adding a tunable 2-local transverse σxσx coupling. We also show the universality and QMA-completeness of spin models with only 1-local σz and σx fields and 2-local σzσx interactions. What are the minimal physical resources required for universal quantum computation? This question is of in- terest in understanding the connections between phys- ical and computational complexity, and for any practi- cal implementation of quantum computation. In 1982, Barahona [1] showed that finding the ground state of the random field Ising model is NP-hard. Such observations fostered approaches to solving problems based on classi- cal [2] and later quantum annealing [3]. The idea of using the ground state properties of a quantum system for com- putation found its full expression in the adiabatic model of quantum computation [4]. This model works by evolv- ing a system from the accessible ground state of an initial Hamiltonian Hi to the ground state of a final Hamilto- nian Hf, which encodes a problem’s solution. The evo- lution takes place over parameters s ∈ [0, 1] as H(s) = (1−s)Hi+sHf, where s changes slowly enough that tran- sitions out of the ground state are suppressed [5]. The simplest adiabatic algorithms can be realized by adding non-commuting transverse field terms to the Ising Hamil- tonian: i hiσ i ∆iσ i,j Jijσ j , (c.f. [6]). However, it is unlikely that the Ising model with trans- verse field can be used to construct a universal adiabatic quantum computer [7]. What then are the simplest Hamiltonians that al- low universal adiabatic quantum computation? For this we turn to the complexity class quantum-Merlin-Arthur (QMA), the quantum analog of NP, and consider the QMA-complete problem k-local Hamiltonian [8]. One solves k-local Hamiltonian by determining if there ex- ists an eigenstate with energy above a given value or be- low another—with a promise that one of these situations is the case—when the system has at most k-local inter- actions. A Yes instance is shown by providing a witness ∗Electronic address: [email protected] †Electronic address: [email protected] eigenstate with energy below the lowest promised value. The problem 5-local Hamiltonian was shown to be QMA-complete by Kitaev [8]. To accomplish this, Ki- taev modified the autonomous quantum computer pro- posed by Feynman [9]. This modification later inspired a proof of the polynomial equivalence between quantum circuits and adiabatic evolutions by Aharonov et al. [10] (see also [11, 12]). Kempe, Kitaev and Regev subse- quently proved QMA-completeness of 2-local Hamil- tonian [14]. Oliveira and Terhal then showed that uni- versality remains even when the 2-local Hamiltonians act on particles in a subgraph of the 2D square lattice [15]. Any QMA-complete Hamiltonian may realize universal adiabatic quantum computation, and so these results are also of interest for the implementation of quantum com- putation. Since 1-local Hamiltonian is efficiently solvable, an open question is to determine which combinations of 2-local interactions allow one to build QMA-complete Hamiltonians. Furthermore, the problem of finding the minimum set of interactions required to build a universal adiabatic quantum computer is of practical, as well as theoretical, interest: every type of 2-local inter- action requires a separate type of physical interaction. To address this question we prove the following theorems: Theorem 1. The problem 2-local ZZXX Hamil- tonian is QMA-complete, with the ZZXX Hamiltonian given as: HZZXX = i + (1) Theorem 2. The problem 2-local ZX Hamiltonian http://arXiv.org/abs/0704.1287v2 mailto:[email protected] mailto:[email protected] is QMA-complete, with the ZX Hamiltonian given as HZX = i + (2) a. Structure In the present paper we briefly review the standard circuit to adiabatic construction to show that 2-local Hamiltonian is QMA-complete when re- stricted to real-valued Hamiltonians. We then show how to approximate the ground states of such 2-local real Hamiltonians by the ZX and ZZXX Hamiltonians. We conclude this work by providing references confirming our claim that the Hamiltonians in Eq. (1) and (2) are highly relevant to experimentalists attempting to build a univer- sal adiabatic quantum computer. I. THE PROBLEM The translation from quantum circuits to adiabatic evolutions began when Kitaev [8] replaced the time- dependence of gate model quantum algorithms with spa- tial degrees of freedom using the non-degenerate ground state of a positive semidefinite Hamiltonian: 0 = H |ψhist〉 = (3) (Hin +Hclock +Hclockinit +Hprop)|ψhist〉. To describe this, let T be the number of gates in the quantum circuit with gate sequence UT · · ·U2U1 and let n be the number of logical qubits acted on by the circuit. Denote the circuit’s classical input by |x〉 and its output by |ψout〉. The history state representing the circuit’s entire time evolution is: |ψhist〉 = T + 1 |x〉 ⊗ |0〉⊗T + U1|x〉 ⊗ |1〉|0〉⊗T−1 + U2U1|x〉 ⊗ |11〉|0〉⊗T−2 + . . . (4) + UT · · ·U2U1|x〉 ⊗ |1〉⊗T where we have indexed distinct time steps by a T qubit unary clock. In the following, tensor product symbols separate operators acting on logical qubits (left) and clock qubits (right). Hin acts on all n logical qubits and the first clock qubit. By annihilating time-zero clock states coupled with classical input x, Hin ensures that valid input state (|x〉 ⊗ |0...0〉) is in the low energy eigenspace: Hin = (11 − |xi〉〈xi|) ⊗ |0〉〈0|1 (5) (11 − (−1)xiσzi ) ⊗ (11 + σz1). Hclock is an operator on clock qubits ensuring that valid unary clock states |00...0〉, |10..0〉, |110..0〉 etc., span the low energy eigenspace: Hclock = |01〉〈01|(t,t+1) (6) (T − 1)11 + σz1 − σzT − σzt σ (t+1) where the superscript (t, t+ 1) indicates the clock qubits acted on by the projection. This Hamiltonian has a sim- ple physical interpretation as a line of ferromagnetically coupled spins with twisted boundary conditions, so that the ground state is spanned by all states with a sin- gle domain wall. The term Hclockint applies a penalty |1〉〈1|t=1 to the first qubit to ensure that the clock is in state |0〉⊗T− at time t = 0. Hprop acts both on logical and clock qubits. It en- sures that the ground state is the history state corre- sponding to the given circuit. Hprop is a sum of T terms, Hprop = t=1Hprop,t, where each term checks that the propagation from time t− 1 to t is correct. For 2 ≤ t ≤ T − 1, Hprop,t is defined as: Hprop,t = 11 ⊗ |t− 1〉〈t− 1| − Ut ⊗ |t〉〈t− 1| − U †t ⊗ |t− 1〉〈t| + 11 ⊗ |t〉〈t|, (7) where operators |t〉〈t− 1| = |110〉〈100|(t−1,t,t+1) etc., act on clock qubits t−1, t, and t+1 and where the operator Ut is the t th gate in the circuit. For the boundary cases (t = 1, T ), one writes Hprop,t by omitting a clock qubit (t− 1 and t+ 1 respectively). We have now explained all the terms in the Hamilto- nian from Eq. (3)—a key building block used to prove the QMA-completeness of 5-local Hamiltonian [8]. The construction reviewed in the present section was also used in a proof of the polynomial equivalence between quantum circuits and adiabatic evolutions [10]. Which physical systems can implement the Hamiltonian model of computation from Eq. (3)? Ideally, we wish to find a simple Hamiltonian that is in principle realizable using current, or near-future technology. The ground states of many physical systems are real-valued, such as the ground states of the Hamiltonians from Eq. (1) and (2). So a logical first step in our program is to show the QMA- completeness of general real-valued local Hamiltonians. A. The QMA-completeness of real-valued Hamiltonians Bernstein and Vazirani showed that arbitrary quan- tum circuits may be represented using real-valued gates operating on real-valued wave functions [17]. Using this idea, one can show that 5-local real Hamiltonian is already QMA-complete—leaving the proofs in [8] oth- erwise intact and changing only the gates used in the circuits. Hin from Eq. (5) and Hclock from Eq. (6) are al- ready real-valued and at most 2-local. Now consider the terms in Hprop from Eq. (7) for the case of self-inverse elementary gates Ut = U Hprop,t = (11 − σz(t−1))(11 + σ (t+1)) (8) (11 − σz(t−1))σ t (11 + σ (t+1)) For the boundary cases (t = 1, T ), define: Hprop,1 = (11 + σz2) − U1 ⊗ (σx1 + σ 2) (9) Hprop,T = (11 − σz(T−1)) − UT ⊗ (σxT − σz(T−1)σ The terms from Eq. (8) and (9) acting on the clock space are already real-valued and at most 3-local. As an ex- plicit example of the gates Ut, let us define a universal real-valued and self-inverse 2-qubit gate: Rij(φ) = (11+σzi )+ (11−σzi )⊗ (sin(φ)σxi +cos(φ)σzj ). The gate sequence Rij(φ)Rij(π/2) recovers the universal gate from [18]. This is a continuous set of elementary gates parameterized by the angle φ. Discrete sets of self inverse gates which are universal are also readily con- structed. For example, Shi showed that a set comprising the C-NOT plus any one-qubit gate whose square does not preserve the computational basis is universal [13]. We immediately see that a universal set of self-inverse gates cannot contain only the C-NOT and a single one-qubit gate. However, the set {C-NOT, X, cosψX + sinψZ} is universal for any single value of ψ which is not a multiple of π/4. A reduction from 5-local to 2-local Hamiltonian was accomplished by the use of gadgets that reduced 3- local Hamiltonian terms to 2-local terms [14]. From the results in [14] (see also [15]) and the QMA-completeness of 5-local real Hamiltonian, it now follows that 2- local real Hamiltonian is QMA-complete and uni- versal for adiabatic quantum computation. We note that the real product σ j , or tensor powers thereof, are not necessary in any part of our construction, and so Hamil- tonians composed of the following pairwise products of real-valued Pauli matrices are QMA-complete and uni- versal for adiabatic quantum computation[29]: {11, 11 ⊗ σx, 11 ⊗ σz , σx ⊗ 11, (10) σz ⊗ 11, σx ⊗ σz, σz ⊗ σx, σx ⊗ σx, σz ⊗ σz}. To prove our Theorems (1) and (2), we will next show that one can approximate all the terms from Eq. (10) using either the ZX or ZZXX Hamiltonians—the Hamil- tonians from Eq. (1) and (2) respectively. We do this using perturbation theory [14, 15] to construct gadget Hamiltonians that approximate the operators σzi σ j and σxi σ i with terms from the ZZXX Hamiltonian as well as the operators σzi σ i and σ j with terms from the ZX Hamiltonian. B. The ZZXX gadget We use the ZZXX Hamiltonian from Eq. (1) to con- struct the interaction σzi σ j from σ xσx and σzσz interac- tions. Let Heff = αijσ j ⊗ |0〉〈0|k, where qubit k is an ancillary qubit and define the penalty Hamiltonian Hp and corresponding Green’s function G(z) as follows: Hp = δ|1〉〈1|k = (11 − σzk) and (11) = (z11 −Hp)−1. Hp splits the Hilbert space into a degenerate low en- ergy eigenspace L− = span{|sisj〉|0〉|∀si, sj ∈ {0, 1}}, in which qubit k is |0〉, and a δ energy eigenspace L+ = span{|sisj〉|1〉|∀si, sj ∈ {0, 1}}, in which qubit k is |1〉. First, we give the ZZXX Hamiltonian which pro- duces an effective σzσz interaction in the low energy subspace. Let Y be an arbitrary ZZXX Hamiltonian acting on qubits i and j and consider a perturbation V = V1 + V2 + V3 that breaks the L− zero eigenspace degeneracy by creating an operator O(ǫ) close to Heff in this space: V1 = [Y +D(σ j + 11)] ⊗ 11k −Aσzi ⊗ |0〉〈0|k V2 = B(σ j + 11) ⊗ σxk (12) V3 = Cσ i ⊗ |1〉〈1|k. The term V2 above allows the mediator qubit k to un- dergo virtual excitations and applies an σx term to qubit j during transitions between the L− and L+ subspaces. During excitation into L+, the term V3 applies a σz term to qubit i. This perturbation is illustrated in figure 1 FIG. 1: The ZZXX gadget used to approximate the operator σzi σ j using only σ xσx and σzσz interactions. The present figure presents a diagrammatic representation of the Pertur- bation Hamiltonian V = V1 + V2 + V3 from Eq. (12) applied to qubits i, j and k. Not shown in the present figure is an overall constant energy shift of D. Let Π± be projectors on L±; for arbitrary operator O we define O±∓ = Π±OΠ∓ (O±± = Π±OΠ±) and let λ(O) denote the lowest eigenvalue of O. One approxi- mates λ(Htarg) of the desired low energy effective 2-local Hamiltonian by a realizable 2-local physical Hamiltonian H̃ = Hp + V , where λ(H̃) is calculated using perturba- tion theory. The spectrum of H̃−− is approximated by the projection of the self-energy operator Σ(z) for real- valued z which has the following series expansion: Σ−−(z) = Hp– + V−− + ︷ ︸︸ ︷ V−+G++(z)V+− (13) + V−+G++(z)V+G++(z)V+− ︸ ︷︷ ︸ ‖V ‖4δ−3 + · · · Note that with our penalty Hamiltonian H−− = 0, and for the perturbing Hamiltonian V = V1 + V2 + V3 only V1 is nonzero in the low energy subspace, V1 and V3 are nonzero in the high energy subspace, and only V2 induces transitions between the two subspaces. The non-zero projections are: V1−− = [Y +Aσ i +D(σ j + 11)] ⊗ |0〉〈0|k V2−+ = B(σ j + 11) ⊗ |0〉〈1|k V2+− = B(σ j + 11) ⊗ |1〉〈0|k (14) V3++ = V3 V+ = (Y + Cσ i +D(σ j + 11)) ⊗ |1〉〈1|k The series expansion of the self-energy follows directly: 1st : (Y −Aσzi +D(σxj + 11)) ⊗ |0〉〈0|k 2nd : z − δ (σxj + 11) 2 ⊗ |0〉〈0|k (15) 3rd : (z − δ)2 (σxj + 11)σ j + 11) ⊗ |0〉〈0|k (z − δ)2 (σxj + 11)Y (σ j + 11) ⊗ |0〉〈0|k (z − δ)2 (σxj + 11) 3 ⊗ |0〉〈0|k The self-energy in the low energy subspace (where qubit k is in state |0〉) is therefore: Σ−−(z) ≃ Ỹ + (z − δ)2 σzi (16) z − δ (z − δ)2 (σxj + 11) (z − δ)2 σzi σ ‖V ‖4δ−3 + · · · Ỹ is the interaction between qubits i and j which is the original physical interaction dressed by the effect of vir- tual excitations into the high energy subspace. Ỹ = Y + (z − δ)2 (σ j + 11)Y (σ j + 11) (17) In practice there will always be some interaction be- tween qubits i and j. We assume Y is a ZZXX Hamil- tonian and express the dressed Hamiltonian Ỹ in terms of modified coupling coefficients. Writing the physical Hamiltonian: Y = hiσ i + hjσ j + ∆iσ i + ∆jσ j + (18) + Jijσ j +Kijσ The new dressed coupling strengths are: hi 7→ hi (z − δ)2 ∆i 7→ ∆i (z − δ)2 (z − δ)2 ∆j 7→ ∆j (z − δ)2 Kij 7→ Kij (z − δ)2 (z − δ)2 with additional couplings: (z − δ)2 ∆j11 + (z − δ)2 j (20) We see that the effect of the gadget on any existing phys- ical interaction is to modify the coupling constants, add an overall shift in energy, and to add a small correction to the σzσz coupling which depends on the strength of the σzi term in Y . If Y is regarded as the net uncontrolled physical Hamiltonian coupling i and j (a source of error) it is only the local σzi field which contributes to an error in the σzσz coupling strength. We make the following choices for our gadget parame- ters A, B, C and D: A = αij (21) D = 2δ1/3Ē2/3 Where Ē is an energy scale parameter to be fixed later. We expand the self-energy (16) in the limit where z is constant (z = O(1) ≪ δ). Writing (z − δ)−1 ≃ − 1 ) gives: Σ(0)−− = Ỹ + αijσ j (22) Ē4/3 (σxj + 11) ‖V ‖4δ−3 + · · · For the self-energy to become O(ǫ) close to Y + j ⊗ |0〉〈0|k, the error terms in (22) must be bounded above by ǫ through an appropriate choice of δ. Define a lower bound on the spectral gap δ as an inverse polynomial in ǫ: δ ≥ Ēǫ−r, where Ē is a constant and integer r ≥ 1. Now bound r by considering the (weak) upper bound on ‖V ‖: ‖V ‖ ≤ ‖Y ‖ + |αij | + 4δ1/3Ē2/3 (23) + 2Ē |αij | The largest term in δ−3‖V ‖4 is O(Ē(Ē/δ)1/3), and so in order that δ−3‖V ‖4 < ǫ we require r ≥ 3. This also bounds the term below fourth order, Ē4/3δ−1/3 = O(Ēǫ) and so for z ≪ δ we obtain ‖Σ−−(z) −Heff‖ = O(ǫ). In fact, Σ(0)−− = Heff + Ēǫ(σ j + 11). Now apply Theorem (3) from [14] and it follows that |λ(Heff)−λ(H̃)| = O(ǫ). It also follows from Lemma (11) of [14] that the ground state wavefunction ofHeff is also close to the ground state of our gadget. The ZZXX Hamiltonian (1) allows for the direct real- ization of all terms in (10) except for σzσx and σxσz interactions. These terms can be approximated with only O(ǫ) error using the gadget in the present section— thereby showing that the ZZXX Hamiltonian can effi- ciently approximate all terms from (10). Similarly, the ZX Hamiltonian allows for the direct realization of all terms in (10) except for σzσz and σxσx interactions. These terms will be approximated with only O(ǫ) er- ror by defining gadgets in the coming sections—showing that the ZX Hamiltonian can also be used to efficiently approximate all terms from (10). C. The ZZ from ZX gadget We approximate the operator βijσ j using the ZX Hamiltonian in Eq. (2) by defining a penalty Hamiltonian as in Eq. (11). The required perturbation is a sum of terms V = V1 + V2: V1 = Y +A|0〉〈0|k (24) V2 = B(σ i − σzj ) ⊗ σxk The non-zero projections are: V1++ = Y ⊗ |1〉〈1|k (25) V1−− = (Y +A11 ⊗ |0〉〈0|k V2+− = B(σ i − σzj ) ⊗ |1〉〈0|k V2−+ = B(σ i − σzj ) ⊗ |0〉〈1|k V1 does not couple the low and high energy subspaces and V2 couples the subspaces but is zero in each subspace. FIG. 2: The ZZ from ZX gadget: The present figure presents a diagrammatic representation of the Perturbative Hamilto- nian V = V1 + V2 from Eq. (24) applied to qubits i, j and k. In addition to these terms shown in the present figure, there is an overall energy shift of A/2. The series expansion of the self-energy follows directly: 1st : (Y +A11) ⊗ |0〉〈0|k 2nd : B2(σzi − σzj )2 (z − δ) ⊗ |0〉〈0|k (26) 3rd : (z − δ)2 (σzi − σzj )Y (σzi − σzj ) ⊗ |0〉〈0|k Note that in this case the desired terms appear at second order in the expansion, rather than at third order as was the case for the ZX from ZZXX gadget. The terms which dress the physical hamiltonian Y coupling qubits i and j appear at third order. The series expansion of the self- energy in the low energy subspace is: Σ(z)−− = (Ỹ +A11) 2B2(1 − σzi σzj ) (z − δ) + O(||V ||4δ−3) where the dressed interaction Ỹ is defined: Ỹ = Y + (z − δ)2 (σ i − σzj )Y (σzi − σzj ) (28) We assume that the physical interaction Y between i and j qubits is a ZX Hamiltonian and express the dressed Hamiltonian in terms of modified coupling constants. Writing the physical Hamiltonian: Y = hiσ i + hjσ j + ∆iσ i + ∆jσ j (29) +Jijσ j +Kijσ We obtain modified coupling strengths: hi 7→ hi + 2B2(hi − hj) (z − δ)2 hj 7→ hi + 2B2(hj − hi) (z − δ)2 . In this case only the local Z field strengths are modified. We choose values for the perturbation interaction strengths as follows: B = and A = βij and ex- pand the self-energy in the limit where z is constant (z = O(1) ≪ δ): Σ(0)−− = Ỹ + βijσ + O(||V ||4δ−3). We again choose δ to be an inverse power in a small parameter ǫ so that δ ≥ Ēǫ−s, and again use the (weak) upper bound on ||V ||: ||V || ≤ ||Y || + βij + 2βijδ (32) The largest term in ||V ||4δ−3 is 4β2ijδ−1, and so in order that ||V ||4δ−3 < ǫ we require r ≥ 1. Using the gadget defined in the present section, the ZX Hamiltonian can now be used to efficiently approximate all terms in (10) except for σxσx interactions. These interactions can also be approximated with only O(ǫ) error by defining an additional gadget in the next section. D. The XX from ZX gadget An σxσx coupling may be produced from the σzσx coupling as follows. We define a penalty Hamiltonian and corresponding Green’s function: (11 − σxk ) = δ|−〉〈−| G++ = z − δ |−〉〈−|k. (33) This penalty Hamiltonian splits the Hilbert space into a low energy subspace in which the ancilla qubit k is in state |+〉 = (|0〉+ |1〉)/ 2 and a high energy subspace in which the ancilla qubit k is in state |−〉 = (|0〉− |1〉)/ The perturbation is a sum of two terms V = V1 + V2, where V1 and V2 are given by: V1 = Y ⊗ 11k +A|+〉〈+|k (34) V2 = B(σ i − σxj )σzk The non-zero projections are: V1++ = |−〉〈−|V1|−〉〈−|k (35) = Y ⊗ |−〉〈−|k V1−− = Y ⊗ |+〉〈+|k +A|+〉〈+|k V2+− = |−〉〈−|V2|+〉〈+|k = B(σxi − σxj )|−〉〈+|k V2−+ = |+〉〈+|V2|−〉〈−|k = B(σxi − σxj )|+〉〈−|k. Once more we see that the perturbation V1 does not cou- ple the subspaces, whereas V2 couples the subspaces but FIG. 3: The XX from ZX gadget: The present figure presents a diagrammatic representation of the Perturbative Hamiltonian V = V1 + V2 from Eq. (34) applied to qubits i, j and k. In addition to the terms shown in the present fig- ure, there is an overall energy shift of A/2. The penalty term applied to qubit k is the σx basis. is zero in each subspace. This perturbation is illustrated in Figure 3. The series expansion of the self-energy follows: 1st : (Y +A11) ⊗ |+〉〈+|k (36) 2nd : B2(σxi − σxj )2 (z − δ) ⊗ |+〉〈+|k 3rd : (z − δ)2 (σxi − σxj )Y (σxi − σxj ) Again we see that the desired term appears at second order, while the third order term is due to the dressing of the physical interaction Y between qubits i and j. In the low energy subspace the series expansion of the self- energy to third order is: Σ(z)−− = (Ỹ +A11) (37) 2B2(11 − σxi σxj ) (z − δ) + O(||V ||4δ−3) where the dressed interaction Ỹ is defined: Ỹ = Y + (z − δ)2 (σxi − σxj )Y (σxi − σxj ). (38) Once more we assume the physical Hamiltonian Y is a ZX Hamiltonian 29 and we describe the effects of dressing to low order in terms of the new dressed coupling strengths: ∆i 7→ ∆i + 2B2(∆i − ∆j) (z − δ)2 ∆j 7→ ∆i + 2B2(∆j − ∆i) (z − δ)2 and in this case only the local X field strengths are mod- ified. Choosing values for our gadget parameters A = γij and B = and expanding the self-energy in the limit where z is constant (z = O(1) ≪ δ) gives: Σ(0)−− = Ỹ ⊗ |+〉〈+|k (40) + γijσ j ⊗ |+〉〈+|k + O(||V ||4δ−3) As before, this self-energy may be made O(ǫ) close to the target Hamiltonian by a bound δ ≥ Ēǫ−1. b. Summary The proof of Theorem (1) follows from the simultaneous application of the ZZXX gadget illus- trated in Fig. 1 to realize all σzσx terms in the target Hamiltonian using a ZZXX Hamiltonian. Similarly, ap- plication of the two gadgets illustrated in Fig. 2 and 3 to realize σxσx and σzσz terms in the target Hamilto- nian proves the first part of Theorem (2). Our result is based on Theorem (3) from [14] which allowed us to ap- proximate (with O(ǫ) error) all the Hamiltonian terms from Eq. (3) using either the ZZXX or ZX Hamiltonians. It also follows from Lemma (11) of [14] that the ground state wavefunction ofHeff is also close to the ground state of our gadget. So to complete our proof, it is enough to show that each gadget satisfies the criteria given in The- orem (3) from [14]. II. CONCLUSION The objective of this work was to provide simple model Hamiltonians that are of practical interest for experi- mentalists working towards the realization of a univer- sal adiabatic quantum computer. Accomplishing such as task also enabled us to find the simplest known QMA- complete 2-local Hamiltonians. The σxσx coupler is re- alizable using systems including capacitive coupling of flux qubits [22] and spin models implemented with po- lar molecules [23]. In addition, a σzσx coupler for flux qubits is given in [24]. The ZX and ZZXX Hamiltonians enable gate model [25], autonomous [26], measurement- based [27] and universal adiabatic quantum computa- tion [10, 14, 15], and may also be useful for quan- tum annealing [28]. 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Mizel, D. A. Lidar, and M. Mitchell (2006), quant-ph/0609067. [13] Y. Shi (2002), quant-ph/0205115. [14] J. Kempe, A. Kitaev, and O. Regev, SIAM J. Computing 35(5), 1070 (2006), quant-ph/0406180. [15] R. Oliveira and B. Terhal (2006), quant-ph/0504050. [16] D. Deutsch and R. Jozsa, Proc. R. Soc. London Ser. A, 439, 553 (1992). [17] E. Bernstein and U. Vazirani, SIAM J. Computing 26, 1411 (1997), quant-ph/9701001. [18] T. Rudolph and L. Grover (2002), quant-ph/0210187. [19] A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head- Gordon, Science 309, 1704 (2005), quant-ph/0604193. [20] D. Nagaj and S. Mozes (2006), quant-ph/0612113. [21] J. Kempe and O. Regev, Quantum Computation and In- formation 3(3), 258 (2003), quant-ph/0302079. [22] D. V. Averin and C. Bruder, Phys. Rev. Lett. 91, 057003 (2003), cond-mat/0304166. [23] A. Micheli, G. Brennen, and P. Zoller, Nature Phys. 2, 341 (2006), quant-ph/0512222. [24] T. P. Orlando, J. E. Mooij, L. Tian, C. H. van der Wal, L. S. Levitov, S. Lloyd, and J. J. Mazo, Phys. Rev. B 60, 15398 (1999). [25] A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, Phys. Rev. Lett. 74, 4083 (1995), quant-ph/9503017. [26] D. Janzing, Phys. Rev. A 75, 012307 (2007), quant-ph/0506270. [27] S. D. Bartlett and T. Rudolph, Phys. Rev. A 74, 040302 (2006), quant-ph/0609002. [28] S. Suzuki, H. Nishimori, and M. Suzuki (2007), quant-ph/0702214. [29] The QMA-completeness of this subset of Hamiltonians was found independently by D. Bacon; preprint, (2007). [30] We thank C.J.S. Truncik, R.G. Harris, W.G. Macready, M.H.S. Amin, A.J. Berkley, P. Bunyk, J. Lamothe, T. Mahon and G. Rose. J.D.B. and P.J.L. completed parts of this work well on staff as D-Wave Systems Inc. http://arxiv.org/abs/cond-mat/0105238 http://arxiv.org/abs/quant-ph/0208135 http://arxiv.org/abs/quant-ph/0411152 http://arxiv.org/abs/cond-mat/0608253 http://arxiv.org/abs/quant-ph/0606140 http://arxiv.org/abs/quant-ph/0405098 http://arxiv.org/abs/quant-ph/0409024 http://arxiv.org/abs/quant-ph/0609067 http://arxiv.org/abs/quant-ph/0205115 http://arxiv.org/abs/quant-ph/0406180 http://arxiv.org/abs/quant-ph/0504050 http://arxiv.org/abs/quant-ph/9701001 http://arxiv.org/abs/quant-ph/0210187 http://arxiv.org/abs/quant-ph/0604193 http://arxiv.org/abs/quant-ph/0612113 http://arxiv.org/abs/quant-ph/0302079 http://arxiv.org/abs/cond-mat/0304166 http://arxiv.org/abs/quant-ph/0512222 http://arxiv.org/abs/quant-ph/9503017 http://arxiv.org/abs/quant-ph/0506270 http://arxiv.org/abs/quant-ph/0609002 http://arxiv.org/abs/quant-ph/0702214

0704.1288

Quantitative size-dependent structure and strain determination of CdSe nanoparticles using atomic pair distribution function analysis

APS/123-QED Quantitative size-dependent structure and strain determination of CdSe nanoparticles using atomic pair distribution function analysis A. S. Masadeh, E. S. Božin, C. L. Farrow, G. Paglia, P. Juhas and S. J. L. Billinge∗ Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824-1116, USA A. Karkamkar and M. G. Kanatzidis Department of Chemistry, Michigan State University, East Lansing, Michigan 48824-1116, USA The size-dependent structure of CdSe nanoparticles, with diameters ranging from 2 to 4 nm, has been studied using the atomic pair distribution function (PDF) method. The core structure of the measured CdSe nanoparticles can be described in terms of the wurtzite atomic structure with extensive stacking faults. The density of faults in the nanoparticles ∼ 50% . The diameter of the core region was extracted directly from the PDF data and is in good agreement with the diameter obtained from standard characterization methods suggesting that there is little surface amorphous region. A compressive strain was measured in the Cd-Se bond length that increases with decreasing particle size being 0.5% with respect to bulk CdSe for the 2 nm diameter particles. This study demonstrates the size-dependent quantitative structural information that can be obtained even from very small nanoparticles using the PDF approach. PACS numbers: 61.46.Df, 61.10.-i, 78.66.Hf, 61.46.-w I. INTRODUCTION Semiconductor nanoparticles are of increasing interest for both applied and fundamental research. Wurtzite- structured cadmium selenide is an important II-VI semi- conducting compound for optoelectronics.1 CdSe quan- tum dots are the most extensively studied quantum nanostructure because of their size-tunable properties, and they have been used as a model system for inves- tigating a wide range of nanoscale electronic, optical, optoelectronic, and chemical processes.2 CdSe also pro- vided the first example of self-assembled semiconductor nanocrystal superlattices.3 With a direct band gap of 1.8 eV, CdSe quantum dots have been used for laser diodes 4, nanosensing 5 , and biomedical imaging.6 In fundamental research, particles with a diameter in the 1- 5 nm range are of particular importance since they cover the transition regime between the bulk and molecular domains where quantum size effects play an important role. Significant deviation from bulk properties are ex- pected for particles with diameter below 5 nm, and were observed in many cases 6,7 as well as in this study. Accurate determination of atomic scale structure, ho- mogeneous and inhomogeneous strain, structural defects and geometrical particle parameters such as diameter and shape, are important for understanding the fundamen- tal mechanisms and processes in nanostructured materi- als. However, difficulties are experienced when standard methods are applied to small nanoparticles. In this do- main the presumption of a periodic solid, which is the ba- sis of a crystallographic analysis, breaks down. Quantita- tive determinations of the nanoparticle structure require methods that go beyond crystallography. This was noted early on in a seminal study by Bawendi et al.8 where they used the Debye equation, which is not based on a crys- tallographic assumption, to simulate semi-quantitatively the scattering from some CdSe nanoparticles. How- ever, despite the importance of knowing the nanoparticle structure quantitatively with high accuracy, this work has not been followed up with application of modern lo- cal structural methods9,10 until recently.11,12,13,14,15,16 In this study we return to the archetypal CdSe nanoparti- cles to investigate the extent of information about size- dependent structure of nanoparticles from the atomic pair distribution function (PDF) method. This is a lo- cal structural technique that yields quantitative struc- tural information on the nanoscale from x-ray and neu- tron powder diffraction data.10 Recent developments in both data collection17,18 and modeling19,20 make this a potentially powerful tool in the study of nanoparticles. Additional extensions to the modelling are necessary for nanoparticles, and some of these have been successfully demonstrated.11,12,21 In this paper, we present a detailed analysis of the structural information available from PDF data on (2- 4 nm) CdSe nanoparticles. The PDF method is demon- strated here as a key tool that can yield precise structural information about the nanoparticles such as the atomic structure size of the core, the degree of crystallinity, lo- cal bonding, the degree of the internal disorder and the atomic structure of the core region, as a function of the nanoparticle diameter. Three CdSe nanoparticle sam- ples with different diameters that exhibit different opti- cal spectra have been studied. The purpose of this paper is not only to explain the PDF data of CdSe nanoparti- cles through a modeling process, but also to systemati- cally investigate the sensitivity of the PDF data to subtle structural modifications in nanoparticles relative to bulk material. The measurement of the nanoparticle size can lead to significantly different results when performed by differ- ent methods, and there is no consensus as to which is the most reliable.22,23 It is also not clear that a single diam- http://arxiv.org/abs/0704.1288v2 eter is sufficient to fully specify even a spherical particle since the presence of distinct crystalline core and disor- dered surface regions have been postulated.8 Powder diffraction is a well established method for structural and analytical studies of crystalline materi- als, but the applicability to such small particles of stan- dard powder diffraction based on crystallographic meth- ods is questionable and likely to be semi-quantitative at best. Palosz et al.24 have shown that the conventional tools developed for elaboration of powder diffraction data are not directly applicable to nanocrystals.24 There have been some reports8,23,25 in the past few years extract- ing nanoparticle diameter from x-ray diffraction (XRD) using the Scherrer formula, which is a phenomenologi- cal approach that considers the finite size broadening of Bragg-peaks.26 This approach will decrease in accuracy with decreasing particle size, and for particle sizes in the range of a few nanometers the notion of a Bragg peak becomes moot.24 At this point the Debye formula27 be- comes the more appropriate way to calculate the scatter- ing.8 The inconsistency between the nanoparticle diame- ter determined from the standard characterization meth- ods and the diameter obtained by applying the Scherrer formula have been observed by several authors.8,23,28 Previous studies of CdSe nanoparticle structure have demonstrated the sensitivity of the XRD pattern to the presence of planar disorder and thermal effects due to nano-size effects.8,29 The diffraction patterns of CdSe nanoparticles smaller than 2.0 nm have been observed to appear markedly different from those of the larger diameters (see Ref. 29 Fig. 11), the large attenuation and broadening in the Bragg reflections in these small nanoparticles, making the distinction between wurtzite and zinc-blende hard using conventional XRD methods. Murray et al.29 reported that the combination of X-ray studies and TEM imaging yields a description of the av- erage CdSe nanoparticle structure. Strict classification of the CdSe nanoparticles structure as purely wurtzite or zinc-blend is potentially misleading.29 Bawendi et al.8 re- ported that CdSe nanoparticles are best fit by a mixture of crystalline structures intermediate between zinc-blend and wurtzite. Here we apply the PDF method to CdSe nanoparticles and refine quantitative structural parame- ters to a series of CdSe nanoparticles of different sizes. Strain in nano systems has been observed before in dif- ferent studies, as well as in this study. Using combined PDF and extended X-ray-absorbtion fine structure (EX- AFS) methods, Gilbertet al.11 observed a compressive strain compared to the bulk in ZnS nanocrystals. Us- ing an electric field-induced resonance method, Chen et 30 detected the enhancement of Young’s modulus of ZnO nanowires along the axial direction when the diam- eters are decreased. Very recently, Quyang et al.31 de- veloped an analytical model for the size-induced strain and stiffness of a nanocrystal from the perspective of thermodynamics and a continuum medium approach. It was found theoretically that the elastic modulus increases with the inverse of crystal size and vibration frequency is higher than that of the bulk.31 Experimentally, the CdQ (Q=S, Se, T e) first-neighbor distances have been studied using both XRD and EXAFS methods.32 The distances were found smaller than those in the bulk com- pounds by less than 1.0%. Herron et al.33 studied CdS nanocrystals and showed a bond contraction of ∼ 0.5% compared to the bulk. Carter et al.34 studied a series of CdSe nanoparticles using the EXAFS method. In the first shell around both the Se and Cd atoms, they found essentially no change in the first-neighbor distance. Chaure et al.35 studied the strain in nanocrystalline CdSe thin films, using Raman scattering and observed a peak shift with decrease in particle size, which was attributed to the increase in stress with decreasing particle size.35 Local structural deviations or disorder mainly affect the diffuse scattering background. The XRD experiments probe for the presence of periodic structure which are reflected in the Bragg peaks. In order to have information about both long-range order and local structure disorder, a technique that takes both Bragg and diffuse scattering need to be used, such as the PDF technique. Here we apply the PDF method to study the structure, size and strain in CdSe nanoparticles as a function of nanoparticle diameter. The core structure of the CdSe nanoparticles can be described by a mixture of crystalline structures intermediate between zinc-blend and wurtzite, which is wurtzite containing a stacking fault density (SFD) of up to ∼ 50%, with no clear evidence of a disordered surface region, certainly down to 3 nm diameter. The structural parameters are reported quantitatively. We measure a size-dependent strain on the Cd-Se bond which reaches 0.5% at the smallest particle size. The size of the well- ordered core extracted directly from the data agrees with the size determined from other methods. II. EXPERIMENTAL DETAILS A. Sample preparation CdSe nanoparticles were synthesized from cadmium acetate, selenium, trioctyl phosphine and trioctyl phos- phine oxide. Sixty four grams of trioctylphosphine ox- ide (TOPO) containing cadmiumacetate was heated to 360◦C under flowing argon. Cold stock solution (38.4 ml) of (Se:trioctylphosphine = 2:100 by mass) was quickly injected into the rapidly stirred, hot TOPO solution. The temperature was lowered to 300◦C by the injection. At various time intervals, 5-10 ml aliquots of the reac- tion mixture were removed and precipitated in 10 ml of methanol. The color of the sample changed from bright yellow to orange to red to brown with time interval vari- ation from 20 seconds to 1200 seconds. Three nanopar- ticle sizes, CdSeI (small), CdSeII (medium) and CdSeIII (large), were used for this study, as well as a bulk CdSe sample for reference. The samples were further purified by dissolving and centrifuging in methanol to remove excess TOPO. This FIG. 1: TEM image of CdSe nanocrystal prepared using the method described in the text. CdSe obtained by 1200 seconds (left) and 15 seconds (right) nucleation. The line-bar is 10 nm in size in both images. FIG. 2: (a) Room temperature UV-vis absorption and (b) photoluminescence spectra from the sample of CdSe nanocrystals. (©) CdSeI, (△) CdSeII, (�) CdSeIII. process also resulted in a narrower particle size distri- bution. The transmission electron micrograph (TEM) images (Fig. 1) show uniformly sized nanoparticles with no signs of aggregation. The ultraviolet visible (UV-vis) absorption and photoluminescence (PL) spectra of the aliquots were recorded by redissolving the nanocrystals in toluene. The spectra are shown in Fig. 2. The band-gap values obtained for the measured sam- ples can be correlated with the diameter of the nanopar- ticles based on the table provided in supplementary infor- mation of Peng et al.36 using the data on exciton peaks TABLE I: CdSe nanoparticle diameter as determined using various methods. CdSeIII CdSeII CdSeI Nucleation time (s) 1200 630 15 Diameter (nm) TEM 3.5(2) 2.7(2) 2.0(2) UV-vis 3.5(4) 2.9(3) ≤ 1.90 PL 3.6(4) 2.9(3) ≤ 2.1 PDF 3.7(1) 3.1(1) 2.2(2) measured with UV-visible light absorption, and photolu- minescence peaks. The particle sizes were measured by TEM as well. The measured values of particle diameter using these various methods are summarized in Table I. B. The atomic PDF method The atomic PDF analysis of x-ray and neutron pow- der diffraction data is a powerful method for studying the structure of nanostructured materials.9,10,37,38,39,40 Recently, it has been explicitly applied to study the structure of discrete nanoparticles.11,12,40,41,42 The PDF method can yield precise structural and size information, provided that special care is applied to the measurement and to the method used for analyzing the data. The atomic PDF, G(r), is defined as G (r) = 4πr [ρ (r)− ρ0] , (1) where ρ(r) is the atomic pair-density, ρ0 is the average atomic number density and r is the radial distance.43 The PDF yields the probability of finding pairs of atoms separated by a distance r. It is obtained by a sine Fourier transformation of the reciprocal space total scattering structure function S(Q), according to G (r) = Q[S(Q)− 1] sinQr dQ, (2) where S(Q) is obtained from a diffraction experiment. This approach is widely used for studying liquids, amor- phous and crystalline materials, but has recently also been successfully applied to nanocrystalline materials.10 C. High-energy x-ray diffraction experiments X-ray powder diffraction experiments to obtain the PDF were performed at the 6IDD beamline at the Ad- vanced Photon Source at Argonne National Labora- tory. Diffraction data were collected using the recently developed rapid acquisition pair distribution function (RAPDF) technique17 that benefits from 2D data col- lection. Unlike TEM, XRD probes a large number of crystallites that are randomly oriented. The powder samples were packed in a flat plate with thickness of FIG. 3: Two dimensional XRD raw data collected using im- age plate detector from (a) CdSe bulk and (b) nanoparticle CdSeIII samples. 1.0 mm sealed between kapton tapes. Data were col- lected at room temperature with an x-ray energy of 87.005 keV (λ = 0.14248 Å). An image plate camera (Mar345) with a diameter of 345 mm was mounted or- thogonally to the beam path with a sample to detec- tor distance of 208.857 mm, as calibrated by using sil- icon standard sample.17 The image plate was exposed for 10 seconds and this was repeated 5 times for a to- tal data collection time of 50 seconds. The RAPDF ap- proach avoids detector saturation whilst allowing suffi- cient statistics to be obtained. This approach also avoids sample degradation in the beam that was observed for the TOPO coated nanoparticles during longer exposures, on the scale of hours, that were required using conventional point-detector approaches. To reduce the background scattering, lead shielding was placed before the sample with a small opening for the incident beam. Examples of the raw 2D data are shown in Fig. 3. These data were integrated and converted to intensity versus 2θ using the software Fit2D,44 where 2θ is the an- gle between the incident and scattered x-ray beam. The integrated data were normalized by the average monitor counts. The data were corrected and normalized9 using the program PDFgetX245 to obtain the total scattering structure function, S(Q), and the PDF, G(r), which are shown in Figs. 4 (a) and (b) respectively. The scattering signal from the surfactant (TOPO) was measured inde- pendently and subtracted as a background in the data reduction. In the Fourier transform step to get from S(Q) to the PDF G(r), the data are truncated at a finite maximum value of the momentum transfer, Q = Qmax. Different values of Qmax may be chosen. Here a Qmax = 25.0 Å was found to be optimal. Qmax is optimized such as to avoid large termination effects and to reasonably mini- mize the introduced noise level as signal to noise ratio decreases with Q value. Structural information was extracted from the PDFs using a full-profile real-space local-structure refinement method46 analogous to Rietveld refinement.47 We used an updated version48 of the program PDFfit19 to fit the experimental PDFs. Starting from a given structure model and given a set of parameters to be refined, PDF- FIG. 4: (a) The experimental reduced structure function F (Q) of CdSe nanoparticle with different diameters and (b) the corresponding PDF, G(r), obtained by Fourier transfor- mation of the data in (a) with Qmax = 25.0 Å −1, from top to bottom: bulk, CdSeIII, CdSeII and CdSeI. fit searches for the best structure that is consistent with the experimental PDF data. The residual function (Rw) is used to quantify the agreement of the calculated PDF from model to experimental data: ω(ri)[Gobs(ri)−Gcalc(ri)]2 ω(ri)G . (3) Here the weight ω(ri) is set to unity which is justified because in G(r) the statistical uncertainty on each point is approximately equal.49,50 The structural parameters of the model were unit cell parameters, anisotropic atomic displacement parameters (ADPs) and the fractional coordinate z of Se/Cd atom. Non structural parameters that were refined were a cor- rection for the finite instrumental resolution, (σQ), low-r correlated motion peak sharpening factor (δ),51,52 and scale factor. When estimating the particle size, a new version of the fitting program with particle size effects in- cluded as a refinable parameter53 was used. The sample resolution broadening was determined from a refinement to the crystalline CdSe and the silicon standard sample and fixed and the particle diameter refined, as described below. Good agreement between these results was ob- tained. III. RESULTS AND DISCUSSION The reduced structure functions for the bulk and nanocrystalline samples are shown plotted over a wide range of Q in Fig 4(a). All of the patterns show signifi- cant intensity up to the highest values of Q, highlighting the value of measured data over such a wide Q-range. All of the diffraction patterns have peaks in similar positions reflecting the similarity of the basic structures, but as the nanoparticles get smaller the diffraction features become broadened out due to finite size effects.26 The PDFs are shown in Fig. 4(b). What is apparent is that, in real-space, the PDF features at low-r are compa- rably sharp in all the samples. The finite size effects do not broaden features in real-space. The finite particle size is evident in a fall-off in the intensity of structural fea- tures with increasing-r. Later we will use this to extract the average particle size in the material. The structure apparent in the G(r) function comes from the atomic or- der within the nanoparticle. The value of r where these ripples disappear indicates the particle core region diam- eter; or at least the diameter of any coherent structural core of the nanoparticle. By direct observation (Fig. 9) we can put a lower limit on the particle diameters to be 3.6, 2.8 and 1.6 nm for CdSeIII, II and I, respectively, where the ripples can be seen to die out by visual inspec- tion. These numbers will be quantified more accurately later. A. Nanoparticle structure Features in the PDF at low-r reflect the internal struc- ture of the nanoparticles. The nanoparticle PDFs have almost the same features as in the bulk in the region below 8.0 Å, reflecting the fact that they share a simi- lar atomic structure on average. In the finite nano-size regime, local structural deviations from the average bulk structure are expected. A large number of semiconductor alloys, especially some sulfides and selenides, do not crystallize in the cu- bic zinc-blende structure but in the hexagonal wurtzite structure54. Both wurtzite and zinc-blende structures are based on the stacking of identical two-dimensional pla- nar units translated with respect to each other, in which each atom is tetrahedrally coordinated with four nearest neighbors. The layer stacking is described as ABABAB... along the [001] axis for wurtzite and asABCABC... along FIG. 5: Fragments from the (a) wurtzite structure, space group (P63mc) and (b) zinc-blende structure, space group (F 4̄3m). the [111] axis for zinc-blende. As can be seen in the Fig. 5, each cadmium and selenium is tetrahedrally coordinated in both structures. However, the next nearest and more distant coordination sequences are different in the two structures. The largest changes in structure are expected in the smallest nanoparticles. In these small nanoparticles, the proportion of atoms on the surface is large making the notion of a well-ordered crystalline core moot. The frac- tion of atoms involved in the surface atoms was estimated as 0.6, 0.45 and 0.35 for 2 nm, 3 nm and 4 nm nanoparti- cle diameters, respectively. This was estimated by taking different spherical cuts from bulk structure, then count- ing the atom with coordination number 4 as core atom and the one with less than 4 as surface atom. For the smallest particles the small number of atoms in the core makes it difficult to define a core crystal structure, mak- ing the distinction between wurtzite and zinc-blende dif- ficult using the conventional XRD methods as nanopar- ticle size decreases.29 The principle difference between these structures is the topology of the CdSe4 connec- tions, which may also be becoming defective in the small nanoparticles. Two structure models wurtzite (space group P63mc) and zinc-blende (space group F 4̄3m), were fit to the PDF data. The results of the full-profile fitting to the PDF data are shown Fig. 6. In this figure we compare fits to the (a) wurtzite and (b) zinc-blende structure mod- els using isotropic atomic displacement factors (Uiso) in both models. The wurtzite structure gives superior fits for the bulk structure. However, for all the nanoparticle sizes, the fits of wurtzite and zinc-blende are comparable as evident from the difference curves in Fig. 6 and the Rw-values reported in Table II. This indicates that clas- sification of the CdSe nanoparticles structure as purely wurtzite or zinc-blend is misleading29 and it is better de- scribed as being intermediate between the two structures, as has been reported earlier8. Introducing anisotropic ADPs (U11 = U22 6= U33) into the wurtzite model, resulted in better fits to the data. The refined parameters are reproduced in Table III and the fits are shown in Fig. 7(a). The values for the nanoparticles are rather close to the values in the bulk wurtzite structure. The model with anisotropic ADPs FIG. 6: (Color online) The experimental PDF, G(r), with Qmax = 19.0 Å −1(blue solid dots) and the calculated PDF from refined structural model (red solid line), with the dif- ference curve offset below (black solid line). PDF data are fitted using (a) wurtzite structure model, space group P63mc and (b) zinc-blende model with space group F 4̄3m. In both models isotropic atomic displacement factors (Uiso) are used. TABLE II: The refined residual (Rw) values obtained from PDF analysis assuming the wurtzite and zinc-blend structure models with space group P63mc and F 4̄3m, respectively. In both models isotropic atomic displacement factors (Uiso) are used. CdSe-bulk CdSeIII CdSeII CdSeI Wurtzite (Rw) 0.16 0.31 0.28 0.31 Zinc-blende (Rw) 0.52 0.32 0.30 0.35 resulted in lower Rw. There is a general increase in the ADPs with decreasing particle size. This reflects inho- mogeneous strain accommodation in the nanoparticles as we discuss below. However, the values of the ADPs along the z-direction for Se atoms (U33) are four times larger in the nanoparticles compared with the bulk where FIG. 7: (Color online) The experimental PDF, G(r), with Qmax = 19.0 Å −1(blue solid dots) and the calculated PDF from refined structural model (red solid line), with the differ- ence curve offset below (black solid line). PDF data are fitted using wurtzite structure model (a) with no stacking fault and (b) with 33% stacking fault density for bulk and 50% for all nanoparticle sizes. In both cases anisotropic atomic displace- ment factors (Uaniso) are used they are already unphysically large. The fact that this parameter is large on the Se site and small on the Cd site is not significant, since we can change the origin of the unit cell to place a Cd ion at the (1/3,2/3,z) position and the enlarged U33 shifts to the Cd site in this case. The unphysically large U33 value on the Se site is likely to be due to the presence of faults in the basal plane stacking. For example, similar unphysical enlargements of perpendicular thermal factors in PDF measurements are explained by the presence of turbostratic disorder in layered carbons55, which is a similar effect to faults in the ABABAB wurtzite stacking. Also, the presence of stacking faults in the nanoparticles has been noted pre- viously.8 It is noteworthy that this parameter is enlarged in EXAFS analyses of bulk wurtzite structures, probably FIG. 8: The the enlargement in the the ADPs along the z- direction for Se site U33, as a function of the stacking fault density. for the same reason.32,56,57 We suspect that the enlarge- ment in this parameter (U33) is related to the stacking fault density present in bulk and that is increasing in the nanoparticles. To test this idea we simulated PDF data using the wurtzite structure containing different stacking fault den- sities. The stacking faults were simulated for different densities (0.167, 0.25, 0.333, and 0.5) by creating wurtzite superlattices with different stacking sequences along the C-axis. The program DISCUS58 was used to create the stacking fault models and PDFgui48 was used to gener- ate the corresponding PDFs. The PDFs were simulated with all the ADPs fixed at Uii = 0.0133 Å 2, the value ob- served in the experimental bulk data collected at room temperature (see Table III). To see if this results in enlarged U33 values we refined the simulated data containing stacking faults using the wurtzite model without any stacking faults. Indeed, the refined Se site U33 increased monotonically with increas- ing stacking fault density. The results are plotted in Fig. 8. Fig. 8 can be considered as calibration curve of stacking fault density in the wurtzite structure, based on the en- largement in the ADPs along the z-direction U33. From this we can estimate a stacking fault density of ∼ 35% for our bulk CdSe sample, and ∼ 50% for each of the nanoparticles. It is then possible to carry out a refinement using a structural model that contains an appropriate stacking fault density. The PDF data of bulk CdSe was therefore fit with a wurtzite model with a 33% density, and the nanoparticle PDF fit with a model with 50% of stacking faults. The refinements give excellent fits, as is evident in Fig. 7(b). The results are presented in Table III. The en- larged U33 parameter on the Se site is no longer present and it is now possible to refine physically reasonable val- ues for that parameter. As well as resulting in physically reasonable ADPs, the quality of the fits to the data are excellent, though the Rw value is slightly larger in the nanoparticles. Attempts to characterize the structure changes using direct measurements such as TEM technique for such small CdSe nanoparticles59 were unsuccessful due to the poor contrast. However, in the present study we were successful in exploring the local atomic structure for CdSe nanoparticles, in real space, at different length scales. The PDF fits clearly indicate that the structure can be described in terms of locally distorted wurtzite structure containing ∼ 50% stacking fault density (i.e., intermediate between wurtzite and zinc-blende) even for the 2 nm diameter particles, Fig. 7. Interestingly, there is little evidence in our data for a significant surface modified region. This surface region is sometimes thought of as being an amorphous-like re- gion. Amorphous structures appear in the PDF with sharp first neighbor peaks but rapidly diminishing and broadening higher neighbor peaks. Thus, in the presence of a surface amorphous region, we might expect to see ex- tra intensity at the first-peak position when the wurtzite model is scaled to fit the higher-r features coming just from the crystalline core. As evident in Fig. 7, this is not observed. Furthermore, as we describe below, the diameter of the crystalline core that we refine from the PDF agrees well with other estimates of nanoparticle size, suggesting that there is no surface amorphous region in these nanoparticles. The good agreement in the intensity of the first PDF peak also presents a puzzle in the op- posite direction since we might expect surface atoms to be under-coordinated, which would result in a decrease in the intensity of this peak. It is possible that the com- peting effects of surface amorphous behavior and surface under coordination perfectly balance each other out, and this cannot be ruled out, though it seems unlikely that it would work perfectly at all nanoparticle diameters. This is also not supported by the nanoparticle size determina- tions described below. B. Nanoparticle size We describe here how we extracted more accurate nanoparticle diameters. This determination is impor- tant since the physical proprieties are size dependent. It is also important to use complementary techniques to determine particle size as different techniques are more dependent on different aspects of the nanoparticle struc- ture, for example, whether or not the technique is sensi- tive to any amorphous surface layer on the nanoparticle. More challenges are expected in accurate size determi- nation as nanoparticle diameter decreases, due to poor contrast near the surface of the nanoparticle. In the literature, CdSe nanoparticles with a diame- ter of 2.0 nm have been considered to be an especially stable size with an associated band edge absorption cen- tered at 414 nm60, that size was observed earlier29,61 with TABLE III: The refined parameters values obtained from PDF analysis assuming the wurtzite structure , space group P63mc, with different stacking fault densities (SFDs). CdSe-bulk CdSeIII CdSeII CdSeI Stacking fault density (%) 0.0 33.0 0.0 50.0 0.0 50.0 0.0 50.0 a (Å) 4.3014(4) 4.3012(4) 4.2997(9) 4.2987(9) 4.3028(9) 4.3015(9) 4.2930(9) 4.2930(8) c (Å) 7.0146(9) 7.0123(9) 7.0145(4) 7.0123(4) 6.9987(9) 6.9975(9) 6.9405(9) 6.9405(7) Se Z-frac. 0.3774(3) 0.3771(3) 0.3761(9) 0.3759(9) 0.3751(6) 0.3747(6) 0.3685(9) 0.3694(9) Cd U11 = U22 (Å 2) 0.0108(2) 0.0102(2) 0.0146(7) 0.0149(7) 0.0149(6) 0.0112(5) 0.0237(9) 0.0213(8) U33 (Å 2) 0.0113(3) 0.0112(3) 0.0262(9) 0.0241(9) 0.0274(9) 0.0271(9) 0.0261(9) 0.0281(9) Se U11 = U22 (Å 2) 0.0109(9) 0.0102(9) 0.0077(7) 0.0138(7) 0.0083(7) 0.0121(7) 0.0110(9) 0.0191(9) U33 (Å 2) 0.0462(9) 0.0115(9) 0.1501(9) 0.02301(9) 0.1628(9) 0.0265(9) 0.1765(9) 0.0311(9) NPa diameter (nm) ∞ ∞ 3.7(1) 3.7(1) 3.1(1) 3.1(1) 2.4(2) 2.2(2) Rw 0.12 0.09 0.20 0.14 0.18 0.15 0.27 0.21 aNP refers to nanoparticle. an estimated diameter of ≤2.0 nm. There are some re- ported difficulties in determining the diameter of such small CdSe nanoparticles. Attempts to characterize the structure changes by TEM and X-ray diffraction tech- niques59 were unsuccessful due to the small diameter of the particles relative to the capping material. If we assume the nanoparticle to have spherical shape (a reasonable approximation based on the TEM in Fig. 1) cut from the bulk, then the measured PDF will look like the PDF of the bulk material that has been attenuated by an envelope function given by the PDF of a homogeneous sphere, as follows62 G (r, d) = G (r) f (r, d) , (4) where G(r) is given in Eq. 1, and f(r, d) is a sphere en- velope function given by f (r, d) = Θ(d− r), (5) where d is the diameter of the homogeneous sphere, and Θ(x) is the Heaviside step function, which is equal to 0 for negative x and 1 for positive. The approach is as follows. First we refine the bulk CdSe data using PDFfit. This gives us a measure of the PDF intensity fall-off due to the finite resolution of the measurement.9 Then the measured value of the finite resolution was kept as an unrefined parameter after that, while all the other structural and non structural param- eters were refined. To measure the PDF intensity fall-off due to the finite particle size, the refined PDF is atten- uated, during the refinement, by the envelope function (Eq. 5) which has one refined parameter, the particle di- ameter. The fit results are shown in Fig. 9 and the result- ing values of particle diameter from the PDF refinement are recorded in Table I. The insets show the calculated and measured PDFs on an expanded scale. The accuracy of determining the nanoparticle size can be evaluated di- rectly from this figure. Features in the measured PDFs that correspond to the wurtzite structure are clearly seen disappearing smoothly attenuated by the spherical PDF envelope function. The procedure is least successful in the smallest nanoparticles, where the spherical particle approximation on the model results in features that ex- tend beyond those in the data. In this case, the spherical approximation may not be working so well. The particle diameters determined from the PDF are consistent with those obtained from TEM, UV-vis and photoluminescence measurements. In particular, an ac- curate determination of the average diameter of the smallest particles is possible in the region where UV-vis and photoluminescence measurements lose their sensitiv- ity.23 In this analysis we have not considered particle size distributions, which are small in these materials. The good agreement between the data and the fits justify this, though some of the differences at high-r may result from this and could contribute an error to the particle size. Several additional fits to the data were performed to test the sphericity of the nanoparticles. Attempts were made to fit the PDF with oblate and prolate spheroid nanopar- ticle form factors. These fits resulted in ellipticities very close to one, and large uncertainties in the refined elliptic- ity and particle diameters, which suggests that the fits are over-parameterized. Another series of fits attempted to profile the PDF with a lognormal distribution of spherical nanoparticles. Allowing the mean nanoparticle diameter and lognormal width to vary resulted in nonconvergent fits, which implies that the particle sizes are not lognor- mal distributed. Therefore, there appears to be little ev- idence for significant ellipticity, nor a significant particle size distribution, as fits assuming undistributed spherical particles give the best results. The simple fitting of a wurtzite structure with ∼ 50% SFD to the data will result in an estimate of the coherent structural core of the nanoparticle that has a structure can be described by a mixture of crystalline structures intermediate between zinc-blend and wurtzite. Compar- ing the nanoparticle core diameter extracted from PDF analysis with the diameter determined from the standard characterization methods yields information about the existence of a surface amorphous region. The agreement between the core diameter extracted from PDF and that determined from the standard methods (Table I), indi- cates that within our measurement uncertainties, there is FIG. 9: (Color online) The experimental PDF, G(r), shown as solid dots. Sphere envelope function (Eq. 5) is used to transform the calculated PDF of bulk CdSe, using wurtzite structure containing 50% stacking fault density, to give a best fit replication of the PDF of CdSe nanoparticles (red solid line). The inset shows on an expanded scale for the high- r region of experimental G(r) on the top of simulated PDF data for different diameters of CdSe nanoparticles (solid line). (a) CdSeIII, (b) CdSeII, (c) CdSeI. Dashed lines are guides for the eye. FIG. 10: (a) The first PDF peak, (•) bulk, (◦) CdSeIII, (�) CdSeII and (△) CdSeI fitted with one Gaussian (—). Dashed line represents the position of first PDF peak in the bulk data. (b)(N) The first PDF peak width vs nanoparticle size, obtained from one Gaussian fit. Dashed line represents the width of first PDF peak in the bulk data. (c) Strain in Cd-Se bond (∆r/r)(%) vs nanoparticle size. (�) Bond values obtained from the local structure fitting and (•) obtained from one Gaussian fit to the first PDF peak. Dotted curves are guides for the eye. no significant heavily disordered surface region in these nanoparticles, even at the smallest diameter of 2 nm (Fig. 9). In contrast with ZnS nanoparticles11 where the heavily disordered surface region is about 40% of the nanoparticle diameter for a diameter of 3.4 nm, the sur- face region thickness being around 1.4 nm.11 C. Internal strain The local bonding of the tetrahedral Cd-Se building unit was investigated vs nanoparticle diameter. The nearest neighbor peaks at r = 2.6353(3) Å come from covalently bonded Cd-Se pairs. The positions and the width of these peaks have been determined by fitting a Gaussian (Fig. 10(a)) and the results presented in Ta- ble IV. The results indicate that there is a significant compressive strain on this near-neighbor bond length, and it is possible to measure it with the PDF with high accuracy. The bond length of Cd-Se pairs shorten as nanoparticle diameter decreases, suggesting the presence of an internal stress in the nanoparticles. The Cd-Se bond lengths extracted from the PDF structural refine- ment are also in good agreement with those obtained from the first peak Gaussian fit, as shown in Fig. 10(c). Thus we have a model independent and a model depen- TABLE IV: The first PDF peak position (FPP) and width (FPW) for different CdSe nanoparticle sizes and the bulk. CdSe-bulk CdSeIII CdSeII CdSeI PDF FPP (Å) 2.6353(3) 2.6281(3) 2.6262(3) 2.6233(3) PDF FPW (Å) 0.1985(09) 0.1990(19) 0.2021(25) 0.2032(25) dent estimate of the strain that are in quantitative agree- ment. The widths of the first PDF peaks have also been extracted vs nanoparticle diameter from the Gaussian fits (Table IV). They remain comparably sharp as the nanoparticles get smaller, as shown in Fig. 10(b). Ap- parently there is no size-dependent inhomogeneous strain measurable on the first peak. However, peaks at higher-r do indicate significant broadening (Fig. 4(b)) suggesting that there is some relaxation taking place through bond- bending. This is reflected in enlarged thermal factors that are refined in the nanoparticle samples. This is sim- ilar to what is observed in semiconductor alloys where most of the structural relaxation takes place in relatively lower energy bond-bending distortions.63,64 IV. CONCLUSION The PDF is used to address the size and structural characterization of a series of CdSe nanoparticles pre- pared by the method mentioned in the text. The core structure of the measured CdSe nanoparticles was found to possess a well-defined atomic arrangement that can be described in terms of locally disordered wurtzite struc- ture that contains∼ 50% stacking fault densit, and quan- titative structural parameters are presented. The diameter of the CdSe nanoparticles was extracted from the PDF data and is in good agreement with the diameter obtained from standard characterization meth- ods, indicating that within our measurement uncertain- ties, there is no significant heavily disordered surface re- gion in these nanoparticles, even at the smallest diame- ter of 2 nm . In contrast with ZnS nanoparticles11 where the heavily disordered surface region is about 40% of the nanoparticle diameter for a diameter of 3.4 nm, the sur- face region thickness being around 1.4 nm.11 Compared with the bulk PDF, the nanoparticle PDF peaks are broader in the high-r region due to strain and structural defects in the nanoparticles. The near- est neighbor peaks at r = 2.6353(3) Å which come from covalently bonded Cd-Se pairs, shorten as nanoparticle diameter decreases resulting in a size-dependent strain on the Cd-Se bond that reaches 0.5% at the smallest particle size. Acknowledgments We would like to acknowledge help from Didier Wer- meille, Doug Robinson, Mouath Shatnawi, Moneeb Shat- nawi and He Lin for help in collecting data. We are grateful to Christos Malliakas for the valuable assistance with the transmission electron microscopy. May thanks to HyunJeong Kim for useful discussion. We are grateful to Prof. Reinhard Neder for the valuable help with the stacking fault simulation. This work was supported in part by National Science Foundation (NSF) grant DMR- 0304391. Data were collected at the 6IDD beamline of the MUCAT sector at the Advanced Photon Source (APS). Use of the APS is supported by the U.S. DOE, Office of Science, Office of Basic Energy Sciences, under Contract No. W-31-109-Eng-38. The MUCAT sector at the APS is supported by the U.S. DOE, Office of Sci- ence, Office of Basic Energy Sciences, through the Ames Laboratory under Contract No. W-7405-Eng-82. ∗ Electronic address: [email protected] 1 G. Hodes, A. Albu-Yaron, F. 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0704.1289

Brane-world Quantum Gravity

arXiv:0704.1289v1 [gr-qc] 10 Apr 2007 Preprint typeset in JHEP style - HYPER VERSION Brane-world Quantum Gravity M.D. Maia∗, Nildsen Silva †and M.C.B. Fernandes‡ Instituto de F́ısica, Universidade de Braśılia, 70919-970, Braśılia, D. F., Brazil Abstract: The Arnowitt-Deser-Misner canonical formulation of general relativity is ex- tended to the covariant brane-world theory in arbitrary dimensions. The exclusive probing of the extra dimensions makes a substantial difference, allowing for the construction of a non-constrained canonical theory. The quantum states of the brane-world geometry are defined by the Tomonaga-Schwinger equation, whose integrability conditions are deter- mined by the classical perturbations of submanifolds contained in the Nash’s differentiable embedding theorem. In principle, quantum brane-world theory can be tested by current experiments in astrophysics and by near future laboratory experiments at Tev energy. The implications to the black-hole information loss problem, to the accelerating cosmology, and to a quantum mathematical theory of four-sub manifolds are briefly commented. Keywords: Quantum Gravity Brane-world Nash Theorem Tomonaga-Schwinger. ∗[email protected] †[email protected] ‡[email protected] http://arxiv.org/abs/0704.1289v1 http://jhep.sissa.it/stdsearch Contents 1. Quantizing the Brane-World 1 2. Covariant Brane-world Gravity 2 3. Canonical Equations 6 4. Tomonaga-Schwinger Quantum States 8 5. Overview and Perspectives 10 1. Quantizing the Brane-World If gravity is to occupy a significant place in modern physics, it can do so only by being qualitatively different from other fields. As soon as we assume that gravity behaves qualitatively like other fields, we find that it is quantitatively insignificant C.W. Misner (1957) After analyzing perturbative quantum gravity, Misner reached the interesting conclu- sion that an effective quantum gravity must have qualities which makes it different from gauge theories [1]. Translating quantitative significance in terms of energy level, Misner’s conclusion suggests that the problem of quantization of the gravitational field should be solved concomitantly with the hierarchy problem of the fundamental interactions. In what follows, we apply this criterium to brane-world gravity. Brane-world gravity is based on a higher dimensional solution of the hierarchy prob- lem. In a seminal paper N. Arkani-Hamed, G. Dvali and S. Dimopolous questioned the currently accepted hypothesis that gravitons are quantitatively relevant only at the Planck scale of energies, essentially because this is an assumption devoid of experimental sup- port. They proposed that the known gauge fields (and hence all ordinary matter) are to be confined within the four-dimensional brane-world, but gravitons can propagate in a higher-dimensional space, the bulk, at the same Tev scale of energies of the gauge fields [2] (For historical papers on the development of the theory see also [3, 4, 5, 6, 7, 8]). Accord- ing to this view, brane-world gravity is qualitatively distinct from, but it is quantitatively equivalent to the gauge fields of the standard model. Brane-world gravity predicts the existence of short lived Tev mini black holes, which in principle can be produced at the laboratory by a high energy proton-proton collision, with implications to the black hole information loss problem at the quantum level. The – 1 – proposed experiment is set in Minkowski space-time, but it ends in a Schwarzschild (or Reissner-Nordstrom) space-time [9]. Therefore, the theory supporting this experiment must be compatible with cross sections of the order of the Schwarzschild radius, and also with an explanation on how the original Minkowski space-time deforms into black hole, and back in a short period of time. Brane-world gravity may also explain the acceleration of the universe (see eg [10] and references therein). In short, due to the presence of the extrinsic curvature, the vacuum in brane-world gravity is richer than the vacuum in general relativity. Besides the cosmological constant, it also contain a conserved geometric tensor built from the extrinsic curvature. Consequently, when studying the quantum fluctuations of such vacuum we may obtain a different estimate for the vacuum energy density as compared with the case of general relativity. Since most of the current research on brane-world theory is based on models defined in a five-dimensional bulk, using specific coordinates and particular symmetries (see eg [11]), we find it necessary to review in the next section the covariant equations of motion of a brane-world defined in an arbitrary bulk, with an arbitrary number of dimensions. Those equations can be found elsewhere [12, 10], but here we have included some details which are required for the quantum description. Readers who are familiar with this may jump to section 3, where the canonical equations of the brane-world with respect to the extra dimensions are discussed. In section 4 we introduce the Tomonaga-Schwinger equation for the brane-world with respect to the extra dimensions and comment on its integrability. 2. Covariant Brane-world Gravity There are essentially three basic postulates in brane-world theory: (1) The bulk geometry is defined by Einstein’s equations; (2) The brane-world is a sub manifold embedded in that bulk ; (3) The gauge fields and ordinary matter are confined to four dimensions, but gravitons propagates along the extra dimensions at Tev energy [2]. The embedding of the brane-world in the bulk plays an essential role on the covariant (that is, model independent) formulation of the brane-world gravity, because it tells how the Einstein-Hilbert dynamics of the bulk is transferred to the brane-world. However, there a are many different ways to embed a manifold into another, classified as local, global, isometric, conformal (or more generally defined by a collineation), rigid, deformable, analytic or differentiable. The choice of one or another depend on what the embedded manifold is supposed to do. In string theory the action principle is defined on the world-sheets, with additional boundary conditions, so that the embedding is necessarily global. Since the world sheets are 2-dimensional they are all conformally flat and their global embedding is not difficult to achieve. However, if higher-dimensional objects such as p-branes are to be considered, then the global embedding may turn out to be difficult to realize in 10 or even in 11 dimensions [13]. Differently from string theory, the Einstein-Hilbert action in brane-world theory is set on the bulk, which is therefore the primary dynamical object. Furthermore, the embedding – 2 – is locally defined, meaning that the bulk is a local fiber bundle whose fibers are the direct sum of the tangent and normal spaces at each point of the brane-world taken as the base space. If we want to draw a picture, the bulk can be seen as as a locally constructed space around each point of the brane-world. A local differentiable embedding requires only that the embedding functions are dif- ferentiable and regular. This follows from Nash’s embedding theorem, an important im- provement over the traditional analytic embedding theorems of Janet and Cartan [14, 15], which demand that the embedding functions are represented by convergent positive power series. Furthermore, Nash’s theorem shows that any sub manifold can be generated by a continuous sequence of small perturbations of an arbitrarily given sub manifold 1. Al- though the theorem was originally demonstrated for the case of an Euclidean bulk, it was later generalized to pseudo Riemannian manifolds [18, 19]. Given a particular Riemannian sub-manifold σ̄4, its local isometric embedding in a certain bulk MD, is given by D = 4 + N differentiable and regular embedding maps X̄A : σ̄4 → MD, such that 2 X̄A,µX̄B,ν GAB = ḡµν , X̄A,µ η̄Bb GAB = 0, and η̄Aa η̄Bb GAB = ḡab (2.1) where η̄Aa are the components of the N linearly independent vector fields in the same coordinates of the bulk where the components GAB of the bulk metric are defined. The vectors {X̄A,µ , η̄Ba } define a Gaussian reference frame called here the embedding frame. The derivatives of the vectors η̄a is expressed in terms of the second and third fundamental forms k̄µνa, Āµab respectively by the Gauss-Weingerten equations [20] η̄Aa,α = ḡ µν k̄αµaX̄A,ν + ḡmnĀαamη̄An (2.2) Without loss of generality we may chose the normal vectors η̄a to be orthogonal to each other, so that ḡab = ǫaδab, where ǫa = ±1 depending on the signature of the bulk [19]. Nash’s perturbative approach to embedding consists in subjecting the fundamental forms of σ̄4 to small parametric deviations along each normal vector. It can be also described by introducing a small perturbation with parameter δya, of the base vectors {X̄A,µ , ηAa } along each normal η̄Aa evaluated on σ̄4, obtaining another set of vectors (no sum on a) ZA,µ = X̄A,µ + (δya£η̄aX̄A),µ = X̄A,µ − δya[X̄ , η̄a]A,µ = X̄A,µ + δyaη̄Aa,µ, (2.3) ηAa = η̄ a + (δy £η̄a η̄a) A = η̄Aa + δy a[η̄a, η̄a] A = η̄Aa (2.4) 1The perturbative approach to the embedding was originally proposed by J. E. Campbell in 1926. However, his result differs from Nash’s theorem because analytic conditions where implicitly used [16, 17]. Since the perturbation procedure is based on regular and differentiable functions, the differentiable embedding is less restrictive to the geometry than the analytic embeddings. 2Capital Latin indices refer to the bulk, which is a Riemannian geometry with metric GAB in arbitrary coordinates. Small case Latin indices refer to the extra dimensions going from 5 to D, and all Greek indices refer to the brane, from 1 to 4. A curly R always denotes bulk curvatures, like in RABCD . Ordinary capital R like in Rµν denotes brane-world curvatures. Covariant derivatives need to be specified, for the bulk or the brane-world metrics. For a vector V A in the bulk its covariant derivative with respect to GAB is denoted as V A;B. On the other hand, from the point of view of the brane-world metric, the components V A behave as a set of N scalar functions as in [20]. For generality we denote G = |det(GAB)|. – 3 – which define a perturbed embedding frame {ZA,µ, ηAa } in the bulk. Admitting that these new functions remain differentiable and regular and that they satisfy the equations similar to (2.1), ZA,µZB,νGAB = gµν , ZA,µηBa GAB = gµa, ηAa ηBb GAB = gab = ḡab (2.5) we obtain a N -parameter local family of submanifolds σ4 generated by local perturbations of σ̄4, by a continuous variations of the parameters δy The next problem is to find a solution of these equations. However, instead of finding the coordinates ZA, it is more convenient to write the perturbed solution in terms of the fundamental forms, expressed in terms of the initial geometry of σ̄4. By direct substitution of Z,µ and ηAa derived from (2.3) in equations (2.5) we obtain gµν(x, y) = ZA,µZB,νGAB = ḡµν− 2δyak̄µνa + δyaδyb[ḡαβ k̄µαak̄νβb + g cdĀµcaĀνdb], (2.6) gµa(x, y) = ZA,µηBa GAB =δybAµab, (2.7) gab(x, y) = η b GAB = ḡab (2.8) kµνa(x, y) = −ηAa,µZB,νGAB = k̄µνa− δybḡαβ k̄µαak̄νβb −gcdδybĀµcaĀνdb, (2.9) Aµab(x, y) = η b GAB=Āµab(x) (2.10) The contravariant components of the perturbed geometry must be consistent with GACGCB = δAB , which can be realized by setting gµρg ρν = δνµ, gacg cb = δba. Since the indices µ and b can never be equal, we must nave gµρg ρb+ gµcg cb = δbµ = 0. After some algebra we see that this corresponds to an identity ymyngabAµamAνbn = −ybyngmaAµ[am]Aν[bn] ≡ 0. Comparing (2.6) and (2.9) we obtain kµνa = − (2.11) Consequently, the local bulk defined in a neighborhood around σ̄4, is foliated by this perturbed geometry, so that the Riemann curvature of the bulk may be expressed in the perturbed embedding frame. For any fields in the bulk ξ and ζ, the covariant derivative Dξζ is defined by the metric affine connection ΓABC , with the Riemann tensor given by R(ξ, ζ) = [Dξ,Dζ ]. Writing the components of this tensor in the embedding frame we obtain the Gauss, Codazzi and Ricci equations, respectively: RABCDZA,αZB,βZC,γZD,δ = Rαβγδ − 2gmnkα[γmkδ]βn (2.12) RABCDZA,αηBb ZC,γZD,δ = kα[γb;δ] − gmnA[γmbkαδ]n (2.13) RABCDηAa ηBb ZC,γZD,δ = −2gmnA[γmaAδ]nb − 2A[γab;δ] − gµνk[γµakδ]νb (2.14) which are the integrability conditions for the embedding. The differentiable embedding occurs when for a given Riemann tensor for the bulk these equations can be solved without appeal to analyticity. A substantial part of Nash’s theorem consists in showing that the solution requires that the functions appearing in the right hand side must be regular. – 4 – The expression (2.11) shows that besides the brane-world gravitational field the extrin- sic curvature kµνa also propagate in the bulk. The implication of this is that the imposition of any restrictive conditions on kµνa also implies on restrictions on the propagation of the gravitational field of the brane-world. On the other hand, from (2.10) it follows that the third fundamental form Aµab does not propagate at all in the bulk, behaving as if it is a confined field. The equations of motion of the brane-world follow directly from the Einstein-Hilbert principle on the bulk and from the integrability conditions (2.12)-(2.14). To see how this works take the trace of the first equation (2.5): gµνZA,µZ ,νGAB = D−N = GABGAB−gabgab, and replace gab from (2.5), obtaining gµνZA,µZ ,ν = GAB − gabηAa ηBb (2.15) The contractions of (2.12) with gµν , and using (2.15) gives the the Ricci tensor and Ricci scalar of the brane-world respectively expressed as Rµν = g cd(gαβkµαckνβd − hckµνd) +RABZA,µZB,ν − gabRABCDηAa ZB,µZC,νηDb (2.16) After another contraction with gµν , using again (2.15), and noting that gadgbcRABCDηAa ηBb ηCc ηDd = 0, we obtain the Ricci scalar R = (K2 − h2) +R− 2gabRABηAa ηBb (2.17) where K2 = gabkµνakµνb. ha = g µνkµνa and h 2 = gabhahb. Therefore the Einstein-Hilbert action for the bulk geometry in D-dimensions can be written as GdDv = R− (K2 − h2) + 2gabRABηAa ηBb GdDv (2.18) where α∗ denotes the fundamental energy scale in the bulk and L∗ is the source Lagrangian. The Euler-Lagrange equations of (2.18) with respect to GAB are Einstein’s equations in D dimensions: RAB − RGAB = α∗T ∗AB (2.19) Here T ∗AB denote the components of the energy-momentum tensor of the sources. The equations of motion of the embedded brane-world can be derived directly from the components of (2.19) written in the embedding frame. The tangent components follow from the contractions of (2.19) with ZA,µZB,ν . After using (2.16) and (2.17) we obtain Rµν − Rgµν −Qµν −Wµν − gabRABηAa ηBb = α∗T ∗µν (2.20) where we have denoted Qµν = g abkρµakρνb − gabhakµνb − (K2 − h2)gµν (2.21) Wµν = g adRABCDηAa ZB,µZC,νηDd – 5 – By a direct calculation we can see that the extrinsic tensor Qµν is an independently con- served quantity with respect to the brane-world metric. The contraction of (2.19) with ZAµ ηBb gives a vectorial equation. Using (2.17) and (2.13) we obtain kρµa;ρ−ha,µ+Aρcakρ cµ −Aµcahc+ 2Wµa = −2α∗(T ∗µa − N + 2 T ∗gµa) (2.22) where we have denoted Wµa = g bdRABCDηAa ηBb ZC,µηDd (2.23) Finally, contracting (2.19) with ηAa η b we obtain N(N +1)/2 scalar equations involving the so called Hawking-Gibbons term Sab = RABηAa ηBb and its trace S = gabSab Sab − Sgab − [R−K2 + h2]gab = α∗T ∗ab (2.24) In its most general form, without assuming extra dimensional matter, the confinement hy- pothesis states that the only non-vanishing components of TAB are the tangent components Tµν representing the confined sources [2]. Therefore we set ABZA,µZB,ν = α∗T ∗µν = −8πGTµν (2.25) ABZA,µηBa = α∗T ∗µa = 0 (2.26) b = α∗T ab = 0 (2.27) Equations (2.20), (2.22) and (2.24) with confinement conditions are sometimes called the gravi-tensor, gravi-vector and gravi-scalars (Usually a single gravi-scalar equation in the 5-dimensional models [21]) equations respectively. These represent generalizations of Ein- stein’s equations of general relativity, in the sense that they describe the evolution of all geometrical components gµν , Aµab and kµνa of the brane-world. Clearly, the usual Ein- stein’s equations are recovered when all elements of the extrinsic geometry are removed from those equations. 3. Canonical Equations The standard ADM canonical quantization of the gravitational field in general relativity was originally intended to describe the quantum fluctuations of 3-dimensional hypersurfaces in a space-time [22]. The space-time metric is decomposed in 3-surface components, plus a shift vector and a lapse function defined in a Gaussian reference frame defined on the 3- dimensional hypersurface. After writing the Einstein-Hilbert Lagrangian in this Gaussian frame, the Euler-Lagrange equations with respect to the shift leads to the vanishing of the Hamiltonian. This is not a real problem because in principle the system could be solved by use of Dirac’s standard procedure for constrained systems. However, as it is well known, the Poisson bracket structure does not propagate covariantly as it would be expected. In spite of all efforts made up to the present, this problem remains unsolved [23, 24, 25, 26]. It is possible to describe a non-constrained canonical system in a special frame defined by a – 6 – 3-dimensional hypersurface orthogonal Gaussian coordinate system. In such special frame the shift vector vanishes and the Hamiltonian constraint does not apply [27]. Nonetheless, this has been regarded as of little value for general relativity itself, essentially because the diffeomorphism group of the theory is one of the fundamental postulates of the theory [28]. The extension of the ADM canonical formulation to the brane-world is straightfor- ward but it requires a few adaptations: First, the bulk is locally foliated by a continuous sequence of brane-worlds propagating along the extra dimensions rather than by a 3-surface propagating along a single time direction. Secondly, the confinement hypothesis implies that the diffeomorphism invariance do not extend to the extra dimensions, otherwise a coordinate transformation in the bulk would have the effect of introducing a component of the energy-momentum tensor of the confined fields and ordinary matter in the bulk. Therefore, in order to maintain the intended solution of the hierarchy problem, the diffeo- morphism of the brane-world must be restricted as a confined symmetry. Actually this can be regarded as one of the merits of brane-world theory, which differentiates it from being just a higher dimensional version of general relativity. However, to deserve such merit the extra dimensions need to be taken seriously as true physical degrees of freedom in the canonical formulation of the theory. The momentum conjugated to the metric field GAB , with respect to the displacement along ηa is defined as usual pAB(a) = where L is the Einstein-Hilbert Lagrangian of the bulk in (2.18). Noting that in the em- bedding frame we can write 2gabRABηAa ηBb = K2−gabha,b, after eliminating the divergence term gabha,b, the Lagrangian can be simplified to L = [R+ (K2 + h2)] G (3.1) Therefore, using (2.11) we obtain the canonically conjugated momenta pµν (a) = ) = − ∂kµνa = −(kµνa + hagµν) G (3.2) pµa(b) = ∂Aµab = 0, pab(c) = ) = 0 (3.3) The last two components are equal to zero because the Lagrangian (3.1) does not depend explicitly on Aµab and on gab,c. Using the above expressions, the Hamiltonian Ha corresponding to the displacement along each orthogonal direction ηa separately, follows from a partial Legendre transforma- tion (no sum on (a)): Ha(g, p(a)) = pAB(a)gAB,a − L = −R G − [(K2 + h2 + 2(K2a + h2a)] – 7 – where we have denoted K2a = k a kµνa, K 2 = gabKaKb, and p(a) = g µνpµν(a). After replacing ha = −p(a) G it follows that Ha = −R pµν(a) (3.4) For a given functional F(gµν , pµν) defined in the phase space of the brane-world, the prop- agation of F along ya is given by the Poisson brackets with each Hamiltonian separately = [F ,Ha] = δ̃gµν δ̃pµν(a) δ̃pµν(a) δ̃gµν (3.5) Here δ̃ denotes the standard functional derivative in phase space. Hamilton’s equations for the brane-world with respect to each extra coordinate ya may be written as δ̃pµν(a) = [gµν ,Ha] = −2kµνa, (3.6) = − δ̃Ha δ̃gµν ,Ha] (3.7) As it can be seen, the differences between the brane-world canonical formulation and the ADM formulation of general relativity follow from the non-vanishing of the Hamiltonians Ha, as a consequence of the brane-world scheme for solving the hierarchy problem. With this result, the ADM quantization program can be retaken, with the difference that the quantum equation should describe the ”states” of four-dimensional submanifolds in the bulk, with respect to the extra dimensions. 4. Tomonaga-Schwinger Quantum States The Tomonaga-Schwinger equation originated from Dirac’s many fingered time formalism for relativistic quantum theory, in which a set of N electrons was associated to N proper times satisfying N Schrodinger’s-like equations [29]. The continuous limit of this equa- tion was formulated by Tomonaga for a relativistic field defined in a region of space-time characterized by an evolving space-like 3-hypersurface σ with a time direction attached to each of its point. This geometric extension of Dirac’s many fingered time, which was soon realized to be equivalent to the interaction representation of quantum mechanics developed by Schwinger [30, 31, 32]. Here, it is more convenient to look at the Tomonaga-Schwinger equation from the geometrical point of view written as = ĤσΨ (4.1) which represents a generalization of Schrodinger’s equation, describing the quantum state functional Ψ of a space-like 3-hypersurface σ embedded in Minkowski space-time. In the – 8 – right hand side, the Hamiltonian operator describes the translational operator along a time- like direction orthogonal to σ. The functional derivative in the left hand side is defined by the limit = lim Ψ(σ′)−Ψ(σ) (4.2) where ∆V denotes the local volume element between two neighboring hypersurfaces σ and The main difficulty of (4.1) is that it is not easily integrable. In the particular case where [Ĥσ, Ĥσ′ ] = 0, the equation can be integrated, but the hypersurfaces σ and σ′ are necessarily flat. In the general case where the hypersurfaces are not flat, the solutions of (4.1) can be determined as an approximate solution after the application of the Yang- Feldmann formalism and Dyson’s expression for the S matrix [32, 33]. The difficulty in solving (4.1) can be traced back to the fact that the limit operation in (4.2) was not defined. In fact, the conditions to decide how close σ and σ′ are were not given previously, and it can be decided only after solving the quantum equation itself using some quantum approximation method. In the application of (4.1) to the brane-world, the limit operation between two four- dimensional brane-worlds σ4 and σ 4 is improved because Nash’s theorem shows at the classical level how to tell the separation between the two sub manifolds. In other words, since each brane-world was generated by classical perturbations of an initial sub manifold σ4, the volume element in (4.2) has been already specified by the parameter δy a of the perturbed geometry. In practice, we may split the bulk volume ∆V between σ4 and σ into a product of the volume ∆v of a a small compact region in σ4, times the variation ∆y of the extra dimensional coordinates ya. Therefore, it sufficient to specify only the limit operation ∆ya → 0 and the functional derivative (4.2) for density functions with compact support on the brane-world, with respect to each extra dimension can be simplified to = lim ∆ya→0 Ψ(σ′)−Ψ(σ) Repeating for all extra dimensions, we find that the Tomonaga-Schwinger equation (4.1) can be extended the brane-world, as a system of N partial equations, one for each extra dimension = ĤaΨa, a = 5..D (4.3) which gives to Ĥa the interpretation of the extra dimensional translational operator. The final quantum state is given by the superposition of the N separates states Ψa as BaΨa. The state functional density Ψ represents the quantum fluctuation of the brane-world sub manifold in the bulk at the (Tev) energy scale, subjected to quantum uncertainties and a state probability given by ||Ψ||2 = Ψ†Ψδyδv – 9 – An observer confined to the brane-world may evaluate the quantum expectation values of the brane-world metric and the extrinsic curvature are given by < Ψ|ĝµν |Ψ >= Ψ†ĝµνΨδyδv, < Ψ|k̂µν |Ψ >= Ψ†k̂µνΨδyδv Since the classical kµνa is the derivative of the metric gµν with respect to ya, we may set boundary conditions on these quantities at the initial brane-world ya = 0 to determine the final solution. 5. Overview and Perspectives We have shown that the Einstein-Hilbert principle applied to the bulk geometry plus the differentiable embedding conditions are sufficient to determine the classical and quantum structures of the brane-world in D-dimensions. In particular, it was shown that Nash’s theorem makes it possible to generate any embedded sub manifold by a continuous se- quence of infinitesimal perturbations of an arbitrarily given embedded geometry along the extra dimensions. Using the classical perturbative embeddings, and the basic principles of the brane-world theory we have obtained a canonical structure very much like the ADM canonical formulation of general relativity, with the exception that the Hamiltonians do not vanish. The definition of the functional derivatives was improved with respect to four-dimensional field theory, by using the previously defined perturbative embedding structure of the brane- world. The quantization of the brane-world was described the Tomonaga-Schwinger equa- tion defined for brane-world sub manifolds, calculated for each extra dimension. Actually, as a result of the the the classical perturbation theory, the Tomonaga-Schwinger equation becomes exactly integrable. In view of current astrophysical observations and the near future high energy exper- iments, there are some applications of the quantum brane-world theory to be detailed in subsequent papers: (a) Brane-world Cosmology Brane-world cosmology offers a possible explanation to the accelerated expansion of the universe, resulting from the modification of the Friedman’s equation by the presence of the extrinsic curvature included in the tensor Qµν given by (2.21) [10]. The presence of this tensor has the meaning that the brane-world vacuum is more complex than the vacuum in general relativity. In fact, for a constant curvature bulk with curvature Λ∗, after eliminating redundant terms, the gravi-tensor vacuum equation becomes Rµν−1/2Rgµν−Qµν+Λ∗gµν = Therefore, the vacuum energy density < ρv > resulting from gravitationally coupled fields must be revaluated, including the extrinsic curvature component. This suggests that in some epochs, say at the early inflationary period, the extrinsic curvature may contribute to the vacuum energy, differently from other periods. The particular case where we have only one extra dimensions (D = 5), has some limita- tions with respect to the differentiable embedding. However, some cosmological models like – 10 – the FRW, deSitter and anti deSitter solutions of Einstein’s equations in four dimensions can be embedded in five dimensional bulks without restrictions, in accordance with the perturbative embedding equations previously shown. Consequently, in such brane-world cosmological models the conditions required for a proper definition of the functional deriva- tives in the Tomonaga-Schwinger equation are well established, and the equation can can be integrated without difficulty. (b) Laboratory production of mini black holes: Brane-world gravity predict the generation of short lived mini black holes produced at the Tev energy in the laboratory, resulting from proton-proton collisions [34]. However, using semi-classical quantum gravity in four dimensions, we have learned that quantum unitarity does not necessarily hold true during the black hole evaporation. On the other hand, using Euclidean path integral, it was shown that the unitarity can be restored with the aid of the ADS/CFT correspondence in the framework of AdS5 × S5 string theory [35]. Since the generation of mini black holes are possible only in the brane-world context, the whole process includes the original Minkowski’s space-time where the experiment is devised. Soon after the collision, the space-time must be deformable into a Schwarzschild or a Reissner-Nordstrom black hole. Finally, after a short period of evaporation the space- time may be back to Minkowski’s configuration or else leaves a curved remnant. The description of such process can start with the classical perturbations in accordance with Nash’s embedded geometries, but the unitarity is has to be decided at the quantum level. In this respect we notice that both Schwarzschild and Reissner-Nordstrom black holes are well defined submanifolds embedded in a six-dimensional flat bulk with signature (4, 2). In this case, the bulk isometry group SO(4, 2) is isomorphic to the conformal group in Minkowski’s space-time, compatible with the ADS/CFT correspondence adapted to the brane-world [36]. Therefore, the quantum unitarity implicitly assumed in the Tomonaga- Schwinger equation, must be consistent with the black hole evaporation theorems in six dimensions. (c) Quantum Four-manifold Theory: The above description of quantum theory of the brane-world is based almost entirely on the general theory of differentiable sub manifolds. This suggests a quantum theory of four-dimensional sub manifolds. It starts with the classical perturbations of embedded geometries, but ends with a quantum version of the embedding theorem, including the fluctuations of the embedding as described by the Tomonaga-Schwinger equation. This quantum theory of submanifolds would be particularly interesting when the bulk has di- mensions greater than five, where the third fundamental form behaves similarly to a gauge field with respect to the extra dimensional group of isometries. The identification of the third fundamental form as a gauge field with the symmetry of the extra dimensions plying the role of the gauge group is old, but it was never taken seriously [37]. 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0704.1290

Isospin diffusion in thermal AdS/CFT with flavor

MPP-2007-42 Isospin diffusion in thermal AdS/CFT with flavor Johanna Erdmenger, Matthias Kaminski, Felix Rust∗ Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 München, Germany We study the gauge/gravity dual of a finite temperature field theory at finite isospin chem- ical potential by considering a probe of two coincident D7-branes embedded in the AdS- Schwarzschild black hole background. The isospin chemical potential is obtained by giving a vev to the time component of the non-Abelian gauge field on the brane. The fluctuations of the non-Abelian gauge field on the brane are dual to the SU(2) flavor current in the field theory. For the embedding corresponding to vanishing quark mass, we calculate all Green functions corresponding to the components of the flavor current correlator. We discuss the physical properties of these Green functions, which go beyond linear response theory. In par- ticular, we show that the isospin chemical potential leads to a frequency-dependent isospin diffusion coefficient. PACS numbers: 11.25.Tq, 11.25.Wx, 12.38.Mh, 11.10.Wx Contents I. Introduction 2 II. Hydrodynamics and AdS/CFT 4 III. Supergravity background and action 5 A. Finite temperature background and brane configuration 5 B. Introducing a non-Abelian chemical potential 6 C. Dirac-Born-Infeld action 7 D. Equations of motion 8 1. Equations for Aa1- and A 2-components 8 2. Equations for Aa0- and A 3-components 9 3. Solutions 9 IV. Isospin diffusion and correlation functions 12 A. Current correlators 12 1. Green functions: Calculation 13 2. Green functions: Results 14 B. Isospin diffusion 17 V. Conclusion 19 ∗Electronic address: [email protected], [email protected], [email protected] http://arxiv.org/abs/0704.1290v2 mailto:[email protected], [email protected], [email protected] Acknowledgments 19 A. Notation 20 B. Solutions to equations of motion 20 1. Solutions for Xα, X̃α and A 2. Solutions for X ′0, X̃ 0 and A 3. Solutions for X ′3, X̃ 3 and A 4. Comparison of numerical and analytical results 23 C. Abelian Correlators 23 D. Correlation functions 24 E. Thermal spectral functions 24 References 25 I. INTRODUCTION Over the past years, there have been a number of lines of investigation for describing QCD- like theories with gravity duals. In this way, considerable progress towards a gauge/gravity dual description of phenomenologically relevant models has been made. One of these lines of investi- gation is the gravity dual description of the quark-gluon plasma obtained by applying AdS/CFT to relativistic hydrodynamics [1, 2, 3, 4, 5, 6, 7]. The central result of this approach is the cal- culation of the shear viscosity from AdS/CFT. More recently, an R charge chemical potential has been introduced by considering gravity backgrounds with R charged black holes [8, 9, 10], and also the heat conductivity has been calculated by considering the R current correlators in these backgrounds [8]. A further approach to generalizing the AdS/CFT correspondence to more realistic field theories is the addition of flavor to gravity duals via the addition of probe branes [11, 12, 13, 14, 15]. This allows in particular for the calculation of meson masses. These two approaches have been combined in order to study the flavor contribution to finite temperature field theories from the gravity dual perspective. This began with [13] where the embedding of a D7-brane probe into the AdS-Schwarzschild black hole background was studied and a novel phase transition was found, which occurs when the D7 probe reaches the black hole horizon. This transition was shown to be of first order in [16] (see [14] for a similar transition in the D4/D6 system), and studied in further detail in [17, 18]. Related phase transitions appear in [19, 20, 21]. Mesons in gravity duals of finite temperature field theories have been studied in [22, 23, 24, 25]. Recently, in view of adding flavor to the quark-gluon plasma, the flavor contribution to the shear viscosity has been calculated in [26, 27], where it was found that ηfund ∝ λNcNfT 3. For a thermodynamical approach in the grand canonical ensemble, the inclusion of a chemical potential and a finite number density is essential. In [28], an isospin chemical potential was in- troduced by considering two coincident D7 probes, and by giving a vev to the time component of the SU(2) gauge field on this probe. This was shown to give rise to a thermodynamical instability comparable to Bose-Einstein condensation, compatible with the field-theoretical results of [29]. For the gauge/gravity dual analysis, a potential generated by an SU(2) instanton on the D7 probe in the gravity background was used [30, 31, 32, 33]. A baryon chemical potential µB is obtained by turning on the diagonal U(1) ⊂ U(Nf ) gauge field on the D7-brane probe [34]. Contributions to the D7-brane action arise from the deriva- tive of this U(1) gauge field with respect to the radial direction. The effects of this potential on the first-order phase transition described above have been studied in [34], where regions of ther- modynamical instability have been found in the (T, µB) phase diagram. – For the D4/D8/D̄8 Sakai-Sugimoto model [15, 35], the phase transitions in presence of a baryon number chemical potential, as well as physical processes such as photoemission and vector meson screening, have been studied in [36, 37, 38]. A related approach has been used to calculate the rate of energy loss of a heavy quark moving through a supersymmetric Yang-Mills plasma at large coupling [39]. In this approach the heavy quark is given by a classical string attached to the D7-brane probe. – A first study of flavors in thermal AdS/CFT beyond the quenched approximation, i.e. with Nf ∼ Nc, was performed in [40]. Here we study finite-temperature field theories with finite isospin chemical potential by consid- ering two coincident D7-brane probes in the Lorentzian signature AdS-Schwarzschild black hole background. As in [28], we introduce an isospin chemical potential by defining , (1.1) for the time component of the SU(2) gauge field on the two coincident D7-branes. This constant chemical potential is a solution to the D7-brane equations of motion and is present even for the D7-brane embedding corresponding to massless quarks. We consider small µ, such that the Bose- Einstein instability mentioned above, which is of order O(µ2), does not affect our discussion here. For simplicity we consider only the D7 probe embedding for which the quark mass vanishes, m = 0. This embedding is constant and terminates at the horizon. We establish the SU(2) non- Abelian action for a probe of two coincident D7-branes. We obtain the equations of motion for fluctuations about the background given by (1.1). These are dual to the SU(2) flavor current Jµa. We find an ansatz for decoupling the equations of motion for the different Lorentz and flavor com- ponents, and solve them by adapting the method developed in [2, 3]. This involves Fourier trans- forming to momentum space, and using a power expansion ansatz for the equations of motion. We discuss the approximation necessary for an analytical solution, which amounts to considering fre- quencies with ω < µ < T . With this approach we obtain the complete current-current correlator. The key point is that the constant chemical potential effectively replaces a time derivative in the ac- tion and in the equations of motion. In the Fourier transformed picture, this leads to a square-root dependence of physical observables on the frequency, ω. This non-linear behavior goes beyond linear response theory. We discuss the physical properties of the Green functions contributing to the current-current correlator. In particular, for small frequencies we find a frequency-dependent diffusion coefficient D(ω) ∝ 1 ω/µ. Whereas frequency-dependent diffusion has – to our knowledge – not yet been discussed in the context of the quark-gluon plasma, it is well-known in the theory of quantum liquids. For instance, for small frequencies the square-root behavior we find agrees qualitatively with the results of [41, 42] for liquid para-hydrogen. Generally, frequency- dependent diffusion leads to a non-exponential decay of time-dependent fluctuations, as discussed for a classical fluid in [43]. Physically, the isospin chemical potential corresponds to the energy necessary to inverting the isospin of a given particle. Within nuclear physics, such a chemical potential is of relevance for the description neutron stars. Moreover, isospin diffusion has been measured in heavy ion reactions [44, 45]. – For two-flavor QCD, effects of a finite isospin chemical potential have been discussed for instance in [46, 47, 48]. The phase diagrams discussed there are beyond the scope of the present paper. We expect to return to similar diagrams in the gauge/gravity dual context in the future. This paper is organized as follows. In section 2 we summarize the AdS/CFT hydrodynamics approach to calculating Green functions, which we use in the subsequent. Moreover we comment on frequency-dependent diffusion within hydrodynamics. In section 3 we establish the D7 probe action in presence of the isospin chemical potential, derive the corresponding equations of motion and solve them. In section 4 we obtain the associated Green functions in the hydrodynamical approximation. We discuss their pole structure and obtain the frequency-dependent diffusion co- efficient. We conclude in section 5 with an interpretation of our results. An explanation of our notation as well as a series of calculations are relegated to a number of appendices. II. HYDRODYNAMICS AND ADS/CFT Thermal Green functions have proven to be a useful tool for analyzing the structure of hydro- dynamic theories and for calculating hydrodynamic quantities such as transport coefficients. For instance, given a retarded current correlation function G(~k)µν in Minkowski space, the spectral function can be written in terms of its imaginary part, χµν(~k) = −2 ImGµν(~k) . (2.1) For the gravity dual approach, this is discussed for instance in [7, 49]. In this paper we use the gauge/gravity dual prescription of [3] for calculating Green functions in Minkowski spacetime. For further reference, we outline this prescription in the subsequent. It is based on the AdS/CFT-correspondence relating supergravity fields A in a black hole background to operators J in the dual gauge theory. The black hole background is asymptotically Anti-de Sitter space and places the dual field theory at finite temperature. This temperature corresponds to the Hawking temperature of the black hole, or more generally speaking, of the black branes. Starting out from a classical supergravity action Scl for the gauge field A, according to [3] we extract the function B(u) (containing metric factors and the metric determinant) in front of the kinetic term (∂uA) Scl = dud4xB(u) (∂uA) 2 + . . . (2.2) Then we perform a Fourier transformation and solve the linearized equations of motion in momen- tum space. This is a second order differential equation, so we have to fix two boundary conditions. The first one at the boundary of AdS at u = 0 can be written as A(u,~k) = f(u,~k)Abdy(~k) , (2.3) where Abdy(~k) is the value of the supergravity field at the boundary of AdS depending only on the four flat boundary coordinates. Thus by definition we have f(u,~k) = 1. For the other boundary, located at the horizon u = 1, we impose the incoming wave condition. This requires that any Fourier mode A(~k) with timelike ~k can travel into the black hole, but is not allowed to cross the horizon in the opposite direction. For spacelike ~k, the components of A have to be regular at the horizon. Then the retarded thermal Green function is given by G(ω, q) = −2B(u) f(u,−~k) ∂u f(u,~k) . (2.4) The thermal correlators obtained in this way display hydrodynamic properties, such as poles located at complex frequencies. Generically, for the R current component correlation functions calculated from supergravity, there are retarded contributions of the form G(ω, q) ∝ iω −Dq2 . (2.5) This may be identified with the the Green function for the hydrodynamic diffusion equation ∂0 J0(t,x) = D∇2 J0(t,x) , (2.6) with J0 the time component of a diffusive current. D is the diffusion constant. In Fourier space this equation reads iωJ0(ω,k) = Dk 2J0(ω,k) . (2.7) In position space, this corresponds to an exponential decay of J0 with time. For the non-Abelian case with an isospin chemical potential, in sections III and IV we will obtain retarded Green functions of the form G(ω, q) ∝ 1 iω −D(ω)q2 . (2.8) Retarded Green functions of this type have been discussed for instance in [43]. (2.8) corresponds to frequency-dependent diffusion with coefficient D(ω), such that (2.7) becomes iωJ0(ω,k) = D(ω)k 2J0(ω,k) . (2.9) In our case, J0 is the averaged isospin at a given point in the liquid. This is a non-linear behavior which goes beyond linear response theory. In particular, when Fourier-transforming back to position space, we have to use the convolution for the product D · J0 and obtain ∂0J0(t,x) +∇2 ds J0(s,x)D(t− s) = 0 (2.10) for the redarded Green function. This implies together with the continuity equation ∂0 J0 +∇·J = 0, with J the three-vector current associated to J0, that J = −∇(D ∗ J0) , (2.11) where ∗ denotes the convolution. This replaces the linear response theory constitutive equation J = −D∇J0. Note that for D(t− s) = Dδ(t− s) with D constant, (2.10) reduces again to (2.6). III. SUPERGRAVITY BACKGROUND AND ACTION A. Finite temperature background and brane configuration We consider an asymptotically AdS5 × S5 spacetime as the near horizon limit of a stack of Nc coincident D3-branes. More precisely, as in [2], our background is an AdS black hole, which is the geometry dual to a field theory at finite temperature. The Minkowski signature background is ds2 = −f(u) dx20 + dx21 + dx22 + dx23 4u2f(u) du2 +R2dΩ25, 0 ≤ u ≤ 1, xi ∈ R, C0123 = (3.1) with the metric dΩ25 of the unit 5-sphere, and the function f(u), AdS radius R and temperature parameter b given in terms of the string coupling gs, temperature T , inverse string tension α ′ and number of colors Nc by f(u) = 1− u2, R4 = 4πgsNcα′2, b = πT. (3.2) The geometry is asymptotically AdS5 × S5 with the boundary of the AdS part located at u = 0. At the black hole horizon the radial coordinate u has the value u = 1. Into this ten-dimensional spacetime we embed Nf = 2 coinciding D7-branes, hosting flavor gauge fields Aµ. The embedding we choose extends the D7-branes in all directions of AdS space and wraps an S3 on the S5. In this work we restrict ourselves to the most straightforward case, that is the embedding of the branes through the origin along the AdS radial coordinate u. This corresponds to massless quarks in the dual field theory. On the brane, the metric in this case simply reduces to ds2 = −f(u) dx20 + dx21 + dx22 + dx23 4u2f(u) du2 +R2dΩ23, 0 ≤ u ≤ 1, xi ∈ R. (3.3) Due the choice of our gauge field in the next subsection, the remaining three-sphere in this metric will not play a prominent role. The table below gives an overview of the indices we use to refer to certain directions and subspaces. AdS5 S coord. names x0 x1 x2 x3 u – µ, ν. . . indices i, j. . . u B. Introducing a non-Abelian chemical potential A gravity dual description of a chemical potential amounts to a non-dynamical time component of the gauge field Aµ in the action for the D7-brane probe embedded into the background given above. There are essentially two different ways to realize a non-vanishing contribution from a chemical potential to the field strength tensor F = 2∂[µAν] + f abcAbµA ν . The first is to consider a u-dependent baryon chemical potential for a single brane probe. The second, which we pursue here, is to consider a constant isospin chemical potential. This requires a non-Abelian probe brane action and thus a probe of at least two coincident D7-branes, as suggested in [28]. Here the time component of the gauge field is taken to be A0 = A a, (3.4) where we sum over indices which occur twice in a term and denote the gauge group generators by ta. The brane configuration described above corresponds to an SU(Nf) gauge group with Nf = 2 on the brane, which corresponds to a global SU(Nf ) in the dual field theory. For Nf = 2, the generators of the gauge group on the brane are given by ta = σ , with Pauli matrices σa. We will see that (3.4) indeed produces non-trivial new contributions to the action. Using the standard background field method of quantum field theory, we will consider the chemical potential as a fixed background and let the gauge fields fluctuate. We single out a par- ticular direction in flavor space by taking A30 = µ as the only non-vanishing component of the background field. From now on we use the symbol Aaν to refer to gauge field fluctuations around the fixed background, Aaν → µδν0δa3 + Aaν . (3.5) We pick the gauge in which Au ≡ 0 and assume that Aµ ≡ 0 for µ = 5, 6, 7. Due to the symmetries of the background, we effectively examine gauge field fluctuations Aµ living in the five-dimensional subspace on the brane spanned by the coordinates x0, x1, x2, x3 and by the radial AdS coordinate u. The magnitude of all components of A and the background chemical potential µ are considered to be small. This allows us to simplify certain expressions by dropping terms of higher order in A and in the chemical potential µ. C. Dirac-Born-Infeld action The action describing the dynamics of the flavor gauge fields in the setup of this work is the Dirac-Born-Infeld action. There are no contributions from the Chern-Simons action, which would require non-zero gauge field components in all of the 4,5,6,7-directions. As mentioned, we con- sider the D7 probe embedding whose asymptotic value at the boundary is chosen such that it cor- responds to vanishing quark mass, m = 0. The metric on the brane is then given by (3.3). Since we are interested in two-point correlators only, it is sufficient to consider the action to second order in α′, SD7 = −T7 (2πα′)2 2π2R3 Tr uh=1∫ du d4x −g gµµ′ gνν′ F aµν F aµ′ν′ , (3.6) where we use the following definitions for the D7-brane tension T7 and the trace over the repre- sentation matrices ta, T7 = (2π) 7g−1s (α ′)−4 , (3.7) tr(ta tb) = Tr δ ab . (3.8) In our case we have Tr = 1/2. The overall factor 2π 2R3 comes from the integration over the 5, 6, 7-directions, which are the directions along the S3. Evaluating the DBI action given in (3.6) with the substitution rule (3.5), we arrive at SD7 = − T7 (2πα′)2 2π2R3 Tr uh=1∫ du d4x −ggµµ′gνν′ 4∂[µA ν] ∂[µ′A ν′] − 8δ0νδ0ν′fabc∂[0Aaµ] Abµ′ µc (3.9) where we use the short-hand notation µc = µδ3c and neglect terms of higher than linear order in µ, and higher than quadratic order in A since both are small in our approach. Up to the sum over flavor indices a, the first term in the bracket in (3.9) is reminiscent of the Abelian super-Maxwell action in five dimensions, considered already for the R charge current correlators in [2]. The new second term in our action arises from the non-Abelian nature of the gauge group, giving terms proportional to the gauge group’s structure constants fabc in the field strength tensor F aµν = 2∂[µA + fabcAbµA D. Equations of motion We proceed by calculating the retarded Green functions for the action (3.9), following the prescription of [3] as outlined in section 2 above. According to this prescription, as a first step we consider the equations of motion obtained from the action (3.9), which are given by 0 = 2∂µ −g gµµ′gνν′ ∂[µ′Aaν′] + fabc −gg00gνν′ µc 0 − 2∂0Abν′ + δ0 ν∂µ −g g00gµµ′Abµ′µc (3.10) It is useful to work in momentum space from now on. We therefore expand the bulk gauge fields in Fourier modes in the xi directions, Aµ(u, ~x) = (2π)4 e−iωx0+ik·xAµ(u,~k). (3.11) As we work in the gauge where Au = 0, we only have to take care of the components Ai with i = 0, 1, 2, 3. For the sake of simplicity, we choose the momentum of the fluctuations to be along the x3 direction, so their momentum four-vector is ~k = (ω, 0, 0, q). With this choice we have specified to gauge fields which only depend on the radial coordinate u, the time coordinate x0 and the spatial x3 direction. 1. Equations for Aa1- and A 2-components Choosing the free Lorentz index in the equations of motion (3.10) to be ν = α = 1, 2 gives two identical differential equations for A1 and A2, 0 = Aaα 2 − fq2 Aaα + 2i Acα, (3.12) where we indicated the derivative with respect to u with a prime and have introduced the dimen- sionless quantities , q = , m = . (3.13) We now make use of the structure constants of SU(2), which are fabc = εabc, where εabc is the totally antisymmetric epsilon symbol with ε123 = 1. Writing out (3.12) for the three different choices of a = 1, 2, 3 results in 0 = A1α 2 − fq2 A1α − 2i A2α , (3.14) 0 = A2α 2 − fq2 A2α + 2i A1α , (3.15) 0 = A3α 2 − fq2 A3α . (3.16) The first two of these equations are coupled, the third one is the same equation that was solved in the Abelian Super-Maxwell case [2]. 2. Equations for Aa0- and A 3-components The remaining choices for the free Lorentz index ν = 0, 3, u in (3.10) result in three equations which are not independent. The choices ν = 0 and ν = u give 0 = Aa0 Aa0 − Aa3 − i Ac3 , (3.17) 0 = wAa0 ′ + qfAa3 ′ + ifabc ′ . (3.18) Solving (3.18) for Aa0 ′, differentiating it once with respect to u and using (3.17) results in equation (3.10) for ν = 3, 0 = Aa3 Aa3 + Aa0 + i Ac0 + 2i Ac3 . (3.19) We will make use of the equations (3.17) and (3.18) which look more concise. These equations of motion for Aa0 and A 3 are coupled in Lorentz and flavor indices. To decouple them with respect to the Lorentz structure, we solve (3.18) for Aa3 ′ and insert the result into the differentiated version of (3.17). This gives 0 = Aa0 ′′′ + (uf)′ 2 − fq2 ′ + 2i ′. (3.20) The equations for a = 1, 2 are still coupled with respect to their gauge structure. The case a = 3 was solved in [2]. We will solve (3.20) for Aa0 ′ and can obtain Aa3 ′ from (3.18). Note that it is sufficient for our purpose to obtain solutions for the derivatives of the fields. These contribute to (2.4), while the functions A = f(u,~k)Abdy(~k) themselves simply contribute a factor of f(u,−~k) which is one at the boundary. 3. Solutions Generally, we follow the methods developed in [2], since our differential equations are very similar to the ones considered there. Additionally, we need to respect the flavor structure of the gauge fields. The equations for flavor index a = 3 resemble the ones analyzed in [2]. However, those for a = 1, 2 involve extra terms, which couple these equations. In our case the equations are coupled not only via their Lorentz indices, but also with respect to the flavor indices. We already decoupled the Lorentz structure in the previous section. As shown below, the equations of motion which involve different gauge components will decouple if we consider the variables Xi = A i + iA X̃i = A i − iA2i . (3.21) Here the A1i , A i are the generally complex gauge field components in momentum space. Note that up to SU(2) transformations, the combinations (3.21) are the only ones which decouple the equations of motion for a = 1, 2. These combinations are reminiscent of the non-Abelian SU(2) gauge field in position space, Ai = A A3i A i − iA2i A1i + iA i −A3i . (3.22) The equations of motion for the flavor index a = 3 were solved in [2]. To solve the equations of motion for the fields Aai with a = 1, 2, we rewrite them in terms of Xi and X̃i. Applying the transformation (3.21) to the equations of motion (3.14) and (3.15) and the a = 1, 2 versions of (3.20) and (3.18) leads to 0 = X ′′α + X ′α + 2 − fq2 ∓ 2mw Xα, α = 1, 2, (3.23) 0 = X ′′′0 + (uf)′ X ′′0 + 2 − fq2 ∓ 2mw X ′0, (3.24) 0 = (w∓m)X ′0 + qfX ′3, (3.25) where the upper signs correspond to X and the lower ones to X̃. We see that some coefficients of these functions are divergent at the horizon u = 1. Such differential equations with singular coefficients are generically solved by an ansatz Xi = (1− u)β F (u), X̃i = (1− u) eβ F̃ (u), (3.26) with regular functions F (u) and F̃ (u). To cancel the singular behaviour of the coefficients, we have to find the adequate β and β̃, the so-called indices, given by equations known as the indicial equations for β and β̃. We eventually get for all Xi and X̃i β = ±iw 1− 2m , β̃ = ±iw . (3.27) Note that these exponents differ from those of the Abelian Super-maxwell theory [2] by a de- pendence on w in the limit of small frequencies (w < m). In the limit of vanishing chemical po- tential m → 0, the indices given in [2] are reproduced from (3.27). In order to solve (3.23), (3.24) and (3.25), we wish to introduce a series expansion ansatz in the momentum variables w and q. In fact, the physical motivation behind this expansion is that we aim for thermodynamical quantities which are known from statistical mechanics in the hydrodynamic limit of small four-momentum ~k. So the standard choice would be F (u) = F0 +wF1 + q 2G1 + . . . . (3.28) On the other hand, we realize that our indices will appear linearly (and quadratically) in the differ- ential equations’ coefficients after inserting (3.26) into (3.23), (3.24) and (3.25). The square root in β and β̃ mixes different orders of w. In order to sort coefficients in our series ansatz, we assume w < m and keep only the leading w contributions to β and β̃, such that β ≈ ∓ , β̃ ≈ ±i . (3.29) This introduces an additional order O(w1/2), which we include in our ansatz (3.28) yielding F (u) = F0 +w 1/2F1/2 +wF1 + q 2G1 + . . . , (3.30) and analogously for the tilded quantities. If we had not included O(w1/2) the resulting system would be overdetermined. The results we obtain by using the approximations (3.29) and (3.30) have been checked against the numerical solution for exact β with exact F (u). These approxima- tions are useful for fluctuations with q,w < 1 (see subsection B 4 in Appendix B). Note that by dropping the 1 in (3.27) we also drop the Abelian limit. Consider the indices (3.27) for positive frequency first. In order to meet the incoming wave boundary condition introduced in section II, we restrict the solution β̃ to the negative sign only. For the approximate β̃ in (3.29) we therefore choose the lower (negative) sign. This exponent describes a mode that travels into the horizon of the black hole. In case of β we demand the mode to decay towards the horizon, choosing the lower (positive) sign in (3.29) consistently. Note that for negative frequencies ω < 0 the indices β and β̃ exchange their roles. Using (3.29) in (3.26) and inserting the ansatz into the equations of motion, we find equations for each order in q2 and w separately. After solving the equations of motion for the coefficient functions F0, F1/2, F1 and G1, we eventually can assemble the solutions to the equations of motion for X as defined in (3.21), X(u) = (1− u)β F (u) = (1− u)β wF1/2 +wF1 + q 2G1 + . . . . (3.31) and a corresponding formula for X̃(u) from the ansatz (3.26). Illustrating the method, we now write down the equations of motion order by order for the function Xα. To do so, we use (3.31) with (3.29) in (3.23) with the upper sign for Xα. Then we examine the result order by order in w and q2, O(const) : 0 = F ′′0 + F ′0 , (3.32) w) : 0 = F ′′1/2 + F ′1/2 − F ′0 − F0 , (3.33) O(w) : 0 = F ′′1 + F ′1 − F ′1/2 − F1/2 −m 4− u(1 + u)2 2uf 2 F0 , (3.34) O(q2) : 0 = G′′1 + G′1 − F0 . (3.35) At this point we observe that the differential equations we have to solve for each order are shifted with respect to the solutions found in [2]. The contributions of order wn in [2] now show up in order wn/2. Their solutions will exhibit factors of order µn/2. Solving the system (3.32) to (3.35) of coupled differential equations is straightforward in the way that they can be reduced to several uncoupled first order ordinary differential equations in the following way. Note that there obviously is a constant solution F0 = C for the first equation. Inserting it into (3.33) and (3.35) leaves us with ordinary differential equations for F ′1/2 and G respectively. Using the solutions of F0 and F1/2 in (3.34) gives one more such equation for F To fix the boundary values of the solutions just mentioned, we demand the value of F (uH = 1) to be given by the constant F0 and therefore choose the other component functions’ solutions such that limu→1 F1/2 = 0, and the same for F1 and G1. The remaining integration constant C is determined by taking the boundary limit u → 0 of the explicit solution (3.31), making use of the second boundary condition X(u) = Xbdy, (3.36) see appendix B. Eventually, we end up with all the ingredients needed to construct the gauge field’s fluctuations X(u) as in (3.31). We solve the equations (3.23) with lower sign for X̃α and (3.24) for X 0 and its tilded partner in exactly the same way as just outlined, only some coefficients of these differential equations differ. The solution for X ′3 is then obtained from (3.25). All solutions are given explicitly in Appendix B together with all other information needed to construct the functions Xα, X̃α, X 0, X̃ 3 and X̃ IV. ISOSPIN DIFFUSION AND CORRELATION FUNCTIONS A. Current correlators In this section we obtain the momentum space correlation functions for the gauge field compo- nent combinations X and X̃ defined in equation (3.21). Recall that the imaginary part of the retarded correlators essentially gives the thermal spectral functions (see also section II). The following discussion of the correlators’ properties is therefore equivalent to a discussion of the corresponding spectral functions. First note that the on-shell action gets new contributions from the non-Abelian structure, SD7 = − T7 (2πα′)2 2π2R3 Tr (4.1) (2π)4 √−gguugjj′ Aaj ′(~q)Aaj′(−~q) − 4iq fabcµc −gg00g33Aa[3Ab0] where j, j′ = 0 , 1 , 2 , 3 and the index u denotes the radial AdS-direction. Up to the sum over flavor indices, the first term in the bracket is similar to the Abelian Super-Maxwell action of [2]. The second term is a new contribution depending on the isospin chemical potential. It is a contact term which we will neglect. The correlation functions however get a structure that is different from the Abelian case. This is due to the appearance of the chemical potential in the equations of motion and their solutions. Writing (4.1) as a function of X and X̃ results in SD7 = − T7 (2πα′)2 2π2R3 Tr (2π)4 −g guugjj′ ′X̃j′ + X̃ + A3j ]∣∣∣∣ (4.2) − 4qµ −gg00g33 X[0X̃3] + X̃[3X0] In order to find the current correlators, we apply the method outlined in section II to (4.2), with the solutions for the fields given in appendix B. As an example, we derive the correlators G0e0 = 〈J0(~q)J̃0(−~q) 〉 and Ge00 = 〈J̃0(~q) J0(−~q)〉 of the flavor current time components J0 and J̃0, coupling to the bulk fields X0 and X̃0, respectively. Correlation functions of all other components are derived analogously. For the notation see appendix A. 1. Green functions: Calculation First, we extract the factor B(u) of (2.2), B(u) = −T7 (2πα′)2 2π2R3Tr −g guu g00 . (4.3) The second step, finding the solutions to the mode equations of motion, has already been per- formed in section III D 3. In the example at hand we need the solutions X0 and X̃0. From (3.31) and from appendix B we obtain ′ =− (1− u) 0 +wqX̃ 2mw+wm ln 2 + q2 1−w1/2 π2 + 3 ln2 2 + 3 ln2(1 + u) + 6 ln 2 ln 1 + u (4.4) +12Li2(1− u) + 12Li2(−u)− 12Li2 + q2 ln ′ = (1− u)−i 0 +wqX 2mw+wm ln 2− q2 1 +w1/2 i π2 + 3 ln2 2 + 3 ln2(1 + u) + 6 ln 2 ln 1 + u (4.5) +12Li2(1− u) + 12Li2(−u)− 12Li2 + q2 ln Note that we need the derivatives to apply (2.4). Now we perform the third step and insert (4.3), (4.4) and (4.5) into (2.4). Our solutions X0 and X̃0 replace the solution f(u,~k) and f(u,−~k) in (2.4). The resulting expression is evaluated at ub = 0, which comes from the lower limit of the u-integral in the on-shell action (4.2). At small u = ǫ → 0, (4.4) and (4.5) give 0 +wqX̃ 2mw+wm ln 2 + q2 − lim 0 +wqX̃ ln ǫ , (4.6) 0 +wqX 2mw+wm ln 2− q2 + lim 0 +wqX ln ǫ . (4.7) In the next to leading order of (4.6) and (4.7) there appear singularities, just like in the Abelian Super-Maxwell calculation [2, equation (5.15)]. However, in the hydrodynamic limit, we consider only the finite leading order. 2. Green functions: Results Putting everything together, for the two Green functions for the field components X0, X̃0 given in (3.21) by X0 = A 0 + iA 0, X̃0 = A 0 − iA20, we obtain G0e0 = 2πT q2 2mw− q2 −wm ln2 , (4.8) Ge00 = 2πT q2 2mw− q2 +wm ln2 . (4.9) These are the Green functions for the time components in Minkowski space, perpendicular to the chemical potential in flavor space. All Green functions are obtained considering hydrodynamic approximations in O(w1/2,w, q2), neglecting mixed and higher orders O(w3/2,w1/2q2, q4). The prefactor in (4.8), (4.9) is obtained using T7 as in (3.7), and carefully inserting all metric factors, together with the standard AdS/CFT relation R4 = 4πgsNcα ′2. As in other settings with flavor [26], we concordantly get an overall factor of Nc, and not N c , for all correlators. Contrary to those approaches, we do not get a factor of Nf when summing over the different flavors. This is due to the fact that in our setup, the individual flavors yield distinct contributions. Most striking is the non-trivial dependence on the (dimensionless) chemical potential m in both correlators. Note also the distinct structures in the denominators. The first one (4.8) has an explicit relative factor of i between the terms in the denominator. In the second correlator (4.9) there is no explicit factor of i. The correlator (4.8) has a complex pole structure for ω > 0, but is entirely real for ω < 0. On the other hand, (4.9) is real for ω > 0 but develops a diffusion structure for ω < 0. So the correlators G0e0 and Ge00 essentially exchange their roles as ω changes sign (see also Fig. 1). We find a similar behavior for all correlators G and Gejl with j, l = 0, 1, 2, 3. This behavior is a consequence of the insertion of O(w1/2) in the hydrodynamic expansion (3.30). We assume m to be small enough in order to neglect the denominator term of order O(wm) ≪ wm, q2). Moreover, using the definitions of w, q and m from (3.13) we may write (4.8) and (4.9) as G0e0 = − ω + q2D(ω) , (4.10) Ge00 = iω − q2D(ω) , (4.11) where the frequency-dependent diffusion coefficient D(ω) is given by D(ω) = . (4.12) We observe that this coefficient also depends on the inverse square root of the chemical potential µ. Its physical interpretation is discussed below in section IV B. In the same way we derive the other correlation functions G3e3 =− ω3/2 (ω − µ) Q̃(ω, q) , Ge33 = ω3/2 (ω + µ) Q(ω, q) , (4.13) G0e3 =− ω q(ω − µ) Q̃(ω, q) , Ge03 = ω q(ω + µ) Q(ω, q) , (4.14) G3e0 =− ω3/2 q Q̃(ω, q) , Ge30 = ω3/2 q Q(ω, q) . (4.15) with the short-hand notation Q(ω, q) = iω − q2D(ω), Q̃(ω, q) = ω + q2D(ω) . (4.16) Note that most of these functions are proportional to powers of q and therefore vanish in the limit of vanishing spatial momentum q → 0. Only the 33-combinations from (4.13) survive this limit. In contrast to the Abelian Super-Maxwell correlators [2] given in appendix C, it stands out that our results (4.10), (4.11) and (4.13) and (4.15) have a new zero at ω = µ or −µ. Nevertheless bear in mind that we took the limit ω < µ in order to obtain our solutions. Therefore the apparent zeros at ±µ lie outside of the range considered. Compared to the Abelian case there is an additional factor ω. The dependence on temperature remains linear. In the remaining X-correlators we do not find any pole structure to order ω, subtracting an O(q2) contribution as in [2], G1e1 =G2e2 = µω , (4.17) Ge11 =Ge22 = − µω . (4.18) As seen from (4.17), Gαeα (with α = 1, 2) are real for negative ω and imaginary for positive ω. The opposite is true for Geαα, as is obvious from the relative factor of i. The correlators of components, pointing along the isospin potential in flavor space (a = 3), are found to be iω −D0q2 , GA3 = GA3 iω −D0q2 , (4.19) = GA3 NcT iω , GA3 iω −D0q2 , (4.20) with the diffusion constant D0 = 1/(2πT ) . Note that these latter correlators have the same structure but differ by a factor 4/Nc from those found in the Abelian super-Maxwell case [2] (see also (C1) and (C2)). In particular the correlators in equation (4.19) do not depend on the chemical potential. To analyze the novel structures appearing in the other correlators, we explore their real and imaginary parts as well as the interrelations among them, ReG0e0(ω ≥ 0) = ReGe00(ω < 0) = − 2µ |ω|+ q2/(2πT ) , (4.21) ReG0e0(ω < 0) = ReGe00(ω ≥ 0) = − 2µ |ω|+ q4/(2πT )2 , (4.22) ImG0e0(ω < 0) = −ImGe00(ω ≥ 0) = 2µ |ω| 2µ |ω|+ q4/(2πT )2 , (4.23) ImG0e0(ω ≥ 0) = 0, ImGe00(ω < 0) = 0. (4.24) Now we see why, as discussed below (4.11), G0e0 and Ge00 exchange their roles when crossing the origin at ω = 0. This is due to the fact that the real parts of all G and Gejl are mirror images of each other by reflection about the vertical axis at ω = 0. In contrast, the imaginary parts are inverted into each other at the origin. Figure 1 shows the real and imaginary parts of correlators G0e0 and Ge00. The different curves correspond to distinct values of the chemical potential µ. The real part shows a deformed resonance behavior. The imaginary part has a deformed interference shape with vanishing value for negative frequencies. All curves are continuous and finite at ω = 0. However due to the square root dependence, they are not differentiable at the origin. Parts of the correlator which are real for positive ω are shifted into the imaginary part by the change of sign when crossing ω = 0, and vice versa. To obtain physically meaningful correlators, we follow a procedure which generalizes the Abelian approach of [6]. In the Abelian case, gauge-invariant components of the field strength ten- sor, such as Eα = ωAα, are considered as physical variables. This procedure cannot be transferred directly to the non-Abelian case. Instead, we consider the non-local part of the gauge invariant trF 2 which contributes to the on-shell action (4.1). In this action, the contribution involving the non-Abelian structure constant – as well as µ – is a local contact term. The non-local contribution however generates the Green function combination . (4.25) We take this sum as our physical Green function. This choice is supported further by the fact that it may be written in terms of the linear combinations (3.21) which decouple the equations of motion. For example, for the time component, written in the variables X0, X̃0 given by (3.21), the combination (4.25) reads (compare to (4.2)) G0e0 +Ge00 +GA30A30 . (4.26) The contribution from GA3 is of order O(µ0), while the combination for the first two flavor directions, G0e0 +Ge00, is of order O(µ). We proceed by discussing the physical behavior of the Green function combinations introduced above. GA3 is plotted in Fig. 2 on the right. Its frequency dependence is of the same form as in the Abelian correlator obtained in [2], as can be seen from (C1). Since we are interested in effects of order O(µ), we drop the third flavor direction a = 3 from the sum (4.26) in the following. It is reassuring to observe that the flavor directions a = 1, 2 which are orthogonal to the chemical potential, combine to give a correlator spectrum qualitatively similar to the one found in [2] for PSfrag replacements −0.05−0.1 0 0.05 0.1 PSfrag replacements −0.05−0.1 0 0.05 0.1 FIG. 1: Real (left plot) and imaginary part (right plot) of the correlator G as a function of frequency ω/T at different chemical potential values µ/T = 0.5 (solid line), µ/T = 0.3 (short-dashed line) and µ/T = 0.1 (long-dashed line). The corresponding plots for the correlator G would look like the mirror image of the ones given. The real part would be reflected about the vertical axis at ω = 0, the imaginary part would be reflected about the origin. All dimensionful quantities are given in units of temperature. The numerical values used for the parameters are q/T = 0.1, Nc = 100. the Abelian Super-Maxwell action (see Fig. 2). However, we discover intriguing new effects such as the highly increased steepness of the curves near the origin due to the square root dependence and a kink at the origin. We observe a narrowing of the inverse resonance peak compared to the form found for the Abelian Super-Maxwell action (and also compared to the form of our GA3 , as is seen from comparing the left with the right plot in Fig. 2). At the origin the real and imaginary part are finite and continuous, but they are not continuously differentiable. However, the imaginary part of GA3 has finite steepness at the origin. The real part though has vanishing derivative at ω = 0. Note that the imaginary part of flavor directions a = 1, 2 on the left plot in Fig. 2 never drops below the real part. In the third flavor direction, as well as in the Abelian solution, such a drop occurs on the positive ω-axis. The correlators G3e3, Ge33, G0e3 and Ge03 have the same interrelations between their respective real and imaginary parts as G0e0 and Ge00. Nevertheless, their dependence on the frequency and momentum is different, as can be seen from (4.13) to (4.15). A list of the 33-direction Green functions split into real and imaginary parts can be found in appendix D. Thermal spectral functions in different directions are compared graphically in appendix E. B. Isospin diffusion The attenuated poles in hydrodynamic correlation functions have specific meanings (for ex- emplary discussions of this in the AdS/CFT setup see e.g. [7], [50]). In our case we observe an attenuated pole in the sum G0e0 + Ge00 at ω = 0. As can be seen from the plots in Fig. 2, our pole lies at Reω = 0. This structure appears in hydrodynamics as the signature of a diffusion pole located at purely imaginary ω. Its location on the imaginary ω-axis is given by the zeros of the denominators of our correlators as (neglecting O(ω , q4 )) ω = −i q . (4.27) PSfrag replacements 0.5 1 1.5−0.5−1−1.5 PSfrag replacements 0.010.005−0.005−0.01 FIG. 2: In the left plot the sum of both correlators in 00-directions is split into its imaginary (dashed line) and real (solid line) part and plotted against frequency. For comparison the right plot shows the corresponding real and imaginary parts for the GA3 . It is qualitatively similar to the Abelian correlator in i = 0 Lorentz direction computed from the Super-Maxwell action in [2]. Note the different frequency scales in the two plots. The curves in a = 1, 2-directions are much narrower due to their square root dependence on ω. Furthermore they have a much larger maximum amplitude. All dimensionful quantities are given in units of temperature. The numerical values used for the parameters are, as in Fig. 1, q/T = 0.1, Nc = 100 and only in the left plot µ/T = 0.2. Squaring both sides of (4.27) we see that this effect is of order O(q4). On the other hand, looking for poles in the correlator involving the third flavor direction GA3 , we obtain dominant contri- butions of order O(q2) and O(µ0) (neglecting O(ω2 , q2 )) ω = −i . (4.28) This diffusion pole is reminiscent of the result of the Abelian result of [2] given in appendix (C). As discussed in section IV A, we consider gauge-invariant combinations G0e0 + Ge00 + GA30A30 . In order to inspect the non-Abelian effects of order O(µ) showing up in the first two correlators in this sum, we again drop the third flavor direction which is of order O(µ0). Motivated by the diffusion pole behavior of our correlators in flavor-directions a = 1, 2 corre- sponding to the combinations X, X̃ (see (4.27)), we wish to regain the structure of the diffusion equation given in (2.9), which in our coordinates (k = (ω , 0 , 0 , q )) reads i ω J0 = D(ω) q 2 J0 . (4.29) Our goal is to rewrite (4.27) such that a term O(ω) and one term in order O(q2) appears. Further- more there should be a relative factor of −i between these two terms. The obvious manipulation to meet these requirements is to multiply (4.27) by ω in order to get ω = −i q2 . (4.30) Comparing the gravity result (4.30) with the hydrodynamic equation (4.29), we obtain the frequency-dependent diffusion coefficient D(ω) = . (4.31) Our argument is thus summarized as follows: Given the isospin chemical potential as in (1.1), (3.5), J0 from (2.9) is the isospin charge density in (4.29). According to (4.29), the coeffi- cient (4.31) describes the diffusive response of the quark-gluon plasma to a gradient in the isospin charge distribution. For this reason we interprete D(ω) as the isospin diffusion coefficient. Near the pole, the strongly-coupled plasma behaves analogously to a diffractive medium with anomalous dispersion in optics. In the presence of the isospin chemical potential, the propaga- tion of non-Abelian gauge fields in the black hole background depends on the square root of the frequency. In the dual gauge theory, this corresponds to a non-exponential decay of isospin fluc- tuations with time. The square root dependence of our diffusion coefficient is valid for small frequencies. As long as ω/T < 1/4, the square root is larger than its argument and at ω/T = 1/4, the differ- ence to a linear dependence on frequency is maximal. Therefore in the regime of small frequen- cies ω/T < 1/4, which is accessible to our approximation, diffusion of modes close to 1/4 is enhanced compared to modes with frequencies close to zero. V. CONCLUSION In this paper we have considered a relatively simple gauge/gravity dual model for a finite tem- perature field theory, consisting of an isospin chemical potential µ obtained from a time component vev for the SU(2) gauge field on two coincident brane probes. We have considered the D7-brane embedding corresponding to vanishing quark mass, for which µ is a constant, independent in particular of the radial holographic coordinate. The main result of this paper is that this model, despite its simplicity, leads to a hydrodynamical behavior of the dual field theory which goes be- yond linear response theory. We find in particular a frequency-dependent diffusion coefficient with a non-analytical behavior. Frequency-dependent diffusion is a well-known phenomenon in condensed matter physics. Here it originates simply from the fact that due to the non-Abelian structure of the gauge field on the brane probe, the chemical potential replaces a time derivative in the action and in the equations of motion from which the Green functions are obtained. Of course the calculation presented has some limitations as far as the approximations made are concerned. This applies in particular to the approximation (3.29) of the so-called indices in the ansatz for solving the equations of motion. Here we have dropped the constant present under the square root and used an expression proportional to the square root of the frequency. This allows for a closed solution without having to use numerics. However using this approximation we have dropped the Abelian limit. This leads ultimately to the square root dependence of the diffusion coefficient on the frequency. This dependence is unphysical for ω → 0, since the diffusion coeffi- cient is expected to be non-zero for zero frequency. We expect physical behavior to be restored if the Abelian limit is included in the calculation. To avoid the approximation described, this requires a numerical approach. A suitable solution method for all momenta and all frequencies has been presented in [7] and in [49]. We are going to study the application of this method to the model presented here in the future. Acknowledgments We are grateful to R. Apreda, G. Policastro, C. Sieg, A. Starinets and L. Yaffe for useful dis- cussions and correspondence. APPENDIX A: NOTATION The five-dimensional AdS Schwarzschild black hole space in which we work is endowed with a metric of signature (−,+,+,+,+), as given explicitly in (3.3). We make use of the Einstein notation to indicate sums over Lorentz indices, and additionally simply sum over non-Lorentz indices, such as gauge group indices, whenever they occur twice in a term. To distinguish between vectors in different dimensions of the AdS space, we use bold symbols like q for vectors in the the three spatial dimensions which do not live along the radial AdS coordinate. Four-vectors which do not have components along the radial AdS coordinates are denoted by symbols with an arrow on top, as ~q. The Green functions G = 〈JĴ〉 considered give correlations between currents J and Ĵ . These currents couple to fields A and  respectively. In our notation we use symbols such as GAa denote correlators of currents coupling to fields Aak and A l , with flavor indices a, b and Lorentz indices k, l = 0, 1, 2, 3. For the gauge field combinations Xk and X̃l given in (3.21), we obtain Green functions G denoting correlators of the corresponding currents. APPENDIX B: SOLUTIONS TO EQUATIONS OF MOTION Here we explicitly write down the component functions used to construct the solutions to the equations of motion for the gauge field fluctuations up to order w and q2. The functions themselves are then composed as in (3.31). The solutions for the components with flavor index a = 3 where obtained in [2]. 1. Solutions for Xα, X̃α and A3α The function Xα(u) solves (3.23) with the upper sign and is constructed as in (3.31) from the following component functions, +O(ω), (B1) F0 = C, (B2) F1/2 = −C 1 + u , (B3) F1 = −C π2 − 9 ln2 2 + 3 ln(1− u) (ln 16− 4 ln(1 + u)) + 3 ln(1 + u) (ln(4(1 + u))− 4 ln u) (B4) Li2(1− u) + Li2(−u) + Li2 1 + u + lnu ln(1 + u) + Li2(1− u) + Li2(−u) , (B5) where the constant C can be expressed it in terms of the field’s boundary value Xbdy = limu→0X(u, k), C = Xbdy ln 2 +mw ln2 2 2 +O(w3/2, q4) . (B6) The solutions of the equations of motion (3.23) with lower sign for the functions X̃α(u) are given by β̃ = −i +O(ω), (B7) F̃0 = C̃, (B8) F̃1/2 = iC̃ 1 + u , (B9) F̃1 = C̃ π2 − 9 ln2 2 + 3 ln(1− u) (ln 16− 4 ln(1 + u)) + 3 ln(1 + u) (ln(4(1 + u))− 4 lnu) (B10) Li2(1− u) + Li2(−u) + Li2 1 + u G̃1 = + ln u ln(1 + u) + Li2(1− u) + Li2(−u) , (B11) with C̃ given by C̃ = X̃bdy ln 2−mw ln2 2 2 +O(w3/2, q4) , (B12) so that limu→0 X̃(u, k) = X̃ The solution for A3α solves (3.16) up to order w and q 2 with boundary value (A3α) . It is A3α = 8 (A3α) (1− u)− iw2 8− 4iw ln 2 + π2q2 1 + i 1 + u + ln u ln(1 + u) + Li2(1− u) + Li2(−u) (B13) 2. Solutions for X ′0, X̃ 0 and A Here we state the solutions to (3.20). This formula describes three equations, differing in the choice of a = 1, 2, 3. The cases a = 1, 2 give coupled equations which are decoupled by transformation from A1,20 to X0 and X̃0. The choice a = 3 gives a single equation. The function X ′0 is solution to (3.24) with upper sign. We specify the component functions as +O(ω), (B14) F0 = C, (B15) F1/2 = −C 1 + u , (B16) F1 = −C π2 + 3 ln2 2 + 3 ln2(1 + u) + 6 ln 2 ln 1 + u (B17) Li2(1− u) + Li2(−u)− Li2 , (B18) G1 = C ln 1 + u , (B19) where the constant C can be expressed in terms of the field’s boundary value Xbdy = limu→0X(u, k), C = − 0 +wqX 2mw+mw ln 2 + q2 . (B20) To get the function X̃ ′0, we solve (3.24) with the lower sign and obtain β̃ = −i +O(ω), (B21) F̃0 = C̃, (B22) F̃1/2 = iC̃ 1 + u , (B23) F̃1 = C̃ π2 + 3 ln2 2 + 3 ln2(1 + u) + 6 ln 2 ln 1 + u (B24) Li2(1− u) + Li2(−u)− Li2 , (B25) G̃1 = C̃ ln 1 + u , (B26) where the constant C̃ can be expressed it in terms of the field’s boundary value X̃bdy = limu→0 X̃(u, k), 0 +wqX̃ 2mw+mw ln 2− q2 . (B27) The solution for (3.20) with a = 3 is the function A30 , given by = (1− u)− 0 + wqA iw− q2 1 + u + q2 ln 1 + u . (B28) PSfrag replacements µ = 0.4 T ω = q = 0.2 T 0.2 0.4 0.6 0.8 1 0.195 0.200 0.205 0.210 PSfrag replacements µ = 0.4 T ω = q = 0.2 T µ = 0.4 T ω = q = 0.2 T Re F (u)/T 0.2 0.4 0.6 0.8 1 0.095 0.100 0.105 0.110 FIG. 3: These plots show the real and imaginary part of the function F (u) which is part of Xα = (1 − u)βF (u). The solid line depicts the analytical approximation, obtained in this paper. As a check we solved the equations of motion for F (u) numerically. They are drawn as dashed lines. In this example we used T = 1. The numerical solution was chosen to agree with the analytical one at the horizon and boundary. 3. Solutions for X ′3, X̃ 3 and A We give the derivatives of X3 and X̃3 as X ′3 = − X ′0 (B29) X̃ ′3 = − X̃ ′0. (B30) The solution for A33 . (B31) 4. Comparison of numerical and analytical results As an example, in Fig. 3 we show the numerical and analytical solutions for the function F (u) in Xα = (1− u)βF (u). Here we compare the numerical result for F (u) obtained from the ansatz (3.26) with (3.27) in (3.23) with the analytically obtained approximation given above in (B1) to (B6). APPENDIX C: ABELIAN CORRELATORS For convenient reference we quote here the correlation functions of the Abelian super-Maxwell theory found in [2]. The authors start from a 5-dimensional supergravity action and not from a Dirac-Born-Infeld action as we do. Therefore there is generally a difference by a factor Nc/4. Note also that here all Nf flavors contribute equally. In our notation Gab11 = G 22 = − iN2c Tω δ + · · · , Gab00 = N2c Tq 2 δab 16π(iω −Dq2) + · · · , (C1) Gab03 = G 30 = − N2c Tωq δ 16π(iω −Dq2) + · · · , Gab33 = N2c Tω 2 δab 16π(iω −Dq2) + · · · , (C2) where D = 1/(2πT ) . APPENDIX D: CORRELATION FUNCTIONS In this appendix we list the real and imaginary parts of the flavor currents in the first two flavor-directions a = 1, 2 and in the third Lorentz-direction coupling to the supergravity-fields X3 and X̃3 (as defined in (3.21)). Re{G3e3(ω ≥ 0)} = Re{Ge33(ω < 0)} = − 2 (ω2 + µ |ω|) 16π2 [2µ |ω|+ q4/(2πT )2] , (D1) Im{G3e3(ω ≥ 0)} = −Im{Ge33(ω < 0)} = − 2µ |ω| (ω2 + µ |ω|) 8π [2µ |ω|+ q4/(2πT )2] , (D2) Re{G3e3(ω < 0)} = Re{Ge33(ω ≥ 0)} = − NcT (ω 2 − µ |ω|) 2µ |ω|+ q2/(2πT ) ] , (D3) Im{G3e3(ω < 0)} = 0, Im{Ge33(ω ≥ 0)} = 0. (D4) APPENDIX E: THERMAL SPECTRAL FUNCTIONS We include here a comparision of the sizes of spectral functions in distinct flavor- and Lorentz- directions (see also (2.1) in section II).PSfrag replacements 0.5 1 1.5 2 PSfrag replacements 0.5 1 1.5 2 FIG. 4: Here the thermal spectral functions in distinct Lorentz- and flavor-directions are plotted against frequency ω/T in units of temperature. In the left plot the chemical potential was chosen to be µ/T = 0.7, in the right one µ/T = 0.2. Flavor-directions a = 1, 2 are summed and displayed as one curve. The frequency-dependence of 00- (solid red line) and 03-Lorentz-directions (short-dashed blue line) is shown. By the dotted line we denote the spectral curve in 11- or 22-directions. This curve was scaled by a factor 100 in order to make it visible in these plots. The third flavor-direction is only plotted for the spectral function in Lorentz-directions 00 (long-dashed green curve). We do not show the 33-direction spectral function which has a square root dependence and is comparable in size with the 11-direction. [1] G. Policastro, D. T. Son, and A. O. Starinets, The shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 87 (2001) 081601, arXiv:hep-th/0104066. [2] G. Policastro, D. T. Son, and A. O. 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Contents Introduction Hydrodynamics and AdS/CFT Supergravity background and action Finite temperature background and brane configuration Introducing a non-Abelian chemical potential Dirac-Born-Infeld action Equations of motion Equations for A1a- and A2a-components Equations for A0a- and A3a-components Solutions Isospin diffusion and correlation functions Current correlators Green functions: Calculation Green functions: Results Isospin diffusion Conclusion Acknowledgments Notation Solutions to equations of motion Solutions for X, X"0365X and A3 Solutions for X0', X"0365X0' and A30' Solutions for X3', X"0365X3' and A33' Comparison of numerical and analytical results Abelian Correlators Correlation functions Thermal spectral functions References

0704.1291

Projective Hilbert space structures at exceptional points

arXiv:0704.1291v4 [math-ph] 12 Nov 2018 Projective Hilbert space structures at exceptional points Uwe Günthera, Ingrid Rotterb and Boris F. Samsonovc1) a Helmholtz Center Dresden-Rossendorf, Bautzner Landstraße 400, D-01328 Dresden, Germany b Max Planck Institute for physics of complex systems, D-01187 Dresden, Germany c Physics Department, Tomsk State University, 36 Lenin Avenue, 634050 Tomsk, Russia E-mail: [email protected] and [email protected] Abstract. A non-Hermitian complex symmetric 2× 2 matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseux- expanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behavior in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase jump behavior are analyzed and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of PT −symmetrically extended Quantum Mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated. PACS numbers: 03.65.Fd, 03.65.Vf, 03.65.Ca, 02.40.Xx published as: J. Phys. A: Math. Theor. 40, 8815 (2007) 1. Introduction A generic property of non-Hermitian operators is the possible occurrence of non-trivial Jordan-blocks in their spectral decomposition [1]. For an operator H(X) depending on a set of parameters X = (X1, . . . , Xm) ∈ M from a space M this means that, in case of a single Jordan block, two or more spectral branches λ1(X), . . . , λk(X) may coalesce (degenerate) at certain parameter hypersurfaces V ⊂ M under simultaneous coalescence of the corresponding eigenvectors Φ1(X), . . . ,Φk(X): λ1(Xc) = · · · = λk(Xc), Φ1(Xc) = · · · = Φk(Xc) ≡ Θ0(Xc) for Xc ∈ V . Spectral points of this type are branch points of the spectral Riemann surface and are called exceptional points (EPs) [1]. At the EPs the set of the originally k linearly independent eigenvectors Φ1(X), . . . ,Φk(X) is replaced by the single eigenvector Θ0(Xc) and k− 1 associated vectors Θ1(Xc), . . . ,Θk−1(Xc) which form a Jordan chain. Together they span the so called k−dimensional algebraic eigenspace (or root space) Sλ(Xc) = span[Θ0(Xc), . . . ,Θk−1(Xc)] [1, 2] so that the total space dimension remains preserved. 1) Deceased 08 November 2012 http://arxiv.org/abs/0704.1291v4 Projective Hilbert space structures 2 The construction extends straight forwardly to the presence of several Jordan blocks for the same eigenvalue λ(Xc). In general, the degeneration hypersurface V ⊂ M consists of components Va of different codimension codimVa = a. Higher order Jordan blocks require a higher degree of parameter tuning (they have a higher codimension) and a correspondingly lower dimension of the component Va. Due to the different dimensions of its components Va the degeneration hypersurface V = Va itself has the structure of a stratified manifold [3]. EPs occur naturally in quantum scattering setups [4, 5] when two or more resonance states coalesce and higher order poles of the S-matrix form. Within the Gamow state approach such S-matrix double poles have been considered in [6, 7, 8, 9], whereas in the Feshbach projection operator formalism (one of the basic approaches to analyze open quantum systems) they naturally occurred in studies of nuclei [10], atoms [11, 12] and quantum dots [13, 14]. EP-related crossing and avoided crossing scenarios have been studied for bound states in the continuum [11, 15, 16, 17] as well as for phase transitions [18, 19, 20, 21]. In asymptotic analyses of quasi-stationary systems EPs show up as hidden crossings [22]. EP-related theoretically predicted level and width bifurcation properties have been experimentally verified in a series of microwave resonator cavity experiments. In [23], the resonance trapping phenomenon (width bifurcation) [10] has directly been proven. The fourfold winding around an EP has been found experimentally [24] in full agreement with the theoretical prediction [19, 25] and related studies [14, 26]. In [27] two-level coalescences have been associated with chiral system behavior. The geometric phase at EPs has been discussed in [14, 26, 27, 28, 29, 30, 31, 32]. EPs play also an important role in the recently considered PT −symmetrically extended quantum models [33, 34, 35]. There they correspond to the phase transition points between physical sectors of exact PT −symmetry and unphysical sectors of spontaneously broken PT −symmetry [36, 37, 38, 39, 40]. Other, non-quantum mechanical examples where EPs play an important role are the optics of bianisotropic crystals [41], acoustic models [42], many hydrodynamic setups where EPs have been studied within pseudo-spectral approaches [43] as well as a large number of mechanical models [44] where they are connected with regimes of critical stability [45]. Recent results on magnetohydrodynamic dynamo models indicate on a close connection between nonlinear polarity reversal mechanisms of magnetic fields and EPs [46]. For completeness we note that the perturbation theory for systems in the vicinity of EPs dates back to 1960 [47] (see also [2]) and that it is closely related to singularity theory, catastrophe theory and versal deformations of Jordan structures [48]. Supersymmetric mappings between EP configurations have been recently considered in [49, 50]. A correct perturbative treatment of models in the vicinity of EPs has to be built on an expansion in terms of root vectors (eigenvectors and associated vectors Θi(Xc)) at the corresponding unperturbed eigenvalue λ(Xc) (see e.g. [2, 44]). For X 6∈ V (away from the EP at Xc and from other EPs) the operator H(X) has a diagonal spectral decomposition with corresponding eigenvectors Φi(X). Choosing the normalization of these eigenvectors away from the EP and without regard to the expansion in terms of root vectors leads to a divergence of the normalization constants in the EP-limit X → Xc. The diagonalization break-down at Xc, the occurrence of the Jordan block structure and the singular behavior of the eigenvector decomposition Projective Hilbert space structures 3 are generic and were many times described in various contexts (see e.g. [51, 52]). The natural question connected with the fitting of the root-vector based normalization and the diagonalizable-configuration normalization (and related controversial discussions on their physical interpretation [51, 52]) is whether and how the singular behavior affects the projective Hilbert space structure of quantum systems. In the present paper we answer this question by resolving the singularity with the help of embedding the original Hilbert space H = Cn into its projective extension n instead of projecting it down to CPn−1 as in standard Hermitian Quantum Mechanics. Diagonal spectral decompositions and decompositions with nontrivial root spaces live then simply in different (and complementary) affine charts of CPn similar like monopole configurations of Hermitian systems have to be covered with two charts (North-pole chart and South-pole chart) of the unit sphere S2 [53]. The basic construction is demonstrated on a simple complex symmetric (non- Hermitian) 2×2−matrix toy model. The consideration of complex symmetric matrices sets no restriction because by a similarity transformation any complex matrix can be brought to a complex symmetric form (see, e.g. [54, 55]). The Hilbert space notations for the 2 × 2−matrix model are fixed in section 2. In section 3, following [32, 44] we derive the leading-order perturbative expansion in the vicinity of an EP at Xc in terms of root vectors and fit it then explicitly with expressions of the diagonal spectral decomposition at X 6= Xc. Combining geometric phase techniques for non-Hermitian systems [28] with projective Hilbert space techniques from [56] we generalize the projective geometric phase techniques of Hermitian systems to paths around EPs (section 4). The corresponding monodromy group is identified as parabolic Abelian subgroup of the special linear group SL(2,C) and evidence is given that vector norm scalings are only due to complex dynamical phases whereas geometrical phases are purely real-valued and norm preserving. In section 5 we consider an instantaneous (stationary type) picture of the system. Within such a picture, we resolve the singular normalization behavior by projectively embedding the Hilbert space H = C2 →֒ CP2. We discuss the necessity for an affine multi- chart covering of CP2 in order to accommodate diagonal-decomposition vectors and root vectors at EPs simultaneously. The usefulness of the phase rigidity as distance measure to EPs is discussed in section 6. In section 7 some EP-related aspects of PT −symmetric quantum models are discussed and a conjecture concerning the quantum brachistochrone problem [34, 57] is formulated. The Conclusions 8 are followed by Appendix A where auxiliary results on Jordan structures of complex symmetric matrices are listed. 2. Setup Subject of our consideration is the behavior of a quantum system near a level crossing point of two resonance states — supposing that for an N−level system the influence of the other N − 2 levels is sufficiently weak. Under this assumption the setup can be modeled by an effective complex symmetric (non-Hermitian) 2×2 matrix Hamiltonian , H = HT . (1) The effective energies ǫ1,2 ∈ C and the effective channel coupling ω ∈ C will in general depend on underlying parameters X = (X1, . . . , Xk) ∈ M from a space M. Projective Hilbert space structures 4 For nonvanishing coupling ω 6= 0 the Hamiltonian can be rewritten as H = E0 ⊗ I2 + ω with I2 denoting the 2× 2 unit matrix and E0 := (ǫ1 + ǫ2), Z := ǫ1 − ǫ2 . (3) In this representation the eigenvalues E± and eigenvectors Φ± of H take the very simple form E± = E0 ± ω Z2 + 1 (4) Z2 + 1 c± , c± ∈ C∗ := C− {0} (5) which makes the branching behavior most transparent2). From the overlap 〈Φ+|Φ−〉 ≡ Φ∗T+ Φ− (6) = c∗+c− 1 + |Z|2 − |Z2 + 1|+ 2Im Z2 + 1 one reads off that 〈Φ+|Φ−〉 = 0 holds only for ImZ = 0 and that for general Z ∈ C the two states Φ+ and Φ− are not orthogonal 〈Φ+|Φ−〉 6= 0. Following standard techniques [44] for non-Hermitian operators, we consider a dual (left) basis Ξ± bi-orthogonal to (H+ − E∗±)Ξ± = 0, 〈Ξk|Φl〉 ∝ δkl, k, l = ± . (7) For complex symmetric H it holds Ξ± ∝ Φ∗± so that the most general ansatz for the right and left basis vectors Φ± and Ξ± can be chosen as Φ± = c±χ±, Ξ± = d ±, c±, d± ∈ C∗ (8) χ± := Z2 + 1 . (9) The bi-orthogonality 〈Ξ±|Φ∓〉 = d±c∓χT±χ∓ = 0 (10) is ensured by the structure of χ± and holds for any value of the parameter Z ∈ C. A normalization 〈Ξ±|Φ±〉 = 1 would set two constraints on the four free scaling parameters c±, d± ∈ C∗ 〈Ξ±|Φ±〉 = d±c±χT±χ± = 1 , (11) so that the system would still have two free parameters which should be fixed by additional assumptions. Subsequently, we will mainly work with the bi-orthogonality properties of the vectors Φ±, Ξ± and fix their normalization only when explicitly required. Due to the arbitrary scaling parameters c±, d± ∈ C∗ of the right and left eigenvectors Φ±,Ξ± ∈ H = C2 (8) it is natural to consider equivalence classes of such vectors defined by corresponding lines π(Φ±), π(Ξ±). These lines form the projective 2) The fact that Φ± depends only on the single parameter Z reflects the property that after rescaling the energy by 1/ω and shifting it by −E0/ω (these transformations do not affect the eigenvectors) the Hamiltonian (2) depends only on Z. Projective Hilbert space structures 5 Hilbert space P(H) = H∗/C∗ = CP1 ∋ π(Φ±), π(Ξ±) [58, 59, 60], whereH∗ := H−{0} denotes the original Hilbert space with the point at origin {0} = (0, 0) deleted to allow for a consistent definition of P(H). The space P(H) is covered by a single chart of homogeneous coordinates (z0, z1) T ∈ H and two complementary charts of affine coordinates U0 ∋ (1, z1/z0), z0 6= 0 and U1 ∋ (z0/z1, 1), z1 6= 0. Comparison with the structure of the auxiliary vectors χ± (9) shows that the χ± can be straight forwardly re-interpreted as points of the projective space CP1 described by the affine coordinate over U0: χ± ≈ π(Φ±). The vectors {Φ±,Ξ±} themselves can be understood as sections of the natural line bundle L = {(p, v) ∈ P(H)×H| v = cp, c ∈ C∗} [53], i.e. as Φ± = π(Φ±)⊗c±, Ξ± = π(Ξ±)⊗d∗±, where π denotes the projection π : H∗ → P(H). The bundle structure is locally trivial π−1(U0) ≈ U0 × C∗ ∋ Φ± [61]3). 3. Jordan structure At an EP, the two eigenvalues E± coalesce E+ = E− = E0. According to (4), this happens for Z2 = −1 and Z = Zc := ±i and via (8) it is connected with a coalescence of the corresponding lines π(Φ+) = π(Φ−) =: π(Φ0) encoded in χ+ = χ− = χ0 := . (12) This means that the eigenvalue E0 has algebraic multiplicity na(E0) = 2 and geometric multiplicity ng(E0) = 1 and by definition the level crossing point is an EP of the spectrum. The bi-orthogonality (10) of Φ± and Ξ∓ is compatible with the coalescence of the lines due to the vanishing bi-norm χT0 χ0 = 0, i.e. the isotropy 4) of χ0, — a generic fact holding for the (geometric) eigenvector at any EP [2, 44]. We note that the coalescence π(Φ+) = π(Φ−) = π(Φ0) still leaves the freedom for the vectors Φ+ = c+χ0 and Φ− = c−χ0 of being two different sections Φ+ 6= Φ− of the same fiber π(Φ0)× C∗ over π(Φ0). The right and left eigenvectors Φ0, Ξ0 at the EP are supplemented by corresponding associated vectors (algebraic eigenvectors) Φ1 and Ξ1 defined by the Jordan chain relations [44] [H(Zc)− E0I2]Φ0 = 0, [H(Zc)− E0I2]Φ1 = Φ0 (13) [H(Zc)− E0I2]+ Ξ0 = 0, [H(Zc)− E0I2]+ Ξ1 = Ξ0 . From the inhomogeneity of these Jordan chain relations it follows immediately that the root vectors Φ0 and Φ1 as well as Ξ0 and Ξ1 scale simultaneously and in a linked way with the same single scale factor c0 and d 0, respectively. This is also visible from their explicit representation (A.9) derived in Appendix A Φ0 = σqc0 , Φ1 = σq Ξ0 = σq ,Ξ1 = σq ∗−1d∗0 , q := 2ω, Zc = ±i =: µi, c0, d0 ∈ C∗ . (15) 3) For completeness we note that the (right) eigenvectors Φ± and the dual (left) ones Ξ± could be understood as elements of a vector bundle P(H)×F and its dual P(H)×F ∗ with pairing in the fibres 〈.|.〉 : F ∗p × Fp −→ C (see, e.g. [58]). The details of this construction will be presented elsewhere. 4) The vector χ0 behaves similar like a vector on the light cone in Minkowski space. Projective Hilbert space structures 6 The simultaneous scaling means that the lines π(Φ0), π(Ξ0) at the EP should be interpreted as the one-dimensional components (projections) of two-dimensional planes which span the root space5) S(E0) [2] and which scale as a whole with a single scale factor. Such a higher-dimensional (complex) plane-structure goes clearly beyond the one-dimensional line structure of the projective space P(H) (mathematically one should extend the natural line bundle of the original projective space to a more general projective flag bundle [62, 63])6) and underlines the fact that the state at an EP itself is not an element of the projective Hilbert space P(H) in its usual understanding. The basis sets {Φ0,Φ1} and {Ξ0,Ξ1} satisfy the well known bi-orthogonality conditions [44] 〈Ξ0|Φ0〉 = 〈Ξ1|Φ1〉 = 0 〈Ξ0|Φ1〉 = 〈Ξ1|Φ0〉 = d0c0 6= 0 . (16) Again, a normalization 〈Ξ0|Φ1〉 = 〈Ξ1|Φ0〉 = 1 would only lead to a constraint d0c0 = 1 on the scale factors, but would not fix them completely. Due to this scaling freedom the single line π(Φ0) of a given Jordan structure, in general, still allows for different sections Φ0,a 6= Φ0,b of the corresponding fiber π(Φ0) × C∗ ∋ Φ0,a,Φ0,b, π(Φ0,a) = π(Φ0,b) = π(Φ0). Let us now consider in detail what happens when the system approaches one of the critical values Zc = ±i. For this purpose we use the well-defined (but completely general and arbitrary) ansatz Z = Zc + ε, |ε| ≪ 1, ε ∈ C (17) and expand the eigenvalues (4) and the line defining vectors χ± (9) in terms of ε. This gives the leading contributions to their Puiseux series representation [2, 44] in ε1/2 as E± = E0 ± ε1/2∆E + o(ε1/2), (18) ∆E := ω 2Zc , ± ε1/2 + o(ε1/2) . (19) Following [32, 44], we expand the eigenvectors Φ±(Z) of the diagonal spectral decomposition in the same local ε1/2 approximation in terms of the Jordan chain (root) vectors Φ0,1 Φ± = Φ0 + ε 1/2(b0Φ0 + b1Φ1) + o(ε 1/2), (20) b0 = ± ∆E, b1 = ±∆E . (21) The coefficients b0,1 are obtained by a two-step procedure. Substituting (17), (18), (20) into the eigenvalue equation and making explicit use of the chain relations (13) yields b1 and leaves b0 still undefined. The coefficient b0 is found by comparing the line structures7) of Φ± with χ± in (19). It remains to match the fiber sections — what can be done in two ways. One may assume a single scaling coefficient c0 of the root space given and consider the 5) In the present simplest model S(E0) fills the whole Hilbert space H = C 6) Indications that all the root vectors of a Jordan chain should scale simultaneously with a single scale factor were given, e.g., in [8] for Gamow vector setups with higher S-matrix poles. 7) The term ε1/2b0Φ0 additionally present in (20) in comparison with the corresponding result in [32] is due to the different choice of the root (Jordan chain) vectors Φ0, Φ1. The chain relation (13) shows that the associated vector Φ1 is defined up to additional Φ0 contributions and can be replaced by any linear combination Φ1 + aΦ0, a ∈ C. Projective Hilbert space structures 7 coefficients c± of the sections Φ± as derived objects. This leads to the identification c+ = c− = σqc0. Apart from this option, one may assume the scaling coefficients c± as primary objects and given so that they may take different values c+ 6= c−. Correspondingly the scaling factor c0 of the root space will then be fitted to c± so that it will take two different values c0,± = c±/(σq) . (22) Both constructions are possible and compatible with the smooth fitting of the line structure encoded in the EP-limiting behavior π(Φ±) → π(Φ0). In a way similar to the above two-step procedure with subsequent fiber fitting the left eigenvectors can be obtained as Ξ∗± = Ξ 0 + ε 1/2(b0Ξ 0 + b1Ξ 1) + o(ε 1/2), (23) d± = σ ∗qZcd0,± . (24) Here, b0 and b1 are the same as in (21) and full compatibility with the bi-orthogonality conditions (7) as well as with (8) is easily verified by direct calculation. In case of a single scaling factor d0 of the dual root space the coefficients d± will coincide d+ = d− = σ ∗qZcd0. Combining (20) and (23) one finds the limiting behavior of the inner products as 〈Ξ±|Φ±〉 = 2b1d0,±c0,±ε1/2 + o(ε1/2) d±c±ε 1/2 + o(ε1/2) . (25) Here, one has to distinguish two normalization schemes. If one assumes the root vector sets {Φ0,Φ1}, {Ξ0,Ξ1} normalized, e.g., with d0,±c0,± = 1 or d0c0 = 1 in (16) then the scalar product 〈Ξ±|Φ±〉 of the eigenvectors in the diagonal spectral decomposition (see (25)) vanishes in the EP-limit. Starting, in contrast, from normalized eigenvector pairs {Φ±,Ξ±} of diagonalizable Hamiltonians as in (11), i.e. with 〈Ξ±|Φ±〉 = 1, then the scale factor products d±c± diverge as d±c± ∝ ε−1/2 for ε → 0. Both normalization schemes are possible and compatible with the smooth limiting behavior π(Φ±) → π(Φ0) of the lines encoded in χ±(ε → 0) → χ0 [cf. (12)]. We see that this special behavior is only related to the fiber sections and not to the fibers (lines) themselves. The two incompatible normalization schemes simply indicate on the need for two complementary charts to cover the whole physical picture in the vicinity of a 2 × 2 Jordan block J2. One of these charts (we will call it the root vector chart) remains regular in the EP-limit, whereas the other (diagonal representation) chart becomes singular. The situation is similar to the two affine charts required to cover the Riemann sphere CP1. Starting from homogeneous coordinates (x0, x1) ∈ CP1 one has the two affine charts U0 ∋ (1, x1/x0), x0 6= 0 and U1 ∋ (x0/x1, 1), x1 6= 0. The mutually complementary affine coordinates z := x1/x0 ∈ C1 and w := x0/x1 ∈ C1 are then related by the well known fractional transformation w = 1/z so that the singular |z| → ∞ limit in the z−chart corresponds simply to the regular w → 0 limit in the w−chart. In other words, the two charts cover the North-pole region and the South- pole region of the Riemann sphere — a construction well known, e.g., from complex analysis and the description of magnetic monopoles [53]. Returning to the two-chart picture of the normalization we see that the original Hilbert space H = C2 should be extended by the set of infinite vectors Φ± what can be naturally accomplished by embedding it into a larger projective space H →֒ CP2. Projective Hilbert space structures 8 Correspondingly the fibers π(Φ±) × C∗ should be extended as π(Φ±) × C∗ →֒ π(Φ±) × CP1. A detailed discussion of these structures will be presented in [64]. An explicit embedding construction for simplified setups with coinciding scale factors d± = c± is given in section 5 below. 4. Geometric phase Following earlier studies [27, 28, 29, 30, 31], geometric phases [65] of eigenvectors of non-Hermitian complex symmetric operators have been recently considered in [32] for paths in parameter space encircling an EP. The results showed full agreement with the phase considerations of [27]. In this section, we combine techniques for non-Hermitian systems [28, 32] with explicit projective space parameterizations for Hermitian systems [56] to provide an explicit projective-space based derivation of the phase representation for non-Hermitian systems. Such an explicit reshaping of the results of [56] to non-Hermitian setups seems missing up to now. Following [28, 29, 30, 32] we consider an auxiliary system with a general non- Hermitian Hamiltonian H(t) depending on a set of non-stationary parameters X(t) = [X1(t), . . . , Xm(t)] ∈ M, H(t) = H [X(t)] and an EP hyper-surface V ⊂ M which is encircled by an appropriate loop Γ in parameter space M ∋ Γ = {X(t), t ∈ [0, T ] : X(0) = X(T )}. The evolution of the quantum system is governed by a usual Schrödinger equation for the right eigenvectors i∂tΦn(t) = H(t)Φn(t), (26) and, due to the time invariance of the bi-orthogonal product 〈Ξm(t)|Φn(t)〉 = δmn , (27) by a complementary evolution law for the left eigenvectors [28] i∂tΞm(t) = H +(t)Ξm(t) . (28) For an adiabatic motion cycle Γ ⊂ M with Hamiltonian H [X(T )] = H [X(0)] the resulting eigenvector Φn(t = T ) of H [X(T )] must lay on the same line as the initial Φn(t = 0), i.e. it can only obtain an additional scaling factor which we parameterize as complex-valued phase Φn(T ) = e iφn(T )Φn(0), φn(T ) ∈ C. (29) Due to the preserved orthonormality (27) the corresponding left eigenvectors evolve Ξm(T ) = e (T )Ξm(0) . (30) The complex phase φn(T ) can be split into a dynamical component [28, 56] ǫn(T ) = − 〈Ξn(t)|H(t)|Φn(t)〉 〈Ξn(t)|Φn(t)〉 dt (31) and the geometric phase γn(T ) = φn(T )− ǫn(T ) . (32) Adapting the techniques of [56] we calculate γn(T ) in terms of explicit projective space coordinates. Setting Φn(t) = cn(t)χn(t) =: [z0(t), z1(t)] T = z0(t)[1, w(t)] Ξm(t) = d m(t)χ m(t) =: [y0(t), y1(t)] T = y0(t)[1, v(t)] T (33) Projective Hilbert space structures 9 (omitting in the projective space coordinates the mode indices m, n) one identifies φn(T ) = −i ln [z0(T )/z0(0)] (34) and obtains from (26), (31) and (32) the differential 1-form of the geometric phase as dγ = − idz0 y∗0dz0 + y y∗0z0 + y 1 + v∗w∗ . (35) Due to the symmetry (8) between left and right eigen-lines this simplifies to dγ = i d ln(1 + w2) . (36) Similar to results on Hermitian systems [56] the differential 1-form (35) is independent of the coordinates z0 and y0 along the fibers and, hence, defines a horizontal connection over the projective Hilbert space of the system. The mere difference in the definitions of the projective structures is in CP1 = S3/S1 for Hermitian systems, whereas 1 = H∗/C∗ for non-Hermitian ones [61]. Let us now apply the general technique to the concrete 2× 2−matrix model (1). Parameterizing the cycle around the EP by (17) with ε = reiα, α ∈ [0, 2π], 0 < r ≪ 1 (37) one reproduces the 1-forms of the geometric phases of [29] dγ± = d ln ε = −1 d ln r . (38) In a similar way one obtains the same 1-forms for the corresponding left eigenvectors Ξ±. Upon integration over a full cycle α(T ) = α(0) + 2π, r(T ) = r(0) one finds γ±(T )− γ±(0) = − . (39) The relation between geometric phases γ± and the cycle phase α can be gained also directly from the structure of the sections Φ±. These sections may be arranged as columns of a diagonalizable 2× 2−matrix Φ(α) := [Φ+(α),Φ−(α)] . (40) The evolution along a cycle is then encoded in the transformation matrix W (α) = Φ(α) [Φ(0)] which for small ε with 0 6= |ε| ≪ 1 can be calculated from the representation (19) as W (α) = 2iZc sin . (41) The elements W (α) form an Abelian parabolic subgroup P of the special linear group SL(2,C) ⊃ P (see, e.g., [62, 63, 66]) W (α+ β) = W (α)W (β) = W (β)W (α) (42) corresponding to the mapping eiα ∈ S1 ≈ U(1) 7→ P ⊂ SL(2,C). For full cycles α = 2πN, N ∈ Z they yield the monodromy transformations [67] W0 := W (0) = I2, W1 := W (2π) = 2iZc i W2 := W (4π) = W 2(2π) = −I2, W3 := W (6π) = −W (2π), W (8π) = I2 = W0 . (43) Projective Hilbert space structures 10 The geometric phase (38) and the monodromy transformations (43) show the typical four-fold covering of the mapping α 7→ γ which was earlier described in [19, 25, 27, 32] and experimentally demonstrated in [24]. A cycle around the EP in parameter space M has to be passed four times in order to produce one full 2π−cycle in the geometric phase. A (non-oriented) eigen-line π(Φ 6= Φ0) ∈ CP2 is already recovered after two cycles π(W2Φ) = π(−Φ) = π(Φ) — similar to the eigenvalue E which for the 2×2−matrix lives on a two-sheeted Riemann surface with the same two branch points Zc = ±i as the line bundle. For the isotropic limiting vector Φ0 at the EP it holds (due to (12)) W (α)Φ0 = e −iα/4Φ0 (44) so that the parabolic subgroup P ∋ W (α) can be identified as invariance group of the projective line at the EP π (W (α)Φ0) = π e−iα/4Φ0 = π(Φ0). (45) We note that despite the non-Hermiticity of the Hamiltonian H the geometric phase is purely real — as for Hermitian systems. Relations (38), (39) show that possible imaginary phase contributions (which would result in a re-scaling of the eigenvectors Φ±) are cancelled by the closed-cycle condition r(T ) = r(0). Hence, the non-preservation of the vector norm in non-Hermitian systems is induced solely by a complex dynamical phase ǫ and requires the presence of the bi-orthogonal basis where a decaying behavior of the right eigenvectors8) Φn ∝ e−iǫnt− t, 〈Φn|Φn〉 = ||Φn||2 ∝ e−Γnt (46) is necessarily connected with increasing vector norms of the dual left eigenvectors Ξm ∝ e−iǫmt+ t, ||Ξm||2 ∝ eΓmt (47) so that indeed 〈Ξm|Φn〉 = δmn. This behavior is well known from resonances and Gamow vector theory (see, e.g. [7]). Comparison of (8) and (46), (47) shows that the formal ansatz Ξm = Φ m for the eigenvectors of the complex symmetric Hamiltonian (cf. [11, 13, 14]) can be used only for instantaneous eigenvectors at a single fixed t = t0 (which formally can be set to t0 = 0) as well as for the subclass of real symmetric matrices (when the system becomes Hermitian and norm-preservation of the eigenvectors holds). In contrast, for explicitly time dependent non-Hermitian setups it only holds Ξm(t) ∝ Φ∗m(t), i.e. the dual basis vectors necessarily live on complex conjugate lines (fibers) π[Ξm(t)] = (π[Φm(t)]) with Ξm(t) 6= Φ∗m(t) for t 6= t0. Aspects of the parameter dependence of the phases and scalings in an instantaneous picture with Ξm = Φ m together with the explicit EP-limit ε → 0 are subject of the next section. 5. Instantaneous picture In modern quantum physics not only the properties of natural systems such as nuclei or atoms are of interest, but rather the design and functionality of artificial quantum- system-based devices plays an essential role. In many cases, for the understanding of 8) For simplicity, we show the relations for stationary non-Hermitian Hamiltonians H with constant complex eigenvalues En = ǫn + i = const ; ǫn,Γn ∈ R. Projective Hilbert space structures 11 the dynamical features of these man-made quantum systems the time dependence is of minor interest. The properties of these systems are mainly governed by the position and number of EPs, i.e. the level crossing points in the complex plane, and their dependence on external control parameters. In this context it appears natural to study the parameter dependence of level energies and widths as well as the corresponding eigenvectors in terms of the instantaneous picture with Ξm = Φ m and c± = d±. This picture is compatible with the Hermitian limit when Imǫ1,2 = Imω = 0 in (1) and the condition Ξm = Φ m is fulfilled by definition We have to distinguish the two possible normalization schemes — the root- vector based normalization (16) with d0c0 = 1 or d0,±c0,± = 1 and the diagonal- representation based normalization (11) with d±c±χ Tχ = 1. In the root-vector based normalization scheme the conditions c± = d± and d0,±c0,± = 1 together with the two relations (22) and (24) imply (in leading-order approximation in ε) c0,± = d0,± and, hence, c0,± = κ with κ = ±1 (independently of the signs± in the index of c0,±) as well as c± = d± = κσq. We see that in leading-order approximation in ε the scaling factors c± = d± are rigidly fixed and independent of ε. A geometric phase (necessarily induced via an ε−dependence) appears incompatible with this normalization. Let us now turn to the diagonal-representation based normalization (11). In the EP-limit ε → 0 the normalization condition (11) for the eigenvectors (5) yields 1 = 〈Ξ±|Φ±〉 = ΦT±Φ± = Z2 + 1 ≈ ∓ 2Zc 2Zcε c ± (48) and we find the expected divergent scaling factors as c2± ≈ ∓2−3/2Z−3/2c ε−1/2 =⇒ c± ∼ ε−1/4 . (49) On the one hand, (49) reproduces the local four-sheeted Riemann surface structure connected with the geometric phase (38), (41), i.e. a fourfold winding around the EP is needed to return to an eigenvector pointing into the same complex direction as a starting vector. (In contrast to the root-vector normalization scheme full compatibility with the geometric phase setup holds.) On the other hand, it leads to divergent vector norms ||Φ±||2 = 〈Φ±|Φ±〉 ≈ 2|c±|2 ≈ |2ε|−1/2 (50) for ε → 0. As it was indicated in section 3, the corresponding singularity can be naturally resolved by embedding the original Hilbert space H ≈ C2 into its projective extension H →֒ CP2 ∋ φ = (u0, u1, u2) so that the set of infinite vectors becomes well defined. Interpreting the two components z0 and z1 of the vector (fiber section) Φ = c(1, w) = (z0, z1) ∈ C2 (51) as affine coordinates on the chart U2 ∋ (u0u2 , , 1), u2 6= 0, U2 ⊂ CP2 Φ = (c, cw) →֒ (c, cw, 1) (52) 9) When compatibility with the Hermitian limit is not required, then the bi-orthonormalization constraints d0,±c0,± = 1 or d±c±χ χ± = 1 fix only two of the four constants c0,±, d0,± or c±, d± and the remaining two can be chosen arbitrarily. For instance, one may set the eigenvector scaling factors as c0,± = C 6= 1 or c± = 1 so that d0,± = C −1 or d± = what would define instantaneous pictures not compatible with the Hermitian limit. Projective Hilbert space structures 12 we can identify Φ with the point φ ∈ CP2 with homogeneous coordinates φ = (u0, u1, u2) = (1, w, c −1). (53) The singularity |c| → ∞ at the EP corresponds then simply to the point φ0 = (1, w, 0) ∈ CP2 with u2 = 0 and we see that the affine chart U2 ∈ CP2 is no longer appropriate for covering φ0. This is in contrast to the root-vector normalization scheme where c is fixed and the chart U2 remains suitable for the covering. Within the present diagonal-representation normalization, instead, φ0 should be parameterized in terms of affine coordinates corresponding to one of the charts10) U0 or U1 with u0 6= 0 or u1 6= 0. Most natural for our representation (51), (53) is the affine chart U0 ∋ (1, u1u0 , ) which can be used for a suitable projective representation of the fibre sections Φ Φ ≈ (1, w, c−1) = (χT , c−1) ≈ (π(Φ), c−1) . (54) Interpreting the normalization condition (48) as constraint on the affine coordinates of Φ in the chart U2 0 = ΦTΦ− 1 = u − 1 (55) one immediately sees that it is equivalent to the conic (singular quadric)11) u20 + u 1 − u22 = 0 (56) in homogeneous coordinates which cover the whole CP2. This conic remains regular at EPs which merely correspond to configurations with u2 = 0. In terms of (χT , c−1)−notations it reads χTχ− c−2 = 0 . (57) It is clear that the conic construction is straight forwardly extendable to Hilbert space embeddings H = Cn →֒ CPn of any dimension n. We arrive at the conclusion that the appropriate state space for open quantum systems in an instantaneous setting will be related to the projective extension CPn of the original Hilbert space H = Cn with states identified with conics k=0 u k − u2n = 0. This is in contrast to Hermitian systems where it is sufficient to project the Hilbert space H∗ = Cn − {0} down to the base space CPn−1, i.e. π : H∗ → P(H∗) ≈ CPn−1. In non-Hermitian setups each fiber π(Φ) × C∗ should be supplemented by ∞. This suggests to extend them to π(Φ) × CP1. From the above construction we see that the singular behavior with regard to the two affine charts is only related to the scale factors c ∈ CP1, whereas π(Φ) behaves smoothly and regular. On its turn, this suggests to reconsider the model dependent physical interpretation of the eigenvector self-orthogonality (isotropy) and the corresponding diverging or non-diverging sensitivity in perturbation expansions like in [51, 52] as result of divergent or non-divergent normalization constants. The Hilbert space extension H = C2 →֒ CP2 together with the observed simultaneous scaling of the whole root space Sλ obtained in section 3, the upper and lower triangular (parabolic subgroup type) structure of the Sλ−related matrices in (A.7), (A.8) and the parabolic subgroup structure (41) at EPs provides one more indication that the natural structure at EPs is connected with projective flags 10) A projective space CPn ∋ (z0, z1, . . . , zn) is covered by n + 1 affine charts Uk ∋ , . . . , , . . . , ) with zk 6= 0 (see, e.g., [59, 61]) in straight forward dimensional extension of the two-chart covering of the Riemann sphere CP1 mentioned in section 3. 11) For conics and quadrics in projective spaces see, e.g. [59, 62, 68]. Projective Hilbert space structures 13 [63]. A study of Jordan chain related flag bundles and the mappings between their complementary affine charts will be presented in [64]. Returning to the ε → 0 limit in (49) we see that → −1 =⇒ c+ =⇒ Φ+ → ±iΦ− , (58) i.e. the two eigenvectors (fiber sections) Φ+, Φ− are phase-shifted one relative to the other by ±i when tending to their common coalescence line at ε = 0: π(Φ+) = π(Φ−) = π(Φ0). We note that this relative ±i phase-shift of the vectors Φ+, Φ− is generic for models in their instantaneous picture and with d± = c± and normalization 〈Ξ±|Φ±〉 = 1. A further result which immediately follows from (49) is the typical distance- dependent phase jump behavior in the vicinity of the EP. In a sufficiently close vicinity of an EP (|ε| ≪ 1) any sufficiently smooth trajectory in an underlying parameter space can be roughly approximated by a straight line segment with an effective parametrization of the type ε = eiα0(ρ+ is), s ∈ [−s0, s0] ⊂ R (59) where α0 =const fixes the direction orthogonal to the effective trajectory in the complex ε−plane and ρ is the minimal distance ρ = |ε(s = 0)| of this trajectory to the EP. The parameter along the path is s ∈ [−s0, s0] ⊂ R. This parametrization gives: [ε(s)] = e−i −iθ(s)|ε(s)|−1/4 |ε(s)| = ρ2 + s2 θ(s) = arctan(s/ρ) ∈ (−π/8, π/8) (60) and we observe that the minimal distance ρ between the parameter trajectory and the EP defines the smoothness of the phase changes. The closer the path approaches the EP the more it will take the form of a Heaviside step function with jump height π/4: θ(s; ρ → 0) → π Θ(s)− 1 . (61) The phase jump behavior can be used as implicit indicator of a possible close location of an EP — a fact especially useful in numerical studies of systems with complicated parameter dependence, but where phases of eigenvectors can be easily extracted. Jumps ±π/4 of wave function phases have been observed numerically in [69] for the model Hamiltonian (1) and in [14] for the special case of a small quantum billiard. According to these results, the phases of the components change smoothly (as a function of a certain control parameter) in approaching the EP and jump by π/4 at the smallest distance from this point. Other phase jump values are possible, but require especially tuned paths. 6. Phase rigidity In numerical studies of man-made open quantum systems depending in a complicated way on several parameters X = (X1, . . . , Xm) ∈ M it is usually important to know how close a given configuration is located to an EP. EPs dominate the system behavior Projective Hilbert space structures 14 also in their vicinities, spectral bands may merge at EPs [51] or the transmission properties of quantum dots (QDs) may become optimal at EPs [70]. A measure for the distance between a given point in parameter space and a closely located EP would provide a convenient tool for adjusting and tuning parameters so that a system may be ’moved’ in parameter space toward to or away from this EP. In [70] it has been shown numerically that within the instantaneous picture (Φ = Ξ∗) an appropriate measure for the detection of EP vicinities is the fraction 〈Φ|Φ〉 . (62) We note that originally similar fractions have been introduced in [71] to describe the transitions between Hamiltonian ensembles with orthogonal and unitary symmetry in Hermitian quantum chaotic systems. There the square modulus |r|2 was dubbed ”phase rigidity”. In our considerations of non-Hermitian systems we use this term in loose analogy for r itself. Decomposing Φ into real and complex components Φ = Φr + iΦi we find from the normalization that ΦTΦ = 1 = ΦTr Φr − ΦTi Φi , ΦTr Φi = 0 (63) and12) hence that the norm is bounded below ||Φ||2 = 〈Φ|Φ〉 = ΦTr Φr +ΦTi Φi = 2ΦTi Φi + 1 ≥ 1 . (64) The phase rigidity can be expressed as ||Φ||2 ∈ [0, 1] (65) where according to (50) for the EP-limit ε → 0 holds r ≈ |2ε|1/2 → 0 . (66) The opposite limit r → 1 is reached when the channel coupling ω in the Hamiltonian (1) vanishes, i.e. when the interaction between the two decaying resonance states tends to zero and any eigenvector can be taken purely real-valued in the instantaneous picture. Finally, we note that for certain quantum dot systems the phase rigidity r is closely related to the transmission properties of these systems. The capability of corresponding numerical studies (including the visualizations of transmission and phase rigidity ’landscapes’ over parameter space) has been recently demonstrated in [70]. 7. PT −symmetric models Toy model Hamiltonians of 2× 2−matrix type have been often used as test ground in PT −symmetrically extended Quantum Mechanics (PTSQM) [33, 34, 35]. They can be obtained from non-Hermitian complex symmetric 2 × 2−matrix Hamiltonians by 12) In equation (63) it can be set ΦTr Φr =: cosh 2 β and ΦTi Φi =: sinh 2 β. This hyperbolic structure shows analogies with the mass shell condition E2 − p2 = m2 of special relativity. The EP-limit ΦTr Φr,Φ i Φi → ∞ corresponds, e.g., to the light-cone limit where the vectors become isotropic — a fact which seems to play an important role in connection with the conjectured Hilbert space worm holes [34] related to the brachistochrone problem of PT −symmetric Quantum Mechanics (PTSQM). Projective Hilbert space structures 15 imposing a PT −symmetry constraint. In a suitable parametrization they have the reiθ s s re−iθ , r, s, θ ∈ R (67) and commute with the operator PT [PT , H ] = 0, P = . (68) Here, P is the parity reflection operator and T — the time inversion (acting as complex conjugation). The eigenvalues of H are E± = r cos(θ)± s2 − r2 sin2(θ) (69) and the corresponding eigenvectors can be represented as [35] |E+〉 = eiα/2 2 cos(α) =: c+χ+ |E−〉 = ie−iα/2 2 cos(α) =: c−χ− (70) where sin(α) = sin(θ) . (71) With regard to the indefinite (Krein space type [38]) PT inner product (u, v) = PT u·v the vectors are normalized as (E±, E±) = ±1, (E±, E∓) = 0 . (72) The indefinite PT inner product is then mapped by the dynamical operator C with [C, H ] = 0 and C = 1 cos(α) i sin(α) 1 1 −i sin(α) (see, e.g., [35]) into the positive definite (Hilbert space type) CPT inner product ((u, v)) = CPT u · v which yields ((E±, E±)) = 1, ((E±, E∓)) = 0 . (74) Let us now reshape the model in terms of the EP-relevant notations of section 2. A simple comparison of (1), (3) with (67), (71) shows that Z = i sin(α) (75) and, hence, that C = 1 cos(α) and that the model is actually one-parametric with essential parameter Z. Together with (2) the parametrization (76) leads to a representation of the Hamiltonian (67) as H = E0I2 + s cos(α)C , E0 = r cos(θ) (77) and [C, H ] = 0 is fulfilled trivially. The compatibility of the PT and the CPT inner products (72), (74) with the bi- orthogonality relations (7) is ensured by the fact that for an eigenvector Φ = c(1, b)T Projective Hilbert space structures 16 exact PT symmetry requires PT Φ ∝ Φ and, hence, c∗b∗(1, 1/b∗)T ∝ c(1, b)T so that |b|2 = 1. For such vectors it holds PT Φ ∝ Ξ∗ and due to the dynamically tuned C also CPT Φ ∝ Ξ∗. As result one finds CPT Φk · Φl ∝ PT Φk · Φl ∝ Ξ+k Φl and full compatibility of the bi-orthogonality with the PT and CPT inner products is established. From (75) we see that possible EPs are solely defined by the value of α. From Zc = ±i we find the corresponding critical αc as αc = ±π/2 + 2Nπ, N ∈ Z . (78) Furthermore, it follows from (71) that a purely Hermitian model with θ = nπ, n ∈ Z corresponds to α = Nπ, N ∈ Z. Exact PT −symmetry is preserved for α ∈ R− {π/2+ πZ}, and the corresponding models are parameterized by elements Z belonging to the purely imaginary straight line segment connecting the two EPs, i.e. by Z ∈ (−i, i), ReZ = 0. According to (70), at the EPs the eigenvectors lay on the same line π(|E+〉) = π(|E−〉) ≈ χ0 = (1, Zc)T and their norms diverge for α → αc like |||E±〉||2 = 〈E±|E±〉 ≈ | cos(α)| → ∞ . (79) The operator C in (73) shows the same singular behavior, i.e. the C−induced mapping between the Krein space and the Hilbert space breaks down at the EPs. In analogy to the singularity resolution presented in section 5 we may map the vectors |E±〉 ∈ C2 into elements from the affine chart U2 ⊂ CP2 corresponding to points e± ∈ CP2 with homogeneous coordinates |E±〉 7→ e± = χT±, c . (80) The original normalization via PT inner product PT |E±〉 · |E±〉 = 1 acts then as generalized conic PT χ± · χ± − T c−1± c−1± = 0 (81) which remains regular in the EP-limit α → αc, but shows the typical EP-related self- orthogonality (isotropy) of the lines PT χ± ·χ± → 0. Again we arrive at the conclusion that the original Hilbert space H = C2 should be projectively embedded into CP2 in order to accommodate EP-related singularities. Finally, we note that the recently uncovered solutions of the PT −symmetric brachistochrone problem with vanishing optimal passage time [34] occurs for α = π/2 what according to (78) can be identified as an EP-regime13). This fact appears compatible with the results of [57] where a vanishing passage time was reported for arbitrary non-Hermitian Hamiltonians. In this regard it is natural to conjecture that a vanishing optimal passage time might be a generic EP-related feature of non-Hermitian systems not necessarily restricted to PTSQM models. 8. Conclusion In the present paper we considered projective Hilbert space structures in the vicinity of EPs. Starting from a leading-order Puiseux-expansion of the bi- orthogonal eigenvectors of a non-Hermitian (complex symmetric) diagonalizable 2×2−matrix Hamiltonian in terms of root vectors (algebraic eigenvectors) at an EP the 13) The corresponding state vector alinement without link to EPs was observed also in [72]. Projective Hilbert space structures 17 normalization divergency of the eigenvectors in the EP-limit has been parameterized. It has been shown that the natural projective line structure related to the eigenvectors of the diagonal Hamiltonian has to be replaced at an EP by a higher dimensional projective structure in which all the root vectors of a Jordan block scale simultaneously with the same single factor. For a simplified setup with left eigenvectors equated to their complex conjugate right counterparts, the normalization divergency has been resolved by embedding the original Hilbert space H = C2 into its projective extension H →֒ CP2. Eigenvectors normalized according to the diagonalizable Hamiltonian and eigenvectors with a normalization inherited from the root vector normalization live then merely in different (complementary) affine charts of CP2. The states themselves can be interpreted as conics in CP2. The line structure of the states behaves smoothly and independently of these charts and their possibly singular transition functions. This indicates on the possibility of a technically efficient description of the global behavior of the non-Hermitian system by factoring the eigenvectors in globally smoothly varying non-singular projective line components π(Φk) and possibly diverging scale factors With the help of the Puiseux expanded eigenvectors it has been shown that the geometric phase obtained on circles around EPs of complex symmetric Hamiltonians is purely real-valued and that the corresponding monodromy transformations are induced by an Abelian parabolic subgroup of SL(2,C). Furthermore the Puiseux expansion has been used to explain phase jumps which in prior work had been numerically observed in the vicinity of EPs. An analytical foundation for the usefulness of the phase rigidity as distance measure to EPs has been provided. Finally, a PT −symmetric model has been studied. It has been shown that the EP-related singularities show up not only in the normalization conditions of the eigenvectors but also in the dynamical symmetry operator C. The normalization singularity has been resolved via a projective extension of the original Hilbert space. From the singularity structure it has been conjectured that the zero passage time effect in the brachistochrone problem of non-Hermitian Hamiltonians might be a generic EP- related artifact. Acknowledgement. We thank Hugh Jones and Andreas Fring for useful comments on [34, 57]. This work has been supported by the German Research Foundation DFG, grant GE 682/12- 3 (U.G.) and by the grants RFBR-06-02-16719, SS-5103.2006.2 (B.F.S.). Appendix A. Jordan normal forms for complex symmetric 2× 2 matrices At the EPs with Zc = ±i =: µi the matrix H(Zc)− E0I2 = ω 1 −Zc =: M (A.1) is related to its Jordan normal form J2(0) = by a similarity transformation M = PRJ2(0)R −1P−1 . (A.2) 14) The question concerning the physical interpretation of diverging or non-diverging normalizations and the corresponding diverging or non-diverging sensitivity in perturbation expansions is highly model dependent (see e.g. [51, 52]) and still requires a detailed investigation. Projective Hilbert space structures 18 From the symmetry properties M = MT , J2(0) = S2J 2 (0)S2 (A.3) with S2 = and P 2 = S2 one finds 1 −iµ −iµ 1 , P = PT = (P−1)+ (A.4) 0 q−1 , q := 2ω . (A.5) The elementary Jordan block J2(0) has right and left root vectors Θ0,Θ1 and Ψ0,Ψ1 satisfying J2(0)Θ0 = 0, J2(0)Θ1 = Θ0 JT2 (0)Ψ0 = 0, J 2 (0)Ψ1 = Ψ0 . (A.6) The explicit solutions of these Jordan chains can be arranged as Toeplitz and Hankel matrices Θ = [Θ0,Θ1] = c0 c1 , Ψ = [Ψ0,Ψ1] = 0 d∗0 d∗0 d (A.7) Ψ̃ := ΨS2 = d∗0 0 d∗1 d . (A.8) From the simplest realization of the bi-orthonormality condition Ψ+Θ = S2, Ψ̃ I2 one finds the parameters c1 = d1 = 0, d0c0 = 1. 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0704.1292

Excitation of the dissipationless Higgs mode in a fermionic condensate

Excitation of the Dissipationless Higgs Mode in a Fermionic Condensate R. A. Barankov1 and L. S. Levitov2 Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801 Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139 The amplitude mode of a fermionic superfluid, analogous to the Higgs Boson, becomes undamped in the strong coupling regime when its frequency is pushed inside the BCS energy gap. We argue that this is the case in cold gases due to the energy dispersion and nonlocality of the pairing interaction, and propose to use the Feshbach resonance regime for parametric excitation of this mode. The results presented for the BCS pairing dynamics indicate that even weak dispersion suppresses dephasing and gives rise to persistent oscillations. The frequency of oscillations extracted from our simulation of the BCS dynamics agrees with the prediction of the many-body theory. The observation of resonance superfluidity in cold atomic Fermi gases [1, 2] at magnetically tunable Fes- hbach resonances [3] opened new avenue of exploring the many-body phenomena. Similar to the earlier work on cold Bose gases which triggered studies of fascinating col- lective phenomena [5, 6], fermionic pairing at Feshbach resonances [7, 8, 9, 10, 11, 12] presents new opportunities. In particular, the high degree of coherence of trapped atoms, and the possibility to control particle interaction in situ on the shortest collective time-scale, the inverse Fermi energy [4], can facilitate exploring new regimes which are difficult to realize in solid state systems. The theory of fermionic pairing predicts two princi- pal collective modes intrinsic to the condensed state. One is the massless Bogoliubov-Anderson mode related to the order parameter phase dynamics. Being a Gold- stone mode, it manifests itself in hydrodynamics in the same way as in Bose systems, and was probed recently in the experiments on gas expansion and oscillation in traps [13, 14, 15, 16]. In addition, there exists a second fundamental elementary excitation [17, 18, 19], related to the dynamics of the order parameter modulus |∆|. No- tably, this excitation is unique to fermionic pairing and has no counterpart in Bose systems [19]. This massive excitation, characterized by a finite frequency, is anal- ogous to the Higgs Boson in particle physics. Like the latter it remained elusive, for a long time evading direct probes, although some indirect manifestations have been discussed [20, 21]. The main obstacle to the detection of the Higgs mode in superconductors is that it is es- sentially decoupled from the phase mode responsible for hydrodynamics and superfluidity. In this work we propose to use the dynamical con- trol of pairing interaction demonstrated in Refs.[1, 2] for parametric excitation of the Higgs mode. We argue that fermion superfluidity in the strong coupling regime real- ized near Feshbach resonance represents a distinct ad- vantage, since in this case the Higgs mode is pushed inside the superconducting gap, h̄ω < 2∆, which elim- inates damping due to coupling to quasiparticles. We demonstrate that this mode can be excited by a time- dependent pairing interaction, as illustrated in Fig. 1. In FIG. 1: Non-decaying Higgs mode in a Fermi gas with energy-dependent pairing interaction excited by the interac- tion switching from gi at t < 0 to g at t > 0. Shown are the time and energy dependence of the pairing amplitude (a), the x-component of the pseudospin vector (b), and the pairing amplitude at the Fermi energy (c) as obtained from the model (2),(3) at g = 0.43, gi = 0.23, a1 = a2 = 0.5, γ/W = 0.01, ∆F /W = 0.016. Note the initial transient of few periods, ex- hibiting some dephasing in pseudospin dynamics (b), followed by synchronized collective oscillations of fermion states. contrast, the BCS theory at weak coupling predicts the Higgs mode frequency right at the edge of the quasiparti- cle continuum, h̄ω = 2∆ [19], which leads to collisionless damping of this mode [22, 28]. The departure from the behavior at weak coupling arises from the change in the character of pairing inter- action in the strong coupling regime, in particular due http://arxiv.org/abs/0704.1292v1 to its finite spatial radius and frequency dispersion. Spa- tial nonlocality of pairing interaction is known to lead to discrete collective modes inside the BCS gap [17, 18]. Similarly, the energy dispersion of the pairing interac- tion and pairing amplitude ∆p that becomes prominent at strong coupling [23], leads to discrete collective mode spectrum (see below). While the exact form of this dis- persion is sensitive to the specifics of the strong cou- pling problem, it is established in the literature that, generally, both effects can occur near Feshbach reso- nance [8, 10, 11, 12, 24]. Although our understanding of the detailed microscopic picture may be hampered by the nonpertubative nature of the strong coupling prob- lem, we shall see that within a simplified model used be- low the inequality h̄ω < 2∆ is fulfilled under very general conditions. The dissipationless BCS dynamics [22, 25] and the pos- sibility to realize it in cold gases [26] attracted much at- tention recently [24, 27, 28]. These investigations, with the exception of Ref. [24], focused on the case of pair- ing interaction which is constant in the entire fermion energy band, concluding [29, 30] that several interesting dynamical states, synchronized and desynchronized (or dephased), can be realized by a sudden change in the interaction strength (see the phase diagram in Ref. [29]). In contrast, as we shall see below, the dephased be- havior is suppressed in the strong coupling regime when due to the energy dispersion of the pairing interaction the Higgs mode falls inside the BCS energy gap. Un- der these conditions an undamped Higgs mode can be excited upon a sudden change in interaction. By analyz- ing the limit when the interaction dispersion disappears we show how the different regimes of Ref. [29] are recov- ered. This correspondence suggests an interpretation of the dephased oscillations discussed in Refs. [22, 29, 30] as a manifestation of the Higgs mode, algebraically de- phased at h̄ω = 2∆. We shall analyze the pairing dynamics in a spatially uniform system using the pseudospin representation [25] of the BCS problem in which spin 1/2 operators s± ± isy describe Cooper pairs (p,−p): H = − λpq(t)s , (1) where ǫp is the free particle spectrum. The interaction λpq(t) that models the energy dispersion at strong cou- pling is taken in the form of a sum of a dispersing and nondispersing parts λpq(t) = (a1 + a2fpfq) , fp = γ2 + ǫ2 , (2) where the dimensionless parameter g(t) specifies the in- teraction time-dependence, the constants a1,2 ≥ 0 satisfy a1 + a2 = 1, and νF is the density of states at the Fermi level. The second term in (2) features dispersion on the energy scale γ. Our motivation for choosing the model (2) was two-fold. Firstly, the form (2) is general enough to provide insight into the role of different features, such as the energy dispersion (which is controlled by the pa- rameter γ) and separability (which is absent unless a1 or a2 vanishes). Secondly, our numerical method utilized the rank two form of (2), allowing for substantial speedup that could not be implemented for a more general inter- action λpq. In addition, the model (2) is physically mo- tivated by the theory of BCS pairing in the simultaneous presence of a retarded and non-retarded interaction [31]. Within the mean-field approximation, the dynamical equations derived from Eq.(1) assume a Bloch form: = 2bp × rp, bp = −(∆ , ǫp), (3) where rp = 2〈sp〉 are Bloch vectors, and the effective magnetic field bp depends on the pairing amplitude ∆p. The latter is defined self-consistently: ∆p = ∆ + i∆y λpq(t) + iry . (4) The interaction time dependence of interest is a step-like change from the initial value gi to the final value g. With- out loss of generality, the phase of the order parameter can be chosen equal zero, allowing us to consider only the x-component of the pairing amplitude, ∆p = ∆ . As an initial state we take the paired ground state (0) = )2 + ǫ2 (0) = )2 + ǫ2 . (5) The equilibrium energy-dependent amplitude ∆p is de- termined by the self-consistency equation , (6) in which λpq is given by (2) with the parameter values gi and g for the initial and final state. The corresponding equilibrium pairing gap values, ∆i and ∆p, are found by numerically solving the integral equation (6). Through- out the paper we use the equilibrium value of the pairing gap at the Fermi level, ∆F , at the final coupling g as a natural energy scale to parameterize the dynamics. We integrate Eqs.(3) using the Runge-Kutta method of the 4-th order with a time step adjusted to achieve sufficient precision of the calculation. In our simulation we use N = 104, 105 equally spaced energy states within bandwidthW , −W/2 < ǫp < W/2, with the level spacing much smaller than all other energy scales in the problem. We analyze the quantity ∆F (t) which at long times oscillates between the maximum and minimum values −2 −1 0 1 Inverse coupling 1/g−1/g 0 0.25 0.5 )−1/2 FIG. 2: Long-time behavior of ∆F (t), the pairing amplitude at the Fermi level, oscillating between ∆+ and ∆−. Shown are two examples of ∆± as a function of the initial state for a non- separable (circles) and a separable (squares) interaction (2). Parameters used: a1 = a2 = 0.5, g = 0.43, ∆F/W = 0.016, γ/W = 0.01, and a1 = 0, a2 = 1, g = 0.61, ∆F /W = 0.005, γ/W = 0.01, respectively. Inset: Linear fit of a sample trace ∆F (t) vs. t −1/2 used to extract ∆±. ∆+ and ∆−. To find the asymptotic values ∆± we em- ploy the numerical procedure sketched in Fig. 2 inset: ∆± are obtained from the linear fits to the maxima and minima of ∆F vs. t −1/2 intersection with the y-axis. The t−1/2 time parameterization is motivated by the de- phasing law δ∆(t) ∝ t−1/2 found in Refs.[22, 28] for the energy-independent interaction. Should the dephasing occur, the asymptotic values would coincide, ∆+ = ∆−. In contrast to the above, for the interaction (2) the dephased behavior is suppressed. Instead, as illustrated in Fig. 2, we observe non-decaying periodic oscillations for a wide range of initial states, both for the initial states close to the normal state (gi ≪ g) as well as for the initial states near equilibrium (gi ≈ g). At increasing gi there is a critical point at which the asymptotic pairing amplitude ∆± becomes zero. To understand the origin of the oscillatory behavior for the dispersive interaction, Eq.(2), we develop perturba- tion theory near the point gi = g. Linearizing the Bloch equations and taking a harmonic variation of the pairing amplitude, δ∆x,y (t) ∝ e−iωtδ∆x,y pω , we find two collective modes for the x and y components of ∆p corresponding to the order parameter amplitude and phase variation (see Ref.[19]). The amplitude (Higgs) mode with fre- quency ω obeys the integral equation λpqδ∆ − ω2/4 , (7) where ∆p is the equilibrium gap obtained from Eq.(6). The equation for δ∆y (the phase mode) is similar to Eq.(7) except for the denominator of the second frac- −2 −1.5 −1 −0.5 0 0.5 1 Inverse coupling 1/g−1/g 0.02 0.04 0.06 0.08 0.1 Interaction dispersion γ/W FIG. 3: (a): The Higgs mode frequency obtained from the simulation with gi ≈ g (circles) and from Eq.(7) (solid line) as a function of the dispersion parameter γ. The quasiparticle energy minimum (dashed line) lies above the collective mode frequency (parameters of the simulation: g = 0.61, a1 = a2 = 0.5). (b): Frequency of the Higgs mode as a function of the initial state for non-separable and separable interactions with the same parameters as in Fig.1. The frequency changes away from gi = g as the amplitude of oscillations increases (Fig.1), indicating unharmonicity of the Higgs mode. tion which is ǫ2 − ω2/4. As expected from Goldstone theorem, the equation for δ∆y is solved by ω = 0. To find the frequency ω of the Higgs mode, we note that for the interaction λpq given by (2), which is an op- erator of rank two, Eq.(7) turns into an algebraic equa- tion involving a 2×2 determinant. Solving it we find that for a2 > 0 the frequency ω lies within the BCS gap, as il- lustrated in Fig.3a. To gain more insight, let us consider a separable interaction, a1 = 0, a2 = 1, which yields − ω2/4 , (8) where ∆q ∝ fq. Balancing the factors under the sum in order to obtain unity on the left hand side, and noting that without the second factor Eq.(8) would be identical to Eq.(6), it is easy to see that ω < 2∆F , i.e. the Higgs mode is discrete. Notably, as Fig.3a illustrates, the frequency obtained from Eq.(7) coincides with the frequency of oscillations in ∆F (t) obtained by simulating BCS dynamics at g ≈ gi, proving that the observed excitation is indeed the Higgs mode. Furthermore, for g away from gi the frequency extracted from ∆F (t) varies with g, decreasing below the value at g ≈ gi and approaching zero at gi ≪ g and gi ≫ g (see Fig.3b). This indicates unharmonicity of the Higgs mode that sets on at a large amplitude of oscillations. To test these ideas further, we considered the regime when the Higgs mode is strictly inside the quasiparticle continuum, which can be realized in the model (2) with −3 −2 −1 0 1 2 3 Inverse coupling 1/g−1/g Onset of dephasing FIG. 4: Quenching of dephasing for weakly dispersing in- teraction. Asymptotic values of the pairing amplitude ∆± for a dispersing (circles) and non-dispersing (red line) inter- action for different initial states. The onset of dephasing is marked by arrows. Parameters used: a1 = a2 = 0.5, g = 0.33, ∆F /W = 0.02, γ/W = 0.1, and a1 = 1, a2 = 0, g = 0.33, ∆F /W = 0.05, respectively. the second term of a repulsive sign, a2 < 0. In this case Eq.(7) has no real-valued solution in the region ω ≤ 2∆. Simulating the BCS dynamics near gi ≈ g we find that ∆(t) exhibits exponentially decaying oscillations of the form e−ηt cos(ω′t+φ) corresponding to a complex-valued frequency ω. For a2 = 0 the collective mode frequency ω = 2∆F lies at the edge of the quasiparticle continuum. This property was linked to algebraic Landau damping of this mode in Refs.[22, 28]. The discrete Higgs mode makes the BCS dynamics un- damped for g near gi even for weakly dispersing interac- tion λpq. It is interesting to connect this behavior to the dephased BCS dynamics found in the case of constant in- teraction. This is illustrated (Fig.4) by the dynamics at weakly dispersing interaction γ ≫ ∆F , where we observe that the region of dephased dynamics shrinks, with the onset of dephasing shifting towards small g < gi. While the oscillation amplitude 1 (∆+−∆−) is now finite, it re- mains small due to dephasing in the transient region (see Fig.1b). This behavior is consistent with the Higgs mode approaching the quasiparticle continuum boundary. In conclusion, we have shown that the energy disper- sion of pairing interaction leads to quenching of dephas- ing of the BCS dynamics, making the Higgs mode of the pairing amplitude discrete. Parametric control of inter- action in the strong coupling regime near a Feshbach res- onance of cold atoms can be used to excite this mode. This research was supported in part by the National Science Foundation under Grant No. PHY05-51164. [1] C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92, 040403 (2004). [2] M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman, and W. Ketterle, Phys. Rev. Lett. 92, 120403 (2004). [3] E. Timmermans, P. Tommasini, M. Hussein, and A. Ker- man, Physics Reports 315, 199 (1999). [4] M. W. Zwierlein, C. H. Schunck, C. A. Stan, S. M. F. Raupach, and W. Ketterle, Phys. Rev. Lett. 94, 180401 (2005). [5] F. Dalfovo, S. Giorgini, L. P. 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Gaebler, C. A. Regal, and D. S. Jin, Phys. Rev. Lett. 97, 220406 (2006). [17] V. G. Vaks, V. M. Galitskii, A. I. Larkin, Zh. Eksp. Teor. Fiz. 41, 1655 (1961) [Sov. Phys. JETP 14, 1177 (1962)]. [18] A. Bardasis and J. R. Schrieffer, Phys. Rev. 121, 1050 (1961). [19] C. M. Varma, J. Low Temp. Phys. 126, 901 (2002). [20] B. I. Ivlev, Pis’ma v Zh. Eksp. Teor. Fiz. 15, 441 (1972) [JETP Lett. 15, 313 (1972)]. [21] P. B. Littlewood and C. M. Varma, Phys. Rev. B 26, 4883 (1982). [22] A. F. Volkov and Sh. M. Kogan, Zh. Eksp. Teor. Fiz. 65, 2038, (1973) [Sov. Phys. JETP 38, 1018 (1974)]. [23] G. M. Eliashberg, Zh. Eksp. Teor. Fiz. 38, 966 (1960) [Sov. Phys. JETP 11, 696 (1960)]. [24] M. H. Szymańska, B. D. Simons, and K. Burnett, Phys. Rev. Lett. 94, 170402 (2005). [25] R. A. Barankov, L. S. Levitov, and B. Z. Spivak, Phys. Rev. Lett. 93, 160401 (2004). [26] A. V. Andreev, V. Gurarie, and L. Radzihovsky, Phys. Rev. Lett. 93, 130402 (2004); R. A. Barankov and L. S. Levitov, Phys. Rev. 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0704.1293

Two characterizations of crooked functions

Two characterizations of crooked functions Chris Godsil∗ Aidan Roy† October 26, 2018 Abstract We give two characterizations of crooked functions: one based on the minimum distance of a Preparata-like code, and the other based on the distance-regularity of a crooked graph. 1 Introduction Highly nonlinear functions over finite vector spaces have attracted much interest in the last several years, for both their applications to cryptography (see [8] for example) and their connections to a variety of different combinatorial structures. The functions that are furthest from linear are called perfect nonlinear; unfortunately, none exist for binary vector spaces, which are the most cryptographically useful. However functions do exist in several lesser categories of nonlinearity, such as almost perfect nonlinear, almost bent, and crooked. We focus on the latter, which is the most specialized of the three. Crooked functions were introduced by Bending and Fon Der Flaass [2], who, build- ing on the graphs of de Caen, Mathon and Moorhouse [9], showed that every crooked function defines a distance-regular graph of diameter 3 with a particular intersection array. Shortly thereafter, van Dam and Fon Der Flaass [14] observed that every crooked function defines a binary code of minimum distance 5, similar to the classical Preparata code. In this paper, we show that the converse of each of these results is also true: crooked functions can be characterized using both Preparata-like codes (Theorem 3) and distance-regular graphs (Theorem 5). Those codes and graphs offer a more combi- natorial way of understanding the nature of nonlinear binary functions. 2 Almost Perfect Nonlinear Functions Before considering crooked functions we need to characterize a more general class, namely almost perfect nonlinear functions. Throughout this article, let V := V (m, 2), Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON N2L 3G1. email: [email protected]. CG is supported by NSERC. Institute for Quantum Information Science, University of Calgary, Calgary, AB, T2N 1N4. email: [email protected]. AR is supported by NSERC and MITACS. http://arxiv.org/abs/0704.1293v1 a vector space of dimension m over F2, with m odd. Given a function f : V → V , consider the following system of equations: x+ y = a f(x) + f(y) = b . (1) Note that solutions to (1) come in pairs: if (x, y) is a solution, then so is (y, x). If f is a linear function, then equation (1) has 2m solutions when b = f(a). We say f is almost perfect nonlinear if, for every (a, b) 6= (0, 0), the system has at most two solutions. Equivalently, f is almost perfect nonlinear if and only if for all a 6= 0 in V , the set Ha(f) := {f(x) + f(x+ a) | x ∈ V } has cardinality 2m−1. We may construct a binary code from a function on V in the following manner. Identify V with the finite field F2m , and let α be a primitive element of F2m . Also let n = 2m− 1, and assume f : V → V is a function such that f(0) = 0. We define a parity check matrix Hf by Hf := 1 α α2 . . . αn−1 f(1) f(α) f(α2) . . . f(αn−1) and define the code Cf to be the kernel of Hf over F2. The code Cf can be thought of as a generalization of the double error-correcting BCH code, which is the specific case of f(α) := α3. It is clear from the parity check matrix that the minimum distance of Cf is at least 3, and it can be shown that the minimum distance is at most 5. The following characterization is due to Carlet, Charpin, and Zinoviev [7, Theorem 5]. Theorem 1. The minimum distance of Cf is 5 if and only if f is almost perfect nonlinear. In this case, the dimension of Cf is k = 2m − 2m− 1. In the next section, we give a similar characterization of crooked functions, which are a special class of almost perfect nonlinear functions. 3 Crooked Functions and Preparata-like Codes A function f : V → V is crooked if the following three conditions hold: 1. f(0) = 0; 2. f(x) + f(y) + f(z) 6= f(x+ y + z) for distinct x, y, and z; 3. f(x) + f(y) + f(z) 6= f(x+ a) + f(y + a) + f(z + a) for all a 6= 0. Condition 2 is equivalent to almost perfect nonlinearity; thus every crooked function is almost perfect nonlinear. Condition 3 states that for every a 6= 0, no three points in Ha(f) are collinear. It follows that f is crooked if and only if f(0) = 0 and Ha(f) is the complement of a hyperplane for all a 6= 0. Note that we are using the original definition of crooked functions given in [2], rather than the generalization appearing in Byrne and McGuire [6] or Kyureghyan [12]. The canonical example of a crooked function is the Gold function. Identify V with F2m for odd m; then f(x) := x +1 is called a Gold function if gcd(k,m) = 1. More gen- erally, f(x) := x2 is crooked provided that gcd(k − j,m) = 1, and Kyureghyan [12] has shown that all crooked power functions have this form. For recent progress in con- structing nonlinear functions which are not equivalent to the Gold functions, see [4, 5, Just as almost perfect nonlinear functions give rise to BCH-like codes, crooked func- tions give to Preparata-like codes. Given f : V → V such that f(0) = 0, let Pf be the code whose codewords are the characteristic vectors of (S, T ), for S ⊂ V ∗ and T ⊂ V , such that the following three conditions hold: • |T | is even, r, and f(r) + f(r). Identifying V with F2m , we get the actual Preparata code when f(x) := x 3 and the generalized Preparata code when f(x) = x2 +1 (see [1]). In general Pf is not linear, and it is easy to verify that Pf always has minimum distance at least 3. The following result is due to Van Dam and Fon Der Flaass [14, Theorem 7]. Theorem 2. If f is crooked, then Pf has minimum distance 5 and size 2 −2m−2. If Pf has minimum distance 5, then it is nearly perfect: it satisfies the Johnson bound [13, Theorem 17.13] with equality. Hence Pf has minimum distance at most 5 for any f . We show the converse of Theorem 2. Theorem 3. If Pf has minimum distance 5, then f is crooked. Proof. We assumed in the definition of Pf that f(0) = 0, so condition 1 of crookedness is satisfied. If Pf has minimum distance 5, then there is no pair (φ, T ) in Pf with |T | = 4. That is, for any distinct w, x, y, z such that w + x+ y + z = 0, f(w) + f(x) + f(y) + f(z) 6= 0. (2) Thus condition 2 of crookedness is also satisfied, and it remains to show condition 3. Since condition 2 is saitsfied, f is almost perfect nonlinear and Cf has dimension 2 2m− 1 by Theorem 1. But Cf is the kernel of Hf , so it follows that Hf has a column space of dimension 2m, namely V × V . This implies that for any (a, b) in V × V , there is a subset S of V ∗ such that = (a, b). (3) Given any x ∈ V , let T = {x, 0}, so that |T | is even and r = x. Then from equation (3), there exists some S ⊂ V ∗ such that = (x, 0) . Choosing S in this way, (S, T ) is in Pf . Now given any y, z and a 6= 0, consider (S′, T ′) := (S ⊕ {y} ⊕ {y + a}, T ⊕ {z} ⊕ {z + a}). This vector is at distance 4 from (S, T ). Since Pf has distance 5, (S ′, T ′) must not be in Pf . But |T ′| is even, and r∈T ′ hence for (S′, T ′) /∈ Pf it must be the case that f(r) + r∈T ′ f(r). This implies f(x+ a) 6= f(r) + f(y) + f(y + a) + f(r) + f(z) + f(z + a), or in other words f(x+ a) 6= f(y) + f(y + a) + f(x) + f(z) + f(z + a). Thus condition 3 of crookedness is satisfied for f . 4 Crooked Graphs As usual, assume f(0) = 0. Define the crooked graph of f , denoted Gf , to have vertex set V ×F2×V with the following adjacency condition: distinct (a, i, α) and (b, j, β) are adjacent if and only if α+ β = f(a+ b) + (i+ j + 1)(f(a) + f(b)). It is not difficult to show that any two vertices in the subset Fai := {(a, i, α) | α ∈ V } are at distance at least three, and that any two distinct subsets Fai and Fbj are joined by a perfect matching. It follows that Gf is a 2 m-cover of the complete graph K2m+1 , and each Fai is a fibre (for background on covers of complete graphs, see [11]). The following theorem is given by Bending and Fon-Der-Flaass [2, Proposition 13]. Theorem 4. If f is crooked, then Gf is an antipodal distance-regular graph with inter- section array {2m+1 − 1, 2m+1 − 2, 1; 1, 2, 2m+1 − 1}. For background on distance-regular graphs, see [3]. Again, we show the converse. Theorem 5. If Gf is distance-regular with intersection array {2m+1 − 1, 2m+1 − 2, 1; 1, 2, 2m+1 − 1}, then f is crooked. Proof. For convenience, consider the graph G′f which consists of Gf with a loop added to every vertex. This can be done by removing the restriction (a, i, α) 6= (b, j, β) from the adjacency condition of Gf . If Gf is distance-regular with a1 = 0 and c2 = 2, then G′ is a graph with the property that any two vertices at distance 1 or 2 have exactly two common neighbours. That is, for any two vertices (a, i, α), (b, j, β) such that (a, i) 6= (b, j), there are exactly two vertices (c, k, γ) such that α+ γ = f(a+ c) + (i+ k + 1)(f(a) + f(c)), (4) β + γ = f(b+ c) + (j + k + 1)(f(b) + f(c)). (5) We restrict our attention to the cases in which i = j, so that a 6= b. Adding (4) and (5) together, there are exactly two pairs (c, k) such that α+ β = f(a+ c) + f(b+ c) + (i+ k + 1)(f(a) + f(b)). Running over all values of α+ β, we see that for fixed (a, b, i), the multiset {f(a+ c) + f(b+ c) + (i + k + 1)(f(a) + f(b)) | c ∈ V, k ∈ F2} = {f(a+ c) + f(b+ c) | c ∈ V } ∪ {f(a+ c) + f(b+ c) + f(a) + f(b) | c ∈ V } (6) contains each element of V exactly twice. Now for some fixed c, consider f(a+ c) + f(b+ c). Letting c′ := c+ a+ b, we have f(a+ c) + f(b+ c) = f(a+ c′) + f(b+ c′). However, the value f(a + c) + f(b + c) only occurs twice in (6), so there is no third solution c′′ 6= c, c′ such that f(a+ c) + f(b+ c) = f(a+ c′′) + f(b+ c′′). In other words, letting x = a+ c, y = b+ c, and z = a+ c′′, we have f(x) + f(y) 6= f(z) + f(x+ y + z) for z 6= x, y. This is condition 2 of crookedness for f . Also because f(a+ c) + f(b+ c) has already occured twice in (6), there is no c′′ such that f(a+ c) + f(b+ c) = f(a+ c′′) + f(b+ c′′) + f(a) + f(b). Setting x = a+ c, y = a+ c′′, z = a and w = a+ b, we have f(x) + f(x+ w) 6= f(y) + f(y + w) + f(z) + f(z + w) for any x, y, z and w, with w 6= 0. This is the condition 3 of crookedness, so f is crooked. References [1] Ronald D. Baker, Jacobus H. van Lint, and Richard M. Wilson. On the Preparata and Goethals codes. IEEE Trans. Inform. Theory, 29(3):342–345, 1983. [2] T. D. Bending and D. Fon-Der-Flaass. Crooked functions, bent functions, and distance regular graphs. Electron. J. Combin., 5(1):Research Paper 34, 14 pp. (electronic), 1998. [3] A. E. Brouwer, A. M. Cohen, and A. Neumaier. Distance-Regular Graphs. Springer- Verlag, Berlin, 1989. [4] Lilya Budaghyan, Claude Carlet, Patrick Felke, and Gregor Leander. An infinite class of quadratic apn functions which are not equivalent to power mappings. In Information Theory, 2006 IEEE International Symposium on, pages 2637–2641, 2006. [5] Lilya Budaghyan, Claude Carlet, and Alexander Pott. New classes of almost bent and almost perfect nonlinear polynomials. IEEE Trans. Inform. Theory, 52(3):1141–1152, 2006. [6] Eimear Byrne and Gary McGuire. On the non-existence of quadratic apn and crooked functions on finite fields. 2005. http://www.maths.may.ie/staff/gmg/APNniceWeilEBGMG.pdf. [7] Claude Carlet, Pascale Charpin, and Victor Zinoviev. Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes Cryptogr., 15(2):125–156, 1998. [8] Florent Chabaud and Serge Vaudenay. Links between differential and linear crypt- analysis. In Advances in cryptology—EUROCRYPT ’94 (Perugia), volume 950 of Lecture Notes in Comput. Sci., pages 356–365. Springer, Berlin, 1995. [9] D. de Caen, R. Mathon, and G. E. Moorhouse. A family of antipodal distance- regular graphs related to the classical Preparata codes. J. Algebraic Combin., 4(4):317–327, 1995. [10] Yves Edel, Gohar Kyureghyan, and Alexander Pott. A new APN function which is not equivalent to a power mapping. IEEE Trans. Inform. Theory, 52(2):744–747, 2006. [11] C. D. Godsil and A. D. Hensel. Distance regular covers of the complete graph. J. Combin. Theory Ser. B, 56(2):205–238, 1992. [12] Gohar Kyureghyan. Crooked maps in finite fields. In 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb ’05), Discrete Mathe- matics & Theoretical Computer Science Proceedings, AE, pages 167–170, 2005. [13] F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-Correcting Codes. North-Holland Publishing Co., Amsterdam, 1977. [14] E. R. van Dam and D. Fon-Der-Flaass. Uniformly packed codes and more distance regular graphs from crooked functions. J. Algebraic Combin., 12(2):115–121, 2000. Introduction Almost Perfect Nonlinear Functions Crooked Functions and Preparata-like Codes Crooked Graphs

0704.1294

A Disciplined Approach to Adopting Agile Practices: The Agile Adoption Framework

Microsoft Word - Final_Agile07_revised2.doc A Disciplined Approach to Adopting Agile Practices: The Agile Adoption Framework Ahmed Sidky, James Arthur ([email protected], [email protected]) Virginia Tech Abstract Many organizations aspire to adopt agile processes to take advantage of the numerous benefits that it offers to an organization. Those benefits include, but are not limited to, quicker return on investment, better software quality, and higher customer satisfaction. To date however, there is no structured process (at least in the public domain) that guides organizations in adopting agile practices. To address this problem we present the Agile Adoption Framework. The framework consists of two components: an agile measurement index, and a 4- Stage process, that together guide and assist the agile adoption efforts of organizations. More specifically, the agile measurement index is used to identify the agile potential of projects and organizations. The 4-Stage process, on the other hand, helps determine (a) whether or not organizations are ready for agile adoption, and (b) guided by their potential, what set of agile practices can and should be introduced. 1. Introduction and Motivation Over the past few years organizations have asked the agile community “Why should we adopt agile practices?” [27]. The numerous success stories highlighting the benefits reaped by organizations that have successfully adopted agile practices provide an answer to this question [49] [41] [9] [8] [34] [32]. As a result, many organizations are now aspiring to adopt agile practices. Once again, however, they are turning to the agile community, but with a different question: “How do we proceed with adopting agile practices?” [27]. Unfortunately, there exists no structured approach (at least in the public domain) for agile adoption. The absence of guidance and assistance to organizations pursuing agility is the main problem addressed by this paper. A major factor contributing to this absence is the number of issues a structured approach must address when providing organizations with guidance for the successful adoption of agile practices. These include, among other issues, determining: (1) the organization's readiness for agility; (2) the practices it should adopt; (3) the potential difficulties in adopting them; (4) and finally, the necessary organizational preparations for the adoption of agile practices. The Agile Adoption Framework introduced in this paper, is an attempt to addresses the issues mentioned above by providing a structured and repeatable approach designed to guide and assist agile adoption efforts. It assists the agile community in supporting the growing demand from organizations that want to adopt agile practices. The Agile Adoption Framework, however, is only one essential ingredient, the other is an agile coach who knows how to apply that framework. Such a person can be an agile consultant hired to facilitate the process, or an in-house employee with sufficient training on agile methods and the use of the framework. The Agile Adoption Framework has two main components: (1) a measurement index for estimating agile potential, and (2) a 4-Stage process that employs the measurement index in determining which, and to what extent, agile practices can be introduced into an organization. Figure 1 illustrates the various components of the framework and the relationships among them. Measurement Index 4-Stage Process Stage 1: Identify Discontinuing Factors Agile Adoption Framework Agile Practices to Adopt Stage 2: Project Level Assessment The 5 Levels Agility populated with Agile Practices Figure 1. Overview of the Agile Adoption Framework Stage 4: Reconciliation Stage 3: Organizational Assessment The first component, the agile measurement index, is a scale the coach uses to identify the agile potential of a project or organization. The agile measurement index is used in the process component of the framework, which consists of four stages working together to guide organizations in identifying agile practices that best fit into their environment. The four stages are: • Stage 1: Identification of Discontinuing Factors. Discovers the presence of any showstoppers that can prevent the adoption process from succeeding. • Stage 2: Project Level Assessment. Utilizes the agile measurement index to determine the target level of agility for a particular project. • Stage 3: Organizational Readiness Assessment. Uses the agile measurement index to assess the extent to which the organization can achieve the target agility level identified for a project. • Stage 4: Reconciliation. Determines the final set of agile practices to be adopted by reconciling the target agile level for a project (from Stage 2) with the readiness of the embodying organization (from Stage 3). Section 2 presents the structure and details of the agile measurement index. Each of the four stages in the process is then presented in detail in Section 3. Section 4 presents industry feedback regarding the framework. Section 5 provides concluding remarks about the Agile Adoption Framework along with comments from the agile community. 2. Agile Measurement Index One of the concerns organizations have when seeking to adopt agile practices is determining how agile they can become [23]. The agile potential (i.e. the degree to which that entity can adopt agile practices) of projects and organizations is influenced by the circumstances surrounding them. To determine the agile potential the coach (or the one conducting the assessment) needs use a measurement index or scale that can assess the agility of an entity. The agile adoption framework refers to this scale as an agile measurement index. The Agile Adoption Framework uses the agile measurement index to determine the agile potential of projects and organizations. The measurement index is composed of four components: 1. Agile Levels 2. Agile Principles 3. Agile Practices and Concepts 4. Indicators. Sections 2.1 through 2.4 introduce each component of the agile measurement index. Section 2.5 focuses on issues related to the tailorability of the index. 2.1. Agile Levels Agile levels, as depicted in Figure 2a, are considered the units of the measurement scale as they enumerate the different possible degrees of agility for a project or organization. The agile potential of a project or organization is expressed in terms the highest agile level it can achieve. The attainment of a particular level symbolizes that the project or organization has realized and embraced the essential elements needed to establish a particular degree of agile effectiveness. For example, when the elements inherent to enhancing communication and collaboration are embodied within the development process, then the Agile Level 1 (Collaborative) is attainted. However, before one can expect to move to Level 2 status, all practices associated with Agile Level 1 must be achieved (or achievable). The 5 Levels of Agility are designed to represent the core qualities of the Agile Manifesto [2], rather than the Agile Principles A B C E D Agile Principles A B C E D (a) Empty Agile Levels (b) Empty Agile Levels with Agile Principles (c) Agile Levels populated with Agile Practices categorized within Agile Principles Agility Increases Figure 2. Components of the 5 Levels of Agility (Indicators are not shown) qualities related to any particular agile method. After careful analysis of the manifesto, five essential agile qualities have been identified. Those qualities comprise the 5 Levels of Agility that are used the agile measurement index: • Level 1: Collaborative. This level denotes the fostering of communication and collaboration between all stakeholders. The dimension of collaboration is the foundation of agile software development [45] [17] [18]. • Level 2: Evolutionary. Evolutionary development is the early and continuous delivery of software. It too is fundamental because every agile method assumes its presence [33]. • Level 3: Effective. The next quality an agile development process must embrace is that of developing high quality, working software in an efficient an effective manner. This quality is needed to prepare the development process so that it can respond to constant change without jeopardizing the software system being developed [29] [18]. • Level 4: Adaptive. This level constitutes establishing the agile quality of responding to change in the process. Defining and responding to multiple levels of feedback is essential in this level [26]. • Level 5: Ambient. The last level concentrates on establishing a vibrant environment needed to sustain and foster agility throughout an organization. Each of the agile levels is composed of a set of agile practices that introduce and sustain the agile quality pertinent to that level. The selection of agile practices and concepts assigned to each agile level is guided by the second component of the measurement index, agile principles. 2.2. Agile Principles Agile principles are the essential characteristics that must be reflected in a process before it is considered Agile. For example, two key agile principles are human centric, which refers to the reliance on people and the interaction between them, and technical excellence, which implies the use of procedures that produce and maintain the highest quality of code possible. The Agile Manifesto outlines 12 principles that characterize agile development processes [13]. After careful grouping and summarization, five agile principles emerge that capture the essence of the 12. These five principles guide the refinement or tailoring of the 5 Levels of Agility: • Embrace change to deliver customer value [12]. The success of a software development effort is based on the extent to which it helps deliver customer value. In many cases the development team, as well as the customer, are in a continuous learning process as to the requirements necessary to realize additional customer value. Hence, an attitude of welcoming and embracing change should be maintained throughout the software development effort. • Plan and deliver software frequently [13] [20] [38]. Early and frequent delivery of working software is crucial, because it provides the customer with a functional piece of the product to review and provide feedback on. This feedback is essential for the process of planning for upcoming iterations as it shapes the scope and direction of the software development effort. • Human centric [17]. The reliance on people and the interactions among them is a cornerstone in the definition of agile software processes. • Technical excellence [26] [31]. Agile developers are committed to producing only the highest quality code possible, because high quality code is essential in fast-paced development environments, such as the ones characterized as agile. • Customer collaboration [13]. Inspired from the original statement of the agile manifesto, there must be significant and frequent interaction between the customers, developers, and all the stakeholders of the project to ensure that the product being developed satisfies the business needs of the customer. In effect, agile principles are used to ensure that the agile levels embody the essential characteristics of agility. Figure 2b illustrates the relationship between agile levels and agile principles. Each agile level should contain agile practices associated with most, if not all, of the agile principles. The principle reflects the approach that the agile practice uses to promote the agile quality pertinent to a level. For example, all the practices in Level 3 (Effective) are promoting the agile objective of developing high quality, working software in an efficient an effective manner. How that objective is achieved though, is determined by the practices associated with agile principles spanning each level. Along the same lines, practices associated with the technical excellence principle will promote its agile objective by focusing on enhancing the technical aspect of the process, while practices associated with the human centric principle promote enhancing the human aspect of the process. The real essence of the 5 Levels of Agility, however, is in the agile practices it enunciates. The next section presents the third component of the agile measurement index – the agile practices. 2.3. Agile Practices Agile practices are concrete activities and practical techniques that are used to develop and manage software projects in a manner consistent with the agile principles. For example, paired programming, user stories, and collaborative planning are all agile practices. Since the agile levels are composed of agile practices (organized along the line of agile principles – see Figure 2c), they are considered the basic building block of the agile measurement index. The attainment of an agile level is achieved only when the agile practices associated with it are adopted. After surveying the agile methods currently used in industry [29] [31] [3], 40 distinct agile practices were chosen to populate the 5 Levels of Agility. These practices, arranged along the lines of the agile levels and principles, are illustrated in Table 1. (The underlining of the practices should be ignored at this point, but is discussed later in the paper.) Although a detailed discussion about each of the agile practices and concepts is outside the scope of this paper, the references associated with each are good starting points to learn more about them. Agile Principles Embrace Change to Deliver Customer Value Plan and Deliver Software Frequently Human Centric Technical Excellence Customer Collaboration Level 5 Ambient Establishing a vibrant environment to sustain agility Low process ceremony [33, 38] Agile project estimation [20] Ideal agile physical setup [33] Test driven development [11] Paired programming [48] No/minimal number of level -1 or 1b people on team [17, 15] Frequent face-to-face interaction between developers & users (collocated) [12] Level 4 Adaptive Responding to change through multiple levels of feedback Client driven iterations [33] Continuous customer satisfaction feedback [35, 42] Smaller and more frequent releases (4-8 weeks) [35] Adaptive planning [33] [20] Daily progress tracking meetings [6] Agile documentation [39, 31] User stories [21] Customer immediately accessible [15] Customer contract revolves around commitment of collaboration [26, 35] Level 3: Effective Developing high quality, working software in an efficient an effective manner Risk driven iterations [33] Plan features not tasks. [20] Maintain a list of all features and their status (backlog) [31] Self organizing teams [33, 38, 31, 18] Frequent face-to-face communication [38, 18, 13] Continuous integration [33] Continuous improvement (refactoring) [31, 12, 24, 5]. Unit tests [28] 30% of level 2 and level 3 people [17, 15] Level 2: Evolutionary Delivering software early and continuously Evolutionary requirements [33] Continuous delivery [33, 31, 26, 12] Planning at different levels [20] Software configuration management [31] Tracking iteration progress [33] No big design up front (BDUF) [4, 12] Customer contract reflective of evolutionary development [26, 35] Level 1: Collaborative Enhancing communication and collaboration Reflect and tune process [35, 42] Collaborative planning [38, 18, 33] Collaborative teams [45] Empowered and motivated teams [13] Coding standards [29, 47, 36] Knowledge sharing tools [33] Task volunteering [33] Customer commitment to work with developing team [13] Table 1. The 5 Levels of Agility populated with Agile Practices and Concepts 2.4. Indicators A set of indicators, or questions, must accompany each agile practice or concept in the measurement index. The agile coach uses these indicators (or questions) to measure the extent to which the organization is ready to adopt an agile practice or concept. The Goal Question Metric approach (GQM) [10] and the Objectives Principles Attributes (OPA) Framework [7] influence the approach used to devise the indicators for each practice. Each indicator is designed to measure a particular organizational characteristic necessary for the successful adoption of the agile practice to which the indicator is related. (This is the goal.) Depending on the question, a manager, developer, or the agile coach is designated to answer it, either subjectively or objectively. For example, assume the coach wants to determine the extent to which an organization is ready to adopt coding standards (Level 1, Technical Excellence). In this respect, two organizational characteristics that need to be assessed are: (1) to what extent do the developers understand the benefits behind coding standards, and (2) how willing are they to conform to coding standards. Several indicators (or questions) are used to assess each of these characteristics. For example, to assess the second (willingness), the assessor might ask the developers to what extent would they abide by coding standards even when under a time constraint. The 5 Levels of Agility contain approximately 300 different indicators for the 40 agile practices. A detailed listing of all the indicators associated with each agile level is found in the framework’s technical documentation [43]. The 5 Levels of Agility shown in Table 1 is one instance of the agile measurement index. Can there be, however, alternative instances? We address that issue in the next section. 2.5. Tailorability of the 5 Levels of Agility The 5 Levels of Agility, along with all their practices and indicators, were presented to members of the agile community. Several of its leaders encouraged us to consider factors that might lead to other instances of the 5 Levels of Agility. These factors are incorporating business values and reorganizing the practices based on experiential success. The two following subsections elaborate on these factors. 2.5.1 Incorporating Business Values. Business values refer to the added benefit realized by an organization after adopting agile practices. For most organizations, the achievement of these business values is the real incentive behind adopting agility. For example, decreasing time to market or increasing product quality are common business values that organizations hope to realize from adopting agile practices. Augustine [40] and Elssamadisy [22] have suggested that the levels of agility might be prioritized according to the business values an organization hopes to realize. This suggestion is both valuable and beneficial to the growth of the framework, because currently, the 5 Levels of Agility are not associated with any business values; instead they are based on the qualities and values of agility. The relationship between agile and business values is parallel to that between the Agile Manifesto (focusing on agile values) and the Declaration of Interdependence (capturing the business values) [2] [1]. 2.5.2 Reorganizing the Practices based on Experiential Success. The agile coaches and consultants Cockburn [16], Cohn [19], and Wake [46], in addition to others, suggest a reorganization of the agile practices based on experiential successes. That is, they advocate that the type of project and the experiences gained from previous adoption efforts can, and should, serve as a basis for formulating a better arrangement of the practices within the agile levels. For example, Cohn suggests that user stories be introduced in the first level of agility, because, from his experience, they enhance collaboration and communication between the stakeholders with regards to requirements. Others suggest that pair programming be in the first level because it helps establish collaboration within teams. This inability to reach a consensus on the position of agile practice emphasizes an important factor in providing guidance in an agile adoption effort: the adherence to agile principles when establishing the levels is paramount, not the positions of the actual practices. The intention behind the levels of agility is to provide a framework to guide the adoption process, not to dictate it Based on the above rationalizations we must conclude that a tailorable measurement index is both desirable and beneficial. However, when tailoring or creating another instance of an agile measurement index, it is important to observe the following guidelines to ensure that the new measurement index has all the necessary components and a valid structure: • Ensure that multiple levels exist. Levels are needed to enumerate the degrees of agility. Without levels, the power of the measurement index, when used in conducting comparative measurements of agility, is diminished. • The measurement index is based on practices and concepts. Foundational to the agile measurement index are agile practices and concepts. The extent to which agile practices and concepts can be adopted determines the agility of a process. • Each practice or concept has indicators. When introducing a new agile practice (other than the 40 identified) to the measurement index, it is important that the practice has an associated set of valid and sufficient indicators. Without indicators, there is no means by which an assessment can be conducted. The next section presents the second component of the Agile Adoption Framework – the 4-Stage Process. This component utilizes the 5 Levels of Agility (i.e., the agile measurement index) to provide structured guidance and assistance to organizations seeking to adopt agile practices. 3. The 4-Stage Process for Agile Adoption The 4-Stage assessment process is the “backbone” of the Agile Adoption Framework. As depicted in Figure 3, it first provides an assessment component that helps determine if (or when) an organization is ready to move toward agility, i.e., make the go/no-go decision. Secondly, the process guides and assists the agile coach in the process of identifying which agile practices the organization should adopt. The four stages are grouped according to the objective they help to achieve: • Objective 1: Make Go/No-Go Decision o Stage 1: Discontinuing Factors • Objective 2: Identify Agile Practices to Adopt o Stage 2: Project Level Assessment o Stage 3: Organizational Readiness Assessment o Stage 4: Reconciliation The next sections explain in detail how each stage of the 4-Stage process contributes to achieving its enunciated objectives. 3.1. Making the Go/No-Go Decision The first objective of the process is to provide organizations with a method for deciding whether or not to proceed with agile adoption initiatives. Since adopting agile practices is essentially a type of Software Process Improvement (SPI), a pre-assessment phase is needed before the decision to start the initiative is made. Traditionally, pre-assessments determine the ability of the organization to undertake an SPI initiative [25]. Organizations lacking the factors necessary for a successful SPI effort are considered “not ready.” In that situation the SPI effort is suspended until the missing factors can be mitigated. Similarly, with respect to agile adoption, pre- assessment helps identify factors in an organization that can prevent the successful adoption of agile practices. If such factors exist, the organization must eliminate them before continuing with the adoption effort. Pre- assessment processes like these are important because they save the organization time, money and effort by identifying missing or existing factors that can cause an SPI initiative to fail [30]. The next section describes how Stage 1 of the process guides and assists organizations in making Go/No-go decisions concerning the adoption of agile practices. This decision is determined by a pre-assessment activity that identifies any discontinuing factors. 3.1.1 Stage 1: Identifying Discontinuing Factors. The intent of Stage 1 is to provide an assessment process that identifies factors which could prevent the successful adoption of agile practices. These are called discontinuing factors, and can vary from one organization to another. Typically, they pertain to an organization’s resources including money, time and effort, as well as the support of its executive leadership. The three discontinuing factors identified by the Agile Adoption Framework are: • Inappropriate Need for Agility: This refers to situations where, from a business or software development perspective, adopting agility does not add any value [44] . • Lack of Sufficient Funds: When funds are unavailable or insufficient to support the agile adoption effort, then an adoption process is not feasible. • Absence of Executive Support: If committed support from executive sponsors is absent, then effective and substantial change in the organization is unlikely to occur [44] [37]. Figure 3. The 4-Stage Process for Agile Adoption No-go Target Agile Level for the Project Target Agile Level for the Organization Suspend Adoption Effort Agile Practices to Adopt Stage 1: Identify Discontinuing Factors Stage 2: Project Level Assessment Stage 3: Organizational Assessment Stage 4: Reconciliation When an organization demonstrates any of these discontinuing factors, it is unprepared to move towards agility and should suspend the adoption process until the environment is more supportive. Indicators focusing on organizational characteristics are used to assess the degree to which a discontinuing factor is present in the organization. The assessor uses one or more indicators to evaluate each organizational characteristic. For example, two organizational characteristics that can be measured to determine whether there is a Lack of Sufficient Funds are (1) the dollar amount allocated to the process improvement effort and (2) the ability to actually spend the funds for agile adoption. An example of a question (indicator) used to assess the ability to spend funds on agile adoption is Can the funds be spent towards any process improvement activity? Another assessment question is Are there any restrictions on the type of activities for which these funds can be used? Over 20 indicators are included in the Agile Adoption Framework to assess the presence of discontinuing factors in organizations [43]. 3.2. Identify Agile Practices to Adopt If Stage 1 indicates that the organization is ready to move towards agility, the journey of introducing agile practices into the development process begins. This involves determining which agile practices and concepts are most suitable for the organization to adopt. Actually, to be more precise, the Agile Adoption Framework first determines the agile practices that a particular project can adopt, not the whole organization. The framework is based on the fundamental belief that each project in an organization can adopt a different degree of agility based on its context. Therefore, the last three stages provide guidelines for identifying the agile practices suitable for a single project: • Stage 2: Project level Assessment: identifies the maximum level of agility the project can reach. This is also known as the target agile level. • Stage 3: Organizational Readiness Assessment: determines the extent to which the organization is ready to accommodate the project’s target agile level. • Stage 4: Reconciliation: settles the differences, if any, between the highest level of agility the project can adopt and the level of agility the organization is ready to embrace, and determines the agile practices that are to be adopted. Sections 3.2.1 through 3.2.3 describe each of these stages, respectively. 3.2.1 Stage 2: Project level Assessment. Stage 2 is the first stage of the adoption process that utilizes the 5 Levels of Agility presented earlier. The objective of this stage is to identify the highest level of agility a project can achieve. This is called the target level, and is one of the 5 agile levels. In theory, all projects should aspire to reach the highest level of agility possible. However, the reality is that circumstances, often outside of the organization’s control, surround each project. These circumstances become constraining factors if they adversely affect the organizations’ ability to adopt an agile practice. Thus, constraining factors limit the level of agility to which a project aspires. For example, frequent face-to-face communication is a desired agile practice at level 3. A factor that is needed to successfully adopt this practice is near team proximity. Assume that the project and organization have no say in changing this project characteristic (i.e. factor), because it is outside of their control. If the project level assessment determines that the factor (near team proximity) is missing for this project, then the highest level of agility for this project will be the same level of agility in which this agile practice is found (which is Level 3 in this case). Because achieving the highest level of agility depends on project circumstances outside of an organization’s control, the first step in Project Level Assessment is to identify those agile practices and concepts that rely on such circumstances. These agile practices are known as limiting agile practices, because if the project characteristics needed to support these practices are not present, the inability to adopt the practice constrains or limits the level of agility attainable by the project. In Table 1, which illustrates the 5 agile levels, the limiting agile practices are underlined. The assessment process defined by Stage 2 focuses on determining the target level of agility for a project. More specifically, it examines only those factors associated with the limiting agile practice, and measures the extent to which they are present. The assessment is conducted using the indicators associated with each limiting agile practice. The process starts by examining the limiting practices at Agile Level 1, and then moves upward on the scale. Once factors needed for the adoption of a limiting practice are found to be missing, the assessment process stops, and the highest level of agility attainable for the project is set to be the level at which that limiting practice is found. In summary, the target level of agility is determined to be the point where the assessment process discovers that one of the project characteristics needed to adopt a limiting agile practice or concept is missing, and neither the project nor organization can do anything to influence or change this circumstance. After the target agile level for the project is identified, the next step in the journey is to conduct an organizational readiness assessment to determine the set of agile practices (for the project) that can be adopted. 3.2.2 Stage 3: Organizational Readiness Assessment. Identifying the target level for a project does not necessarily mean that that level is achievable. To determine the extent to which that target level can be achieved, the organization must be assessed to determine whether it is ready to adopt each of the agile practices and concepts associated up to, and including, the target level. Investing time and effort in this type of pre-adoption assessment of each agile practice increases the probability of success for the overall transition to agility [14], because it significantly reduces the risks associated with the agile adoption process. Similar to Stage 2, Stage 3 of the process also relies on the 5 Levels of Agility. Again, the indicators play a critical role in determining the extent to which the target level can be achieved. To save time and money during this assessment stage, instead of assessing how ready the organization is relative to adopting the practices in all 5 agile levels, only those within the target agile level and below are used. The assessor uses the set of indicators (questions) associated with the agile practices to measure the extent to which the requisite organizational characteristics are present. For example, Collaborative Planning is an agile concept in Level 1. To assess the readiness of the organization to adopt this concept, the following are some of the organizational characteristics that need to be present: (a) collaborative management style, (b) management buy-in to adopt the agile practice, (c) transparency of management, (d) small power-distance in the organization, and (e) developers buy-in to adopt the agile practice Agile practices Organizational characteristic needed NA PA LA FA Reflect and tune ….. Transparency of management X Small power-distance in the organization X Developers buy-in X Collaborative management style X Collaborative planning Management buy-in X Coding standards ….. NA: Not Achieved (0%-35%) LA: Largely Achieved (65%-85%) PA: Partially Achieved (35%-65%) FA: Fully Achieved (85%-100%) Table 2. Organizational Assessment Results Each of these organizational characteristics is assessed using a number of different questions. Depending on the question, a manager or developer within the organization, or the assessor himself or herself answers it. The 5 Levels of Agility incorporate approximately 300 indicators to measure the various organizational characteristics related to agile practices and concepts [43]. The result of the organizational assessment stage is a table that depicts the extent to which each organizational characteristic is achieved (see Table 2). This format for displaying results is beneficial to executives and decision makers as it draws attention to the characteristics of the organization that can cause problems in adopting a practice. Resembling project level assessment, determining the highest agile level an organization is capable of achieving is dependent on the organization’s readiness to adopt the practices in that agile level. If the organizational characteristics needed for a practice are found to be not achieved or only partially achieved, then this is an indication that the organization is not ready to adopt that practice. As a result, the highest level of agility the organization can reach becomes the level at which a necessary organizational characteristic is missing. For example, in Table 2 since collaborative planning is in Agile Level 1, and since two of the characteristics that it needs are deficient, the highest level of agility for that organization is Level 1. 3.2.3 Stage 4: Reconciliation. Following the organizational readiness assessment, the agile level achievable by the organization is known. Prior to that, Stage 2 had identified the agile level that the project aspires to adopt. Therefore, the final step, reconciliation, is necessary to determine the agile practices the project will adopt. During this phase the differences between the projects’ target level and the organization’s readiness level are resolved to determine the final set of agile practices that will be adopted/employed. Three different scenarios are possible during this stage: • Organization Readiness Level > Project Target Level: No reconciliation is needed and all the practices within the project’s agile level and below become the chosen agile practices for adoption. This is a rare case because the project environment is usually contained with the organization. • Organization Readiness Level = Project Target Level: No reconciliation is needed and all the practices within the project’s agile level and below become the chosen agile practices for adoption. This is the ideal case since the project is achieving 100% of its agile potential. • Organization Readiness Level < Project Target Level: Reconciliation is necessary. As discussed below, the framework provides two options for reconciling this situation. (1) The first option relies on the how ready and willing the organization is for changes and improvements. The results of the organizational assessment have identified exactly which characteristics are hindering the organization from reaching higher levels of agility (i.e. the project’s target level). If changing any of these characteristics is within the control of the organization, then the organization can undertake the necessary steps to improve them. If all of the recommended changes have been successfully made, then the organization can support agile practices at the project’s target level. Otherwise, the projects’ target level must be lowered accordingly. (2) The second option is suitable for organizations that are unwilling to invest time, effort or money towards change, and only wants to adopt those agile practices that are within their current capacity. In this case, it is recommended to adopt only the agile practices the organization is ready for. The obvious downside to this approach is that the project is restricted to operating at a lower level of agility than its potential. This reconciliation stage helps the organization in realistically identifying the agile practices it can adopt. At the same time, if the organization is able and willing to improve, then this stage guides it as to where the improvements need to occur so that the project can operate at its full agile potential. Moreover, by utilizing this approach, the organization prepares itself sufficiently before starting the process of introducing agile practices into the development process. The next section provides a brief overview of the feedback gathered from presentations of the Agile Adoption Framework to members of the agile community. 4. Quantitative Feedback about the Agile Adoption Framework The Agile Adoption Framework was presented to 28 members of the agile community. The feedback was gathered during 90-minute personal visits to the participants (or a group of them) in which the framework was presented and then discussed. After the presentation, the participants filled out a survey eliciting their feedback. In this section the results of the participants’ feedback are examined from two perspectives, the first being the role or position of the participant, and the second being their years of experience. Additionally the feedback for the 5 Levels of Agility is presented separately from that of the 4-Stage process, since they were gathered through separate questionnaires. 4.1. Results for the 5 Levels of Agility. The questionnaire concerning the 5 Levels of Agility focused on gathering feedback about its comprehensiveness, practicality, necessity, as well as whether the practices were placed at appropriate levels. Figure 4 illustrates that, in general, the participants were mostly in agreement with regard to comprehensiveness, practicality and necessity. However, some variability is observed among the participants concerning relevance. The most prominent concern was the position of the agile practices within the levels. We conjecture that this is due to the fact that each participant has different experiences, depending on their role, years of experience and the projects in which they have been involved. As a result, each participant places a different priority on the use of practices as reflected in their experiences. These beneficial insights and feedback have led us to recognize the utility of, and need for, the flexibility to tailor the 5 Levels of Agility to fit experiences and perhaps business goals. When examining the results classified by role, it is important to note that agile coaches and consultants had more positive feedback, in general, than the other positions. The results from the comprehensiveness, practicality and necessity show that there is in need for structure and guidance on how to organize these agile practices and concepts – this is exactly what the 5 Levels of Agility is intended to provide. 4.2. Results for the 4- Stage Process. Figure 5 shows the feedback obtained relative to the 4-Stage assessment process. The feedback focused on the understandability of the process, its practicality, necessity, completeness, and effectiveness. As compared to the feedback on the 5 Levels of Agility, the feedback on the 4-Stage process is even more encouraging. Note that the agreement level is proportional to the years of experience and the roles of the individuals: the more experience and direct involvement with agile adoption, the higher the agreement rating. All of the highly experienced people strongly agreed that the process is clear and easy to understand. This can be expected, because the process is designed to model their particular activities. The completeness of the 4-Stage process had the lowest agreement percentage when compared to the other aspects of the process. We conjecture that a major factor contributing to this was the process used to gather the feedback. More specifically, only 90 minutes were allotted for presenting the framework to the participants, having follow-up discussions, and conducting the survey. We expect that this timeframe was too short for the participant (or anyone) to fully grasp the essence of the complete framework and the substantial set of relationships among its constituent components. This expectation is somewhat confirmed by the participants that returned the questionnaires at a later time (and having the time to reflect on the presentation and supporting material) – they both strongly agreed that the 4-Stage process is complete. 5. Conclusion The Agile Adoption Framework is a first step toward addressing the need for providing organizations with a structured and repeatable approach to guide and assist them in the move toward agility. The framework is independent of any one particular agile method or style. Therefore, there are no restrictions on using XP or SCRUM or any other agile style within the framework. Moreover, the framework has two levels of assessment: one at the project level and another on an organizational level. Hence, it accommodates the uniqueness of each project, and at the same time, recognizes that each project is surrounded by, and is part of, an overall organization that must be ready to adopt the requisite agile practices. We view the Agile Adoption Framework as an initial contribution towards answering the complex question of how to adopt agile practices. In summary, we propose this framework as an approach to guide and assist organizations in their quest to adopt agile practices. Through identifying and assessing the presence of discontinuing factors, organizations can make a go/no-go decision regarding the move toward agility. By determining the target level for a project and then assessing the organization to determine the extent to which it is ready to achieve that target level of agility, the framework manages to provide coaches with a realistic set of agile practices for the project to adopt. The 4-Stage process assessment, through its utilization of the 5 Levels of Agility, provides an Figure 4. Results of 5 Levels of Agility grouped by role and experience Grouped by Years of Experience Grouped by Role/Position 1-2 Years of Experience (11 Participants) COMP PRAC NESS RELV Developers (8 Participants) COMP PRAC NESS RELV 3-5 Years of Experience (9 Participants) COMP PRAC NESS RELV Management / Administrative (8 Participants) COMP PRAC NESS RELV 6-12 Years of Experience (8 Participants) COMP PRAC NESS RELV Agile Coaches/Consultants (12 Participants) COMP PRAC NESS RELV Abbreviations COMP: Comprehensiveness PRAC: Practicality NESS: Necessity RELV: Relevance Strongly Agree Slightly Agree Neither Agree nor Disagree Slightly Disagree Strongly Disagree extensive outline of the areas within the organization that need improvement before the adoption effort starts. While we recognize that the framework has significant room for improvement, we are encouraged by the comments given about the Agile Adoption Framework from members of the agile community: • “I think this is fantastic (work)” –Agile consultant with 12 years experience • “This is the RIGHT time for this work! Excellent Job” – Agile consultant with 8 years experience • “Overall this is first-class work and I endorse this work as legitimate in its interest and merit to our industry” (paraphrased due to length) – XP Coach with 6 years experience 6. References [1] Declaration of Interdependence, http://pmdoi.org/, 2005. [2] Manifesto for Agile Software Devleopment, www.agilemanifesto.org, Utah, Feb 2001. [3] P. Abrahamsson, S. Outi, J. Ronkainen and J. Warsta, Agile software development methods - Review and analysis, VTT Electronics Finland, 2002, pp. 112. [4] S. Ambler, Agile Modeling: Effective Practices for Extreme Programming and the Unified Process Wiley, 2002. [5] S. W. Ambler and P. J. Sadalage, Refactoring Databases: Evolutionary Database Design, Addison-Wesley Professional, 2006. [6] L. Amy and C. Raylene, Effects of agile practices on social factors, ACM Press, St. Louis, Missouri, 2005. [7] J. Arthur and R. Nance, Managing Software Quality: A Measurement Framework for Assessment and Prediction, Springer, 2002. Grouped by Years of Experience Grouped by Role/Position 1-2 Years of Experience (11 Participants) UNDER PRAC NESS COMP EFEC Developers (8 Participants) UNDER PRAC NESS COMP EFEC 3-5 Years of Experience (9 Participants) UNDER PRAC NESS COMP EFEC Management / Administrative (8 Participants) UNDER PRAC NESS COMP EFEC 6-12 Years of Experience (8 Participants) UNDER PRAC NESS COMP EFEC Agile Coaches/Consultants (12 Participants) UNDER PRAC NESS COMP EFEC Abbreviations UNDER: Understandability PRAC: Practicality NESS: Necessity COMP: Completeness EFEC: Effectiveness Strongly Agree Slightly Agree Neither Agree nor Disagree Slightly Disagree Strongly Disagree Figure 5. Results of the 4-Stage Process grouped by role and experience [8] L. Barnett, Agile Survey Results: Solid Experience And Real Results Agile Journal, 2006. [9] L. Barnett and C. Schwaber, Adopting Agile Development Processes; Improve Time-To-Benefits For Software Projects Forrester Research, 2004. [10] V. Basili, Software modeling and measurement: the Goal/Question/Metric paradigm, University of Maryland at College Park, 1992, pp. 24. [11] Beck, Test Driven Development: By Example, Addison-Wesley Longman Publishing Co., Inc., 2002. [12] K. Beck, Extreme Programming Explained: Embrace Change, Addison-Wesley, 2000. [13] K. Beck, R. C. Martin, A. Cockburn, M. Fowler and J. Highsmith, Manifesto for Agile Software Devleopment, www.agilemanifesto.org, Utah, Feb 2001. [14] B. Boehm and R. Turner, Management challenges to implementing agile processes in traditional development organizations, Software, IEEE, 22 (2005), pp. 30-39. [15] B. W. Boehm and R. Turner, Balancing Agility and Discipline, Addison-Wesley Professional, Boston, 2003. [16] A. Cockburn, Personal Communication Salt Lake City, UT, November 2006. [17] A. Cockburn, Agile Software Development Pearson Education, Indianapolis, 2001. [18] A. Cockburn and J. Highsmith, Agile Software Development: The People Factor, Computer, Volume 34 (2001), pp. Pages: 131 - 133 [19] M. Cohn, Personal Communication, Dayton, OH, October 2006. [20] M. Cohn, Agile Estimating and Planning Prentice Hall PTR, 2005. [21] M. Cohn, User Stories Applied, Addison -Wesley, Boston, 2004. [22] A. Elssamadisy, Personal Communication, Amherst, MA, October 2006. [23] A. Elssamadisy, Getting Beyond "It Depends!" Being Specific But Not Prescriptive About Agile Practice Adoption Agile Journal, 2006. [24] M. Fowler, K. Beck, J. Brant, W. Opdyke and D. Roberts, Refactoring: Improving the Design of Existing Code, Addison Wesley, Reading, Massachusetts, 1999. [25] R. B. Grady, Successful software process improvement, Prentice-Hall, Inc., 1997. [26] J. Highsmith, Agile Software Development Ecosystems, Pearson Education, Indianapolis, 2002. [27] J. Highsmith, Agile: From Rogue Teams to Enterprise Acceptance Cutter Consortium: Business Technology Trends and Impacts, 2006. [28] A. Hunt and D. Thomas, Pragmatic Unit Testing in C\# with NUnit, The Pragmatic Programmers, 2004. [29] J. Hunt, Agile software construction Springer, London 2006. [30] P.-H. Jan and J. Jorn, AIM - Ability Improvement Model, 2005. [31] A. S. Koch, Agile software development : evaluating the methods for your organization, Artech House, Boston, MA 2005. [32] S. Kuppuswami, K. Vivekanandan, P. Ramaswamy and P. Rodrigues, The effects of individual XP practices on software development effort, SIGSOFT Softw. Eng. Notes, 28 (2003), pp. 6-6. [33] C. Larman, Agile and Iterative Development, Pearson Education, Boston, 2004. [34] A. Law and R. Charron, Effects of agile practices on social factors, Proceedings of the 2005 workshop on Human and social factors of software engineering, ACM Press, St. Louis, Missouri, 2005. [35] R. C. Martin, Agile Software Development, Principles, Patterns, and Practices, Prentice Hall 2002. [36] J. W. Newkirk and R. C. Martin, Extreme Programming in Practice Prentice Hall, 2001. [37] A. Pukinskis, 5 stumbling blocks for new corporate agile projects, the agile blog, 2005. [38] D. Rosenberg, M. Stephens and M. Collins-Cope, Agile development with ICONIX process : people, process, and pragmatism Apress Berkeley, CA 2005. [39] A. Rueping, Agile Documentation : A Pattern Guide to Producing Lightweight Documents for Software Projects John Wiley & Sons 2003. [40] A. Sanjiv, Personal Communication, Reston, VA, October 2006. [41] B. Schatz and I. Abdelshafi, Primavera gets agile: a successful transition to agile development, Software, IEEE, 22 (2005), pp. 36-42. [42] K. Schwaber and M. Beedle, Agile Software Development with SCRUM, Prentice Hall,, 2002. [43] A. Sidky and J. Arthur, Agile Adoption Process Framework - Indicators Document, CORR - cs.SE/0612092 2006. [44] M. K. Spayd, Evolving agile in the enterprise: implementing XP on a grand scale, 2003, pp. 60- [45] J. Tabaka, Collaboration Explained; Facilitation Skills for Software Project Leaders, Addison- Wesley, 2005. [46] W. Wake, Personal Communication, Richmond, VA, September 2006. [47] W. C. Wake, Extreme Programming Explored, Addison-Wesley Professional, 2001. [48] L. Williams and R. Kessler, Pair Programming Illuminated, Addison-Wesley Longman Publishing Co., Inc., 2002. [49] L. Williams, R. R. Kessler, W. Cunningham and R. Jeffries, Strengthening the case for pair programming, Software, IEEE, 17 (2000), pp. 19-

0704.1295

Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16

Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 B Wiendlocha, J Tobola and S Kaprzyk Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059 Cracow, Poland E-mail: [email protected] Abstract. Electronic structure of a novel superconducting noncentrosymmetric compound Mg10Ir19B16 was calculated using the Korringa-Kohn-Rostoker method. Electronic part of the electron-phonon coupling constant, McMillan- Hopfield parameters, were calculated using the rigid-muffin-tin approximation (RMTA). The magnitude of the electron-phonon coupling constant λ, analysing atomic contributions, is discussed. Our results show, that superconductivity in Mg10Ir19B16 is presumably mediated by electron-phonon interaction. PACS numbers: 74.25.Jb 1. Introduction The interest in noncentrosymmetric structures, exhibiting superconductivity, has grown up in the past years. There are only a few examples of this type, which belong to various classes of materials, e.g. an antiferromagnetic heavy fermion system CePt3Si [1], ferromagnetic uranium compound UIr, superconducting under pressure [2], or non-magnetic ternary borides Li2Pd3B [3] and Li2Pt3B [4]. The main reason, why these systems are especially attracting, is related to the role of inversion symmetry in electron pairing. The absence of inversion symmetry may suppress the triplet pairing or mix singlet and triplet symmetry [5, 6, 7]. Very recently, Klimczuk and co-workers [8] synthesised a new type of intermetallic light-element based compound Mg10Ir19B16, exhibiting superconductivity near 5 K. This novel material also belongs to the rare noncentrosymmetric structures, and crystallises in a large and rather complex bcc cell (space group I-43m) [8]. Mg10Ir19B16 is partly similar to the Li2(Pd,Pt)3B system, since these structures contain an alkali metal (Li, Mg), boron, and heavy transition metal (Pd, Pt, Ir). In this work we intend to start the discussion on superconductivity mechanism in this unusual compound, analysing the electronic structure and the strength of electron- phonon coupling (EPC). Assuming the BCS-type behaviour in Mg10Ir19B16, we study whether superconductivity is driven by light boron sublattices, like e.g. in MgB2, or by heavy transition metal atoms, as suggested for Li2Pd3B [9]. 1.1. Computational details Electronic structure calculations were performed using the Korringa-Kohn-Rostoker (KKR) multiple scattering method [10, 11]. The crystal potential was constructed in http://arxiv.org/abs/0704.1295v1 Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 2 the framework of the local density approximation (LDA), using von Barth and Hedin formula [12] for the exchange-correlation part. For all atoms angular momentum cut-off lmax = 3 was set, k-point mesh in the irreducible part of Brillouin zone (IRBZ) contained over 200 points. Densities of states (DOS) were computed using the tetrahedron k-space integration technique, generating over 600 tetrahedrons in IRBZ. Due to the high atomic number of iridium (Z = 77) semi-relativistic calculations were performed, but neglecting the spin-orbit coupling, which is commented at the end of this paper. Figure 1. Unit cell of Mg10Ir19B16. The lack of inversion centre is clearly seen e.g. along the main diagonal, where B1 and Mg1 atoms break the inversion symmetry. Generated by XCRYSDEN [27]. As far as the crystal structure is concerned, experimental lattice constant a = 10.568 Å and atomic positions [8] were accounted for the computation (for clarity also shown in table 2). Atoms in the unit cell were surrounded by muffin-tin (MT) spheres with following radii: RMg = 2.82, RIr = 2.50, RB = 1.40 (in atomic units), filling about 60% of the cell volume. In the primitive cell of this system, 45 atoms occupy 7 inequivalent sites, which all are listed in table 2. The noncentrosymmetricity of this system is important, and cannot be regarded as the effect of a lattice distortion, as observed e.g. in UIr [9]. The inversion symmetry is broken by both boron sublattices (B1 and B2), as well as Mg1 and Ir3 sites, thus the crystal has a half of symmetry operations of the cubic group Oh, i.e. only 24 operations. Among of all sublattices, the positions of iridium atoms are the closest to have inversion symmetry. Since Ir3 occupies (x, z, z) sites, with x ≃ 0.07, the full cubic symmetry is restored after shifting this position to (0, z, z). The analysis of superconducting properties is based on the computed McMillan- Hopfield (MH) parameters η [13, 14] which determine electronic part of the electron- phonon interaction, and directly enter the formula for the electron-phonon coupling constant λ: . (1) Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 3 -1.0 -0.75 -0.5 -0.25 0.0 0.25 Energy (Ry) Total DOS -0.04 -0.02 0.0 0.02 0.04 Energy (Ry) Total DOS Figure 2. (a) Total DOS of Mg10Ir19B16. (b) Zoom near the Fermi level (EF = 0) with atomic contributions. In equation (1), ηi is the MH parameter for each nonequivalent atom i which is characterised by the atomic massMi and averaged squared vibrational frequency 〈ω This equation divides the electron-phonon coupling constant into site-dependent parts, thus it allows to justify which sublattice gives the most important contribution to the total λ. Calculations of MH parameters were preformed using the rigid muffin tin approximation (RMTA) [15, 16, 17, 18], which gives the following expression for η at each atomic site i: (2l+ 2)ni (EF )n (EF ) (2l + 1)(2l+ 3)N(EF ) dr r2Ril(r) dVi(r) Ril+1(r) . (2) Here, l is the angular momentum number, nil(EF ) is the partial (angular-decomposed) density of states per spin at the Fermi energy (EF ), R (r) is a regular solution of the radial Schrödinger equation, normalised to unity inside the MT sphere of radius RiMT , and Vi(r) is the self-consistent, spherically-symmetric potential. N(EF ) is the total DOS at EF per spin and per cell. For more detailed discussion of approximations involved in this method, see e.g. [19, 20] and references therein. 2. Results and discussion The calculated total and atomic densities of states functions are presented in figure 2. As one could expect, electronic structure is dominated by iridium 5d states. The total DOS curve may be characterised as a collection of numerous van Hove singularities, reflecting the large number of atoms in the unit cell and various interatomic distances. The site-decomposed DOS are shown in figure 3, and their values at the Fermi level are gathered in table 1. The total DOS at Fermi level is about N(EF ) ≃ 150 Ry −1 per formula unit. The densities at EF , calculated per atom, are rather low (average value per atom: 3.3 Ry−1), and generally EF is located outside of dominating peaks of DOS, on a small decreasing slope of Ir DOS. Strong hybridisation of d states of Ir and p states of B and Mg is manifested in the separation of bonding and anti-bonding states, Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 4 Table 1. Densities of states n(EF ), l-decomposed DOS nl (Ry −1/spin per atom) McMillan-Hopfield parameters η, MH parameters for each scattering channel ηl,l+1 (mRy/a.u. 2 per atom), and summary MH parameters for each site wi × ηi (mRy/a.u.2 per site). wi is the number of atoms occupying site in primitive cell. Atom wi n(EF ) ns np nd nf ηi η wi × ηi Ir1 1 2.77 0.15 0.88 1.86 0.03 2.8 0.1 1.4 1.3 2.8 Ir2 6 3.16 0.07 0.34 2.71 0.03 3.3 0.0 0.8 2.5 19.8 Ir3 12 2.41 0.06 0.34 1.97 0.03 2.6 0.0 0.6 2.0 31.2 B1 4 0.40 0.03 0.35 0.01 0.00 1.2 0.0 1.2 0.0 4.8 B2 12 0.27 0.04 0.22 0.01 0.00 0.6 0.0 0.6 0.0 7.2 Mg1 4 0.98 0.17 0.69 0.10 0.03 0.7 0.5 0.2 0.0 2.8 Mg2 6 0.67 0.09 0.41 0.14 0.03 0.2 0.1 0.1 0.0 1.2 and EF is located in the DOS valley (figure 2 and figure 3). The interatomic distances, listed in table 2, supports the enhanced p-d hybridisation, especially between Ir and B atoms (the smallest distances, 2.1 - 2.2 Å). It is also interesting to compare the computed site-decomposed Ir densities, to the DOS of metallic fcc iridium, which is presented in figure 3(d). The shape of lower part of DOS in Ir2 and Ir3 is quite similar to the case of Ir-fcc. This is probably due to the fact, that the Ir2-Ir3 and Ir3-Ir3 coordination, as well as interatomic distances, are very close to the fcc phase (Ir-Ir distance is ∼ 2.7 Å in the aforementioned cases, see table 2). However, the n(EF ) values on Ir atoms are much lower, comparing to fcc structure (n(EF ) ≃ 6.3 Ry −1/spin in Ir-fcc), being the effect of enhanced hybridisation near EF . Noteworthy, Ir1 has a quite different atomic coordination, with respect to Ir2 and Ir3 positions, being surrounded practically only by 4 boron atoms (B1). It is clearly reflected by the apparently different DOS shape below EF . Ir1a) Ir2b) Ir3c) fcc Ird) -0.6 -0.3 0.0 0.3 Energy (Ry) -0.6 -0.3 0.0 0.3 Energy (Ry) -0.6 -0.3 0.0 0.3 Energy (Ry) Mg1g) -0.6 -0.3 0.0 0.3 Energy (Ry) Mg2h) Figure 3. Site-decomposed DOS of Mg10Ir19B16 (EF = 0). The total DOS of fcc iridium metal is given for comparison in panel (d). Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 5 Table 2. The smallest interatomic distances between atoms (in Å) and atomic positions in Mg10Ir19B16. The Ir-Ir distance in Ir-fcc is about 2.7 Å, as between Ir2-Ir3 and Ir3-Ir3. Ir1 Ir2 Ir3 B1 B2 Mg1 Mg2 Atomic position Site Ir1 9.1 5.9 3.9 2.1 5.0 3.1 3.6 (0, 0, 0) (2a) Ir2 3.8 2.7 4.5 2.2 4.0 3.1 (0, 0.25, 0.5) (12d) Ir3 2.7 2.1 2.2 4.0 2.9 (0.0702, 0.2525, 0.2525) (24g) B1 3.4 3.3 3.0 3.0 (0.3331, 0.3331, 0.3331) (8c) B2 3.3 2.6 2.5 (0, 0, 0.3473) (24g) Mg1 5.0 3.1 (0.1127, 0.1127, 0.1127) (8c) Mg2 3.1 (0.1639, 0.1639, 0.4140) (12e) Electronic dispersion curves near the Fermi level are presented in figure 4. The bands located below −0.1 Ry (partly showed) are very flat, and form the narrow large DOS peaks, seen already in figure 2. Conversely, the bands that cross EF are quite dispersive, which results in the low DOS in this energy range. Noteworthy, there is an energy gap along the P-N direction (parallel to the kz axis in the reciprocal space). This may suggest some anisotropic transport properties of this compound. The calculated McMillan-Hopfield factors for all sites, with contributions from each scattering channel (l → l+1), are presented in table 1. Estimation of the electron- phonon coupling constant λ, using MH parameters, requires also the knowledge of average phonon frequencies. For such a large structure, containing 90 atoms in the cubic unit cell, phonon spectra calculations are difficult to be carried out. Nevertheless we can try to investigate the strength of electron-phonon interaction assuming reasonable values of 〈ω2i 〉 and studying their influence on estimated λ values. Debye frequencies Θ of monoatomic crystals of iridium (fcc), boron (rhomboedral) and magnesium (hcp) may be helpful for choosing sensible phonon frequency range for our discussion. At first we assume, that the same type of atoms, occupying different sites in the cell, have similar average vibrational frequencies. Iridium, as the heaviest element, is expected to have the lowest 〈ω2i 〉, Debye frequency of metallic Figure 4. Dispersion curves near the EF = 0 in Mg10Ir19B16. Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 6 15 20 25 30 35 40 (meV) 0.3 (a) Ir1 40 50 60 70 80 90 100 (meV) 15 20 25 30 35 40 (meV) 2 ΘB/ 2 ΘMg/ Figure 5. Contributions to the electron-phonon coupling constant λ in Mg10Ir19B16 from iridium (a), boron (b) and magnesium (c) as a function of a root of average square frequency. Top curve on each graph is a sum of contributions from particular sites. Vertical lines mark the value of Θi/ iridium is about ΘIr ≃ 36 meV (420 K) [21]. In contrast, the lightest boron is certainly expected to have the highest phonon frequencies, and for crystalline boron ΘB ≃ 100 meV (1200 K) [22]. Finally, magnesium average frequencies are expected to locate between the values of iridium and boron. Debye frequency of crystalline Mg is rather low: ΘMg ≃ 34 meV (400 K) [21]. We may also recall, that average square phonon frequency is often estimated as 〈ω2〉 ≃ 1 Θ2, which is a good approximation in monoatomic structures. In our case, these values may also be helpful for choosing feasible range of 〈ω2i 〉. Figure 5 presents estimated electron-phonon coupling constant λ, associated with particular crystal sites, plotted as a function of average square phonon frequency. For each type of atoms, a wide frequency range was chosen, to illustrate the changeability of λ. In the case of iridium, the largest contribution to λ comes from Ir3 sublattice, due to the large population of this site. Among boron and magnesium sites, the B2 and Mg1 atoms provide the largest contributions. However, one has to remember that this comparison is valid only if we assume identical frequencies for the same atoms at different sites. In this simplified analysis, we may also plot the overall coupling constant for constituent atoms, by adding the contributions from each site, which is illustrated by solid lines in figure 5. As we can see, the obtained partial coupling constants are not high for each atom. At the moment, we are not aware of any experimental findings of the EPC constant in this compound. However, we can try to estimate the range of ”experimental” λ, analysing the magnitude of the observed critical temperature. If we assume, that we are dealing with BCS-type superconductivity, we may substitute the experimental value of TC = 4.5 K into the McMillan formula for TC [13]: 1.04(1 + λ) λ− µ⋆(1 + 0.62λ) . (3) However, because the Debye temperature Θ of Mg10Ir19B16 is not known yet, we plot λ in figure 6 as a function of Θ, for typical values of Coulomb pseudopotential parameter µ⋆. The resulting EPC constant λ varies between 0.5 and 0.75, and for e.g. Θ = 250 K we obtain λ ≃ 0.60, whereas for Θ = 350 K we get λ ≃ 0.55 (for µ⋆ = 0.11), as one can see in the figure 6. Thus, if Mg10Ir19B16 is treated as Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 7 200 250 300 350 400 450 500 Debye temperature (K) = 0.11 = 0.13 = 0.15 Figure 6. ”Experimental” value of the electron-phonon coupling constant λ, evaluated from the McMillan formula, using observed TC = 4.5 K, plotted as a function of Debye temperature Θ for three values of Coulomb pseudopotential parameters µ⋆. a conventional superconductor, the lower limit of λ is expected to be 0.5. In order to get this value from our RMTA calculations, relatively low average frequencies for all constituent atoms are required. If we take the following 〈ω2i 〉: 14 meV for Ir, 50 meV for B and 20 meV for Mg, we obtain λIr ≃ 0.3, λB ≃ 0.1 and λMg ≃ 0.1, which gives the expected lower limit of EPC constant (λ ≃ 0.5). If higher values of λ are experimentally observed, it will indicate either additional contributions, neglected within the RMTA framework, or even lower phonon frequencies, than the values used in the aforementioned estimations. Especially iridium contribution is sensitive to the change of 〉 parameter, i.e. increasing it to 25 meV (probably an upper limit) results in decrease of λIr to 0.1, that is to the value found for other sublattices. Generally, our calculations indicates, that iridium sublattices seem to be the most important for the onset of superconductivity in Mg10Ir19B16. Finally, we shortly comment the possible influence of the lack of inversion symmetry and spin-orbit (SO) interaction on the electronic structure and superconductivity. The absence of a centre of inversion in a superconductor affects the symmetry of superconducting state, allowing for an admixture of singlet and triplet components [6, 7]. Because the singlet pairing is based on the time-reversal symmetry [23], which is present as long as the compound is not magnetic, mainly the triplet channel is affected by the lack of inversion, and the superconductivity may be even suppressed, see e.g. [5, 6, 7]. The SO interaction, which mixes the initial spin-up and spin-down electronic states, plays an additional role. It was found, that it may control the mixing of parity of the superconducting state [7]. This seems to be the case of Li2Pd3B and Li2Pt3B compounds, where specific heat [24] and NMR [25] measurements strongly support phonon-mediated isotropic superconductivity, while penetration depth measurements suggests an admixture of spin-singlet and triplet components in the superconducting energy gap [26], with larger triplet component in the Pt-case. This kind of experimental study for Mg10Ir19B16 should be prior to the theoretical discussion of gap symmetry in this compound. As far as the band structure of Mg10Ir19B16 is concerned, the modifications due to the SO interaction are not expected to significantly affect the obtained values of DOS and MH parameters. For metallic fcc iridium, our value of MH parameter, Electronic structure of the noncentrosymmetric superconductor Mg10Ir19B16 8 calculated also neglecting SO interaction, ηIr ≃ 135 mRy/(a.u.) 2, gives the correct magnitude of the electron-phonon coupling constant λ = 0.32, comparing to the observed λ = 0.34 [13]. Here, the average square frequency was estimated from the formula 〈ω2〉 ≃ 1 Θ2, using the Debye temperature Θ = 420 K. 3. Summary and conclusions The results of LDA electronic structure calculations of new Mg10Ir19B16 superconductor were presented. The main contributions to densities of states near EF are provided by iridium atoms. The electron-phonon coupling constant λ was roughly estimated, using the calculated McMillan-Hopfield parameters and qualitative discussion of average phonon frequencies. We discussed the relation of the experimental transition temperature and the magnitude of EPC coupling. Within the rigid-muffin-tin approximation, the main contribution to λ comes from iridium, with smaller contributions from boron and magnesium. If any information about the dynamic properties of atoms in this compound become available, our analysis presented of figure 5 will allow to find better theoretical estimation of λ. The location of EF on the slope of Ir DOS peak leads to rough suggestion, that hole doping on iridium sites, e.g. with rhodium, may increase the densities and MH parameters. Acknowledgments We would like to thank dr Tomasz Klimczuk for helpful discussions. This work was partly supported by the Polish Ministry of Science and Higher Education (PhD grant). 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B 72 104506 [25] Nishiyama M, Inada Y and Zheng G 2005 Phys. Rev. B 71 220505(R) [26] Yuan H Q, Agterberg D F, Hayashi N, Badica P, Vandervelde D, Togano K, Sigrist M and Salamon M B 2006 Phys. Rev. Lett. 97 017006 [27] A. Kokalj 1999 J. Mol. Graphics Modelling 17, 176. Code available from http://www.xcrysden.org/. http://www.xcrysden.org/ Introduction Computational details Results and discussion Summary and conclusions

0704.1296

Prospects of using simulations to study the photospheres of brown dwarfs

Convection in Astrophysics Proceedings IAU Symposium No. 239, 2007 F. Kupka, I.W. Roxburgh & K.L. Chan, eds. c© 2007 International Astronomical Union DOI: 00.0000/X000000000000000X Prospects of using simulations to study the photospheres of brown dwarfs Hans-Günter Ludwig1 1CIFIST, GEPI, Observatoire de Paris-Meudon, 92195 Meudon, France email: [email protected] Abstract. We discuss prospects of using multi-dimensional time-dependent simulations to study the atmospheres of brown dwarfs and extrasolar giant planets, including the processes of convec- tion, radiation, dust formation, and rotation. We argue that reasonably realistic simulations are feasible, however, separated into two classes of local and global models. Numerical challenges are related to potentially large dynamic ranges, and the treatment of scattering of radiation in multi-D geometries. Keywords. hydrodynamics, convection, radiative transfer, methods: numerical, stars: atmo- spheres, stars: low-mass, brown dwarfs 1. Introduction The increasing number of brown dwarfs and extrasolar planets of spectral class L and later discovered by infrared surveys and radial velocity searches has spawned a great deal of interest in the atmospheric physics of these objects. Their atmospheres are sub- stantially cooler than, e.g., the solar atmosphere, allowing the formation of molecules, or even liquid and solid condensates – in astronomical parlance usually referred to as “dust”. Convection is a ubiquitous phenomenon in these atmospheres shaping their ther- mal structure and the distribution of chemical species. Hydrodynamical simulations of solar and stellar granulation have become an increasingly powerful and handy instru- ment for studying the interplay between gas flows and radiation. In this paper we discuss the prospects of developing similar multi-dimensional and time-dependent simulations of very cool atmospheres. The most important additional process – in view of previous developments for hotter atmospheres – that one needs to tackle is the dust formation coupled to the hydrodynamic transport processes and radiative transfer. In the following, we shall take a slightly broader point of view than just considering brown dwarf (hereafter BD) atmospheres and include also the atmospheres of extrasolar giant planets (hereafter EGPs) in the discussion since their atmospheric dynamics is controlled by similar processes as brown dwarfs atmospheres. As we shall later see, computational limitations do not allow to address the problem of the atmospheric dynamics as a whole, i.e., studying the global atmospheric circulation together with convective flows taking place at relatively small spatial scales given by the the pressure scale height at the stellar surface. Consequently modelers have addressed the problem of the local and global circulation separately. Since many BDs and EGPs are rapid rotators rotation constitutes another important process which one needs to take into consideration for the global circulation. We would like to emphasize that we are discussing the perspectives for multi-D time- dependent simulations. Considerable insight has already been gained by the develop- ment of one-dimensional, hydrostatic, time independent model atmospheres (hereafter http://arxiv.org/abs/0704.1296v1 2 H.-G. Ludwig “standard” models) for L- and T-type objects (e.g., Burrows et al. 2006, Tsuji 2002, Allard et al. 2001, Ackerman & Marley 2001), and we draw from this work. Simulations will augment our understanding of BD and EGP atmospheres by adding information about the detailed cloud meteorology on the local scale of convective cells, as well as on the scale of the global wind circulation pattern. This includes further characterization of the effects of irradiation in close-in EGPs. The simulation of the atmospheric dynamics might also add to our knowledge about local dynamo action in substellar objects, and acoustic activity contributing to the heating of chromospheres. 2. Micro-physical input In order to perform realistic simulations micro-physical input data must be available – radiative opacities, equation-of-state (EOS), and a kinetic model describing the formation of dust grains. The requirements are similar to those for standard models, and conse- quently in simulation work one can usually take recourse to the descriptions developed for 1D models – largely on the same level of sophistication. In all three before-mentioned areas substantial progress has been made over the last decade, spawned by the discov- ery of the first brown dwarf and EGP in 1995. In particular, since the early work of Rossow 1978 kinetic models describing the nucleation, growth, and evaporation of dust grains under conditions characteristic of brown dwarf atmospheres have been developed, see Helling et al. 2004 and references therein. Hence, the present input data allow to set-up simulations on a sufficiently realistic level. 3. Time scales: convection, radiation, dust, rotation, & numerics To obtain insight into potential challenges one faces in simulations of the dynamics of brown dwarf and EGP atmospheres it is illuminating to take a look at the characteristic time scales of the governing physical processes. Figure 1 depicts these time scales in a representative brown dwarf model atmosphere at Teff=1800K and log g=5.0 of solar chemical composition. The model comprises the stellar photosphere and the uppermost layers of the convective stellar envelope. Since it is expected that the cloud decks are located in vicinity of the boundary of the convective envelope (in this model located at a geometrical height of 23 km) it also contains the layers in which the dust harboring layers are expected. We emphasize that the model structure is taken from an experimental hydrodynamical simulation in which dust formation was not taken into account. Since here we are interested in order of magnitude estimates only this is not a critical issue. In figure 1 the line labeled “C-F-L” computed as the sound crossing time over a pres- sure scale height depicts the upper limit of the time step which is allowed in an explicit hydrodynamical scheme due to the Courant-Friedrichs-Levy stability criterion. Depend- ing on the actual resolution of the numerical grid this number may be one to two orders of magnitude smaller than indicated. The line labeled “convection” depicts the modulus of the Brunt-Vaiäsälä period providing a measure of the time scale on which the convective flow evolves. Two lines labeled “radiation” indicate the time scale on which radiation changes the thermal energy content of the gas. The dashed-dotted line is computed using Rosseland mean opacities which give the correct behavior in the optical thick layers, the solid line is based on Planck mean opacities which give a better representation in the optically thin layers. For the rotational period depicted by the dashed line labeled “rotation” we took a representative value close to one day. For three different dust grain diameters of 1, 10, and 100µm we plotted the sedimentation (“rain out”) time scale taking as drifting time over a pressure scale height. The drift velocities were taken from Prospects for simulations of brown dwarf photospheres 3 0 20 40 60 80 Geometrical height [km] 2 1 0 −1 −2 −3 radiation convection rotation 100µm rain out C−F−L Figure 1. Characteristic time scales of various processes in a brown dwarf atmosphere of Teff=1800K as a function of geometrical height. The tick marks close to the abscissa indicate the (log Rosseland) optical depth. For details see text. the work of Woitke & Helling 2003. We did not depict the formation time scale of the dust grains in the figure: for grains of 100µm diameter it is of the same order as the sed- imentation time scale. Consequently, it is unlikely that larger grains can stay in brown dwarf atmospheres. The formation time scale becomes rapidly shorter for smaller grains so that they can be considered being essentially formed in quasi-static phase-equilibrium (see also Helling 2005. Computing resources available today typically allow to simulate a dynamical range in time of 104. . . 105, and 102. . . 103 per dimension (for 3D models) in space. From figure 1 we conclude that it should be feasible to include convection, radiative transfer effects, and dust formation in a simulation of a BD/EGP atmosphere. The simultaneous inclusion of rotation is beyond reach, in particular if one takes into consideration that one would like to simulate many rotational periods to obtain a statistically relaxed state. However, the substantial difference between the time scales on which rotation and convection operate moreover indicates that rotation is dynamically not relevant for the surface granulation pattern in BD/EGPs. A rather strong modeling limitation comes about by the large spatial scale separation between the typical size of a convective cell and the global scale of a BD or EGP of about 104, at best reduced to 103 for the case of young, low mass EGPs. Hence, typical BD/EGP conditions are hardly within reach with 3D models, and the steep increase of the computational cost with spatial resolution (for explicite numerical schemes with (∆x)4) makes it likely that this situation prevails during the nearer future. We expect that 3D simulations will for some time be either tailored to simulate the global meteorology, or will be restricted to local models simulating the convective flow in detail. Figure 2 illustrates the kinematics of the flow in a local BD simulation analogous to figure 1. The horizontal root-mean-square of the vertical velocity component is depicted by the diamond symbols. The key-point to note is that the convective motions proper are largely confined to the convectively unstable layers. The velocities in the convectively stable layers with log τ < 0 are almost exclusively related to sound waves. As essentially oscillatory motions they are ineffective for mixing so that they provide little updraft to keep dust grains aloft in the atmosphere. The green line is illustrating an estimate 4 H.-G. Ludwig 0 20 40 60 80 Geometrical height [km] 100.0 1000.0 locit RMS vertical hydro velocity ix h ctive 2 1 0 −1 −2 −3 Figure 2. Characteristic velocities in brown dwarf atmosphere analogous to figure 1. of the effective mixing velocity provided by the convective motions. The decline of the amplitude is rather steep – in the test model with a scale height of about 1/3 of the local pressure scale height. Comparing the mixing velocities with typical grain sedimentation velocities indicates that the kinematics could support cloud decks in the convective zone and a thin adjacent overshooting layer at its top boundary. Fitting observed spectra Burrows et al. 2006 find a preference for a rather large grain size of ≈ 100µm in BD atmospheres. This would make the grain sedimentation velocities comparable to convective velocities which is numerically uncritical. More demanding would be small grain sizes. The distribution of small grains would hinge on the capability of a numerical code to deal with large velocity ranges. Any non-physical diffusivity in a code can artificially extend the region over which clouds of small grains could exist. 4. The multi-D story so far A number of simulations of BD/EGP atmospheres have been already conducted in 2D and 3D geometry. Here, the problem of the circulation between the day- and night-side of close-in EGPs (“hot Jupiters”) achieved particular attention. However, to our knowledge none of the studies has addressed the coupled problem of hydrodynamics, dust formation, radiation, and rotation, but rather have focused on different parts of the overall problem. We would like to refer the interested reader to Showman & Guillot 2002, Cho et al. 2003, Burkert et al. 2005, including the follow-up works of these groups. 5. Serendipity In the previous sections we summarized expectations about the insights one might gain, and challenges one might face when trying to construct multi-D models for BD/EGP atmospheres. We added figure 3 as a reminder that of course the unforeseen results are the most interesting ones. Figure 3 illustrates a slight but distinct change of the granulation pattern between the familiar solar granulation and granulation in an M- dwarf. E.g., similarly and perhaps more drastically the formation of dust in BD/EGP atmospheres might modulate the convective dynamics in unexpected ways – who knows. Prospects for simulations of brown dwarf photospheres 5 Figure 3. Grey-scale images of the vertical velocity component of a solar hydrodynamical model (top row) and a M-dwarf model (bottom row). From left to right, the velocities are depicted at (Rosseland) optical depth unity as well as one and two pressure scale heights below that level in the respective models. The absolute image scales are 5.6 × 5.6Mm2 for the solar and 0.25× 0.25Mm2 for the M-dwarf model. 6. Conclusions Reasonably realistic local or global models of brown dwarf and extrasolar giant planet atmospheres coupling hydrodynamics, radiation, dust formation, and rotation are numer- ically in reach at present. However, “unified” models spanning all spatial scales from the global scale down to scales resolving the flow in individual convective cells are stretch- ing the computational demands beyond normally available capacities. Hence, we expect that a separation between local and global models will prevail during the nearer future. Whether this will turn out to be a severe limitation remains to be seen. While appar- ently subtle we would like to point to the solar dynamo problem where the still not fully satisfactory state of affairs might be related to the lack of the inclusion of small enough scales when modeling the global dynamo action. In BD/EGP atmospheres it is perceiv- able that the local transport of momentum by convective and acoustic motions might alter the global flow dynamics – in the simplest case by adding turbulent viscosity. If the sizes of dust grains in BD/EGP atmospheres turn out to be small, and the grains consequently exhibit low sedimentation speeds, numerical simulations must have the ability to accurately represent the large dynamic range between grain and convec- tive/acoustic velocities. Overly large numerical diffusivities artificially enlarge the height range over which cloud decks can persist. Standard model atmospheres are treating the wavelength-dependence of the radiation field commonly in great detail which is not possible in the more demanding multi-D geometry of simulation models. An approximate multi-group treatment of the radiative transfer has been developed for simulations of stellar atmospheres which has also been 6 H.-G. Ludwig proven to provide reasonable accuracy at acceptable computational cost in cooler (M- type) atmospheres. We expect that the scheme also works for even cooler atmospheres. However, one simplification usually made is treating scattering as true absorption. De- pending on the specific dust grain properties this approximation might need to be re- placed by a more accurate treatment of scattering. Hence, another challenge a modeler might face is to device a computationally economic scheme to treat scattering in the time-dependent multi-D case. References Ackerman, A.S., Marley, M.S, 2001, ApJ 556, 872 Allard, F., Hauschildt, P.H., Alexander, D.R., Tamanai, A., Schweitzer, A., 2001, ApJ 556, 357 Burkert, A., Lin, D.N.C., Bodenheimer, P.H., Jones, C.A., Yorke, H.W., 2005, ApJ618, 512 Burrows, A., Sudarsky, D., Hubeny, I., 2006, ApJ 640, 1063 Cho,J. Y.-K., Menou,K., Hansen, B.M.S., Seager, S., 2003, ApJ 587, 117 Helling, Ch., 2005, in: Proceeding of he workshop on Interdisciplinary Aspects of Turbulence, Garching: Max-Planck-Institut für Astrophysik, eds.: F. Kupka, W. Hillebrandt, p. 152 Helling, Ch., Klein, R., Woitke, P., Nowak, U., Sedlmayr, E., 2004, A&A423, 657 Rossow, R.W., 1978, Icarus, 36, 1 Showman, A.P., Guillot, T., 2002, A&A385, 166 Tsuji, T., 2002, ApJ 575, 264 Woitke, P., Helling, Ch., 2003, A&A399, 297 Discussion C. Helling: Your wish list implies that no progress has been made in the brown dwarf modeling. Additionally, I am convinced that we will need to work on both sides: on 1D models which are fast and applicable, not only on 3D models though they will play an important role. Ludwig: My wish list was intended as overall collection of things we would like to understand about brown dwarf atmospheres. Progress related to the various points has indeed already been made. Concerning the mutual role of 1D and 3D models, I fully agree. 3D models should address crucial aspects that are in principle not accessible in 1D. Insight emerging from 3D models should then be transferred to 1D models. F. Kupka: Considering the complexity of molecular opacity I am actually surprised how robust the opacity binning seems to be. Ludwig: Tests have been performed for M-type stars where the effect of many millions of – primarily molecular – lines is captured quite well. As for brown dwarfs: the dust opacity has a rather smooth functional dependence on wavelength. Hence, it should be easy, but scattering is a problem. I.W. Roxburgh: You said overshooting was small, could you quantify this in terms of local scale height? Ludwig: The velocity amplitude declines exponentially with a scale height of about 1/3 of the local pressure scale height. For comparison: in solar models the scale height of decline is about six times larger. However, keep in mind that the hydrodynamical model presented here is experimental, in particular does not include any effects of dust formation. Introduction Micro-physical input Time scales: convection, radiation, dust, rotation, & numerics The multi-D story so far Serendipity Conclusions

0704.1297

The exceptionally extended flaring activity in the X-ray afterglow of GRB 050730 observed with Swift and XMM-Newton

Astronomy & Astrophysics manuscript no. grb050730 c© ESO 2018 November 1, 2018 The exceptionally extended flaring activity in the X–ray afterglow of GRB 050730 observed with Swift and XMM-Newton M. Perri1,2, D. Guetta2, L.A. Antonelli1,2, A. Cucchiara3, V. Mangano4, J. Reeves5, L. Angelini5, A.P. Beardmore6, P. Boyd5, D.N. Burrows3, S. Campana7, M. Capalbi1,2, G. Chincarini7,8, G. Cusumano4, P. Giommi1,9, J.E. Hill5, S.T. Holland5,10, V. La Parola4, T. Mineo4, A. Moretti7, J.A. Nousek3, J.P. Osborne6, C. Pagani3, P. Romano7, P.W.A. Roming3, R.L.C. Starling6, G. Tagliaferri7, E. Troja4,6, L. Vetere1,3 and N. Gehrels5 1 ASI Science Data Center, Via Galileo Galilei, I-00044 Frascati, Italy 2 INAF – Astronomical Observatory of Rome, Via Frascati 33, I-00040 Monte Porzio Catone (Rome), Italy 3 Department of Astronomy & Astrophysics, Pennsylvania State University, University Park, PA 16802, USA 4 INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica, Sezione di Palermo, Via La Malfa 153, I-90146 Palermo, Italy 5 NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA 6 Department of Physics & Astronomy, University of Leicester, Leicester LE1 7RH, UK 7 INAF – Astronomical Observatory of Brera, Via Bianchi 46, I-23807 Merate, Italy 8 Università degli Studi di Milano-Bicocca, Dipartimento di Fisica, Piazza delle Scienze 3, I-20126 Milano, Italy 9 Agenzia Spaziale Italiana, Unità Osservazione dell’Universo, Viale Liegi 26, I-00198 Roma, Italy 10 Universities Space Research Association, 10211 Wincopin Circle, Suite 500, Columbia, MD, 21044-3432, USA Received: 11 August 2006 / Accepted: 27 March 2007 ABSTRACT Aims. We observed the high redshift (z = 3.969) GRB 050730 with Swift and XMM-Newton to study its prompt and afterglow emission. Methods. We carried out a detailed spectral and temporal analysis of Swift and XMM-Newton observations. Results. The X–ray afterglow of GRB 050730 was found to decline with time with superimposed intense flaring activity that extended over more than two orders of magnitude in time. Seven distinct re-brightening events starting from 236 s up to 41.2 ks after the burst were observed. The underlying decay of the afterglow was well described by a double broken power-law model with breaks at t1 = 237 ± 20 s and t2 = 10.1 −2.2 ks. The temporal decay slopes before, between and after these breaks were α1 = 2.1 ± 0.3, α2 = 0.44 +0.14 −0.08 and α3 = 2.40 +0.09 −0.07, respectively. The spectrum of the X–ray afterglow was well described by a photoelectrically absorbed power-law with an absorbing column density NzH=(1.28 +0.26 −0.25) × 10 22 cm−2 in the host galaxy. Evidence of flaring activity in the early UVOT optical afterglow, simultaneous with that observed in the X–ray band, was found. Strong X–ray spectral evolution during the flaring activity was present. The rise and decay power-law slopes of the first three flares were in the range 0.8–1.8 using as zero times the beginning and the peak of the flares, respectively. In the majority of the flares (6/7) the ratio ∆t/tp between the duration of the event and the time when the flare peaks was nearly constant and ∼ 0.6–0.7. We showed that the observed spectral and temporal properties of the first three flares are consistent with being due both to high-latitude emission, as expected if the flares were produced by late internal shocks, or to refreshed shocks, i.e. late time energy injections into the main afterglow shock by slow moving shells ejected from the central engine during the prompt phase. The event fully satisfies the Ep–Eiso Amati relation while is not consistent with the Ep vs. Ejet Ghirlanda relation. Key words. gamma rays: bursts – X–rays: individual (GRB 050730) 1. Introduction The successful launch on 2004 November 20 of the Swift Gamma–ray Burst Explorer (Gehrels et al. 2004) has opened a new era in the study of Gamma Ray Bursts (GRBs). The autonomous and rapid slewing capabilities of Swift allow the prompt (1–2 minutes) observation of GRBs, discovered and localised by the wide-field gamma–ray (15–350 keV) Burst Alert Telescope (BAT, Barthelmy et al. 2005a), with the two Send offprint requests to: e-mail: [email protected] co-aligned narrow-field instruments on-board the observatory: the X–Ray Telescope (XRT, Burrows et al. 2005a), operating in the 0.2–10 keV energy band, and the Ultraviolet/Optical Telescope (UVOT, Roming et al. 2005), sensitive in the 1700– 6000 Å band. The Swift unique fast pointing capability is cru- cial in the X–ray energy band, where the reaction times of other satellites are limited to time scales of several hours. With Swift, thanks also to the XRT high sensitivity, it is possible for the first time to study in detail the evolution of the X–ray afterglows of GRBs during their very early phases. http://arxiv.org/abs/0704.1297v1 2 M. Perri et al.: The exceptionally extended flaring activity in the X–ray afterglow of GRB 050730 Indeed, one of the main results of Swift is the identification of unexpected and complex features in the early X–ray afterglows. In particular, three distinct phases are observed in the majority of GRB light curves: an early (t<500 seconds from the trigger) steep decline, with a power-law index of ∼ 3, a second (t<10 ks) very shallow phase with a slope of ∼ 0.5, a third phase characterized by a more conventional decay slope of ∼ 1 (e.g. Tagliaferri et al. 2005; Cusumano et al. 2006a; Nousek et al. 2006; O’Brien et al. 2006). In a few cases (GRB 050315, Vaughan et al. 2006; GRB 050318, Perri et al. 2005; GRB 050505, Hurkett et al. 2006; GRB 050525A, Blustin et al. 2006; GRB 060614, Mangano et al. 2007) a further steepening with a decay slope of ∼ 2, consistent with a jet break, is observed. Moreover, Swift had detected in about one half of the bursts strong flaring activity in the X–ray energy band superimposed on the afterglow decay (e.g. Burrows et al. 2005b; Romano et al. 2006; Falcone et al. 2006). The understanding of the origin of these bright X–ray flares is intensively discussed in the literature. A mech- anism proposed as responsible for the flaring activity is late internal shocks (Burrows et al. 2005b; Fan & Wei 2005; Zhang et al. 2006; Liang et al. 2006). In this scenario the X– ray flares are produced by the same internal dissipation pro- cesses which cause the prompt emission, likely internal shocks within the expanding fireball occurring before it is decel- erated in the external medium (e.g. Rees & Mészáros 1994). This model requires that the GRB central engine is still ac- tive after the end of the prompt emission and various mech- anisms providing such extended internal activity have been put forward (e.g. King et al. 2005; Perna et al. 2006). An al- ternative scenario has been recently considered by Guetta et al. (2007) who, based on a detailed analysis of the X–ray flaring activity observed in the afterglow of GRB 050713A, interpreted the X–ray flares as due to refreshed shocks, i.e. late time collisions with the main afterglow shock of slow- moving shells ejected from the central engine during the prompt phase (Rees & Mészáros 1998; Kumar & Piran 2000; Sari & Mészáros 2000). In this paper we present a detailed analysis of Swift and XMM-Newton observations of GRB 050730, focusing on the intense and extended X–ray flaring activity that characterizes its afterglow. In Section 2 the observations and the data re- duction are presented, in Section 3 we describe the tempo- ral analysis and Section 4 is dedicated to the spectral anal- ysis. The results are discussed in Section 5 and finally in Section 6 we summarize our findings. Throughout this pa- per errors are quoted at the 90% confidence level for one pa- rameter of interest (∆χ2 = 2.71) unless otherwise specified. We adopted the standard ΛCDM cosmological parameters of Ωm = 0.27, ΩΛ = 0.73 and H0 = 70 km s −1 Mpc−1. Times are referenced from the BAT trigger T0 while temporal and spec- tral indices are written following the notation F(t, ν) ∝ t−αν−β. Results on the optical spectrum of the afterglow of this GRB are reported by Starling et al. (2005), Chen et al. (2005a) and Prochaska et al. (2006). Multi-wavelength observations of the afterglow of GRB 050730 are presented by Pandey et al. (2006). −100 0 100 200 300 Time since BAT trigger (s) Swift BAT Fig. 1. BAT 20–150 keV background subtracted light curve of the prompt emission of GRB 050730. Data are binned to 10 seconds resolution and errors are at the 1σ level. The horizontal dashed line indicates the 0 level. 2. Observations and data reduction 2.1. Swift BAT The BAT detected and located GRB 050730 at T0=19:58:23 UT on 2005 July 30 (Holland et al. 2005). On the basis of the refined ground analy- sis (Markwardt et al. 2005), the BAT position is RA(J2000)=212.◦063, Dec(J2000)=−3.◦740, with a 90% containment radius of 3′. The BAT prompt emission light curve (Fig. 1) is characterized by a duration of T90 = 155±20 s. The emission starts ∼ 60 s before the trigger, peaks at ∼ 10 s after the trigger and declines out to ∼ 180 s after the trigger. The time averaged (over T90) spectrum in the 15–150 keV energy band is well described by a power-law model with en- ergy index βBAT = 0.5 ± 0.1 and χ r = 0.71 (with 56 degrees of freedom, d.o.f.). The total fluence in the 15–150 keV band is (2.4 ± 0.3) × 10−6 erg cm−2. Assuming as redshift z = 3.969 (Chen et al. 2005b, see Sect. 2.2), the isotropic-equivalent radi- ated energy in the BAT bandpass (74.5–745.4 keV in the burst rest frame) is EBATiso = (8.0 ± 1.0) × 10 52 erg. 2.2. Ground-based Observatories Following the identification of the optical counterpart by UVOT (Holland et al. 2005), the field of GRB 050730 was ob- served by numerous ground-based telescopes. The afterglow detections in the R (Sota et al. 2005), r′ (Gomboc et al. 2005), I and J (Cobb et al. 2005) bands were soon distributed via the GRB Circular Network (GCN). Observations in the optical band with the MIKE echelle spectrograph on Magellan II led to the GRB redshift measurement z = 3.969 (Chen et al. 2005b) M. Perri et al.: The exceptionally extended flaring activity in the X–ray afterglow of GRB 050730 3 based on the detection of a strong hydrogen Lyα absorption line and of several absorption lines from other elements. The red- shift was later confirmed with observations by the ISIS spectro- graph on the William Herschel Telescope (Rol et al. 2005), the IMACS imaging spectrograph on the Magellan Observatory Baade Telescope (Holman et al. 2005a), FORS1 and UVES on the Very Large Telescope (D’Elia et al. 2005). The optical afterglow decay at later times was followed by several optical telescopes. In particular, measurements were made in the R band up to ∼4 days after the trig- ger (Holman et al. 2005a, 2005b; Burenin et al. 2005; Klotz et al. 2005; Damerdji et al. 2005; D’Elia et al. 2005; Bhatt & Sahu 2005; Kannappan et al. 2005). Finally, a radio afterglow with a flux density Fr = 145 ± 28 µJy at 8.5 GHz was detected about 2 days after the trigger using the Very Large Array (Cameron 2005). 2.3. Swift UVOT Swift UVOT began to observe the field of GRB 050730 at 20:00:22 UT, 119 seconds after the BAT trigger. The first 100 seconds exposure in the V band led to the identifi- cation of the optical afterglow at RA(J2000)=14h08m17.s09, Dec(J2000)=-03◦46′18.′′9 (Holland et al. 2005). We refined the preliminary photometric analysis (Blustin et al. 2005) processing all UVOT data using the standard UVOT software package (Swift software v. 2.1 in- cluded in the HEAsoft package v. 6.0.2). The flux in all filters was estimated by integrating over a 3.5′′ region. A background region for subtraction of the sky contribution to the flux has been selected in a relatively empty part of the field of view. The results are listed in Table 1. No significant detection was found in the U and UV filters, which is consistent with the high redshift measured for this GRB. All the magnitudes are corrected for Galactic extinction (E(B − V) = 0.049, Schlegel et al. 1998). 2.4. Swift XRT The XRT observations of the GRB 050730 field started at 20:00:28 UT, 125 seconds after the BAT trigger, with the in- strument in Auto State. After a first exposure in Image Mode (see Hill et al. 2004 for a description of readout modes) dur- ing which no on-board centroid was determined, the instrument switched into Windowed Timing (WT) mode for the entire first Swift orbit from T0+133 s to T0+794 s. Starting from the sec- ond orbit (T0+4001 s), the instrument was in Photon Counting (PC) mode for 26 consecutive orbits until 11:49:22 UT on 2005 August 1 (T0+143.4 ks). The field of GRB 050730 was re- observed with the XRT from August 3 starting at 15:28:11 UT (T0+329.4 ks) until August 5 13:57:02 UT (T0+496.7 ks). The XRT data were processed with the XRTDAS soft- ware (v. 1.7.1) included in the HEAsoft package (v. 6.0.4). Event files were calibrated and cleaned with standard filter- ing criteria with the xrtpipeline task using the latest calibration files available in the Swift CALDB distributed by HEASARC. Events in the energy range 0.3–10 keV with grades 0–12 (PC mode) and 0–2 (WT mode) were used in the analysis (see Burrows et al. 2005a for a definition of XRT event grades). After the screening, the total exposure time for the first XRT observation was 649 seconds (WT) and 58480 seconds (PC), while for the follow-up observation the PC exposure time was 34669 seconds. In the 0.3–10 keV PC image of the field a previ- ously uncatalogued X–ray source was visible within the BAT error circle with coordinates RA(J2000)=14h08m17.s2, Dec(J2000)=−03◦46′19′′. This position, derived using data not affected by pile-up (orbits 5–26, see below), has a 90% uncertainty of 3.5′′ using the latest XRT bore- sight correction (Moretti et al. 2006) and is consistent with the position of the optical counterpart (Holland et al. 2005; Jacques & Pimentel 2005). For the WT mode data, events for temporal and spectral analysis were selected using a 40 pixel wide (1 pixel corre- sponds to 2.36′′) rectangular region centered on the afterglow and aligned along the WT one dimensional stream in sky co- ordinates. Background events were extracted from a nearby source-free rectangular region of 50 pixel width. For PC mode data, the source count rate during orbits 2–4 was above ∼ 0.5 counts s−1 and data were significantly affected by pile-up in the inner part of the Point Spread Function (PSF). After comparing the observed PSF profile with the analytical model derived by Moretti et al. (2005), we removed pile-up effects by excluding events within a 5 pixel radius circle centered on the afterglow position and used an outer radius of 30 pixels. From orbit 5 the afterglow brightness was below the pile-up limit and events were extracted using a 10 pixel radius circle, which encloses about 80% of the PSF at 1.5 keV, to maximize the signal to noise ratio. The background for PC mode was estimated from a nearby source-free circular region of 50 pixel radius. Source count rates for temporal analysis were corrected for the fraction of PSF falling outside the event extraction regions. Moreover, the loss of effective area due to the presence of 2 CCD hot columns within the extraction regions was properly taken into account. The count rates were then converted into unabsorbed 0.3–10 keV fluxes using the conversion factor derived from the spectral analysis (see Sect. 4). For the spectral analysis, ancillary response files were gen- erated with the xrtmkarf task applying corrections for the PSF losses and CCD defects. The latest response matrices (v. 008) available in the Swift CALDB were used and source spectra were binned to ensure a minimum of 20 counts per bin in order to utilize the χ2 minimization fitting technique. 2.5. XMM-Newton XMM-Newton follow-up observations of GRB 050730 started 26.4 ks (for the two EPIC-MOS cameras) and 29.4 ks (for the EPIC-PN) after the initial BAT Trigger. The XMM- Newton ODF (Observation Data Files) data were pro- cessed with the epproc and emproc pipeline scripts, using the XMM-Newton SAS analysis package (v. 6.5). A bright rapidly decaying source was detected near the aim-point of 4 M. Perri et al.: The exceptionally extended flaring activity in the X–ray afterglow of GRB 050730 Fig. 2. Swift XRT (filled circles) and XMM-Newton (filled squares) 0.3–10 keV light curve of the X–ray afterglow of GRB 050730. The BAT 20–150 keV prompt emission light curve, extrapolated to the 0.3–10 keV band, is also shown as filled triangles. UVOT optical data in the V and B bands, arbitrarily scaled for comparison with the X–ray band, are indicated with open squares and open triangles, respectively. The solid line is the best fit model to the XRT and XMM-Newton light curve. The dashed line represent the underlying double broken power-law decay (see Sect. 3.1). The dotted line is a power-law model with temporal decay index αV = 0.3 normalized to fit data points in the V band. Errors are at the 1σ level. all three EPIC detectors and the afterglow was localized at RA(J2000)=14h08m17.s3, Dec(J2000)=-03◦46′18.′′5. The dura- tion of the XMM-Newton follow-up observation was 33.7 ks (MOS) and 30.4 ks (PN). After screening out times with high background flaring, the dead-time corrected net exposures were 25.0 ks (MOS) and 17.9 ks (PN). All three EPIC cameras (PN and 2 MOS) were used in Full Window Mode, with PN and MOS2 using the “Thin” filter and MOS1 using the “Medium” optical blocking filter. Source spectra and light curves for all 3 EPIC cameras were extracted from circular regions of 30′′ radius centered on the afterglow. Background data were taken from a 60′′ radius cir- cle on the same chip as the afterglow, but free of any back- ground X–ray sources. The data were further screened by in- cluding only good X–ray events (using the selection expres- sion flag=0 in evselect), by including events with single and double pixel events (pattern<=4) for the PN and by selecting single to quadruple pixel events (pattern<=12) for the MOS. Data below 300 eV and above 10 keV were also removed. For the temporal analysis, we adopted the light curve from the MOS data, primarily because the PN data was heavily af- fected by background flares towards the end of the observation, while the MOS covered a wider duration at the beginning of the observation. The data from the two MOS detectors were com- bined and the count rate to 0.3–10 keV unabsorbed flux conver- sion factor was calculated from the best fit absorbed power-law spectrum (see Sect. 4). For the spectral analysis, ancillary and redistribution re- sponse files for fitting were generated with the SAS tasks ar- fgen and rmfgen, respectively. Moreover, source spectra were binned to a minimum of 25 counts per bin. 3. Temporal analysis The background subtracted 0.3–10 keV Swift XRT and XMM-Newton light curves of the X–ray afterglow of GRB 050730 are shown in Fig. 2. The same figure also shows the BAT 20–150 keV prompt emission light curve, converted in the 0.3–10 keV energy band using the BAT spectral best fit model which is valid also in the XRT bandpass (see Sect. 4.1). The UVOT optical light curves in the V and B bands, in arbitrary units, are also plotted (see Sect. 2.3). M. Perri et al.: The exceptionally extended flaring activity in the X–ray afterglow of GRB 050730 5 Fig. 3. Swift XRT 0.3-10 keV light curve of the GRB 050730 X–ray afterglow during the first orbit. The solid line is the best fit model to the data obtained considering a linear rise exponen- tial decay for the three flares (see Sect. 3.1). The dashed vertical lines delimit the seven time intervals considered for the spectral analysis. Data are binned to 8 seconds resolution and errors are at the 1σ level. 3.1. X–ray afterglow The X–ray afterglow of GRB 050730 is characterized by a very complex structure. The first Swift orbit (from T0+133 s to T0+794 s), after an initial steep decay phase that joins well with the end of the BAT prompt emission, is dominated by three bright X–ray flares peaking at about 235, 435 and 685 seconds after the BAT trigger. Taking into account cosmological time dilation these times correspond to about 47, 88 and 138 sec- onds in the GRB rest frame. A flaring episode is also observed in the second orbit peaking at about T0+4500 s. While the un- derlying decay of the afterglow during the first two orbits is shallow, starting from the third orbit (T0+10 ks) the afterglow light curve shows a much steeper decline with superimposed flaring activity. We first modeled the X–ray light curve of the afterglow with a double broken power-law model with slopes α1, α2, α3 and temporal breaks t1, t2, describing the underlying power-law decay of the afterglow, plus seven Gaussian functions model- ing the flaring episodes. We found for the first power-law a decay index α1 = 2.1 ± 0.3 followed, after a first time break at t1 = 237±20 s, by a shallower decay with index α2 = 0.44 +0.14 −0.08. A second temporal break is found at t2 = 10.1 −2.2 ks after which a steep decay with index α3 = 2.40 +0.09 −0.07 is observed. This model did not provide a good fit (χ2r = 1.73, 143 d.o.f.), mostly due to short time scale fluctuations and to deviations of the three first bright flares from a symmetric Gaussian shape. We thus considered a different functional form for the three X–ray flares, namely a linear rise exponential decay: F(t) ∝ (t − t0)/(tp − t0) for times between the flare start time t0 and Fig. 4. Swift XRT 0.3–1.5 keV (upper panel) and 1.5–10 keV (middle panel) light curve of the GRB 050730 X–ray afterglow during the first Swift orbit. In the lower panel the corresponding hardness ratio is plotted. Data are binned to 12 seconds reso- lution and error bars indicate statistical uncertainties at the 1σ level. peak time tp, F(t) ∝ exp[−(t − tp)/tc] for t > tp where tc is the exponential decay time. The model improves significantly the fit with χ2r = 1.43 (140 d.o.f.) and F-test chance probability of 1.3 × 10−6. The best fit parameters of the overall underlying double broken power-law model were unchanged with respect to the previous fit listed above. The linear rise exponential de- cay best fit parameters for the first three flares are given in Table 2 while Table 3 reports the Gaussian best fits for the other four flares. As Fig. 3 illustrates, the second brightest flare (referred to as flare 2 in the following) has a flux variation of amplitude ∼ 3.6 and is characterized by a steep rise, lasting ∼ 90 seconds, followed by a slower decay with duration ∼ 170 seconds. An asymmetric shape, with a steep rise followed by a shallower decay, was also observed for the other two flares (flare 1 and 3) and for these episodes the flux variation was ∼ 1.7 and ∼ 2.6, respectively. The ratio between the duration (∆t) and the peak time (tp) of the X–ray flares was calculated using the linear rise exponential decay and Gaussian best fit parameters for the first three and the last four flares, respectively. For both analytical models the duration of a flare was defined as the time interval during which the flare intensity was above 5% of its peak value. The results are given in Tables 2 and 3. The rise and decay portions of the first three flares were also fit with single power-laws. For the estimation of power- law indices, times were expressed from the onset of the flares (t0) for the rise, from the peak time (tp) for the decay, and the contribution of the underlying afterglow decay was taken into account. The power-law temporal indices for the rising (αr) and decaying (αd) portions of the three flares are reported in Table 6 M. Perri et al.: The exceptionally extended flaring activity in the X–ray afterglow of GRB 050730 The temporal behavior of the afterglow during the first three X–ray flares was also studied in different energy bands. In Fig. 4 we show the 0.3–1.5 keV (upper panel) and 1.5–10 keV (middle panel) light curves of the early X–ray afterglow together with the corresponding hardness ratio (lower panel). From the figure it is apparent that i) in the harder band the pro- file of the flares is sharper and ii) a peak time shift, with flares in the hard band peaking at earlier times, is observed. We note that these temporal properties as a function of energy are the same observed for the prompt emission pulses of GRBs. As already mentioned, late flaring activity of the X–ray af- terglow was observed at ∼ 4.5, 10.4, 18.7 and 41.2 ks after the trigger. Due to the Swift orbital gaps the temporal cover- age of flares 4, 5 and 6 is poor and these episodes could not be well studied, but the last flare was entirely covered by the XMM-Newton follow-up observation allowing a detailed tem- poral analysis. In Fig. 5 the XMM-Newton (MOS1+MOS2) light curve is shown together with the XRT curve covering the same time interval. A very good agreement between the two curves is found and the ∼ 5% higher normalization of the XRT data points is of the same order of the uncertainties in the abso- lute flux calibration of the instrument (Campana et al. 2006). We first fit the XMM-Newton light curve with a single power-law decay and obtained an extremely poor fit to the light curve, with a fit statistic of χ2r = 8.81 (32 d.o.f.) and a decay index of α = 2.10 ± 0.04. The light curve was then parameterized with a long duration flare super-imposed on a steep power-law decay. We obtained an underlying decay in- dex α = 2.45 ± 0.15, while the flare could be adequately mod- eled with a Gaussian function; the fit statistic was acceptable (χ2r = 1.22, 29 d.o.f.). The long duration flare peaked at 41 ks (or 8.2 ks in the GRB rest frame), with a σ of 6.8 ks (1.4 ks rest frame). The total fluence of the flare when parameterized this way was 8.3 × 1050 erg (1.5-50 keV band in the GRB rest frame), which represents 20% of the integrated afterglow emis- sion over the XMM-Newton observation. The rise and decay phases of flare 7 were also fit with single power-laws express- ing times from the onset and from the peak time of the flare, respectively. The measured temporal indices αr and αd are also reported in Table 4. 3.2. Optical band The UVOT optical light curve of the afterglow of GRB 050730 in the V and B bands is also characterized by a complex behavior, as is illustrated in Fig. 2. The early (T0+100 s - T0+800 s) UVOT V light curve re- veals flaring activity. A re-brightening is in fact observed at ∼ T0+500 s, almost simultaneously with the brightest X–ray flare observed with XRT. Due to the relatively poor sampling of the UVOT light curve, a detailed temporal analysis is not possi- ble, however we note that the amplitude of the flux variation in the V band (∼ 3) is of the same order of the one measured in the X–ray energy band. The decay of the afterglow in the V band, up to about T0+12 ks and excluding the re-brightening episode, is shallow (αV ∼ 0.3) and in agreement with the one measured in the X–ray band (αX = 0.44 +0.14 −0.08). Hints of a steepening of Fig. 5. XMM-Newton (MOS1+MOS2, filled circles) and Swift XRT (open circles) 0.3–10 keV light curve of the late (∼ T0+12 hours) X–ray afterglow of GRB 050730. The solid line is the best fit model to the XMM-Newton light curve. The dashed line represent the underlying power-law decay (see Sect. 3.1). Data points errors are at the 1σ level. the V light curve at ∼ T0+12 ks are also found, but the limited temporal coverage and the large statistical uncertainties char- acterizing the UVOT data points do not allow us to constraint the late optical afterglow decay. The UVOT light curve in the B band is characterized by a flat power-law decay (αB ∼ 0.3) without re-brightening events. However, it should be noted that i) the curve is very sparsely sampled, with only three data points during the first Swift orbit, and ii) in the B band there is a strong flux reduction due to the Lyman break and thus large statistical uncertainties affect the data points. Accurate optical observations of the afterglow of GRB 050730 in the R and I bands are presented by Pandey et al. (2006). The authors report an early time flux decay with indices αR = 0.54± 0.05 and αI = 0.66± 0.11 followed, after a temporal break at about 9 ks after the trigger, by a strong steep- ening with decay indices αR = 1.75±0.05 and αI = 1.66±0.07. 4. Spectral analysis For the spectral analysis of Swift XRT and XMM-Newton data we used the XSPEC package (v. 11.3.2, Arnaud 1996) included in the HEAsoft package (v. 6.0.4). 4.1. XRT As a first step, the 0.3–10 keV XRT average spectrum during the first Swift orbit (WT mode, from T0+133 s to T0+794 s) was fit adopting a single power-law model with the neutral hydrogen-equivalent absorption column density fixed at the Galactic value in the direction of the GRB (NGH = M. Perri et al.: The exceptionally extended flaring activity in the X–ray afterglow of GRB 050730 7 3.0 × 1020 cm−2, Dickey & Lockman 1990). The fit obtained was poor (χ2r = 1.28, 311 d.o.f.). From the inspection of residuals a strong deficit of counts at low energies, likely due to the presence of an absorbing column density in excess to the Galactic one, was found. Indeed, the addition of a col- umn density NzH using Solar metallicity redshifted to the rest frame of the GRB host (z = 3.969) free to vary in the spec- tral fit (zwabs model in XSPEC) resulted in a more acceptable fit with a column density of NzH = (1.28 +0.26 −0.25) × 10 22 cm−2 (see Table 5). This value is consistent with the neutral hydrogen column densities derived from the optical spectra reported by Starling et al. (2005) and Chen et al. (2005a). From Fig. 4 it is apparent that strong spectral evolution takes place during the intense flaring activity observed in the first Swift orbit. We thus split the WT observation in seven time intervals to study the spectra during the rise and the de- cay portions of each flare (see Fig. 3). The time-resolved spec- tral best fits (see Table 5) clearly show evidence for spectral variation during the flares and an overall softening of the spec- tra with time associated with a decrease of the rest frame col- umn density. The time-resolved spectral analysis was also per- formed adopting a broken power-law model to investigate the possible presence of spectral breaks (e.g. Falcone et al. 2006; Guetta et al. 2007). Also in this case an additional absorption column density NzH at the rest frame of the GRB host was con- sidered. For all segments we did not find evidence for spectral breaks within the XRT energy band. The XRT late 0.3–10 keV spectrum (PC mode, from T0+4.0 ks to T0+143.4 ks) was also fit using a single power- law model with a fixed Galactic absorption column density and an additional absorbing column at the burst rest frame. The ob- servation was divided in two time intervals (4.0–18.1 ks and 21.3–143.4 ks from the trigger). The results are listed in Table 5 where we see that spectral softening between the two time in- tervals was found. 4.2. XMM-Newton The PN and MOS 0.3–10 keV spectra were fit jointly al- lowing the cross normalization between the detectors, which is consistent within < 5%, free to vary . The two MOS spectra and responses were combined to maximize the signal to noise, after first checking that they were consistent with each other. As in the case of the XRT spectrum, the PN and MOS spec- tra were first fit with a single power-law model with the neu- tral hydrogen-equivalent absorption column fixed at the known Galactic value. The fit obtained was not acceptable (χ2r = 1.40, 489 d.o.f.), while the energy index was β = 0.76 ± 0.02. The addition of a neutral absorption column in the GRB host galaxy frame at z = 3.969 resulted in more acceptable fit (χ2r = 1.14, 489 d.o.f.) with an excess absorption column above the Galactic column of (6.8 ± 1.0) × 1021 cm−2, while the con- tinuum energy index was now β = 0.87 ± 0.02. The XMM-Newton afterglow spectra were also sliced into three segments of approximately 10 ks in duration, in order to search for any spectral evolution within the XMM-Newton ob- servation. A small change in the continuum parameters was found, the spectrum evolved from hard to soft; the energy index changed from β = 0.87 ± 0.03 in the first 10 ks, to β = 0.99 ± 0.05 during the final segment. No evidence was found for a change in the column density, which was subse- quently fixed at NH = 6.8 × 10 21 cm−2 in all the segments. The spectral best fit parameters are shown in Table 5. We also searched for any evidence of emission lines, either in the mean spectrum, or in the three segments. No statistically significant lines were found, at the level of > 99% confidence. As the redshift of the burst is known, then we can set an upper- limit to the equivalent width of any emission lines. Over the range of 0.4–8 keV and using the mean spectrum, we found a < 100 eV upper limit to any emission lines. More specifically we can set a limit on the iron Kα line (e.g. the H-like line at 6.97 keV rest frame, 1.40 keV observed frame) of < 30 eV. 5. Discussion 5.1. Early X–ray light curve This GRB, with its rather high redshift, z = 3.969, gives us the possibility to investigate the X–ray and optical light curves soon after the trigger and thus to study in detail the soft tail of the prompt emission and the very beginning of the afterglow phase. Indeed, due to cosmological time dilation the XRT and UVOT observations started, in the rest frame of the burst, only 27 seconds after the trigger. Other examples of such early observations of high redshift bursts are GRB 050904 (z = 6.29, Cusumano et al. 2006b) and GRB 060206 (z = 4.0, Monfardini et al. 2006). The XRT light curve shows at the very beginning (133– 205 s from the BAT trigger) a rapidly decaying emission that joins quite nicely with the BAT flux when converted to the XRT bandpass, a feature that has been observed in various Swift bursts (Tagliaferri et al. 2005, Barthelmy et al. 2005b; Nousek et al. 2006; O’Brien et al. 2006). For GRB 050730, the steep temporal decay index (α1 = 2.1 ± 0.3, see Sect. 3.1) characterizing the early X–ray light curve sug- gests that the observed emission is likely associated with the tail of the prompt emission rather than to the beginning of the shocked inter-stellar medium afterglow phase. Indeed, in the internal shock model scenario for the prompt γ–ray emission (e.g. Rees & Mészáros 1994), the cessation of the emissivity is characterized by a rapid decay due to the de- layed arrival of the high angular latitude prompt emission of the shocked surface (high-latitude or curvature emission, Kumar & Panaitescu 2000; Dermer 2004). The high-latitude emission predicts a relationship between the temporal decay in- dex αd and the spectral index βd during the decay given by the equation αd = 2+βd (Kumar & Panaitescu 2000) where the de- cay slope is measured using as zero time (t0) the beginning of the prompt emission (F(t, ν) ∝ (t − t0) −αdν−βd ). We have tested this prediction by fitting the early XRT light curve (segment “0” in Fig. 3) with the above constraint using the spectral index measured during the decay (βd = 0.42 ± 0.08, see Table 5) and leaving the zero time as a free parameter (see Liang et al. 2006 for a description of the method). We found t0 = −31 −31 s, i.e. 8 M. Perri et al.: The exceptionally extended flaring activity in the X–ray afterglow of GRB 050730 the zero time is located at the rising segment of the BAT prompt emission light curve (see Fig. 1), as expected in the framework of the high-latitude emission. This result indicates that the steep decay observed in the early XRT light curve is most likely the soft tail of the BAT emission. This hypothesis is also supported by the BAT and early (segment “0” in Fig. 3) XRT spectra. We found very similar spectral slopes (βBAT = 0.5 ± 0.1 and βXRT = 0.42 ± 0.08, respectively), likely indicating that the early X–ray and γ–ray emissions are produced by the same emission mechanism. 5.2. Flaring activity The exceptionally extended flaring activity of the X–ray af- terglow of GRB 050730 allows us to study this phenomenon over more than two orders of magnitude in time. The XRT early steep decay is followed by three bright X–ray flares peak- ing at 236, 437 and 685 seconds after the BAT trigger (see Fig. 3). These flares were under the sensitivity of the BAT instrument and thus were not detected in the hard X–ray en- ergy band. These flares show, as other strong flares observed with Swift (e.g. Romano et al. 2006; Falcone et al. 2006; Pagani et al. 2006; Godet et al. 2006; Chincarini et al. 2006), a clear spectral evolution with the hardness ratio that mimics the variation of the light curve. A phase lag, with the harder (E > 1.5 keV) light curve peaking at earlier times with respect to the softer energy band is also found (see Fig. 4). So far, most of the X–ray flares observed in Swift X– ray light curves have been interpreted as due to late in- ternal shocks (e.g. Burrows et al. 2005b; Romano et al. 2006; Falcone et al. 2006; Godet et al. 2006). In this scenario the GRB central engine is active far beyond the end of the prompt γ–ray emission phase requiring new mechanisms ca- pable of powering new relativistic outflows at late-time (e.g. King et al. 2005; Perna et al. 2006). A diagnostic to check if the re-brightenings are due to late internal shocks has been re- cently proposed by Liang et al. (2006). In the internal-origin scenario for X–ray flares the decay emission of the flaring episodes should be dominated by high-latitude emission with the decay temporal index related to the decay spectral index as αd = βd + 2 (Kumar & Panaitescu 2000). We have checked this hypothesis for the first three bright X–ray flares observed in the GRB 050730 afterglow. The spectral indices measured during the decays were used (segments “1b”, “2b” and “3b”, see Fig. 3 and Table 5) and we found best fit zero times t0,1 =193 −11 s, t0,2 =341 −42 s and t0,3 =592 −93 s, respectively. We can see that for all three episodes the zero time values are located at the be- ginning of the corresponding flare and thus the observed decay slopes are consistent with being due to high-latitude emission as predicted by the late internal shock scenario. Another possibility is that the X–ray flares are pro- duced by refreshed shocks (Rees & Mészáros 1998; Kumar & Piran 2000; Sari & Mészáros 2000). In the standard internal-external fireball model (e.g. Rees & Mészáros 1994; Sari & Piran 1997) re-brightening episodes in the afterglow light curve are explained as slow shells ejected from the central engine during the prompt phase that catch up with the main afterglow shock after it has decelerated in the external inter-stellar medium. Indeed, Guetta et al. (2007) have recently interpreted the X–ray flares of the afterglow of GRB 050713A as due to refreshed shocks. Here we applied the diagnostic proposed for GRB 050713A to the case of GRB 050730. First, to estimate the rise and decay slopes of the early flares we operationally selected as zero time the beginning and the peak of the flare, respectively. This assumption implicitly means that the flare is completely independent from the main event generating the prompt emission and the forward shock light curve. We found temporal indices in the 0.8–1.8 range (see Table 4). Much steeper slopes (∼ 3–6) are obtained if instead times are referenced from the BAT trigger, as usually done for most of the Swift X–ray flares (e.g. Burrows et al. 2005b; Romano et al. 2006; Falcone et al. 2006). Second, we checked if the temporal and the spectral indices during the decay phase are related as predicted by the standard afterglow model (Sari, Piran & Narayan 1998). We restricted the analysis to the episodes which have enough statistics to allow a detailed temporal and spectral study, i.e. the first three flares and the last one. For the decay phases we expect the following relations to hold (Sari, Piran & Narayan 1998; Dai & Cheng 2001): Fν ∝ (νm/νc) −1/2(ν/νm) −p/2Fν,max ∝ t −3p/4+1/2 (for p > 2) and Fν ∝ (νm/νc) (p−1)/2(ν/νc) −p/2Fν,max ∝ t −(3p+10)/16 (for 1 < p < 2). From the best fit spectral energy indices during the decay of the four flares (see Table 5) we derived the predicted temporal slopes α1d = 0.93 ± 0.05, α d = 0.89 ± 0.03, α3d = 1.02 ± 0.15, and α d = 1.40 ± 0.05, which are consistent with the measured values of the decay indices listed in Table 4. We note that for the last flare, peaking at 41.2 ks after the trigger, we used the afterglow model relations for the slow cooling case (Fν ∝ (ν/νm) −(p−1)/2Fν,max ∝ t 3(1−p)/4) which very likely applies at these late times (Sari, Piran & Narayan 1998). Important information on the mechanism producing the flares can also be obtained from the comparison between the observed variability timescale and the time at which the flare is observed (Ioka et al. 2005). We calculated, for all the 7 re- brightenings of the GRB 050730 afterglow, the ratio between the duration of the flare (∆t) and the time when the flare peaks (tp) (see Tables 2 and 3). We found ∆t/tp ∼ 0.3 for the first flare and ∆t/tp ∼ 0.6 − 0.7 for the others, in agreement with the ∆t/tp > 0.25 limit discussed by Ioka et al. (2005) for the refreshed shock scenario. Remarkably, the ∆t/tp ratio is nearly constant for all flares, with the exception of the first one which occurred during the bright and steep tail of the prompt emis- sion (see Fig. 3) and most likely we are observing only the tip of the flare. A duration of the flare proportional to tp, as ob- served in the afterglow of GRB 050730, is explained both in the refreshed shock scenario (Kumar & Piran 2000) and in the late internal shock model (Perna et al. 2006). We note that the last four flares have ∆t/tp values in the range 0.7–0.9, while for flares 2 and 3 we observe slightly lower ratios (0.5–0.6), possibly indicating a moderate temporal evolution of ∆t/tp. However, due to the relatively large error bars, the measured values are also consistent with being constant and ∼ 0.6 − 0.7. Flaring activity is also observed in the optical afterglow of GRB 050730. Indeed, the UVOT V early light curve (see Fig. 2) shows a re-brightening at ∼ T0+500 s, almost simul- M. Perri et al.: The exceptionally extended flaring activity in the X–ray afterglow of GRB 050730 9 taneously with the brightest X–ray flare observed with the XRT peaking at T0+437 s. Moreover, the optical re-brightening amplitude (∼ 3) is of the same order of the flux variation observed in the X–ray energy band. Strong indications of correlated variability between the X–ray and optical energy bands are also present in the NIR/Optical light curves re- ported by Pandey et al. (2006): a significant re-brightening at about T0+4 ks (R filter), close to the X–ray flare peaking at T0+4.5 ks (see Fig. 2), is observed. Moreover, a bump in the J, I, R and V light curves at about T0+10 ks is observed (Pandey et al. 2006), again simultaneous with the X–ray flare peaking at T0+10.4 ks. Although the X–ray flaring activity is not uniformly covered by optical observations, we find several indications of simultaneous re-brightening events in the X–ray and optical bands, in agreement with the refreshed shock model (e.g. Granot et al. 2003). 5.3. Evidence of a jet break The X–ray afterglow light curve shows a clear temporal break around 10 ks after the trigger (see Sect. 3.1). At about the same time a strong steepening of the I and R afterglow light curves, although with a shallower post-break slope, is also observed (see Pandey et al. 2006 for a detailed analysis on the discrepancy between the X–ray and optical slopes). Due to the achromatic nature of the temporal break it is very likely that a jet break is occurring when the bulk Lorentz factor γ of the col- limated relativistic outflow becomes lower than the inverse of the jet opening angle θjet (e.g., Rhoads 1997, Sari et al. 1999), as also reported by Pandey et al. (2006). In this framework, the jet opening angle can be determined through the equation θjet = 0.161[tb/(1 + z)] 3/8(nη/Eiso) 1/8 (e.g., Bloom et al. 2003) where θjet is in radians, the jet temporal break tb in days, the total isotropic-equivalent energy Eiso in units 10 52 erg, the den- sity n of the circumburst medium in cm−3 and η is the efficiency of conversion of the outflow kinetic energy in electromagnetic radiation. An accurate estimation of the bolometric isotropic- equivalent energy radiated by GRBs requires the knowl- edge of their intrinsic spectrum over a broad energy band (Bloom et al. 2001; Amati et al. 2002). For GRB 050730, the spectrum is well fit by a single power-law with energy index βBAT = 0.5 ± 0.1 up to the high energy limit of the BAT sen- sitivity bandpass (150 keV) indicating that i) we are observing the low energy tail of the Band model (Band et al. 1993) gen- erally used to describe GRB spectra and ii) the νF(ν) spectrum peak energy Ep is above ∼ 750 keV (i.e. (1 + z) × 150 keV) in the burst rest frame. A lower limit to the bolometric isotropic- equivalent radiated energy Eiso is given by the observed radi- ated energy in the BAT bandpass (EBATiso = (8.0±1.0)×10 52 erg, see Sect. 2.1). An upper limit to Eiso is obtained in the most conservative case where the peak energy Ep is equal to 10 4 keV (Amati et al. 2002). By integrating the best fit BAT power-law spectrum in the whole 1–104 keV rest frame energy band we thus derived an upper limit of 4.5 × 1053 erg to Eiso. Taking the central value of the interval derived above we obtained for GRB 050730 Eiso = (2.6 ± 1.9) × 10 53 erg. With z = 3.969, tb = 10.1 −2.2 ks (see Sect. 3.1) and the Eiso range above derived, we find θjet = 1.6 −0.2 deg, for η = 0.2 (Frail et al. 2001) and assuming the value of circumburst den- sity n = 10 cm−3 discussed by Bloom et al. (2003). With this value of the jet opening angle, the inferred collimation- corrected bolometric radiated energy is Ejet = (1.0 −0.8) × 1050 erg. Taking into account the Eiso and Ejet values derived above and the lower limit to the peak energy (Ep > 750 keV rest frame), we find that GRB 050730 is consistent with the Ep vs. Eiso relation found by Amati et al. (2002) and recently updated in Amati (2006). We also find that GRB 050730 is inconsistent, even taking into account the 3σ scatter around the best fit correlation, with the Ep vs. Ejet relation found by Ghirlanda et al. (2004) and with its updated version presented by Nava et al. (2006). In order to make this GRB consistent with the Ep-Ejet relation a much higher circumburst density (n ∼ 105 cm−3) would be required. GRB 050730 is inconsis- tent with the model-independent Eiso-Ep-tb correlation found by Liang & Zhang (2005). 6. Summary and conclusions We have presented a detailed temporal and spectral analy- sis of the afterglow of GRB 050730 observed with Swift and XMM-Newton. The most striking feature of this GRB is the in- tense and exceptionally extended, over more than two order of magnitude in time, X–ray flaring activity. Superimposed to the afterglow decay we observed seven distinct re-brightening events peaking at 236 s, 437 s, 685 s, 4.5 ks, 10.4 ks, 18.7 ks and 41.2 ks after the BAT trigger. The underlying decline of the afterglow was well described with a double broken power-law model with breaks at t1 = 237± 20 s and t2 = 10.1 −2.2 ks. The temporal decay slopes before, be- tween and after these breaks were α1 = 2.1±0.3,α2 = 0.44 +0.14 −0.08 and α3 = 2.40 +0.09 −0.07, respectively. Strong spectral evolution during the flares was present to- gether with an overall softening of the underlying afterglow with the energy index varying from β = 0.42 ± 0.08 dur- ing the early (133–205 s) steep decay to β = 0.99 ± 0.05 at much later (50-60 ks) times. An absorbing column density NzH = (1.28 +0.26 −0.25) × 10 22 cm−2 in the host galaxy is observed during the early (133–781 s) Swift observations while a lower column density NzH = (0.68 ± 0.10) × 10 22 cm−2 is measured during the late (29.4–50.8 ks) XMM-Newton follow-up obser- vation, likely indicating photo-ionization of the surrounding medium. Evidence of flaring activity in the early UVOT op- tical afterglow, simultaneous with that observed in the X–ray band, was found. From the temporal analysis of the first three bright X–ray flares we found that the rise and decay power-law slopes are in the range 0.8–1.8 if the beginning and the peak of the flares are used as zero time, respectively. We also found that, with the exception of the first flare, for all episodes the ratio between the duration of the flare (∆t) and the time when the flare peaks (tp) is nearly constant and is ∆t/tp ∼ 0.6 − 0.7. We showed that the observed properties of the first three flares are consistent with being due to both high-latitude emis- 10 M. Perri et al.: The exceptionally extended flaring activity in the X–ray afterglow of GRB 050730 sion, as expected for flares produced by late internal shocks, or to late time energy injection into the main afterglow shock by slow moving shells (refreshed shocks). An analysis of a larger sample of bursts would help in understanding what are the physical mechanisms responsible for the X-ray flaring ac- tivity. We interpreted the X–ray temporal break at around 10 ks as a jet break and derived a cone angle of ∼ 2 deg and a radi- ated energy Ejet = (0.2 − 3.3) × 10 50 erg against an isotropic- equivalent energy Eiso = (0.7 − 4.5) × 10 53 erg. GRB 050730 satisfies the Ep vs. Eiso Amati relation while is inconsistent with the Ep vs. Ejet Ghirlanda relation. Acknowledgements. We are grateful to the referee for his/her useful comments and suggestions. We also thank C. Guidorzi for a very care- ful reading of the paper and F. Tamburelli and B. Saija for their work on the XRT data reduction software. 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Perri et al.: The exceptionally extended flaring activity in the X–ray afterglow of GRB 050730 11 Table 1. UVOT detections of GRB 050730 in the V and B filters. Column (1) gives the image mid time in seconds since the BAT trigger, column (2) the net exposure time, column (3) the filter used, and column (4) the afterglow magnitude with 1σ error. T(mid) Exposure Filter Magnitude (s) (s) 170 99 V 17.4 ± 0.2 270 10 B 18.4 ± 0.4 299 9.7 V 17.2 ± 0.5 383 9.7 V 17.3 ± 0.5 468 9.7 V 16.4 ± 0.3 524 30 B 19.0 ± 0.4 552 9.7 V 16.3 ± 0.3 637 9.7 V 17.0 ± 0.4 721 9.7 V 16.8 ± 0.4 734 20 B 19.2 ± 0.5 10164 900 B 20.2 ± 0.3 11947 837 V 18.6 ± 0.2 23519 835 V 19.5 ± 0.3 34990 843 V 19.9 ± 0.4 Table 2. Temporal best fit parameters of the first three bright X–ray flares of GRB 050730 using a linear rise exponential decay model. The corresponding ∆t/tp values for the three flares are also indicated (see Sect. 3.1). Flare t0 tp tc K × 10 −10 ∆t/tp (s) (s) (s) ( erg cm−2 s−1) 1 207+5 −5 236 −4 17 −5 8 ± 1 0.3 ± 0.1 2 344+5 −5 437 −4 54 −6 12.2 ± 0.7 0.58 ± 0.04 3 614+ 9 −10 685 −6 87 −21 6.1 ± 0.7 0.5 ± 0.1 Table 3. Temporal best-fit parameters of the X–ray flares 4, 5, 6 and 7 of GRB 050730 using Gaussian functions. An asterisk indicates a frozen parameter. The corresponding ∆t/tp values are also indicated (see Sect. 3.1). Best-fit parameters for flare 7 were derived from XMM-Newton data only. Flare tp σ K × 10 −11 ∆t/tp (s) (s) ( erg cm−2 s−1) 4 4484+124 −235 641 −149 11 ± 2 0.7 ± 0.2 5 10391+330 −315 1500 ∗ 5.5 ± 1.0 0.71 ± 0.02 6 18714+952 −1156 3638 −789 1.5 ± 0.4 0.9 ± 0.2 7 41244+691 −744 6170 −718 0.12 ± 0.02 0.7 ± 0.1 Table 4. Best fit temporal indices of the rising (αr) and decaying (αd) portions of the X–ray flares 1, 2, 3 and 7 using a single power-law model. Flare αr αd 1 −1.56 ± 0.69 1.25 ± 0.32 2 −1.84 ± 0.30 1.31 ± 0.44 3 −1.46 ± 0.62 0.75 ± 0.35 7 −0.89 ± 0.34 1.61 ± 0.44 12 M. Perri et al.: The exceptionally extended flaring activity in the X–ray afterglow of GRB 050730 Table 5. Results of single power-law spectral fits to the 0.3–10 keV spectrum of the afterglow of GRB 050730. A local (z = 0) absorption column fixed at the known Galactic value of NGH = 3.0 × 10 20 cm−2 (Dickey & Lockman 1990) was used in the fits. An asterisk indicates a frozen parameter. segment time interval NzH × 10 22 β χ2r (d.o.f.) (s) (cm−2) WT (all) 133–781 1.28+0.26 −0.25 0.70 +0.03 −0.03 1.01 (310) WT (0) 133–205 1.8+0.9 −0.8 0.42 +0.08 −0.08 0.86 (76) WT (1a) 205–233 1.6+2.5 −1.6 0.29 +0.16 −0.16 0.86 (24) WT (1b) 233–313 3.1+1.3 −1.1 0.82 +0.12 −0.12 1.30 (51) WT (2a) 313–433 2.1+0.8 −0.7 0.71 +0.08 −0.08 1.10 (93) WT (2b) 433–601 0.9+0.5 −0.5 0.70 +0.07 −0.07 1.11 (111) WT (3a) 601–681 0.7+0.8 −0.7 0.77 +0.12 −0.12 0.87 (46) WT (3b) 681–781 1.0+0.6 −0.6 1.01 +0.10 −0.10 0.81 (65) PC (1) 4001–18149 1.1+0.4 −0.4 0.61 +0.04 −0.04 0.95 (224) PC (2) 21288–143438 1.0+0.7 −0.6 0.81 +0.08 −0.08 1.16 (86) XMM (all) 29436–59811 0.68+0.10 −0.10 0.87 +0.02 −0.02 1.14 (489) XMM (1) 29436–40000 0.68∗ 0.87+0.03 −0.03 0.88 (345) XMM (2) 40000–50000 0.68∗ 0.93+0.03 −0.03 0.98 (243) XMM (3) 50000–59811 0.68∗ 0.99+0.05 −0.05 0.77 (135) Introduction Observations and data reduction Swift BAT Ground-based Observatories Swift UVOT Swift XRT XMM-Newton Temporal analysis X–ray afterglow Optical band Spectral analysis XMM-Newton Discussion Early X–ray light curve Flaring activity Evidence of a jet break Summary and conclusions

0704.1298

The obscured quasar population from optical, mid-infrared, and X-ray surveys

**FULL TITLE** ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** **NAMES OF EDITORS** The obscured quasar population from optical, mid-infrared, and X-ray surveys C. Vignali Dipartimento di Astronomia, Università degli Studi di Bologna, Italy INAF – Osservatorio Astronomico di Bologna, Italy A. Comastri INAF – Osservatorio Astronomico di Bologna, Italy D. M. Alexander Department of Physics, Durham University, UK Abstract. Over the last few years, optical, mid-infrared and X-ray surveys have brought to light a significant number of candidate obscured AGN and, among them, many Type 2 quasars, the long-sought after “big cousins” of lo- cal Seyfert 2 galaxies. However, despite the large amount of multi-wavelength data currently available, a proper census and a panchromatic view of the ob- scured AGN/quasar population are still missing, mainly due to observational limitations. Here we provide a review of recent results on the identification of obscured AGN, focusing primarily on the population of Type 2 quasars selected in the optical band from the Sloan Digital Sky Survey. 1. Introduction: the X-ray and mid-infrared hunt for obscured quasars The quest for the identification of luminous and obscured Active Galactic Nuclei (AGN), the so-called Type 2 quasars predicted by unification schemes of AGN (e.g., Antonucci 1993) and required by many synthesis models of the X-ray background (XRB; e.g., Gilli, Comastri & Hasinger 2007), has been the topic of numerous investigations over the past few years. Although moderate-depth and ultra-deep X-ray surveys (see Brandt & Hasinger 2005 for a review) have proven effective to reveal a large number of Type 2 quasar candidates at high redshift (e.g., Alexander et al. 2001; Mainieri et al. 2002; Mignoli et al. 2004), the optical counterparts of these sources are typically faint and therefore represent challenging targets to obtain spectroscopic redshifts and reliable classifications based on “standard” optical emission-line ratio techniques. Furthermore, there is evidence that despite the fact that ≈ 80% of the XRB has been resolved into discrete sources by ultra-deep X-ray surveys in the 2–8 keV band (e.g., Bauer et al. 2004; Hickox & Markevitch 2006), only ≈ 60% of the XRB has been resolved above ≈ 6 keV (Worsley et al. 2004, 2005), indicating that while ultra- deep X-ray surveys provide efficient identification of AGN activity, they do not provide a complete census of the obscured AGN population. It is plausible that http://arxiv.org/abs/0704.1298v1 2 Vignali, Comastri & Alexander one of the “missing” XRB components is related to the population of Compton- thick AGN (i.e., sources with column densities > 1/σT ≈ 10 24 cm−2, where σT is the Thomson cross section; see Comastri 2004 for a review on Compton-thick AGN) which are required by the AGN synthesis models of the XRB (Gilli et al. 2007) but, as being heavily obscured, are difficult to discover and identify in the X-rays by the current generation of X-ray telescopes; see, e.g., Tozzi et al. (2006) for the Compton-thick AGN candidate selection in the Chandra Deep Field-South. In the presence of obscuration, the nuclear emission is expected to be re- emitted at longer wavelengths and hence mid-infrared (MIR) observations can be crucial to reveal obscured AGN emission. Recently, numerous attempts have been made in this direction, fully exploiting the capabilities of the detectors on- board Spitzer (e.g., Alonso-Herrero et al. 2006; Polletta et al. 2006; Donley et al. 2007). Unfortunately, the AGN locus in the color-color diagrams obtained from Spitzer photometry (e.g., Lacy et al. 2004; Stern et al. 2005) is often contaminated by starburst galaxies, therefore further investigations and adjust- ments are required to efficiently distinguish the obscured and elusive AGN from the less intriguing unobscured population. The present limitations of Spitzer diagnostic diagrams to select obscured AGN may be overcome either by refining the selection criteria (e.g., Mart́ınez-Sansigre et al. 2005, 2006) or targeting the optically faint or invisible source population with, e.g., MIR spectroscopy (e.g., Houck et al. 2005; Weedman et al. 2006a). The next obvious step where most of the observational efforts will be con- centrated in the years to come is to compare the obscured AGN selection criteria adopted at different wavelengths and compute their efficiency in the detection and identification of the most heavily obscured quasars. Parallel to these kinds of studies, the analyses of the multi-wavelength (from MIR to X-rays) proper- ties of obscured AGN and quasars will be crucial to investigate their emission accurately (e.g., Weedman et al. 2006b), refine the current torus models and templates (e.g., Silva, Maiolino & Granato 2004), and derive some fundamental parameters such as the masses of the super-massive black holes (SMBHs) resid- ing in these sources and their Eddington ratios (e.g., Pozzi et al. 2007). Finally, the co-evolution of galaxies and SMBHs will be investigated up to high redshifts, where a significant fraction of MIR–submillimeter-selected obscured AGN will probably be found (e.g., Alexander et al. 2005a; Mart́ınez-Sansigre et al. 2005). 2. Are any optically selected Type 2 quasars out there? Although primarily designed, in the AGN research field, for the discovery of broad-line (Type 1) objects, the Sloan Digital Sky Survey (SDSS; York et al. 2000) has provided a sample of 291 high-ionization narrow emission-line AGN in the redshift range 0.3–0.83 (Zakamska et al. 2003; small filled circles in Fig. 1), many of which are identified as candidate Type 2 quasars on the basis of their [O iii]5007Å luminosities. In the following, we summarize the main results obtained over the last three years for a sub-sample of these optically selected Type 2 quasars with ROSAT, Chandra, and XMM-Newton observations. From the analysis of primarily archival ROSAT observations, Vignali, Alexander & Comastri (2004, hereafter V04) were able to place constraints The obscured quasar population 3 Figure 1. Logarithm of the measured L[O III] luminosities vs. redshifts for all of the sources in the original sample of Type 2 quasar candidates from Zakamska et al. (2003; small filled circles). At the right of the panel, the 2–10 keV luminosities, estimated using the correlation between the [O iii] and 2–10 keV flux (Mulchaey et al. 1994), are shown. The key provides a description of the X-ray observations. The grey region defines the locus of the sources which still lie in the quasar regime (i.e., above 1044 erg s−1) even taking into account the dispersion in the L[O III]–L2−10 keV correlation. After completion of the Chandra AO8 observing cycle, X-ray information will be available for the most extreme radio-quiet Type 2 quasars from the Zakamska et al. (2003) sample. on the X-ray emission of 17 SDSS Type 2 quasar candidates (open circles in Fig. 1). Using the [O iii] line luminosity to predict the intrinsic X-ray power of the AGN (following the correlation of Mulchaey et al. 1994), V04 found that at least 47% of the observed sample shows indications of X-ray absorption, including the four highest luminosity sources with predicted unobscured luminosities of ≈ 1045 erg s−1, hence well above the typically adopted threshold of 1044 erg s−1 in the 2–10 keV band for Type 2 quasars. In Vignali, Alexander & Comastri (2006, hereafter V06), the most up- to-date results on the SDSS Type 2 quasar population were presented. Using a combination of Chandra and XMM-Newton pointed and serendip- itous observations (for a total of 16 sources; filled triangles in Fig. 1), selected predominantly among the most luminous in [O iii], V06 detected X-ray emission from ten sources. For seven of these AGN, basic/moderate- quality X-ray spectral analyses constrained the column densities in the range ≈ 1022 – a few 1023 cm−2 (filled triangles in Fig. 2). Once their ob- served X-ray luminosities are corrected for the effect of absorption, there is indication that the X-ray luminosity predictions based on the Mulchaey et 4 Vignali, Comastri & Alexander Figure 2. Comparison of the 2–10 keV luminosity computed from the com- pilation of V06 with that predicted assuming the Mulchaey et al. (1994) cor- relation. The dotted line shows the 1:1 ratio between the two luminosities. For the seven sources for which X-ray spectral fitting was possible using either Chandra or XMM-Newton data, the X-ray luminosity has been de-absorbed assuming the best-fit column density (filled triangles), while for the remain- ing X-ray fainter AGN (open circles), some of which undetected, the X-ray luminosity is derived from the X-ray flux with no correction for the unknown absorption. Leftward arrows indicate upper limits on the observed X-ray luminosity; the grey region shows the locus of heavily obscured, candidate Compton-thick quasars, where the observed luminosity is less than 1% of the predicted one. al. (1994) correlation are consistent with the values obtained from X-ray spectral fitting (all these seven sources lie close to the 1:1 line in Fig. 2). Having calibrated the [O iii] line luminosity as an indicator of the intrinsic X-ray emission on the seven sources with good X-ray photon statistics, there are indications that the X-ray undetected sources and the sources with a limited number of counts (open circles in Fig. 2) are possibly more obscured than those found absorbed through direct X-ray spectral fitting, as pointed out also by Ptak et al. (2006). This would imply that up to ≈ 50% of the population is characterized by column densities in excess to 1023 cm−2, with a sizable number of Compton-thick quasars possibly hiding among the X-ray faintest sources (see the quasars located in the grey region in Fig. 2). This possibility is also suggested by the comparison of the X-ray-to-[O iii] flux ratios of our sources vs. those obtained from a large sample of mostly nearby Seyfert 2 galaxies (see Guainazzi et al. 2005 and Fig. 6 of V06). Given the highly inhomogeneous selection and incompleteness of the sam- ple presented by Zakamska et al. (2003), the number density of SDSS The obscured quasar population 5 selected Type 2 quasars can be derived only roughly. In an attempt to provide a first-order estimate, V06 obtained a value of ≈ 0.05 deg−2, while Gilli et al. (2007) XRB synthesis models predict a surface density of Type 2 quasars of ≈ 0.15 deg−2 in the ≈ 0.3–0.8 redshift range. This comparison indicates that the Zakamska et al. (2003) selection is relatively efficient at finding obscured quasar activity; however, a combination of blank-field X-ray surveys and optical selection techniques will provide a more com- plete census. Given the accurate analyses carried out by Zakamska et al. (2003) in the original selection of the obscured SDSS AGN sample, it seems unlikely that a significant population of starburst galaxies or low-luminosity AGN is hid- ing among the sources with the highest [O iii] luminosity in the V06 sample. Hence, it is likely that some of the Zakamska et al. (2003) quasars are obscured by Compton-thick material in the X-ray band. Although effective and compar- atively complete, the optical selection and requirement for [OIII] in the SDSS spectra clearly limit obscured quasar searches to z < 1. Due to the likely optical faintness of obscured quasars at z > 1, a multi-wavelength selection process in- cluding targeted follow-up observations is likely to be required for a comparably complete census of obscured quasar activity at high redshift. 3. What’s next in the study of SDSS Type 2 quasars? The observations approved for Chandra Cycle 8 (10 ks pointing for 12 tar- gets) will allow us to have an almost complete coverage of the most extreme SDSS Type 2 quasar candidates (i.e., above a predicted 2–10 keV luminosity of 1044 erg s−1 also taking into account the dispersion in the L[O III]–L2−10 keV correlation; open squares in Fig. 1). While direct X-ray spectral fitting will be possible only for the most X-ray luminous SDSS Type 2 quasars, for the newly observed faint X-ray sources, basic spectral analysis will be carried out by means of the hardness-ratio technique. For the few-count or undetected sources, it will be possible to derive the average source properties through stacking analysis; use of this has been prevented thus far by the limited number of sources with either Chandra or XMM-Newton constraints and the paucity of X-ray counts. Luckily, it will be possible also to estimate the column density distribution of SDSS Type 2 quasars and, by stacking the X-ray source spectra in different NH bins, to search for faint spectral features, as shown by Alexander et al. (2005b) for a sample of submillimeter galaxies. Acknowledgments. The authors acknowledge support by the Italian Space Agency (contract ASI–INAF I/023/05/0; CV and AC) and the Royal Society (DMA). The authors wish to thank the people involved in the HELLAS2XMM survey for useful discussions. References Alexander, D. M., Brandt, W. N., Hornschemeier, A. E., Garmire, G. P., Schneider, D. P., Bauer, F. E., & Griffiths, R. E. 2001, AJ, 122, 2156 Alexander, D. M., Smail, I., Bauer, F. E., Chapman, S. C., Blain, A. W., Brandt, W. N., & Ivison, R. J. 2005a, Nat, 434, 738 6 Vignali, Comastri & Alexander Alexander, D. M., Bauer, F. E., Chapman, S. C., Smail, I., Blain, A. W., Brandt, W. N., & Ivison, R. J. 2005b, ApJ, 632, 736 Alonso-Herrero, A., et al. 2006, ApJ, 640, 167 Antonucci, R. 1993, ARA&A, 31, 473 Bauer, F. E., Alexander, D. M., Brandt, W. N., Schneider, D. P., Treister, E., Horn- schemeier, A. E., & Garmire, G. P. 2004, AJ, 128, 2048 Brandt, W. N., & Hasinger, G. 2005, ARA&A, 43, 827 Comastri, A. 2004, Astrophysics and Space Science Library, Vol. 308: Supermassive Black Holes in the Distant Universe, 245 Donley, J. L., Rieke, G. H., Perez-Gonzalez, P. G., Rigby, J. R., & Alonso-Herrero, A. 2007, ApJ, in press (astro-ph/0701698) Gilli, R., Comastri, A., & Hasinger, G. 2007, A&A, 463, 79 Guainazzi, M., Fabian, A. C., Iwasawa, K., Matt, G., & Fiore, F. 2005, MNRAS, 356, Hickox, R. C., & Markevitch, M. 2006, ApJ, 645, 95 Houck, J. R., et al. 2005, ApJ, 622, L105 Lacy, M., et al. 2004, ApJS, 154, 166 Mainieri, V., Bergeron, J., Hasinger, G., Lehmann, I., Rosati, P., Schmidt, M., Szokoly, G., & Della Ceca, R. 2002, A&A, 393, 425 Mart́ınez-Sansigre, A., Rawlings, S., Lacy, M., Fadda, D., Marleau, F. R., Simpson, C., Willott, C. J., & Jarvis, M. J. 2005, Nat, 436, 666 Mart́ınez-Sansigre, A., Rawlings, S., Lacy, M., Fadda, D., Jarvis, M. J., Marleau, F. R., Simpson, C., & Willott, C. J. 2006, MNRAS, 370, 1479 Mignoli, M., et al. 2004, A&A, 418, 827 Mulchaey, J. S., Koratkar, A., Ward, M. J., Wilson, A. S., Whittle, M., Antonucci, R. R. J., Kinney, A. L., & Hurt T. 1994, ApJ, 436, 586 Polletta, M. d. C., et al. 2006, ApJ, 642, 673 Pozzi, F., et al. 2007, A&A, submitted Ptak, A., Zakamska, N. L., Strauss, M. A., Krolik, J. H., Heckman, T. M., Schneider, D. P., & Brinkmann, J. 2006, ApJ, 637, 147 Silva, L., Maiolino, R., & Granato, G. L. 2004, MNRAS, 355, 973 Stern, D., et al. 2005, ApJ, 631, 163 Tozzi, P., et al. 2006, A&A, 451, 457 Vignali, C., Alexander, D. M., & Comastri, A. 2004, MNRAS, 354, 720 (V04) Vignali, C., Alexander, D. M., & Comastri, A. 2006, MNRAS, 373, 321 (V06) Weedman, D. W., et al. 2006a, ApJ, 651, 101 Weedman, D. W., et al. 2006b, ApJ, 653, 101 Worsley, M. A., Fabian, A. C., Barcons, X., Mateos, S., Hasinger, G., & Brunner, H. 2004, MNRAS, 352, L28 Worsley, M. A., et al. 2005, MNRAS, 357, 1281 York, D. G., et al. 2000, ApJ, 120, 1579 Zakamska, N. L., et al. 2003, ApJ, 126, 2125 http://arxiv.org/abs/astro-ph/0701698

0704.1299

The Realm of the First Quasars in the Universe: the X-ray View

**FULL TITLE** ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** **NAMES OF EDITORS** The Realm of the First Quasars in the Universe: the X-ray View C. Vignali (1), W.N. Brandt (2), O. Shemmer (2), A. Steffen (2), D.P. Schneider (2), S. Kaspi (3,4) (1) Dipartimento di Astronomia, Università degli Studi di Bologna, Italy; (2) Department of Astronomy & Astrophysics, Pennsylvania State University, University Park, USA; (3) Wise Observatory, Tel Aviv University, Israel; (4) Physics Department, Technion, Haifa, Israel. Abstract. We review the X-ray studies of the highest redshift quasars, fo- cusing on the results obtained with Chandra and XMM-Newton. Overall, the X-ray and broad-band properties of z > 4 quasars and local quasars are similar, suggesting that the small-scale X-ray emission regions of AGN are insensitive to the significant changes occurring at z≈0–6. 1. Introduction In recent years, optical surveys (e.g, the Sloan Digital Sky Survey and the Digital Palomar Sky Survey) have discovered a large number (≈ 1000) of quasars at z > 4. From the pioneering study of Kaspi et al. (2000; see Fig. 1a), the number of X-ray detected AGN at z > 4 has increased to more than 110 (Fig. 1b), mostly thanks to exploratory observations with Chandra (e.g., Vignali et al. 2001, 2005; Brandt et al. 2002; Bassett et al. 2004; Lopez et al. 2006; Shemmer et al. 2006a) and longer exposures with XMM-Newton (e.g., Shemmer et al. 2005). At the very faint X-ray fluxes, X-ray surveys have provided detection of several z > 4 AGN and quasars (e.g., Schneider et al. 1998; Silverman et al. 2002; Vignali et al. 2002). Here we provide a summary of some of the main recent results: • X-ray emission is a universal property of AGN. The X-ray properties of high- redshift AGN and quasars (derived from either stacked or individual X-ray spec- tra) are similar to those of local quasars, with no evidence for widespread absorp- tion. For radio-quiet quasars (RQQs), a photon index of Γ ≈1.9–2.0 is obtained (e.g., Vignali et al. 2005; Shemmer et al. 2005), also at z > 5 (Shemmer et al. 2006a), while for “moderate” radio-loud quasars (RLQs) and blazars, Γ ≈1.7 and Γ ≈1.5 are obtained (Lopez et al. 2006), respectively. • The comparison with the lower redshift (luminosity) Palomar-Green quasars observed by XMM-Newton (Piconcelli et al. 2005) indicates that the photon index does not vary significantly with redshift and luminosity, but seems to depend primarily on the accretion rate (i.e., steeper X-ray slopes are associated with higher Eddington ratio sources; Shemmer et al. 2006b). • Following X-ray studies of early ’80 and ’90, the relation between X-ray and longer wavelength emission has been investigated by means of the point-to-point spectral slope between 2500 Å and 2 keV in the source rest frame (αox). Any http://arxiv.org/abs/0704.1299v1 2 Vignali et al. Figure 1. Observed-frame, Galactic absorption-corrected 0.5–2 keV flux versus AB1450(1+z) magnitude for z > 4 AGN and quasars. (a) The situ- ation after the Kaspi et al. (2000) work using ROSAT data; (b) the up- dated census of X-ray observations of z > 4 AGN, including the results from moderate-depth and ultra-deep X-ray surveys. changes in the accretion rate over cosmic time might lead to changes in the fraction of total power emitted as X-rays. Using 333 AGN at z≈0–6.3 (88% X-ray detections), Steffen et al. (2006) confirmed that log L 2500 Å correlates with log L2 keV with an index < 1, and αox depends upon log L2500 Å (with the slope perhaps depending on L 2500 Å The research field related to z > 4 AGN still offers plenty of opportunities. In particular, the detection of X-ray variability in some z > 4 quasars over time scales of month-year (Shemmer et al. 2005) needs further investigations to check the possibility that quasars are more variable in the early Universe. Furthermore, detailed X-ray spectra of z > 4 RLQs filling the observational gap between “moderate” RLQs and blazars are still needed, as well as studies of “peculiar” quasars and faint AGN population at the highest redshifts. References Bassett, L.C., et al. 2004, AJ, 128, 523 Brandt, W.N., et al. 2002, ApJ, 569, L5 Kaspi, S., Brandt, W.N., & Schneider, D.P. 2000, AJ, 119, 2031 Lopez, L.A., et al. 2006, AJ, 131, 1914 Piconcelli, E., et al. 2005, A&A, 432, 15 Schneider, D.P., et al. 1998, AJ, 115, 1230 Shemmer, O., et al. 2005, ApJ, 630, 729 Shemmer, O., et al. 2006a, ApJ, 644, 86 Shemmer, O., Brandt, W., Netzer, H., Maiolino, R., & Kaspi, S. 2006b, ApJ, 646, L29 Silverman, J.D., et al. 2002, ApJ, 569, L1 Steffen, A.T., et al. 2006, AJ, 131, 2826 Vignali, C., et al. 2001, AJ, 122, 2143 Vignali, C., et al. 2002, ApJ, 580, L105 Vignali, C., Brandt, W.N., Schneider, D.P., & Kaspi, S. 2005, AJ, 129, 2519

0704.1300

The obscured X-ray source population in the HELLAS2XMM survey: the Spitzer view

The obscured X-ray source population in the HELLAS2XMM survey: the Spitzer view Cristian Vignali∗,†, Francesca Pozzi∗, Andrea Comastri†, Lucia Pozzetti†, Marco Mignoli†, Carlotta Gruppioni†, Giovanni Zamorani†, Carlo Lari∗∗, Francesca Civano∗, Marcella Brusa‡, Fabrizio Fiore§ and Roberto Maiolino§ ∗Dipartimento di Astronomia, Università di Bologna, Via Ranzani 1, I–40127 Bologna, Italy †INAF – Osservatorio Astronomico di Bologna, Via Ranzani 1, I–40127 Bologna, Italy ∗∗INAF – Istituto di Radioastronomia (IRA), Via Gobetti 101, I–40129 Bologna, Italy ‡Max Planck Institut für Extraterrestrische Physik (MPE), Giessenbachstrasse 1, D–85748 Garching bei München, Germany §INAF – Osservatorio Astronomico di Roma, Via Frascati 33, I–00040 Monteporzio-Catone (RM), Italy Abstract. Recent X-ray surveys have provided a large number of high-luminosity, obscured Active Galactic Nuclei (AGN), the so-called Type 2 quasars. Despite the large amount of multi-wavelength supporting data, the main parameters related to the black holes harbored in such AGN are still poorly known. Here we present the preliminary results obtained for a sample of eight Type 2 quasars in the redshift range ≈ 0.9–2.1 selected from the HELLAS2XMM survey, for which we used Ks-band, Spitzer IRAC and MIPS data at 24 µm to estimate bolometric corrections, black hole masses, and Eddington ratios. Keywords: Galaxies: active – Galaxies: nuclei – X-rays: galaxies PACS: 98.54.-h, 98.58.Jg INTRODUCTION Over the last six years, the X-ray surveys carried out by Chandra and XMM-Newton (e.g., [1, 2, 3]; see [4] for a review) have provided remarkable results in resolving a significant fraction of the cosmic X-ray background (XRB; [5, 6]), up to ≈ 80% in the 2–8 keV band (e.g., [7, 8]). Despite the idea that a large fraction of the accretion-driven energy density in the Universe resides in obscured X-ray sources has been widely sup- ported and accepted (e.g., [9, 10]), until recently only limited information was available to properly characterize the broad-band emission of the counterparts of the X-ray ob- scured sources and provide a reliable estimate of their bolometric output. In this context, Spitzer data have provided a major step forward the understanding of the broad-band properties of the X-ray source populations. If, on one hand, Spitzer data have allowed to pursue the “pioneering” studies of [11] on the spectral energy distributions (SEDs) of broad-line (Type 1), unobscured quasars at higher redshifts (e.g., [12]), on the other hand they have produced significant results in the definition of the SEDs of narrow-line (Type 2), obscured AGN (e.g., [13]). In this work we aim at providing a robust determination of the bolometric luminosity for hard X-ray selected obscured AGN. This result can be achieved by effectively http://arxiv.org/abs/0704.1300v1 disentangling the nuclear emission related to the active nucleus from the host galaxy starlight, which represents the dominant component (at least for our obscured sources) at optical and near-infrared (near-IR) wavelengths. SAMPLE SELECTION AND Ks-BAND PROPERTIES The sources presented in this work were selected from the HELLAS2XMM survey ([3]) which, at the 2–10 keV flux limit of ≈ 10−14 erg cm−2 s−1, covers ≈ 1.4 square degrees of the sky using XMM-Newton archival pointings ([14]). Approximately 80% of the HELLAS2XMM sources have a spectroscopic optical classification in the final source catalog (see [15] for details). In particular, we selected eight sources from the original sample of [16] which are characterized by faint (23.7–25.1) R-band magnitudes and bright Ks-band counterparts (≈ 17.6–19.1); all of our sources are therefore classified as extremely red objects (EROs, R − Ks > 5 in Vega magnitudes). From the good- quality Ks-band images, [16] were able to study the surface brightness profiles of these sources, obtaining a morphological classification. While two sources are associated with point-like objects, the remaining six sources are extended, showing a profile typical of elliptical galaxies. In this latter class of sources, the active nucleus, although evident in the X-ray band, appears hidden or suppressed at optical and near-IR wavelengths, where the observed emission is clearly dominated by the host galaxy starlight. The relatively good constraints on the nuclear emission in the near-IR represent a starting point for the analysis of the Spitzer IRAC and MIPS data. Due to the faint R-band magnitudes of our sources, optical spectroscopy was not feasible even with the 8-m telescope facilities; however, the bright near-IR counterparts of our sources allowed us to obtain spectroscopic redshifts in the Ks band with ISAAC at VLT for two sources: one point-like AGN is classified as a Type 1.9 quasar at a redshift of 2.09, while one extended source has line ratios typical of a LINER at z=1.35 (see [17] for further details on these classifications). For the remaining sources, the redshift has been estimated using the optical and near-IR magnitudes, along with the morphological information, as extensively described in §5.1 of [16]; all of the redshifts are in the range ≈ 0.9–2.1. The large column densities [≈ 1022 – a few×1023 cm−2] and the 2–10 keV luminosities [≈ (1 − 8)× 1044 erg s−1, once corrected for the absorption] place our sources in the class of the high-luminosity, obscured AGN, the so-called Type 2 quasars (see, e.g., [18] and references therein). SPITZER DATA For our sample of eight sources, we obtained IRAC observations of 480 s integration time and MIPS observations at 24 µm for a total integration time per position of ≈ 1400 s. All of the sources are detected in the four IRAC bands and in MIPS; the faintest source in MIPS has a 24 µm flux density of ≈ 150 µJy (≈ 5σ detection; see [19] for further details on data reduction and cleaning procedures). ANALYSIS OF THE TYPE 2 QUASAR SPECTRAL ENERGY DISTRIBUTIONS A reliable determination of the bolometric output of our AGN sample requires that the nuclear component, directly related to the accretion processes, is disentangled from the emission of the host galaxy, which provides a dominant contribution in the optical and Ks bands (in the case of extended sources, see [16]). To achieve this goal, we constructed SEDs for all our sources over the optical, near- and mid-IR range. At the same time, we used Spitzer data to improve our previous estimates on the source redshift when possible. In the following, we consider the sample of six extended sources and two point-like objects separately, since a different approach has been adopted for the two sub-samples. Extended sources As already pointed out, from the Ks-band morphological analysis carried out by [16], we know that at least up to 2.2 µm (observed frame) the stellar contribution is mostly responsible for the emission of these sources. At longer wavelengths, the emission of the active nucleus is expected to arise as reprocessed radiation of the primary emission, while the emission from the galaxy should drop significantly, assuming reasonable elliptical templates. Although many models have been developed in the past to deal with circum-nuclear dust emission (including the effects of the torus geometry and opening angle, grain size distribution and density), in our study we adopted a more phenomenological approach. To reproduce the observed data, we used a combination of two components, one for the host galaxy and another related to the reprocessing of the nuclear emission. For the galaxy component, we adopted a set of early-type galaxy templates obtained from the synthetic spectra of [20], assuming a simple stellar population spanning a large range of ages (see [19] for details). For the nuclear component, we adopted the templates of [21], which are based on the interpolation of the observed nuclear IR data (at least, up to ≈ 20 µm) of a sample of local AGN through the radiative transfer models of [22]. The strength of such an approach is that the nuclear templates depend upon two quantities, the intrinsic 2–10 keV luminosity (which provides the normalization of the SED) and the column density (responsible for the shape of the SED), and these are known directly from the X-ray spectra ([23]), once the redshift is known. We also used all the available information, extended over the Spitzer wavelength range, to place better constraints on the source redshift than those reported in [16]. Overall, we find a good agreement with the redshifts presented in [16], although Spitzer allows us to provide estimates with lower uncertainties; only for one source the redshift is significantly lower (z ≈ 1 instead of ≈ 2) and likely more reliable. The data are well reproduced by the sum of the two components; the emission from the galaxy progressively becomes less important at wavelengths above ≈ 4 µm (in the source rest frame), where the nuclear reprocessed emission starts emerging significantly (see Fig. 1, left panel), being dominant in MIPS at 24 µm. Furthermore, the latter is fully consistent with the upper limits provided in the Ks band by [16]. FIGURE 1. Rest-frame SEDs for two representative Type 2 quasars of the current sample: an obscured AGN hosted by an elliptical galaxy (on the left) and a point-like AGN (on the right). (Left) The observed data (filled circles) are reproduced by summing up (solid line) the contribution of an early-type galaxy template (dot-dashed line) to the reprocessed nuclear component (dashed line). The dotted line shows the nuclear component obtained from the templates of [21], normalized using the X-ray luminosity and column density (i.e., without fitting the data; see text and [19] for details). The downward-pointing arrow indicates the constraint on the nuclear emission derived from the Ks-band data ([16]). The combination of the two templates is also consistent with the R-Ks color. (Right) The observed data (filled circles) are well reproduced by the red quasar template from [13] (solid line). Point-like sources For the two point-like sources, we adopted a different strategy, since their emission in the near-IR is dominated by the unresolved AGN. To reproduce their observed SEDs, we extincted a Type 1 quasar template from [11] with several extinction laws, but we were not able to find a satisfactory solution. Then we used the recently published red quasar template from [13] and found good agreement with the data (Fig. 1, right panel), consistently with the results obtained for some obscured AGN in the ELAIS-S1 field ([24]). As in the AGN sub-sample described above, most of the uncertainty lies in the far-IR, where a proper study of the SEDs would require MIPS data at 70 and 160 µm. BOLOMETRIC CORRECTIONS The determination of the SEDs is meant to be the first step toward the estimate of the bolometric luminosities (Lbol) of obscured AGN. The bolometric luminosities can be estimated from the luminosity in a given band by applying a suitable bolometric correction kbol; typically, to convert the 2–10 keV luminosity into Lbol, kbol≈ 30 is assumed, although this value was derived from the average of few dozens of bright, mostly low-redshift Type 1 quasars ([11]). For obscured sources, only few estimates are present in literature (e.g., [13]). We derived kbol by integrating the quasar SEDs over the X-ray (0.5–500 keV) and IR (1–1000 µm) intervals; in the X-ray band, we converted the 2–10 keV luminosity assuming a power law with photon index Γ = 1.9 (typical for AGN emission) and the observed column density ([23]). To derive the bolometric corrections, we accounted for both the covering factor of the absorbing material (i.e., the opening angle of the torus) and the anisotropy of the IR emission. According to unification models of AGN, the former effect should be directly related to the observed fraction of Type 2/Type 1 AGN which, in the latest models of [6], is ≈ 1.5 in the luminosity range of our sample. Furthermore, the torus is likely to re-emit a fraction of the intercepted radiation in a direction which does not lie along our line-of-sight; the correction for this anisotropy, according to the templates of [21], is ≈ 10–20% (given the column densities of our sources). Once these corrections are taken into account, we obtain 〈kbol〉 ≈ 35 (median kbol≈ 26), similar to the average value of [11]; the Type 1.9 quasar at z=2.09 has the highest kbol (≈ 97); see [19] for a discussion on the uncertainties in these estimates. BLACK HOLE MASSES AND EDDINGTON RATIOS For the six AGN hosted by elliptical galaxies, we can derive both the galaxy and black hole masses. Since the near-IR emission is dominated by the galaxy starlight, we computed the rest-frame LK assuming the appropriate SED templates and then the galaxy masses using M⋆/LK ≈ 0.5− 0.9 ([20]); all of our AGN are hosted by massive galaxies (≈ 1−6×1011 M⊙). To estimate the black hole masses, we used the local MBH–LK ([25]) which, along with the M⋆/LK values, provides a MBH–M⋆ relation. Despite several attempts in the recent literature to investigate whether and how the black hole mass vs. stellar mass relation evolves with cosmic time, there is no consensus yet. In this work, we assume the findings of [26], who found that in the redshift range covered by our sources, the MBH-M⋆ relation evolves by a factor of ≈ 2 with respect to the local value; see [19] for an extensive discussion. Under this hypothesis, we obtain black hole masses for the six obscured quasars hosted by elliptical galaxies of ≈ 2.0×108 −2.5×109 M⊙; these values are broadly consistent with the average black hole masses obtained by [27] for the Sloan Digital Sky Survey (SDSS) Type 1 quasars (using optical and ultra-violet mass scaling relationships) in our redshift range (≈ 3.5×108 −8.6×108 M⊙). As a final step, we derived the Eddington ratios, defined as Lbol/LEdd, where LEdd is the Eddington luminosity. We note that the uncertainties related to these estimates are clearly large, due to the uncertainties of the approach adopted to derive the bolometric luminosities (through the templates of [21]) and the black hole masses (see above). The average Eddington ratio is ≈ 0.05, suggesting that our obscured quasars may have already passed their rapidly accreting phase and are reaching their final masses at low Eddington rates. The Eddington ratios of our sources are significantly lower than those derived for the SDSS Type 1 quasars in the same redshift range (≈ 0.3–0.4, see [27]). SUMMARY We used optical, near-IR, and Spitzer IRAC and MIPS (at 24 µm) data to unveil the re- processed nuclear emission of eight hard X-ray selected Type 2 quasars at z ≈ 0.9−2.1. From proper modelling of the nuclear SEDs, we derived a median (average) bolometric correction of ≈ 26 (≈ 35). For the six obscured sources dominated by the host galaxy starlight up to near-IR wavelengths, we also derived black hole masses of the order of 2.0×108−2.5×109 M⊙ and relatively low Eddington ratios (≈ 0.05), suggestive of a low-activity accretion phase. ACKNOWLEDGMENTS The authors acknowledge partial financial support by the Italian Space Agency under the contract ASI–INAF I/023/05/0. REFERENCES 1. R. Giacconi et al., ApJS 139, 369–410 (2002). 2. D. M. Alexander et al., AJ 126, 539–574 (2003). 3. F. Fiore et al., A&A 409, 79–90 (2003). 4. W. N. Brandt and G. Hasinger, ARA&A 43, 827–859 (2005). 5. A. Comastri, G. Setti, G. Zamorani and G. Hasinger, A&A 296, 1–12 (1995). 6. R. Gilli, A. Comastri and G. Hasinger, A&A 463, 79–96 (2007). 7. F. E. Bauer, D. M. Alexander, W. N. Brandt, D. P. Schneider, E. Treister, A. E. Hornschemeier and G. P. Garmire, AJ 128, 2048–2065 (2004). 8. R. C. Hickox and M. Markevitch, ApJ 645, 95–114 (2006). 9. A. J. Barger, L. L. Cowie, R. F. Mushotzky, Y. Yang, W.-H. Wang, A. T. Steffen and P. Capak, AJ 129, 578–609 (2005). 10. P. F. Hopkins, L. Hernquist, T. J. Cox, T. Di Matteo, B. Robertson and V. Springel, ApJS 163, 1–49 (2006). 11. M. Elvis et al., ApJS 95, 1–68 (1994). 12. G. T. 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Dunlop, MNRAS 352, 1390–1404 (2004). astro-ph/0612023 arXiv:0704.0735 Introduction Sample selection and Ks-band properties Spitzer data Analysis of the Type 2 quasar spectral energy distributions Extended sources Point-like sources Bolometric corrections Black hole masses and Eddington ratios Summary

0704.1301

IRAS 18317-0757: A Cluster of Embedded Massive Stars and Protostars

IRAS 18317−0757: A Cluster of Embedded Massive Stars and Protostars T.R. Hunter, Q. Zhang, T.K. Sridharan Harvard-Smithsonian Center for Astrophysics, MS-78, 60 Garden St., Cambridge, MA 02138 [email protected] ABSTRACT We present high resolution, multiwavelength continuum and molecular line images of the massive star-forming region IRAS 18317−0757. The global infrared through millimeter spectral energy distribution can be approximated by a two temperature model (25K and 63K) with a total luminosity of approximately log(L/L⊙) = 5.2. Previous submillimeter imaging resolved this region into a cluster of five dust cores, one of which is associated with the ultracompact H II region G23.955+0.150, and another with a water maser. In our new 2.7mm continuum image obtained with BIMA, only the UCH II region is detected, with total flux and morphology in good agreement with the free-free emission in the VLA centimeterwave maps. For the other four objects, the non-detections at 2.7mm and in the MSX mid-infrared bands are consistent with cool dust emission with a temperature of 13-40K and a luminosity of 1000-40000 L⊙. By combining single-dish and interferometric data, we have identified over two dozen virialized C18O cores in this region which contain ≈ 40% of the total molecular gas mass present. While the overall extent of the C18O and dust emission is similar, their emission peaks do not correlate well in detail. At least 11 of the 123 infrared stars identified by 2MASS in this region are likely to be associated with the star-forming cluster. Two of these objects (both associated with UCH II) were previously identified as O stars via infrared spectroscopy. Most of the rest of the reddened stars have no obvious correlation with the C18O cores or the dust continuum sources. In summary, our observations indicate that considerable fragmentation of the molecular cloud has taken place during the time required for the UCH II region to form and for the O stars to become detectable at infrared wavelengths. Additional star formation appears to be ongoing on the periphery of the central region where up to four B-type (proto)stars have formed amongst a substantial number of C18O cores. http://arxiv.org/abs/0704.1301v1 – 2 – Subject headings: stars:formation — ISM: individual (IRAS 18317−0757) — ISM: individual (G23.95+0.15) — infrared: stars — ISM: individual (AFGL2194) 1. Introduction The formation mechanism of massive stars is a topic of active research. Because massive star formation regions typically lie at distances of several kiloparsecs, the identification of high mass protostars requires both good sensitivity and high angular resolution. As a conse- quence of their presumed youth, ultracompact HII regions (UCH II regions) provide a good tracer of current massive star formation (Wood & Churchwell 1989) and may be expected to be accompanied by protostars in earlier evolutionary stages. Indeed, recent high-resolution millimeterwave images of UCH II regions have revealed the high-mass equivalent of “Class 0” protostars. Examples include the young stellar object IRAS 23385+6053 (Molinari et al. 1998), the compact methyl cyanide core near the G31.41+0.31 UCH II (Cesaroni et al. 1994), the proto-B-star G34.24+0.14MM (Hunter et al. 1998), the G9.62+0.19-F hot core (Testi et al. 2000), and the protocluster G24.78+0.08 (Furuya et al. 2002). To identify these deeply-embedded objects requires an optically thin tracer in order to probe through the large extinction toward the giant molecular cloud cores that harbor them. Submillimeter contin- uum emission from cool dust is a good tracer of protostars because it remains optically thin at high column densities (NH . 10 25cm−2) (Mezger 1994). Similarly, spectral line emission from C18O is a good optically thin tracer that can reveal areas of high molecular gas column density. Our target in this study, IRAS 18317−0757, is a luminous infrared source (log LFIR = 5.2) at a kinematic distance of 4.9 kpc (vLSR = 80 km s −1). Based on its IRAS colors, it has been identified as a massive protostellar candidate (Chan, Henning, & Schreyer 1996). Pre- vious single-dish radio frequency studies of this region have revealed hydrogen recombination line emission (Kim & Koo 2001; Lockman 1989; Wink, Wilson, & Bieging 1983) and water maser emission (Genzel & Downes 1977; Churchwell, Walmsley, & Cesaroni 1990). The cen- timeterwave continuum emission shows both extended components (up to 13′) (Kim & Koo 2001; Becker et al. 1994) and a UCH II region (Wood & Churchwell 1989). The region has been detected in various dense gas tracers including the NH3(1,1), (2,2) and (3,3) transi- tions (Churchwell, Walmsley, & Cesaroni 1990) and CS(7-6) (Plume, Jaffe, & Evans 1992), though it was not detected in a methyl cyanide search (Pankonin et al. 2001), nor in a 6 GHz hydroxl maser search (Baudry et al. 1997), nor in two 6.7 GHz methanol maser searches (Szymczak, Hrynek, & Kus 2000; Walsh et al. 1997). The CO(1-0) line shows a – 3 – complex profile which has prevented the identification of high-velocity outflow emission in large-beam (1′) surveys (Shepherd & Churchwell 1996). Also known as AFGL2194, compact infrared emission was first detected from the ground in theK, L, andM bands by Moorwood & Salinari (1981) and later by Chini, Krügel, & Wargau (1987). Airborne observations of far-infrared continuum and fine-structure lines (S, O, N, and Ne) yield an electron density of 3500 cm−3 for the UCH II region and indicate a stellar type of O9 to early B (Simpson et al. 1995). Complete infrared spectra (2.4− 195µm) have been recorded by the ISO SWS and LWS spectrometers (Peeters et al. 2002). At higher angular resolution, the region has been independently observed as part of two submillimeter continuum imaging surveys. In both cases, the emission is resolved into several components (Hunter et al. 2000; Mueller et al. 2002). Recent near-infrared imaging and spectroscopy has revealed the presence of a small cluster of stars associated with the UCH II region, including an O7 star (with N III emission and He II absorption) whose ionizing flux can account for all of the compact radio continuum emission (Hanson, Luhman, & Rieke 2002). These develop- ments have prompted the higher angular resolution millimeterwave observations which are presented in this paper in hopes of understanding this active site of massive star formation. 2. Observations With the Berkeley-Illinois-Maryland-Association (BIMA) Millimeter Array (Welch et al. 1996), IRAS 18317−0757 was simultaneously observed in 110 GHz continuum and C18O (1- 0). The continuum bandwidth was 600 MHz. The spectral resolution for the line data was 0.2 MHz (0.53 km s−1). The phase gain calibrator was the quasar 1741−038. The bandpass calibrator was 3C273. The absolute flux calibration is based on 3C273 and Uranus. A single track in B-configuration was obtained on 1998 October 10, and in C-configuration on 1999 February 5. The synthesized beam for the combined data imaged with robust weighting is 4.4′′ by 2.3′′ at a position angle of −2◦. To recover the missing flux from extended structures in the interferometer spectral line data, a single-dish map of C18O (1-0) was recorded at the NRAO1 12 Meter telescope on 16 June 2000. A regular grid of 7x7 points was observed, with a spacing of 25′′ providing full sampling of the telescope’s 58′′ beam. The system temperature was 200K and the on- source integration time was 2.8 minutes per point. The data were combined as zero-spacing information with the BIMA data in the MIRIAD (Multichannel Image Reconstruction, Image 1The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. – 4 – Analysis and Display) software package. The resulting beamsize in the final datacube (with natural weighting and UV tapering applied) is 10.3′′ × 7.1′′ at a position angle of −18◦. To complement the millimeter data, infrared images and point source information for this region in J , H and K bands were obtained from the 2 Micron All Sky Survey (2MASS) (Cutri et al. 2003) and in the mid-infrared bands from the Midcourse Space Experiment (MSX) archives (Egan et al. 1999) and HIRES-processed IRAS data (Hunter 1997). Radio continuum images were also retrieved from the VLA galactic plane survey of Becker et al. (1994). 3. Results 3.1. Millimeter and radio continuum The 2.7mm continuum image from our BIMA observations is shown in grayscale in Fig- ure 1, along with overlays of the 6cm and 20cm contours from the VLA galactic plane survey images (Becker et al. 1994). At all three wavelengths, the source structure consists of a bright rim on one edge of a partially-complete shell. Both the IRAS point source and the MSX point source positions lie close to the middle of the shell region. The bright, compact component was identifed as an irregular/multiple-peaked UCH II region by Wood & Churchwell (1989). At the two longest radio wavelengths, additional faint emission extends to the northwest. 3.2. Submillimeter continuum The 350 micron continuum image from the survey of Hunter et al. (2000) is shown in grayscale in Figure 2. For comparison, the position of the IRAS and MSX point sources are indicated along with the single-dish water maser position. Five independent submillimeter sources can be identified and their coordinates and flux densities are given in Table 1. The two dominant sources are SMM1 and SMM2. The peak of SMM1 coincides with the UCH II position. SMM2 lies 22′′ to the west, and likely coincides with the water maser emission, whose position is uncertain to ±10′′ (no interferometric observations exist). The water maser is apparently quite variable over time: 60 Jy in 1976 (Genzel & Downes 1977) to 0.7 Jy in 1989 (Churchwell, Walmsley, & Cesaroni 1990) to undetected in 2002 (H. Beuther 2003, private communication). The three other sources have no known counterpart at other wavelengths. – 5 – Fig. 1.— In all panels, the grayscale is the 2.7 mm continuum imaged with a 4.4′′ by 2.3′′ synthesized beam. Contours: upper right (2.7 mm): 3.6 mJy/beam × (4 to 36 by 4); lower left (6 cm): 1.27 mJy/beam × (-3, 3, 6, 12, 24, 48, 96); lower right (20 cm): 0.35 mJy/beam × (-3, 3, 6, 12, 24, 48, 96, 192, 384, 768). Both centimeter maps were generated from data originally published by Becker et al. (1994). In the upper left panel, the dashed circle marks the BIMA primary beam at half-maximum response. In the upper right panel, the letters A, C, D and E correspond to the fitted position of the peak emission in the corresponding MSX band (8.3µm, 12.1µm, 14.7µm, 21.4µm) all of which are contained in the IRAS PSC error ellipse. The six-pointed stars indicate the positions of 2MASS star 86 (nearest to the 2.7mm peak) and star 92. They are classified as O8.5 and O7, respectively (Hanson, Luhman, & Rieke 2002). The line marks the axis used to produce Figure 10. – 6 – 3.3. Mid-infrared continuum Each of the MSX images of this field show that the mid-infrared emission is dominated by the UCH II region associated with SMM1. Contour plots of two of the bands (8.3 and 14.7µm) are shown in Figure 3. The positions of the five submillimeter sources are indicated in both panels. The fitted positions of a two-dimensional Gaussian model in each of the four MSX bands all agree to within 2′′ and fall within the IRAS PSC error ellipse. Of the five submillimeter sources, the peak in each band lies closest to SMM1. HIRES-processed images provide additional high resolution information from the IRAS data (Aumann, Fowler, & Melnyk 1990). The 20-iteration contour maps at 25 and 60µm are shown in Figure 4, again with the five submillimeter sources marked. The emission remains essentially unresolved in each band, though there is some hint that the two westernmost submillimeter sources (SMM3 and 4) are detected in the contour extensions at 25 and 60µm. 3.4. Spectral Energy Distributions Using the flux density data from Table 2, the mid-infrared through radio wavelength spectral energy distribution (SED) for the entire region is shown in Figure 5. The flux density measurements at wavelengths longward of 21 µm have been fit with a simple two-temperature modified blackbody dust model plus a free-free component, summarized in Table 3. The flux density measurement at 1.3mm from the literature (Chini et al. 1986) should be considered a lower limit as it was obtained with a single element detector with a 90′′ beam centered on the IRAS position, which misses most of SMM3 and SMM4. The temperature of the cold component of dust (25K) agrees quite well with the kinetic temperature (25.8K) de- rived from ammonia (1,1) and (2,2) observations with a 40′′ beam centered on the UCH II position (Churchwell, Walmsley, & Cesaroni 1990). It is interesting to note that the warm component of dust dominates the total luminosity of the region, which is in contrast to more isolated high-mass protostellar objects (Sridharan et al. 2002) and even many other UCH II regions. Using the grain emissivity index (β) along with the temperature and optical depth derived from the fit, one can calculate the number of cold and warm grains required to explain the observed flux density (Lonsdale-Persson & Helou 1987; Hildebrand 1983). The corresponding mass of dust can then be calculated for each clump and for the extended emis- sion. As is typical, the cold grains dominate the mass of dust. Assuming a gas to dust mass ratio of 100 (Sodroski et al. 1997), the total gas mass of each clump is listed in column six of Table 1 and the total mass of the region is ≈ 7400 M⊙. Using the individual gas masses and source diameters (from the angular diameter and distance), we compute the column density of hydrogen (NH = NHI + 2NH2) toward each clump in column seven of Table 1. – 7 – Fig. 2.— 350 micron continuum image of IRAS 18317−0757 observed with an 11′′ beam (Hunter et al. 2000). The UCH II position is marked by the star symbol. The cross marks the water maser position uncertainty (Genzel & Downes 1977). The square contains the fitted peak of the single point source seen in all four MSX bands, which is contained by the IRAS error ellipse. The dashed circle marks the BIMA primary beamsize, for reference to Figures 3, 7 and 11. The circles and dotted line defines the point sources and extended emission region listed in Table 1. – 8 – Fig. 3.— MSX images of IRAS 18317−0757. Contour levels are 0.005, 0.01, 0.02, 0.04, 0.08, 0.16, 0.32 and 0.64 erg cm−2 s−1 steradian−1. The crosses mark the position of the submil- limeter sources SMM1-5 (Table 1). The dashed circle marks the BIMA primary beamsize, for reference to Figures 2, 7 and 11. – 9 – Fig. 4.— IRAS HIRES-processed images of IRAS 18317−0757. Gaussian fits to the HIRES restoring beams are denoted by dotted ellipses. The crosses mark the position of the sub- millimeter sources SMM1-5 (Table 1). The contour levels are in MegaJy steradian−1: 25µm (190, 381, 762, 1523, 3047, 6093, 12186), 60µm (258, 516, 1032, 2064, 4128, 8257, 16513). – 10 – Finally, we have estimated the visual and infrared (K band) extinctions toward each clump by using the conversion formula of AV ∼ NH/(2×10 21) derived from observations of the ISM at UV (Whittet 1981; Bohlin, Savage & Drake 1978) and X-ray wavelengths (Ryter 1996; Predehl & Schmitt 1995), followed by the relation AK = 0.112AV from Rieke & Lebofsky (1985). The extinction values listed in columns eight and nine of Table 1 have been further reduced by a factor of two to more accurately estimate the extinction toward a young star at the center of the clump, rather than behind it. As the SED model predicts, the free-free emission mechanism still dominates over the dust emission at frequencies as high as 110 GHz. In fact, the image at this frequency is nearly identical to the centimeter images. The 110 GHz flux densities for the SMM1-5 are listed in Table 1. Nearly all of the 110 GHz flux can be associated with SMM1, with the rest of the emission sitting just outside the 22′′ aperture used to define this object in the 350µm map. By contrast, we have not detected any emission for SMM2-5. Each of these upper limits is consistent with an SED proportional to ν4, corresponding to dust emission with β = 2. For the case of SMM3 and SMM4, they lie sufficiently far from the main source that useful upper limits can be obtained from both the IRAS and MSX data which provide a constraint on the individual properties of these dust cores. To visualize this constraint, the spectral energy distributions of SMM3 and SMM4 are shown in Figure 6 along with the two most extreme models consistent with the data. The corresponding dust temperature and luminosity upper and lower limits are summarized in Table 4. Although the luminosity remains uncertain to within a factor of 30-60, the lower limits (∼ 1000L⊙) indicate that these objects may be powered by individual massive stars or protostars. 3.5. C18O (1-0) images The integrated C18O (1-0) line emission (75-85 km s−1) is shown as grayscale in Figure 7. The dotted circles denote the positions of the submillimeter continuum clumps from Figure 2. The C18O emission has been clipped (set to zero) at all points below 2.5σ (0.2 Jy beam−1). There are five major peaks of emission, four of which agree roughly with the submillimeter continuum sources SMM1-4. The strongest component peaks very close (offset: ∆α,∆δ = +3.0′′,−2.6′′) to a small cluster of stars identified by Hanson, Luhman, & Rieke (2002) that are associated with the UCH II emission and SMM1. Assuming optically-thin line emission with T = 25K, we have computed the total column density of C18O listed in column 5 of Table 5. These values have been converted to visual extinction AV using the relationship of Hayakawa et al. (1999) for the Chamaeleon I dark cloud: N(C18O)(cm−2) = 3.5× 1014AV − 5.7 × 1014. Next, the values of AV have been converted AK (see section 3.4) and these – 11 – Fig. 5.— Global SED of IRAS 18317−0757. The flux density measurements are summarized in Table 2 and the components of the dust and free-free emission models are described in Table 3. The cold dust component is the dashed line, the warm dust component is the dotted line, and the free-free component is the dash-dot line. The sum of all three components is the solid line. – 12 – Fig. 6.— Individual SEDs of the submillimeter sources SMM3 and SMM4. The two longest wavelength flux density measurements are summarized in Table 1. The infrared upper limits are from the MSX and IRAS HIRES images. The solid lines indicate the warmest dust model consistent with the data, while the dashed lines indicate the coolest dust model (17-41K for SMM3, 14-41K for SMM4). – 13 – are listed in column 6 of Table 5. Assuming relative abundances of NH : NCO = 10 4 and NCO : NC18O = 490, the gas mass of each clump has been computed and listed in column 7 of Table 5. Likewise, the mass of gas associated with each of the continuum sources SMM1-5 is listed in Table 6. The total gas mass (including the extended emission and all the clumps) is 7300M⊙. Although this value is in good agreement with the mass derived independently from the total 350µm dust emission, the fraction of mass in the extended emission (outside of SMM1-5) is 70% in C18O but only 40% in dust. To study the C18O emission in greater detail, channel maps of C18O are shown in Figure 8. We have analyzed these maps in two ways: we first inspected the maps visually, then used an objective computer algorithm. In the visual method, we manually identified 26 cores in position-velocity space. Shown in Figure 9 is a grid of spectra constructed by integrating the emission in a 10′′ aperture (0.24 pc) centered at the position of each core. The mass contained within these apertures represents about 40% of the total gas mass. We next attempted to objectively analyze the C18O date cube by running the “clumpfind” program (Williams, de Geus & Blitz 1994). This algorithm contours the data, locates the peaks and follows them to the low intensity limit without any constraint on the shape of the resulting clump. It was designed to operate on large scale maps of GMCs in which the emission is well separated into distinct clumps. Our data do not fit this description, as the cores are embedded in significant extended emission. Nevertheless, we proceeded and used the recommended contour levels by setting both the starting contour and the contour interval to be twice the RMS of the individual channel maps (0.6 Jy km s−1). The program identified 17 clumps, three of which were weak and centered slightly outside the primary beam (which we reject). The largest seven clumps range in mass from 200-1400 M⊙, while the smallest seven range from 18-94 M⊙. The fraction of mass placed into these 14 clumps is 45% of the total emission, which is quite similar to our visual technique. The emission from the several cores in the western ridge (2,4,5,6,7,8,10) were merged together by clumpfind into a single large clump in the late stages of the execution when the lowest contour levels are being examined. In a few other cases, two initial clumps merged into one. This merging effect of the algorithm explains the fewer number (but larger mass) of clumps found. The rest of the clumps are in good general agreement with our visual identification technique. A Gaussian line profile has been fit to each C18O clump, and the corresponding velocity, amplitude and linewidth is given in Table 5. Using the linewidth (δv) and aperture radius (r), we compute the virial mass from the formula: M = 210r(pc) δv2 (km2 s−2) (Caselli et al. 2002). In most cases, and in the overall sum, the virial masses of the clumps are quite similar to their C18O-derived masses, suggesting that the clumps are in hydrostatic equilibrium. In only three clumps does the C18O-derived mass exceed the virial mass by more than 50%. The – 14 – highest excess (76%) is seen in clump 18, associated with the UCH II region. Two of these clumps (17 and 18) lie near the UCH II region and also exhibit the steepest spatial profiles, possibly suggesting an unstable condition. A cut along position angle 80◦ in the velocity channel centered at 80.1 km s−1 is shown in Figure 10. The minimum in C18O emission corresponds to the presence of the free-free continuum emission along the southern portion of the shell structure seen in Figure 1, thus the steep profile may be due to interaction with the UCH II region. In any case, considering the uncertainties in the C18O mass calculations, the good agreement between the C18O mass and the virial mass is analagous to the results found in a survey of 40 lower-mass C18O clumps in the Taurus complex (Onishi et al. 1996). 3.6. Near-infrared point sources In the 2MASS All-Sky Data Release Point Source Catalog there are 123 objects within a 1′ radius of the UCH II position. Listed in Table 7 are the 44 of these stars that are detected in all three bands. The K band image is shown in Figure 11 with the position of the C18O clumps indicated by dotted circles. In general, the non-coincidence between the two phenomena is striking. As listed in column 6 of Table 5, the extinction at K band through the C18O clumps ranges from 2.3 to 12.5 magnitudes, with a median value of 8.1. The faintest star detected has K magnitude of 14.44 while the brightest upper limit has magnitude 10.11. Thus, even the brightest K band star observed in the field (star 112 with MK = 7.35) would be undetected if placed behind the typical clump. This fact may account for the lack of stars seen toward the C18O clumps. A (J −H) vs. (H −K) color-color diagram of the 2MASS stars is shown in Figure 12. The solid line marks the locus of main sequence stars and the dashed lines denote the reddening vector which is annotated in magnitudes of visual extinction. We see that 19 of the stars exhibit more than 10 magnitudes of visual extinction. Eleven of these 19 are located within the lowest C18O contours (see Figure 13) and are likely to be associated with the star-forming material of the cluster. For example, associated with the UCH II region is a small cluster of five stars. Of these five stars, star 92 is the object identified as an O8.5 star on the basis of its infrared spectrum (with weak HeI emission) (Hanson, Luhman, & Rieke 2002). It lies near the peak of the millimeter continuum map and is one of the few stars that reside within any of the C18O clumps. The next closest star, number 86, lies close to the center of the shell structure seen in the millimeter continuum. Due to the presence of N III emission, it is classified as an O7 star. The ratio of He I to Brγ confirms the level of ionizing flux expected from such a star. To within a factor of two, it can account for all the Lyman continuum flux from the centimeter emission and probably explains the shell-like symmetry. – 15 – Fig. 7.— The integrated emission from C18O (1-0) is shown in grayscale. The dashed circle marks the BIMA primary beam at half-maximum response. The dotted circles mark the apertures defining submillimeter continuum sources SMM1-5. The cross marks the water maser position uncertainty from Genzel & Downes (1977). The star marks the peak of the UCH II region. The ellipse is the IRAS position uncertainty and the square contains the MSX point source position. – 16 – Fig. 8.— Channel maps of C18O (1-0). The LSR velocity (in km s−1) of the center of the channel is given in the upper right corner of each panel. The top left panel shows the integrated intensity map, along with the synthesized beamsize. The crosses mark the positions of the C18O cores identified and listed in Table 5. Contour levels are +/ − 0.2 × (3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33) Jy beam−1. – 17 – Fig. 9.— Spectra from each C18O (1-0) clump position listed in Table 5, integrated over a 10′′ diameter aperture. A Gaussian fit to each spectrum is overlaid in dotted lines. The fit parameters and the corresponding virial masses are listed in Table 5. – 18 – Fig. 10.— Solid line: spatial profile of C18O along the position angle +80◦ passing through clump 18 in the channel map for velocity 80.1 km s−1. Dotted line: same profile for the 2.7mm continuum image. The axis of this strip is indicated in Figure 13 and Figure 1. The vertical scale on the left side describes the line data, while the right side describes the continuum. – 19 – Besides stars 86 and 92, three additional stars (4, 47 and 106) exhibit excess near-infrared emission (i.e. they lie to the right of the reddening vector in Figure 12) which may indicate the presence of circumstellar disks. Star 106 lies only 4′′ from the center of SMM5, while star 47 lies at the edge of C18O clump 12. Star 4 sits just outside the BIMA primary beam where very little C18O has been detected. The brightest star in the field, number 112, has J magnitude 8.7 and the colors of an M2 star consistent with about 3 magnitudes of visual extinction. It could be a foreground giant at 1.8 kpc. There is no reference to it in the SIMBAD database. 4. Discussion 4.1. A protocluster of massive stars? With the exception of the two O-stars associated with SMM1, and star 106 possibly associated with SMM5, none of the near-infrared stars are associated with the other sub- millimeter continuum or C18O cores. The “starless” continuum objects SMM2-SMM4 may harbor embedded ZAMS stars or protostars that are not yet visible in the near-infrared, while the C18O cores have not yet formed protostars. If so, they may resemble the lower mass prestellar cores detected by ISO (Bacmann et al. 2000). To examine this hypothesis, we can compare the limiting K band magnitude of the 2MASS image with the expected brightness of an embedded ZAMS star with total luminosity equal to the dust luminosity of each core. It is difficult to estimate the individual luminosities of these cores due to the limited, coarse-resolution imaging data available on the mid-infrared side of their SEDs. However, using upper limits obtained from the MSX and IRAS images (as listed in Table 4) the luminosities of SMM3 and 4 are constrained to be in the range ≈ 1000− 40000 L⊙, de- pending on the dust temperature. Assuming the temperature of 25K derived for the region as a whole, yields a typical luminosity of 2 × 104 L⊙ consistent with a B0 star. A B0 star has absolute visual magnitude MV = −4.1 (Allen 1976) and V − K = −0.85 (Koornneef 1983), yielding MK = −3.25. At a distance of 4.9 kpc, this would be reduced to an apparent K magnitude of 8.65. If such a star was placed at the center of the SMM4 dust cloud, a K extinction of 6.4 magnitudes (see Table 1) would result, yielding a final K magnitude of ∼ 15. By comparison, the faintest star detected in the 2MASS image has a K magnitude of 14.44. Thus we cannot rule out the possibility that each submillimeter source (SMM2-5) may contain a ZAMS or main sequence star rather than a protostar. Deeper imaging in K, L or M band would improve the constraints. At present, our best reasonable conclusion is that SMM2-5 contain some number of young stellar objects or main sequence stars with luminosities equivalent to at least a B-type star. – 20 – Fig. 11.— Near-infrared K band image of IRAS 18317−0757 from the 2MASS database. The dotted circles represent the position of the 26 C18O (1-0) clumps identified in Table 5 and plotted in Figure 8. The line indicates the axis used to produce the emission profiles in Figure 10. – 21 – Fig. 12.— Color-color diagram (J −H vs. H −K) for the stars detected in all three bands of the 2MASS PSC (numbers correspond to Table 7). The solid line marks the main se- quence (Koornneef 1983). The dashed lines marks the reddening band (Rieke & Lebofsky 1985) with visual extinction levels marked (for an O7-O9 star). The circled numbers are the two stars associated with the peak centimeter through submillimeter continuum emis- sion. Of these, star 92 is most closely associated with the UCH II region and exhibits an infrared spectrum consistent with a spectral type of O8.5, while star 86 is classified as O7 (Hanson, Luhman, & Rieke 2002). – 22 – Fig. 13.— The integrated emission from C18O (1-0) is shown in grayscale and contours. The dashed circle marks the BIMA primary beam at half-maximum response. Point sources from the 2MASS catalog (detected in all three bands) are indicated by their number from Table 7 (with font size proportional to K band brightness). The cross marks the water maser position uncertainty from Genzel & Downes (1977). The line indicates the axis used to produce the emission profiles in Figure 10. Contour levels are 20% to 90% of the peak emission (15.8 Jy beam−1). – 23 – With the exception of SMM5, as one moves from east to west across the region, the general trend is for objects in the cluster to exhibit fewer signs of compact, energetic phe- nomena. SMM1 is associated with the well-developed UCH II region. The dust core (SMM2) associated with the water maser probably traces an intermediate stage indicative of outflow or disk activity from the protostar. The next two dust cores (SMM3 and SMM4) exhibit no maser activity or ionized gas. We note that the dust-derived masses for these two objects exceed but remain in reasonable agreement with the C18O-derived masses (within 27% and 45% respectively). In contrast, the dust-derived mass of the faintest submillimeter source (SMM5) is a factor of 3 larger than the C18O-derived mass. Unfortunately, the uncertainties in the mass estimates are too large for us to interpret this difference in physical terms (such as a depletion of CO, which has been seen in objects such as B68 by Bergin et al. (2002)). The other molecular cores not seen in continuum may be the youngest features in the region on their way to forming stars. Or they could simply be colder, inactive objects where the accompanying dust emission is below the detection threshold. Deeper and higher reoslution submillimeter observations are needed to explore these possibilities. 4.2. Fragmented structure The C18O emission of IRAS 18317−0757 is distributed in clumps aligned roughly along an east-west ridge. Evidence of periodic density structure has been previously observed in C18O in giant molecular clouds, specifically Orion A (Dutrey et al. 1991). The typical spatial wavelength they find is 1 parsec, and the fragment masses range from 70-100 M⊙. More recently, fragmentation has been seen to extend to even smaller spatial scales in Orion from VLA observations of NH3 (Wiseman & Ho 1998). Similarly, new observations of the mini- starburst W43 in submillimeter dust continuum reveal about 50 fragments with typical size of 0.25 pc and mass of 300 M⊙ (Motte, Schilke, & Lis 2003). In comparison, C 18O maps of the Taurus complex reveal 40 dense cores of a similar typical size as those in W43 (0.23 pc) but with a smaller typical mass of 23 M⊙ (Onishi et al. 1996). In IRAS 18317−0757 the typical spacing we find between the major C18O cores is roughly 24′′ (0.5 pc), i.e. intermediate between Orion and W43, while our fragment masses range from 35-187M⊙, i.e. intermediate between W43 and Taurus. The fraction of total mass that resides in C18O cores is 40% which is somewhat larger than the value of 20% seen in NH3 cores in W3OH (Tieftrunk et al. 1998), and the 19% seen in CS cores in Orion B (Lada, Bally & Stark 1991). In any case, it is becoming clear that high-mass star formation regions, like their low-mass counterparts, contain a wealth of information on the mass spectrum of protostellar fragmentation. Whether these objects will all form stars remains unclear. A combination of single-dish and sensitive interferometric studies will be needed to better quantify the picture, especially down to the – 24 – low mass end of the distribution. 4.3. Future work High resolution mid and far infrared imaging is needed to accurately determine the temperature and size of the individual dust cores presently identified, and to search for lower mass objects. Deeper imaging in the near-infrared is needed to search for additional ZAMS stars at high extinction levels within the dust and C18O cores. Also, narrow band imaging in H2 lines and (sub)millimeter interferometric imaging of SiO transitions may help distinguish which of the cores show jets and outflows. Submillimeter interferometry with higher spectral resolution in other optically-thin tracers less affected by depletion would be useful to search for evidence of active infall toward the C18O cores identified in this work. Finally, interferometric observations of the 22 GHz or submillimeter water maser transitions would be quite useful to better localize the maser activity to SMM2 or one of the C18O cores2. 5. Conclusions Our high angular resolution observations of the luminous (log(L/L⊙)=5.2), massive star-forming region IRAS 18317−0757 have revealed a complex field of objects likely to be in various stages of star formation. Of the five submillimeter dust cores, one is associated with the UCH II region G23.955+0.150, and another with a water maser. The 2.7mm continuum is completely dominated by free-free emission from the UCH II region, with total flux and morphology in agreement with VLA centimeterwave maps. For the other four objects, the upper limits found at 2.7mm and in the MSX mid-infrared band are consistent with pure optically-thin dust emission at temperatures of 13-40 K and a dust grain emissivity index β = 2. Three out of four of these objects have no associated 2MASS star, and they are each likely to contain at least one (proto)star of luminosity 1000-40000 L⊙. In addition, we have identified two dozen C18O cores in this region which contain ≈ 40% of the total molecular gas mass (7300M⊙) present. Their typical size is 0.25 pc and linewidth is 2-3 km s −1. While the overall extent of the C18O and dust emission is similar, most of the emission peaks do 2VLA observations undertaken by the authors on 2004 January 08 (project AH833) in B-configuration have resolved a pair of 22 GHz water maser spots: one (at 18:34:23.99, -07:54:48.4) coincident with the submillimeter continuum source SMM 2 and the other one (at 18:34:24.49, -07:54:47.5) coincident with the molecular gas clump number 16. – 25 – not correlate well in detail. Compared to the dust emission, a greater fraction of the C18O emission exists in extended features. At least 11 of the 123 infrared stars identified by 2MASS in this region are likely to be embedded in the star-forming material, including two O stars powering the UCH II emission. Most of the rest of the reddened stars anti-correlate with the position of the dust and C18O cores and are likely visible simply due to the relatively lower extinction. In summary, our observations indicate that considerable fragmentation of the molecular cloud has taken place during the time required for the UCH II region to form and for the O stars to become detectable at infrared wavelengths. Additional star formation appears to be ongoing throughout the region with evidence for up to four B-type (proto)stars scattered amongst more than two dozen molecular gas cores. We thank Yu-Nung Su for obtaining the 12-Meter data for us, Robert Becker for pro- viding freshly-prepared VLA survey images, and Ed Churchwell for providing valuable com- ments on the manuscript. Several expedient suggestions and corrections to this paper were provided by the anonymous referee. This research made use of data products from the Mid- course Space Experiment. Processing of the data was funded by the Ballistic Missile Defense Organization with additional support from NASA Office of Space Science. 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S., & Churchwell, E. 1989, ApJS, 69, 831 This preprint was prepared with the AAS LATEX macros v5.2. – 29 – Table 1: Observed emission properties of submillimeter clumps Coordinates (J2000) Flux densitya (Jy) Massd log(NH) Source R.A. Decl. 350µm 2.7mm M⊙ cm −2 mag mag SMM1 18:34:25.4 −07:54:49 63± 6 1.47 ± 0.03 1360 ± 120 23.75 ± 0.04 140 15.7 SMM2 18:34:24.0 −07:54:50 67± 6 < 0.033b 1400 ± 120 23.76 ± 0.04 144 16.1 SMM3 18:34:21.8 −07:55:05 30± 6 < 0.033b 640± 120 23.42 ± 0.08 66 7.4 SMM4 18:34:21.7 −07:54:44 27± 6 < 0.045b 560± 120 23.36 ± 0.09 57 6.4 SMM5 18:34:26.5 −07:54:36 24± 6 < 0.033b 500± 120 23.32 ± 0.09 52 5.8 Extended 140 ± 20 0.25 ± 0.03 2900 ± 300 Totalc 350 ± 60 1.72 ± 0.06 7400 ± 900 aWithin 22′′ aperture, as shown in Figure 2 bUpper limits are 3σ cTotal flux including extended emission within dotted region of Figure 2 dAssuming T=25K and grain emissivity Q=1.0E-04 at 350µm (Hildebrand 1983) eAssuming uniform emission of diameter of 11′′ (0.25 pc) fExtinction to a star at the center of the dust core, i.e. half the column density – 30 – Table 2. Summary of flux density measurements of IRAS 18317−0757 Frequency (GHz) Wavelength (µm) Flux (Jy) Aperture/beamsize Instrument Reference 3.8 0.885a 15′′ CFHT 1 4.6 0.902a 15′′ CFHT 1 8.3 34.4 ± 1.7 21′′ MSX this work 12 66.3 ± 6.6 62′′ × 27′′ IRAS 2 12.1 61± 2 22′′ MSX this work 14.7 61± 2 22′′ MSX this work 21.3 245 ± 15 23′′ MSX this work 25 395 ± 99 84′′ × 29′′ IRAS 2 33.5 1400 ± 100 44′′ KAO 3 36 1700 ± 100 44′′ KAO 3 51 1900 ± 100 44′′ KAO 3 57.3 2100 ± 100 44′′ KAO 3 60 2285 ± 708 144′′ × 62′′ IRAS 2 88.4 2400 ± 100 44′′ KAO 3 100 3339 ± 902 152′′ × 137′′ IRAS 2 857 352 320 ± 64 CSO/SHARC 4 857 352 350 ± 60 98′′ × 44′′ CSO/SHARC this work 230 1.3 mm 4.6± 0.5 90′′ IRTF 2 110 2.725 mm 1.31 ± 0.07 54′′ NRAO 12m 5 110 2.735 mm 1.72 ± .04 BIMA this work 86 3.486 mm 1.32 ± 0.20 78′′ NRAO 11m 6 14.94 2.007 cm 0.536 b VLA-B 7 14.8 2.0 cm 1.56 ± 0.08 60′′ Effelsberg 100m 6 10.3 2.91 cm 2.11 ± 0.04 160′′ NRO-45m 8 8.875 3.378 cm 1.85 ± 0.09 84′′ Effelsberg 100m 6 4.9 6.1 cm 1.290 ± 0.003 9′′ × 4′′ VLA-B 9 4.875 6.15 cm 2.32 ± 0.12 2.6′ Effelsberg 100m 6 4.875 6.15 cm 2.50 ± 0.13 2.6′ Effelsberg 100m 10 4.860 6.17 cm 0.227 b VLA-A/B 7 1.527 19.6 cm 1.508 ± 0.005 VLA-B 9 1.527 19.6 cm 1.608 ± 0.040 7′′ × 4′′ VLA-B 11 1.425 21.0 cm 2.13 ± 0.01 VLA-DnC 12 References: (1) Chini, Krügel, & Wargau (1987); (2) Chini et al. (1986); (3) Simpson et al. (1995); (4) Mueller et al. (2002); (5) Wood, Churchwell, & Salter (1988); (6) Wink, Altenhoff, & Mezger (1982); (7) Wood & Churchwell (1989); (8) Handa et al. (1987); (9) Becker et al. (1994); (10) Altenhoff et al. (1979); (11) Garwood et al. (1988); (12) Kim & Koo (2001). – 31 – aaperture did not cover the UCH II position bmeasurements suffer from missing flux – 32 – Table 3: Parameters of the global SED model Component T (K) β τ125µm L (L⊙) M (M⊙) cold dust 25 2.0 0.38 50,000 7200 warm dust 63 2.0 0.062 110,000 70 total dust 160,000 7300 free-free Fν(Jy) = 2.3ν −0.117 Table 4: Range of parameters for the SED models of SMM3 and SMM4 Component Infrared limit T (K) βa τ125µm L (L⊙) M (M⊙) SMM3 < 696Jy at 60µm 17-41 2.0 0.30-0.049 1400-39000 270-1600 SMM4 < 2.3Jy at 21.3µm 14-41 2.0 0.48-0.044 630-38000 240-2600 aValue fixed in fit – 33 – Table 5. Position and mass of C18O (1-0) clumps R.A. Dec Emission log(N(C18O)) AK a MH2 b Velocity Peak flux Linewidth Mvirial # J2000 J2000 Jy km s−1 cm−2 mag. M⊙ km s −1 Jy beam−1 km s−1 M⊙ 1 18:34:21.183 -07:55:13.78 14.3 16.39 8.1 119 80.3± 0.1 6.9± 0.6 2.1± 0.2 110 2 18:34:21.583 -07:54:53.17 16.7 16.46 9.5 139 80.0± 0.1 6.3± 0.6 2.3± 0.2 120 3 18:34:21.602 -07:55:23.45 14.3 16.39 8.1 120 80.4± 0.2 6.2± 0.6 2.1± 0.2 110 4 18:34:21.689 -07:55:05.58 17.9 16.49 10.2 150 80.1± 0.1 7.8± 0.6 2.2± 0.2 115 5 18:34:21.760 -07:54:42.88 14.8 16.40 8.3 123 80.0± 0.1 5.5± 0.5 2.6± 0.3 136 6 18:34:22.090 -07:54:32.96 11.3 16.29 6.5 94 79.9± 0.2 3.9± 0.5 2.8± 0.4 147 7 18:34:22.097 -07:55:13.66 18.9 16.51 10.6 158 80.2± 0.1 8.3± 0.6 2.3± 0.2 120 8 18:34:22.321 -07:54:52.93 14.7 16.40 8.3 123 79.5± 0.1 6.1± 0.6 2.2± 0.2 115 9 18:34:22.470 -07:54:09.96 3.8 15.82 2.3 32 78.7± 1.0 4.4± 0.9 0.8± 0.2 42 10 18:34:22.539 -07:55:04.57 17.3 16.48 9.9 145 79.7± 0.1 7.1± 0.6 2.3± 0.2 120 11 18:34:23.098 -07:54:55.95 22.1 16.58 12.5 185 80.0± 0.1 6.6± 0.5 3.2± 0.2 168 12 18:34:23.381 -07:55:16.93 10.7 16.27 6.2 90 78.8± 0.3 3.1± 0.5 3.4± 0.5 154 13 18:34:23.457 -07:53:59.41 10.3 16.25 5.9 86 78.8± 0.3 3.0± 0.5 3.2± 0.6 168 14 18:34:23.558 -07:54:43.66 11.3 16.29 6.5 94 79.7± 0.2 3.9± 0.5 2.7± 0.4 122 15 18:34:23.750 -07:55:00.31 18.5 16.51 10.6 155 80.3± 0.2 5.6± 0.5 3.3± 0.3 161 16 18:34:24.341 -07:54:46.89 14.7 16.40 8.3 123 80.0± 0.2 4.8± 0.5 2.9± 0.3 131 17 18:34:24.570 -07:54:59.70 17.1 16.47 9.7 143 80.0± 0.1 6.4± 0.5 2.5± 0.2 113 18 18:34:25.749 -07:54:49.53 21.0 16.56 11.9 176 79.3± 0.1 9.1± 0.6 2.0± 0.2 100 19 18:34:25.968 -07:55:29.42 7.4 16.11 4.3 62 78.9± 0.1 5.2± 0.8 1.2± 0.2 36 20 18:34:26.017 -07:55:00.15 18.3 16.50 10.4 153 79.1± 0.1 7.8± 0.6 2.2± 0.2 115 21 18:34:26.812 -07:55:14.18 8.5 16.17 5.0 71 79.0± 0.1 5.0± 0.7 1.4± 0.3 49 22 18:34:27.061 -07:54:32.24 8.6 16.17 5.0 72 77.9± 0.2 4.4± 0.6 1.8± 0.3 81 23 18:34:27.319 -07:54:46.83 13.0 16.35 7.4 108 78.3± 0.1 6.0± 0.6 2.1± 0.2 110 24 18:34:27.843 -07:54:53.72 12.0 16.31 6.8 100 78.6± 0.1 6.1± 0.6 1.9± 0.2 81 25 18:34:22.880 -07:54:28.50 8.5 15.25 5.0 71 78.9± 0.3 2.7± 0.6 2.1± 0.6 110 26 18:34:23.292 -07:54:21.00 5.0 15.02 3.0 42 79.3± 0.3 2.6± 0.6 1.8± 0.6 81 Total clump component 352 2930 2915 Extended component 527 4410 Total emission 879 7340 aAV computed from N(C 18O) using formula from Hayakawa et al. (1999), and converted to AK using Rieke & Lebofsky (1985). bMass computed using formula from Scoville et al. (1986) assuming optically-thin gas at 25K and [12CO/C18O]=490 and [H2/CO]=10 yielding a conversion factor of 8.35M⊙ (Jy km s −1)−1. – 34 – Table 6: Mass of C18O (1-0) emission associated with SMM objects Emissiona Massb # Jy km s−1 M⊙ SMM1 53.5 450 SMM2 72.7 610 SMM3 60.1 500 SMM4 45.9 380 SMM5 20.3 170 Extended 620 5200 Total 870 7300 aNot corrected for primary beam attenuation bMass computed using formula from Scoville et al. (1986) assuming optically-thin gas at 25K and [12CO/C18O]=490 and [H2/CO]=10 – 35 – Table 7. Stars detected in all three 2MASS bands # 2MASS PSC 1 18342011-0755050 4a 18342057-0755207 5 18342064-0754497 6 18342066-0754341 7 18342067-0754593 8a 18342072-0755099 9 18342080-0754450 13 18342105-0755076 18 18342147-0754360 20 18342159-0755177 22a 18342166-0754189 23a 18342173-0754530 27a 18342201-0754594 29 18342216-0754251 31 18342246-0754219 37 18342269-0755357 39a 18342276-0755213 42a 18342290-0754516 43 18342298-0754369 47a 18342326-0755195 48a 18342333-0754504 49a 18342337-0754304 50 18342348-0754172 51a 18342349-0754103 53 18342371-0754156 55 18342385-0754347 59 18342399-0755140 67a 18342438-0755341 72a 18342463-0755311 76 18342487-0753549 79a 18342503-0754140 86a 18342523-0754455 92a 18342551-0754473 – 36 – Table 7—Continued # 2MASS PSC 96 18342566-0755162 99 18342580-0754214 106a 18342624-0754379 107a 18342627-0755288 108a 18342633-0755004 111 18342652-0754227 112 18342673-0755285 118 18342720-0755108 119 18342730-0754170 122 18342752-0755014 123 18342753-0755053 aStars with AV > 10, according to the color- color diagram shown in Figure 12 Introduction Observations Results Millimeter and radio continuum Submillimeter continuum Mid-infrared continuum Spectral Energy Distributions C18O (1-0) images Near-infrared point sources Discussion A protocluster of massive stars? Fragmented structure Future work Conclusions

0704.1302

Photometry of the SW Sex-type nova-like BH Lyncis in high state

Astronomy & Astrophysics manuscript no. 4530 c© ESO 2018 November 5, 2018 Photometry of the SW Sex-type nova-like BH Lyncis in high state⋆ V. Stanishev1,2⋆⋆, Z. Kraicheva2⋆⋆, and V. Genkov2⋆⋆ 1 Department of Physics, Stockholm University, Albanova University Center, 106 91 Stockholm, Sweden 2 Institute of Astronomy, Bulgarian Academy of Sciences, 72 Tsarighradsko Shousse Blvd., 1784 Sofia, Bulgaria Received ; accepted ABSTRACT Aims. We present a photometric study of the deeply eclipsing SW Sex-type nova-like cataclysmic variable star BH Lyn. Methods. Time-resolved V-band CCD photometry was obtained for seven nights between 1999 and 2004. Results. We determined 11 new eclipse timings of BH Lyn and derived a refined orbital ephemeris with an orbital period of 0.d155875577(14). During the observations, BH Lyn was in high-state with V ≃ 15.5 mag. The star presents ∼ 1.5 mag deep eclipses with mean full-width at half-flux of 0.0683(±0.0054)Porb . The eclipse shape is highly variable, even changing form cycle to cycle. This is most likely due to accretion disc surface brightness distribution variations, most probably caused by strong flickering. Time- dependent accretion disc self-occultation or variations of the hot spot(s) intensity are also possible explanations. Negative superhumps with period of ∼ 0.d145 are detected in two long runs in 2000. A possible connection between SW Sex and negative superhump phe- nomena through the presence of tilted accretion disc is discussed, and a way to observationally test this is suggested. Key words. accretion, accretion discs – binaries: eclipsing – stars: individual: BH Lyn – novae, cataclysmic variables 1. Introduction BH Lyncis is an eclipsing novalike (NL) cataclysmic variable (CV) with an orbital period of ∼ 3.h74 (Andronov et. 1989). Thorstensen et al. (1991a), Dhillon et al. (1992), and Hoard & Szkody (1997) have shown that spectral behavior of BH Lyn resembles that of SW Sex-type novalikes. SW Sex stars are spectroscopically defined sub-class of novalikes (Thorstensen et al. 1991b). Most of them are eclipsing, but show single- peaked emission lines contrary to the expected double-peaked from high-inclined accretion discs. Other distinctive character- istics are high-velocity emission components, narrow absorp- tion components superimposed over emission lines around or- bital phase 0.5, and a large phase offset of the emission line ra- dial velocities, with respect to the photometric conjunction. The eclipse profiles are V-shaped rather that U-shaped, and the ac- cretion discs brightness temperature distribution derived from eclipse mapping is much flatter than expected for a steady-state accretion disc (e.g. Rutten et al. 1992). Patterson (1999) re- ports that most of the SW Sex stars show both negative and positive superhumps. Besides, some of the members show low states (Honeycutt et al. 1993). Currently, there is no widely ac- cepted model of SW Sex stars. In most of the CVs, the accre- tions stream from secondary hits on the outer disc edge, and a hot spot is formed at the impact. In the most elaborated model of the SW Sex stars, Hellier (1998) suggested that part of the gas in the stream does not stop in the vicinity of the hot spot. Instead, it continues moving above the disc surface, hits the disc close to the white dwarf, and thus forms a second spot. Recently, Rodriguez-Gil et al. (2001) discovered variable circular polariza- tion in LS Peg and suggested that SW Sex stars are intermediate polars with the highest mass accretion rates. ⋆ Based on observations obtained at Rozhen National Astronomical Observatory, Bulgaria ⋆⋆ E-mail: vall,#zk,#[email protected] (VS,#ZK,#VG) Table 1. V band observations of BH Lyn. The eclipse timings are also given. UT date HJD Start Duration HJD mid-eclipse -2451000 [hour] -2451000 Feb. 20, 1999 230.3833 3.23 230.45114 Jan. 08, 2000 552.2748 5.61 552.33385 552.48983 Jan. 09, 2000 553.2240 9.16 553.26945 553.42513 553.58089 Mar. 12, 2000 616.2509 4.21 616.39900 Feb. 28, 2003 1699.3052 7.44 1699.42318 1699.57916 Dec. 19, 2003 1993.5075 3.77 1993.56007 Jan. 18, 2004 2023.5165 3.86 2023.64410 The object of this study, BH Lyn, is mostly studied spectro- scopically, and the existing photometric data are generally used to obtain the eclipse ephemeris and to supplement the spectral observations. In this paper, we report the results of our photom- etry of BH Lyn obtained in 1999-2004. 2. Observations and data reduction The photometric observations of BH Lyn were obtained with the 2.0-m telescope in the Rozhen Observatory. A Photometrics 10242 CCD camera and a Johnson V filter were used. The CCD camera was 2×2 pixels binned, which resulted in ∼13 s of read- out dead-time. In total, 7 runs of photometric data were obtained between 1999 and 2004. The exposure time used was between 30 and 60 s. Some details of the observations are given in Table 1. After bias and flat-field corrections, the photometry was done with the standard DAOPHOT aperture photometry procedures (Stetson 1987). The magnitude of BH Lyn was measured relative http://arxiv.org/abs/0704.1302v1 2 V. Stanishev et al.: Photometry of the SW Sex-type nova-like BH Lyncis in high state 230.4 230.5 02-20-1999 552.3 552.4 552.5 01-08-2000 553.3 553.4 553.5 553.6 01-09-2000 616.3 616.4 03-12-2000 1699.3 1699.4 1699.5 1699.6 02-28-2003 HJD-2451000 1993.5 1993.6 12-19-2003 2023.5 2023.6 01-18-2004 Fig. 1. V-band observations of BH Lyn. The solid line shows the sinusoidal fit with the period of the superhumps detected in the 2000 data. to the star BH Lyn-5 (V = 14.47), and BH Lyn-4 (V = 15.30) served as a check (Henden & Honeycutt 1995). The runs are shown in Fig. 1, and it appears that BH Lyn was in high state during all observations. 3. Results The eclipse timings given in Table 1 were determined by fitting a parabola to the lower half of the eclipses. To refine the orbital ephemeris of BH Lyn, we also used the eclipse timings collected by Hoard & Szkody (1997). The O −C residuals with respect to the best linear ephemeris are shown in Fig. 2. Clearly, the linear ephemeris does not describe the eclipse times well and, as Hoard & Szkody (1997) point out, this is mainly due to the anoma- lously large, positive residual of the first eclipse timing. Hoard & Szkody (1997) suggested that the first eclipse timing was in error and calculated a linear ephemeris without it. The O − C residuals of our new eclipse timings are rather large, ∼0.d006, and increasingly positive. Together with the first two timings, whose O − C residuals are also positive, this suggests the pres- ence of a curvature in the O−C residuals. The dashed line is the second-order polynomial fit to all eclipse timings. The quadratic term is 7.6 × 10−12 and implies that the orbital period of BH Lyn increases on a time scale of ∼ 4.2 × 106 yrs. In most of the CVs, the mass donor star is the less massive one, and hence, if the mass transfer is conservative, the orbital period of the sys- tem will increase. For plausible component masses in BH Lyn, MWD ∼ 0.73 and M2 ∼ 0.33 (Hoard & Szkody 1997), the mass transfer rate should be Ṁ ∼ 5 × 10−8 M⊙ yr −1 to be compati- ble with the putative orbital period increase. However, there are several arguments against this scenario. First, there is a bulk of evidence that CVs evolve toward shorter orbital periods due to the angular momentum loss of the secondary by magnetic break- ing (Warner 1995). Second, Ṁ ∼ 5 × 10−8 M⊙ yr −1 is probably too high and generally not typical for CVs. Third, the eclipse timings presented by Andronov et al. (1989) have been deter- mined by the phase folding of observations with photographic plates with rather long exposure times of 8, 12, and 30 min. It is not surprising then, that those timings exhibit relatively large -60000 -40000 -20000 0 20000 40000 Cycle number -0.01 Fig. 2. O − C residuals of the minima with respect to the best linear ephemeris. The second-order polynomial fit to the O − C residuals is also shown. The solid line is our best linear ephemeris. The filled circles show our new timings. scatter (the timings with cycle numbers ∼ 3000). The second timing has been determined from plates with exposure time 30 min, only slightly shorter than the total eclipse duration, and its large positive O − C of this timing may be a statistical fluctua- tion. Because the first two timings are the ones that determine the curvature in the O − C residuals, one may question whether the curvature is real. Future observations may prove that the or- bital period of BH Lyn increases, however, our opinion is that only two timings determined from patrol plates do not provide enough evidence for this. We therefore determined an updated linear ephemeris without using the first two timings: HJDmin = 2447180.33600(28)+ 0. d155875577(14)E. (1) This ephemeris is shown by the solid line in Fig. 2. It is very similar to the ephemeris of Hoard & Szkody (1997); the orbital period is only slightly larger and the reference times differ by ≤ 1 min. V. Stanishev et al.: Photometry of the SW Sex-type nova-like BH Lyncis in high state 3 0 5 10 Frequency [cycle/day] Jan. 8 & 9, 2000 Porb Psh=0 d.145 Fig. 3. Periodogram of the January 2000 data. The negative su- perhump and the orbital periods are indicated. The light curves show prominent humps whose maxima oc- cur at different orbital phases in the different runs (Fig. 1). We interpret this as an indication of the presence of superhumps. Because our data are sparse, they are clearly not enough for an in-depth study of superhumps in BH Lyn. After removing the data during eclipses, we computed the Lomb-Scargle peri- odogram (Scargle 1982) of the two January 2000 series only (Fig. 3). The strongest peak around the expected frequency of the superhumps corresponds to a period of ∼0.d1450±0.0065, which is close to the negative superhumps period 0.d1490±0.0011 re- ported by Patterson (1999). The least-squares fit gives the semi- amplitude of the signal of 0.084±0.005 mag. We have also searched all runs for periodic variations on the minute time-scale. The power spectra show many peaks with fre- quencies below ∼ 150 cycle day−1, but the attempts to fit the runs with periods corresponding to any of the peaks in the peri- odograms were not satisfactory. Thus, most probably no coher- ent oscillations are present. The individual power spectra show a typical red noise shape characterized by a power-law decrease of the power with frequency P( f ) = f γ. The mean power spectrum of BH Lyn has power-law index γ = −1.77. Because the red noise processes have strong low-frequency variability, it is most likely that the peaks in the periodograms are due to the red noise. Nevertheless, the peak at ∼ 32 cycles day−1 is present in most periodograms, and it is also noticeable in the mean power spec- trum (Fig. 4). This might indicate the presence of quasi-periodic oscillations like the ones discussed by Patterson et al. (2002), however, a study based on more data is needed to confirm this. The red noise in the power spectra of CVs is a result of flickering (Bruch 1992). BH Lyn light curves show strong flick- ering activity; flickering peaks with typical durations of 5–10 min and amplitudes reaching ∼ 0.2 mag can be recognized in Fig. 1. The mean standard deviation in the light curves after the low-frequency signals have been subtracted is ∼ 0.06 mag. This value is consistent with the standard deviation found in the light curves of the NLs TT Ari, MV Lyr and PX And (Kraicheva et al. 1999a,b; Stanishev et al. 2002). The depth of the eclipses in BH Lyn during our observa- tions is ∼ 1.5 mag, and their average full-width at half-flux is 0.0683(±0.0054)Porb. The out-of-eclipse magnitudes were fit- ted with low-order polynomial functions to account for bright- ness variations that are not due to the eclipse, but most prob- ably arise from the superhumps. The eclipses were normalized to the fits and are shown in Fig. 5. As can be seen, there is a substantial variability of the eclipse shape, even during a sin- gle night. The variations are most notable in the upper half of the eclipse profiles. Half of the eclipses appear to be fairly sym- 1 10 100 1000 Frequency [cycle/day] 0.001 0.010 0.100 1.000 Psh=0 d.145 QPOs ∼ 47 min? P(f)∝ f-1.77 Fig. 4. The mean power spectrum of BH Lyn light curves. 02-20-1999 01-08-2000 No.1 01-08-2000 No.2 01-09-2000 No.1 01-09-2000 No.2 -0.1 0.0 0.1 01-09-2000 No.3 03-12-2000 02-28-2003 No.1 -0.1 0.0 0.1 02-28-2003 No.2 Orbital phase 12-19-2003 -0.1 0.0 0.1 01-18-2004 Fig. 5. Normalized eclipses of BH Lyn. The dashed lines are guide to the eye to see the difference of the eclipse profile eas- ier. The symbols used for the eclipses in Fig. 6 are shown in the lower left corners. metric, while the rest are clearly asymmetric. More interestingly, though, the egress of the eclipses on Mar. 3, 2000 and No.1 on Jan. 9, 2000, and possibly the ingress of some other eclipses, are not monotonic. To highlight the differences, in Fig. 6 we show all the eclipses together. Except for the single eclipse in 1999, the ingress of all eclipses are very similar. The egress of the eclipse are however very different, and the eclipses could be split into three sequences. In Fig. 6, each of these groups is plotted with a different symbol. 4 V. Stanishev et al.: Photometry of the SW Sex-type nova-like BH Lyncis in high state -0.1 0.0 0.1 Orbital phase Fig. 6. The three eclipse groups plotted together with different symbols. 4. Discussion Because of the large variability of the eclipse profiles in BH Lyn, we are reluctant to attempt eclipse mapping or to try to estimate the system parameters from the eclipse width. Clearly, such an- alyzes could give false results. The rather rapid changes in the eclipse profiles, even during a single night, could be explained by temporal variations of the AD surface brightness distribution. Large flickering peaks can be seen before or after some of the eclipses (Fig. 5). If such a peak occurs during an eclipse, it could alter its shape, even to cause the eclipse not to be monotonic. Another explanation could be that the amount of overflowing gas varies, and as a consequence the intensity of the two hot spots could also change, causing variations in the eclipse profile. Variations of the area of the eclipsing body with time will also cause variations of the eclipses. Given the time scale of the observed changes, the secondary is ruled out. On the other hand, the SW Sex stars most likely possess very complex accre- tion structures, and it may be that the AD is self-occulting. Self- occultation seems to be the most reasonable explanation of the UV observations of another SW Sex star, DW UMa (Knigge et al. 2000), hence giving support for this in BH Lyn. Variations of the effective area of the occulting parts may cause the observed eclipse profile changes. The presence of negative superhumps in eclipsing SW Sex stars is very interesting. The origin of negative superhumps is still a puzzle, but they are believed to be caused by a retro- grade precession of an accretion disc (AD) that is tilted with respect to the orbital plane (Bonnet-Bidaud et al. 1985). If neg- ative superhumps do arise from the precession of tilted ADs, then the accretion stream overflow would easily occur (Patterson et al. 1997). Therefore, the SW Sex and negative superhumps phenomena should have the same origin. Due to the presence of precessing tilted AD, the amount of gas in the overflowing stream will be modulated on the negative superhump period. Hence, the intensity of the second hot spot will change and may produce superhumps (Patterson et al. 1997; Stanishev et al. 2002). This scenario can be observationally tested. In this model, the negative superhumps should manifest themselves in spectra in two ways: 1) the intensity of the high-velocity emis- sion components in spectra, which are thought to arise from the second spot, should be modulated with the superhumps pe- riod; 2) since the orientation of the tilted disc with respect to the observer will change over the precession cycle, at certain precession phases, the SW Sex signatures should disappear. To test these predictions, time-resolved high signal-to-noise spec- trophotometry over several consecutive nights is needed, since the precession periods are of the order of a few days. We en- courage such studies. Acknowledgements. The work was partially supported by NFSR under project No. 715/97. References Andronov, I.L., Kimeridze G.N., Richter G.A., & Smykov, V.P. 1989, IBVS, Bonnet-Bidaud, J.M., Motch, C., & Mouchet, M. 1985, A&A, 143, 313 Bruch, A. 1992, A&A, 266, 237 Dhillon V.S., Jones D.H., Marsh T.R., & Smith R.C. 1992, MNRAS, 258, 225 Hellier C. 1998, PASP, 110, 420 Henden, A.A., & Honeycutt, R.K. 1995, PASP, 107, 324 Hoard D.W, & Szkody, 1997, ApJ, 481, 433 Honeycutt R.K., Livio M., & Robertson J. W. 1993, PASP, 105, 922 Knigge, C., Long, K.S., Hoard, D.W., Szkody, P., & Dhillon, V.S. 2000, ApJ, 539, L49 Kraicheva Z., Stanishev V., Genkov V., & Iliev L. 1999a, A&A, 351, 607 Kraicheva Z., Stanishev V., & Genkov V. 1999b, A&AS, 134, 263 Patterson J. 1999, in Disk Instabilities in Close Binary Systems, ed. S. Mineshige, & J. C. Wheeler, (Tokyo: Universal Academy Press), 61 Patterson J., Thorstensen J.R., Kemp J., et al. 2002, PASP, 114, 1364 Patterson, J., Kemp, J., Saad, J., et al. 1997, PASP, 109, 468 Rodriguez-Gil P., Casares J., Martinez-Pais I.G., Hakala P., & Steeghs D. 2001, ApJ, 548, L49 Rutten R.G.M., van Paradijs J., & Tinbergen J. 1992, A&A, 260, 213 Scargle J.D. 1982, ApJ, 263, 835 Stanishev V., Kraicheva Z., Boffin H. M. J., Genkov V., Papadaki C., & Carpano, S. 2004, A&A, 416, 1057 Stanishev V., Kraicheva Z., Boffin H., & Genkov V. 2002, A&A, 394, 625 Stetson, P. 1987, PASP, 99, 191 Thorstensen J.R., Davis M.K., & Ringwald, F.A. 1991a, ApJ, 327, 248 Thorstensen, J.R., Ringwald, F.A., Wade, R.A., Schmidt, G,D., & Norsworthy, J.E. 1991b, AJ, 102, 272 Warner, B. 1995, The cataclysmic variables stars, (Cambridge University Press, Cambridge) List of Objects ‘BH Lyn’ on page 1 ‘BH Lyn’ on page 1 ‘BH Lyn’ on page 1 ‘BH Lyn’ on page 1 ‘BH Lyncis’ on page 1 ‘BH Lyn’ on page 1 ‘SW Sex’ on page 1 ‘SW Sex’ on page 1 ‘SW Sex’ on page 1 ‘LS Peg’ on page 1 ‘SW Sex’ on page 1 ‘BH Lyn’ on page 1 ‘BH Lyn’ on page 1 ‘BH Lyn’ on page 1 ‘BH Lyn’ on page 1 ‘BH Lyn’ on page 1 ‘BH Lyn’ on page 2 ‘BH Lyn’ on page 2 ‘BH Lyn’ on page 2 ‘BH Lyn’ on page 2 ‘BH Lyn’ on page 2 ‘BH Lyn’ on page 2 ‘BH Lyn’ on page 3 V. Stanishev et al.: Photometry of the SW Sex-type nova-like BH Lyncis in high state 5 ‘BH Lyn’ on page 3 ‘BH Lyn’ on page 3 ‘TT Ari’ on page 3 ‘MV Lyr’ on page 3 ‘PX And’ on page 3 ‘BH Lyn’ on page 3 ‘BH Lyn’ on page 3 ‘BH Lyn’ on page 3 ‘BH Lyn’ on page 4 ‘DW UMa’ on page 4 ‘BH Lyn’ on page 4 Introduction Observations and data reduction Results Discussion

0704.1303

General Doppler Shift Equation and the Possibility of Systematic Error in Calculation of Z for High Redshift Type Ia Supernovae

All- Angle Doppler Shift Equation and High-Redshift Type Ia Supernova General Doppler Shift Equation and the Possibility of Systematic Error in Calculation of Z for High Redshift Type Ia Supernovae Steven M Taylor [email protected] Abstract Systematic error in calculation of z for high redshift type Ia supernovae could help explain unexpected luminosity values that indicate an accelerating rate of expansion of the universe. Introduction The general form of the relativistic Doppler shift equation is )cos1(0 ' θβγνν −= , (1) where = and =β with u being velocity of source. With an emission angle the general form reduces to the familiar °= 0θ = . (2) The condition corresponds to an emission antiparallel to the source’s velocity vector and is typically assumed for astronomical purposes. °= 0θ With the assumption redshift parameter is defined as °= 0θ =z 1 . (3) Evidence of Accelerating Universe and Possible Systematic Error The primary evidence of an accelerating rate of expansion of the Universe is that measurements of apparent magnitude of some high-z, type Ia supernovae are fainter than would be expected for non-accelerating cosmological models. [1] Perlmutter and Schmidt of the Cosmology Supernovae Project have noted that along with other possible sources of systematic errors, gravitational lensing may contribute to a change in luminosity of high-redshift supernovae. Citing several authors, they note that mailto:[email protected] as radiation traverses the large scale structure from where it is emitted and where it is detected, it could be lensed as it encounters fluctuations in gravitational potential. Some images could be demagnified as their light passes through under-dense regions. It is also noted that it would also be possible for a light path to encounter denser regions magnify the image. It is noted that such an effect may limit the accuracy of luminosity distance measurements.[2] A lower luminosity in relation to z is to date the strongest evidence of an accelerated expansion rate for the Universe. In the same sense that a change in luminosity due to reasons other than distance, such as gravitational lensing could produce systematic error, so could a false z measurement. Lowered luminosity would be consistent with a false z measurement if that measurement was less redshifted due to reasons extraneous to the expansion rate of the universe as presented by cosmological models. Whether by gravitation or other effect, any canting in angle of emission of light from a receding source will cause an increase in frequency as seen by an observer. Taking the derivative of the general relativistic Doppler shift equation with respect to θ yields: θγβν . (4) Since is an absolute minimum, any deviation from that angle results in a higher°= 0θ 'ν . Example According to Perivolaropoulos, the Gold supernova data set of 157 points show that transition from a decelerating towards and accelerating universe to be at z= 0.46 + 0.13. Using a graph of apparent magnitude vs. redshift based on the Gold data, a supernova with an approximate 44 apparent magnitude and measured redshift parameter of , lies on the curve for an accelerating universe. If it did have a redshift parameter of , the supernova would lie on curve consistent with a decelerating universe.[3] 95.0≅z 30.1≅z Using equation (3) a measured parameter of 95.0≅z corresponds to 58.0=β , and likewise a parameter of corresponds to30.1≅z 68.0=β . Assuming, for the sake of argument, that the Universe is decelerating and the supernova does have a parameter of , we can calculate the canting in angle of emission from that allows an observer to measure a parameter of 30.1≅z °= 0θ 95.0≅z . Using 0ν =1 Hz to simplify, and 68.0=β (corresponding to z =1.30), equation 2 yields ,ν = 0.4364 Hz. Likewise with 0ν =1 Hz to simplify, and 58.0=β (corresponding to z =0.95), equation 2 yields ,ν = 0.5156 Hz. 0792.0' =Δv Hz (5) Inserting 68.0=β into equation 4, yields )68.01( sin68.0 . (6) Using 0.0792 Hz for (5) into equation 6 yields an emission angle of . °≅ 90.4θ Conclusion Given the large distances that light from high z supernovae travel, and the modest canting from in emission angle required to help explain decreased luminosity for high redshift supernovae, the possibility of systematic error in z measurement for high redshift supernova should be further investigated. °= 0θ References 1. S Perlmutter, B Schmidt, Measuring Cosmology with Supernova, arxiv:astro- ph/0303428 v1 (2003) 2. ibid 3. L. Perivolaropoulos, Accelerating Universe: Observational Status and Theoretical Implications, arXiv:astro-ph/0601014 v2 (2006)

0704.1304

The Reverse Shock of SNR 1987A

The Reverse Shock of SNR 1987A Kevin Heng JILA, University of Colorado, Boulder, CO 80301-0440 Abstract. The reverse shock of supernova remnant (SNR) 1987A emits in Hα and Lyα , and comes in two flavors: surface and interior. The former is due to direct, impact excitation of hydrogen atoms crossing the shock, while the latter is the result of charge transfer reactions between these atoms and slower, post-shock ions. Interior and surface emission are analogous to the broad- and narrow-line components observed in Balmer-dominated SNRs. I summarize a formalism to derive line intensities and ratios in these SNRs, as well as a study of the transition zone in supernova shocks; I include an appendix where I derive in detail the ratio of broad to narrow Hα emission. Further study of the reverse shock emission from SNR 1987A will allow us to predict when it will vanish and further investigate the origins of the interior emission. Keywords: Atomic processes and interactions ; physical processes (kinematics) ; supernova rem- nants PACS: 95.30.Dr; 98.38.Am; 98.38.Mz INTRODUCTION: SNR 1987A For the past 20 years, supernova remnant (SNR) 1987A has provided a wonderful op- portunity to study emission mechanisms, radiative transfer and a myriad of physics for conditions unattainable on Earth. One such sub-field is the study of high Mach num- ber, collisionless shocks. The impact of the supernova (SN) blast wave upon ambient medium sets up a double shock structure consisting of a forward and a reverse shock. In SNR 1987A, the ejecta comprising mostly neutral hydrogen (which exists due to adia- batic expansion cooling) crosses the reverse shock at ∼ 12,000 km s−1; the excitation and subsequent radiative decay of the atoms result in Hα and Lyα emission, readily measured by instruments such as the Space Telescope Imaging Spectrograph (STIS) onboard the Hubble Space Telescope (see [1] and references therein). In the most recent study of the reverse shock [1], it was found that both Hα and Lyα emission exist in two flavors: surface and interior. In a young, pre-Sedov-Taylor remnant such as SNR 1987A, the freely-streaming debris has a unique velocity for a given radial distance from the SN core, exactly analogous to Hubble flow in an expanding universe. The projected velocity of the atoms crossing the reverse shock is proportional to the line-of-sight depth of the shock surface from the supernova mid-plane. It follows that upon impact excitation, the wavelength of the emitted photon is uniquely related to this depth, and the emission streaks in Figure [1] trace out the surface of the reverse shock, thereby warranting the term “surface emission”. If one believes this interpretation, then it is apparent from Figure [1] that there is both Hα and Lyα emission emerging from beneath the surface of the reverse shock, since at any given frequency or wavelength, flux appears at radial distances smaller than the radius of the shock. On this basis, we coin the term “interior emission”. http://arxiv.org/abs/0704.1304v1 +10000 km/s +5000 km/s -10000 km/s0 -5000 km/s FIGURE 1. STIS data of reverse shock emission from SNR 1987A and accompanying schematic, taken from [1]. (a) Hα surface emission from the reverse shock isolated by masks. (b) Lyα surface emission with the same masks applied. (c) Schematic representation of the supernova debris with the boundary being defined by the reverse shock. For freely-expanding debris, there is a unique correspondence between velocity and the origin of the emission along the line of sight. The shock velocity of SNR 1987A is ∼ 8000 km s−1, since it is the velocity of the atoms in the rest frame of the reverse shock, moving at ∼ 4000 km s−1. Strong shock jump conditions dictate that the ions are then at a velocity of ∼ 6000 km s−1 in the observer’s frame. Thus, the fast atoms are being converted into slow ions at the reverse shock. In addition to impact excitation, atoms may also donate their electrons to ions in the shocked plasma (i.e., charge transfer), thereby producing a population of slow atoms. The subsequent excitation (or charge transfer to excited states) of these atoms results in lower velocity Hα and Lyα emission, creating the illusion that these photons originate from beneath the reverse shock surface — “interior” emission. Both interior and surface emission originate from the same location, but the spectral-spatial mapping is no longer unique. BALMER-DOMINATED SUPERNOVA REMNANTS Dick McCray and I puzzled over the origins of the interior emission — he came up with the charge transfer idea, while I sat down and worked out the mathematical details. Deep into creating a formalism to compute the line intensities and ratios, I stumbled upon an old problem, namely the study of Balmer-dominated SNRs ([2], [3] and [4]). (I call the problem “old” because it was posed in the same year I was born.) These objects are typically much older than SNR 1987A, and are observationally characterized by blast wave reverse shock fast H I SNR 1987ABalmer-Dominated SNRs blast wave Stationary H I FIGURE 2. Contrasting the physical situations in “normal" Balmer-dominated SNRs and SNR 1987A, taken from [5]. two-component, Balmer line profiles consisting of a narrow (∼ 10 km s−1) and a broad (∼ 1000 km s−1) line. The former comes from the direct, impact excitation of stationary hydrogen atoms by the SN blast wave, while the latter is a result of charge transfer reactions of these atoms with post-shock ions. The terms “fast” and “slow” are solely a matter of one’s frame of reference. In the frame of the observer, the situation of fast atoms and slow ions in SNR 1987A now gets switched to slow atoms and fast ions in these Balmer-dominated SNRs (Figure [2]). The interior and surface emission of the former are the broad and narrow components of the latter. Nevertheless, the physics of the problem remain the same. I suddenly realized that I now had the mathematical machinery not only to model the emission lines in SNR 1987A, but to treat this broader class of objects as well. We generalized the methods of [3] — we asked the question: can one exhaustively track the fate of a hydrogen atom as it engages in charge transfer and excitation, eventually culminating in impact ionization? It turns out that we can if we make certain fairly accurate approximations, allowing us to find simple, analytical formulae for the rate coefficients of these reactions, weighted by how many times the atom undergoes charge transfers; each such event changes the nature of the atomic velocity distribution [4]. By knowing how to compute these rate coefficients, we can in turn compute the probability for each reaction occurring, thereby obtaining the composite velocity distribution. These distributions are intermediate be- tween a beam and a Maxwellian, and we thus named them “skewed Maxwellians”. We call atoms in such a skewed Maxwellian “broad neutrals”. The full width at half- maximum (FWHM) of these velocity distributions is then uniquely related to the shock velocity, provided one knows the temperatures of the electrons and ions. I have included an appendix describing in detail the derivation of the broad and narrow Hα rate coef- ficients (§), since the ratio of broad to narrow Hα emission is extensively studied in FIGURE 3. Ratio of the broad to narrow Hα emission, Ib/In, versus shock velocity, vs. The theoretical predictions by [3] (denoted “CKR80”) and [4] (denoted “HM06”) are plotted against several data points from various SNRs. Models N and F represent calculations for β = 0.25 and 1, respectively, where β ≡ Te/Tp is the ratio of electron to proton temperatures. Balmer-dominated SNRs. We can set theoretical bounds on the ratio of broad to narrow Hα emission, Ib/In, as shown in Figure [3]. Generally, our predictions agree quite well with observations, but a glaring discrepancy persists: the theoretical prediction of Ib/In ∼ 0.1 (i.e., interior-to- surface Hα ratio) in SNR 1987A is lower by an order of magnitude compared to the observed ratio. This points to two possibilities: there is a mechanism for interior emis- sion we have not yet modeled (C. Fransson, Aspen talk, 2007); and/or the assumption of thin shock fronts in these SNRs is a flawed one. THE SHOCK TRANSITION ZONE What if these shock fronts that we have been modeling as mathematical discontinuities all along do indeed have a finite width? We decided to investigate this issue, resolving the atomic physics while keeping the plasma kinetics unresolved [6]. In a system of pre-shock atoms and post-shock ions, there must exist a transition zone in which one population is converted into the other, via charge transfers and ionizations. This “shock transition zone” has a width on the order of the mean free path of atoms passing through the ionized gas, lzone ∼ 10 15n−10 cm, where n0 is the pre-shock ionic density (in cm Velocity atomic beam pre-shock protons Density residual atoms width ~ 1/n(σ post-shock protonsShock Frame beginning of transition zone end of transition zone FIGURE 4. Schematic diagram of the shock transition zone, in the case of a strong shock, taken from [6]. The width of the zone is on the order of the mean free path of interactions (charge transfer and ionization). The velocity of the ions goes down to 1/4 of its pre-shock value almost immediately, according to the Rankine-Hugoniot jump condition. The ionic density first jumps by a factor of 4 to conserve momentum, then eventually evolves to a value which depends on the pre-shock ionic density. The results are surprising — for a strong (∼ 1000 km s−1) shock, the ions are shocked immediately. There is no velocity structure within the shock transition zone (Figure 4), thus validating the thin shock assumptions of [3] and [4]. The ionic velocities are decelerated to 1/4 of their pre-shock values at the beginning of the zone, while the ionic densities jump by a factor of 4 to conserve momentum, consistent with the Rankine- Hugoniot jump conditions. There is, however, structure in both the atomic and ionic densities, which is relevant to the study of Lyα resonant scattering in young SNRs (pre- Sedov-Taylor phase). The mean free path for the scattering of Lyα photons is much less than lzone; photons are produced in the zone but scatter in a distance much less than its width. There is evidence for Lyα resonant scattering in SNR 1987A [1]. THE FUTURE As SNR 1987A enters its third decade, many questions regarding its fate abound. A central one concerning the reverse shock is: when will it disappear? There is a competition between pre-shock atoms crossing the shock — and ultimately emitting the Hα and Lyα photons we observe — and post-shock (ultraviolet and X-ray) photons diffusing upstream. These photons are capable of ionizing the atoms before they have a chance to undergo impact excitation (or charge transfer). If the flux of ionizing photons exceeds that of the atoms, the reverse shock emission will vanish. In [7], we predict this event to occur between about 2012 and 2014; we are currently planning further observations to finetune this prediction (PI: J. Danziger). These observations may yet shed light on the origins of the interior emission. One thing is for certain — SNR 1987A will provide current and future, young generations of astronomers/astrophysicists (such as myself) with an abundance of rich problems to ponder over. ACKNOWLEDGMENTS I am deeply grateful to Richard McCray for being a wonderful advisor, and to the Aspen Center for Physics for the hospitality of their support staff, and the generosity of both financial support and the Martin and Beate Block Prize (awarded at the conference). I thank Roger Chevalier, John Danziger, Eli Dwek, Alak Ray, Dick Manchester, Bryan Gaensler, Brian Metzger, Claes Fransson, Dieter Hartmann, Peter Lundqvist, Karina Kjaer, Alicia Soderberg, Lifan Wang, Philipp Podsiaklowski, Shigeyama Nagataki, Avi Loeb, Bob Kirshner, Saurabh Jha, Jason Pun, Andrew MacFayden, Jeremiah Murphy and Chris Stockdale for intriguing conversations and/or wonderful company during the conference. I apologize if I have left out anyone who belongs to the preceding list. APPENDIX: DERIVING Ib AND In IN HENG & MCCRAY (2007) In this section, I derive in more detail the Hα broad- and narrow-line rate coefficients, denoted Ib(Hα) and In(Hα) respectively, and simply stated in [4]. For simplicity, we refer to them just as Ib and In. For narrow Hα line emission, atoms are found in a beam and may be excited an arbitrary number of times until it gets transformed into a broad neutral via charge transfer or destroyed by ionization. Let the probability of excitation be PE0 , where the “0” means that the atom has undergone zero charge transfers prior to excitation. Let the rate coefficient for excitation to the atomic level n be RE0,n. Considering multiple excitations yield: RE0,n 1+PE0 +P +P3E0 + ... = RE0,n PiE0 = RE0,n 1−PE0 , (1) since 0 < PE0 < 1. One can consider excitations up to some level m, depending on the atomic data available. Ignoring collisional de-excitation, the rate coefficient for the narrow Hα line 1−PE0 RE0,nCn3, (2) where Ci j is the probability that an atomic excited to a state i will transit to a state j < i via all possible cascade routes; it is thus called the “cascade matrix”. Let us next derive the rate coefficient for the broad Hα line. We first account for charge transfer to excited states directly from the atomic beam to the level n, which has a rate coefficient RT ∗0 ,n. Accounting for multiple excitations before such a charge transfer, we have RT∗0 /(1−PE0). Next, we need to account for the creation of broad neutrals and the multiple charge transfers they are capable of undergoing: 1−PE0 + ... 1−PE0 1−PE0 The 1/(1−PE0) and 1/(1−PE) terms account for repeated excitations prior to engaging in charge transfer. As in [4], we make the approximation that the rate coefficients and probabilities are approximately unchanged after the first charge transfer, and thus they do not possess a subscript (e.g., PE versus PE0). Physically, these are reactions involving broad neutrals. Charge transfer to excited states and excitation of the broad neutrals are given by RT∗,n/(1−PE) and RE,n/(1−PE), respectively. Putting everything together and summing excitations to some level m, we get: 1−PE0 (RE,n +RT ∗,n)+RT ∗0 ,n Cn3. (4) REFERENCES 1. K. Heng et al., Astrophysical Journal 644, 959–970 (2006). 2. R. Chevalier, and J. Raymond, Astrophysical Journal 225, L27–L30 (1978). 3. R. Chevalier, R. Kirshner, and J. Raymond, Astrophysical Journal 235, 186–195 (1980). 4. K. Heng, and R. McCray, Astrophysical Journal 654, 923–937 (2007). 5. K. Heng, Celestial Outbursts & their Effects on Ambient Media, Ph.D. thesis, University of Colorado, Boulder (2007), available soon on astro-ph. 6. K. Heng, M. van Adelsberg, R. McCray, and J. Raymond, in preparation for submission to the Astrophysical Journal 0, 0–0 (2007). 7. N. Smith, S. Zhekov, K. Heng, R. McCray, J. Morse, and M. Gladders, Astrophysical Journal 635, L41–L44 (2005). Introduction: SNR 1987A Balmer-Dominated Supernova Remnants The Shock Transition Zone The Future Appendix: Deriving Ib and In in Heng & McCray (2007)

0704.1305

Carbon Nanostructures as an Electromechanical Bicontinuum

Carbon Nanostructures as an Electromechanical Bicontinuum Cristiano Nisoli∗, Paul E. Lammert∗, Eric Mockensturm† and Vincent H. Crespi∗ Department of Physics and Materials Research Institute Department of Mechanical and Nuclear Engineering The Pennsylvania State University, University Park, PA 16802-6300 (Dated: November 2, 2018) A two-field model provides an unifying framework for elasticity, lattice dynamics and electrome- chanical coupling in graphene and carbon nanotubes, describes optical phonons, nontrivial acoustic branches, strain-induced gap opening, gap-induced phonon softening, doping-induced deformations, and even the hexagonal graphenic Brillouin zone, and thus explains and extends a previously dis- parate accumulation of analytical and computational results. PACS numbers: 62.25.+g, 81.05.Tp, 63.22.+m, 77.65.-j, 46.05.+b Vibrations in carbon nanostructures such as tubes, fullerenes, or graphene sheets [1, 2, 3] have a ubiquitous influence on electronic, optical and thermal response: scattering from optical phonons limits charge transport in otherwise ballistic nanotube conductors [4, 5]; twist deformations gap metallic tubes [6, 7]; ballistic phonons transport heat in nanotubes with great efficiency [8, 9, 10]; resonant Raman spectroscopy can unambiguously identify a tube’s wrapping indices (n,m) [11, 12, 13, 14]; electron-phonon interactions may ultimately limit the electrical performance of graphene [15, 16]. Computa- tionally intensive atomistic models of lattice dynamics often lack simplified model descriptions that can facili- tate insight, yet traditional analytical continuum mod- els [1, 2, 17, 18], while very useful and important, cannot describe atomistic phenomena without phenomenologi- cal extensions [19, 20, 21]. Although continuum models are restricted to long-wavelength physics, they have been used to describe atomic-scale phenomena in bulk binary compounds by incorporating a separate continuum field for each sublattice [23]: in graphene, two fields are neces- sary. Here we present an analytical “bicontinuum” model that represents the full atomistic detail of the graphenic lattice, including optical modes, nonlinear dispersion of in-plane phonons, electromechanical effects and even the hexagonal graphenic Brillouin zone, a construct generally held to be exclusively atomistic. Graphene decomposes into the two triangular sublat- tices of Fig. 1. We describe in-plane deformations of the sublattices via two fields, ui(x), vi(x), i = 1, 2, and their strain tensors uij = ∂(iuj) and vij = ∂(jvi). The density of elastic energy contains direct and cross terms: V [u, v] = d[u] + d[v] + c[u, v]. (1) Six-fold symmetry of the sublattices implies isotropy of the direct terms [24]: d[u] = µ′ uijuij + j. (2) Symmetry dictates the form of the cross term FIG. 1: The two sublattices (circles and squares) of graphene and the three unit vectors ê(l) used in the text. φ, z are cylindrical coordinates of a tube, while Ψ = π/6− θc with θc the chiral angle. Also, anisotropic (uxx = uxy = 0, uyy = 2γ, qx = ℓ γ), shear (uxx = uyy = 0, uxy = η, qy = −ℓ η) strains. c[u, v] = 2 µuijvij + λu + α (u− v)2 (3) − β eijk uij + vij uk − vk The tensor eijk, which is invariant under C3v, can be represented by the three unit vectors {ê(l)} of Fig. 1: eijk = k . (4) Only the last term in Eq. 4 is not invariant under gen- eral rotation. (In nanotubes, it depends on the helical angle θc: eφφφ = −eφzz = − sin(3θc), ezzz = −eφφz = − cos(3θc), where φ, z are defined in Fig. 1). This elastic energy density, the lowest-order approximation in both derivatives and fields, contains six parameters: µ′ and λ′, being confined to one sublattice, describe next-neighbor interactions; the cross terms µ and λ describe nearest- neighbor interaction; α describes the stiffness against rel- ative shifts of the sublattices; β determines the strength of rotational symmetry breaking and so carries the point group symmetry of graphene. These parameters are nor- malized to the sublattice surface density σs, so that the elastic energy is W = σs V d Taking 1 u̇2 + v̇2 as the surface density of kinetic http://arxiv.org/abs/0704.1305v2 energy, the equations of motion read üi = ∂jσ ui − vi + β e ilm vlm + ulm v̈i = ∂jσ ui − vi − β e ilm vlm + ulm with the sublattice 2-D stress tensors = 2µ′ uij + λ′ δijukk + 2µ v ij + λ δijvkk −β eijk uk − vk = 2µ′ vij + λ′ δijvkk + 2µu ij + λ δijukk −β eijk uk − vk As expected, α determines the frequency of two degener- ate k = 0 optical modes: ωΓ 2 = 4α. First, we briefly show that the usual macroscopic elas- tic energy of graphene and its Lamé coefficients can be obtained from V by considering a static, uniform solu- tion of Eqs. 5 with identical deformations on both lattices with an internal displacement 2qi ≡ ui − vi : 2qi = ℓ e ilm u lm = ℓ e ilm v lm, (7) where ℓ = β/α is a characteristic length. Anisotropic (2γ = uxx−uyy) and shear (η = uxy) strains produce in- ternal displacements qx = ℓγ and qy = −ℓη (Fig. 1). The elastic energy for uniform deformations Wu = Vuσg d then simplifies to Vu[u, q] = uijuij + + 2αq2 − 2 β eijkuijqk, (8) where σg = 2σs = 2.26 g cm −2 is the surface density of graphene, µR ≡ µ + µ′ − β , λR ≡ λ + λ′ + β the measurable Lamé coefficients [24]. Macroscopic prob- lems do not distinguish between the two sublatices; elim- inating qi in Eq. 8 through Eqs. 4 and 7 we obtain the familiar, isotropic, macroscopic energy for graphene, Vu = µRu ijuij + λRu j/2. In the long wavelength limit Eqs. 5 returns the familiar longitudinal and trans- verse speeds of sound in terms of the Lamé coefficients: v2L = 2µR + λR, v T = µR. The out-of-plane displacements u⊥(x) and v⊥(x) do not couple with the in-plane ui, vi in the harmonic limit: invariance under simultaneous sign change of u⊥ and v⊥ prevents it, for flat sheets. Introducing 2p⊥(x) = u⊥(x)+ v⊥(x) and 2q⊥(x) = u⊥(x)−v⊥(x), V⊥ must be invariant under p⊥ → p⊥ + L(x), L(x) a linear function in the plane, and thus, can contain only second (and higher) derivatives in p⊥. Symmetry dictates (cf. Appendix) V⊥ = 4α⊥q ⊥ − 4α′⊥∂iq⊥∂iq⊥ + 4β⊥eijk ∂kq⊥∂ijp⊥ + 2µ+ ∂ijp⊥∂ ijp⊥ + λ ∂iip⊥∂ − 2µ− ∂ijq⊥∂ ijq⊥ − λ−⊥∂ iq⊥. (9) The frequency of the k = 0 out-of-plane optical mode is α⊥, and the out-of-plane acoustic branch is quadratic at small wave-vector, as expected. FIG. 2: Bicontinuum phonons compared to EELS data (dia- monds [26] and squares [27]), fitting either to the entire Bril- louin zone (top) or just around Γ along Γ → M . The bicontinuum phonons are much more richly struc- tured than in a traditional continuum model: they in- clude all the optical branches, show nonlinear dispersion at large wavevector, and even display the main features of the Brillouin zone, all without sacrificing the advan- tages of a continuum framework. Plane-wave solutions of Eqs. 5 returns an analytically solvable fourth-order secu- lar equation in ω(k), yielding two acoustic and two opti- cal branches. The longitudinal branches cross at the ver- tices of a hexagon. Since the two-field elastic energy den- sity respects the point group symmetry of the graphene lattice, this hexagon is oriented just as the graphene Bril- louin zone; although the model, unlike in the envelope function approach [25], has no built-in length scale, the elastic parameters can be constrained so that the crossing point coincides with the K point of graphene. A simi- lar argument holds for the out-of-plane modes: strikingly one can construct the correct Brillouin zone within a con- tinuum model. Fig. 2 shows the bicontinuum phonons fit to electron-energy-loss spectroscopy (EELS) data [26, 27] for parameters fitted either to the full Brillouin zone or just around Γ [28]. The bicontinuum provides a unified framework for nan- otube mechanics which can describe all current compu- tational results on the coupling of nanotube phonons to static structural distortions, to each other (e.g. breathing-to-Raman or longitudinal-to-transverse modes in helical tubes) and to the tube electronic structure. In a cylindrical geometry with coordinates {r, φ, z}, a coupling between the tangential displacements ui, and the radial ur = u⊥ appears in V of Eq. 1 via u φ + ur /r (and similarly for v); this accounts for the emergence of the Radial Breathing Mode (RBM) [29]. We consider uniform solutions: u = uoe −iωt, v = voe −iωt. The tube’s helicity can be subsumed into new axes {ξ, ζ} (ξ = φ cos 3θc + z sin 3θc ζ = −φ sin 3θc + z cos 3θc) ro- tated by an angle 3θc with respect to the base of the tube. In terms of p, q we obtain pξ, pζ = 0 and ω2 − 4α pr = 0 ω2 − v +β2/α qζ = 0 ω2 − 4α ω2 − 4α⊥ + 2µ−2µ ′+λ−λ′ . (10) Unlike standard elasticity [17], which cannot describe op- tical modes, or standard atomistic descriptions, which cannot be solved analytically, the two-field continuum model enables an exact analytical solution for the cou- pling between the RBM and the graphite-like optical mode through the first two of Eqs. in (10); the RBM induces a shear in the sublattices, uφφ = vφφ = ur/r, which couples with the internal displacement through β, and vice versa. Thus, the RBM is not purely ra- dial, but has a longitudinal component qzB ∼ ℓ2r cos 3θc, as previously seen in a numerical calculation[30]. Ex- pansion of the RBM frequency in powers of l/r re- veals a correction to the the standard continuum re- sult vL/r [17]: ωB = . The graphite-like optical modes of chiral tubes are ωξ = ωζ/ωξ = 1 + , also of mixed longitudi- nal/transverse character except for armchair and zig-zag nanotubes, while the out-of-plane optical mode ω⊥ = 4α⊥ − 2µ−2µ ′+λ−λ′ is purely radial. A density functional theory calculation of the breathing mode [31] reports different frequencies with (ωB) and without (ω̃B) coupling to optical modes. We predict r2 ω̃2B − ω2B β2/α as r → ∞: using ref [31] data for ω̃B, ωB we ob- tain ℓ ≡ β/α = 0.25 Å (0.27 Å) for non metallic zig-zag (armchair) tubes, in good agreement with the parameters from our fit to the graphene phonons [28]. The bicontinuum can also describe electron-lattice cou- pling to both acoustic and optical modes, by incorporat- ing a tight-binding model whose nearest neighbor hop- ping integrals t(1), t(2), t(3) are modulated by the in-plane elastic deformations: dt(l) = −τ ê(l)i ê ij + τ ê i/e (11) where e is the inter-atomic distance and τ a parameter to be determined [32]. For example, lattice deformations open gaps in metallic tubes, and these gaps in turn affect vibrational frequencies. If ǫc, ǫv are the conduction and valence bands, we have to nearest neighbors ǫc(k) 2−ǫv(k)2 = t(l)+2 t(l)t(m) cos(k·a(n)), (12) where a(n) ≡ e(l) − e(m), n(l,m) is cyclic in {1, 2, 3} (e.g. a(3) ≡ e(1) − e(2)) and {e(i)} connects nearest neighbors. From Eqs. 11,12 we find the band gap opened by strain in a metallic nanotube to be uijuij − eijku eijkφ̂ kuij φ̂hq h − 1 eijku ij φ̂k In the second line of equation (13) the symmetry of the honeycomb lattice is broken by the unit vectors φ̂i, ẑi of the cylindrical coordinates. In terms of 2γ′ ≡ uφφ − uzz, η′ ≡ uφz, qz , equation (13) reads ∆ = 3τ |qz/e+ γ′ cos(3θc) + η′ sin(3θc)| , (14) which corrects and extends a well known previous result within a one-field continuum model [7] that neglected the inner displacement (i.e. qi = 0). Opening bandgaps in metallic nanotubes causes several shifts in observed quantities. The term proportional to q2z in Eq. (13) show that longitudinal optical modes open a bandgap in metallic tubes of any helicity; the elastic en- ergy lowers by a term proportional to the square of the bandgap, leading to a the softening of longitudinal optical frequency in metallic nanotubes, as revealed by a recent DFT study [33]. Eq. (13) predicts also a softening of the RBM in metallic nanotubes δωB = −A cos2 (3θc), high- est for zig-zag tubes as seen in DFT [31], and relates it to the optical softening, with A = (1−ℓ/e)ωoptδωopte2/4v2L, ωopt the graphite-like optical mode, and δωopt its soften- ing in metallic tubes (A ≃ 2%). Other shifts can be predicted: the speed of sound for the twist mode softens by ∆ct = − v A sin2 (3θc), or ≃ 2.2% in armchair tubes. Doping-induced structural deformations can also be studied by minimizing the total energy (elastic plus doped electrons). Subtle phenomena absent in other models [22] can be accessed within the bicontinuum framework. Going to next-nearest-neighbor in the hop- ping integrals (dt 1 = −τ1 â ij [32]), we find that at first order in both a/r and the number of dopant elec- trons per atom ρe, semiconducting (n, 0) nanotubes show doping-induced changes in tube length (dL/L = uzz) and axial bond-length (dbax = eu zz − qz): dL/L = ρeτ + 3τ1 2µR+λR µR+λR dbax = ± ρeτ2mCω2opte . (15) where mC is the mass of the carbon atom. The sign is positive (negative) for r = n mod 3 = 2 (n mod 3 = 1). Recent DFT results [34] indeed show shrinking or stretch- ing of bax for n = 16, 13 or n = 14, 11 tubes respec- tively, as predicted by Eq. 15. In DFT, the overall tube lengthens in the second case (n = 14, 11), again in ac- cord with the bicontinuum; the lengthening found for r = 2, is less than for r = 1, perhaps a consequence of the change in sign in Eqs. 15. Finally the shrinking of the axial bond determines an up-shift in the longitudinal graphite-like optical mode and might explain recent Ra- man results that point toward anomalous bond contrac- tion under doping in semiconducting nanotubes [35, 36]. In summary, a symmetrized two-field continuummodel of graphene and carbon nanotubes provides the first uni- fied analytical treatment for a wide range of vibrational and electromechanical phenomena including nonlinear dispersion of in-plane phonons, zone-edge degeneracies and optical modes. A full range of vibrational-electronic- mechanical couplings, which were absent from previous continuummodels or happened upon in an ad hoc fashion in computational work, can now be understood within a single unified analytical framework. Extending the for- malism to include higher-order effects arising from curva- ture or metallic character (i.e. symmetry breaking terms containing φ̂i, ẑi, as in Eq. 13), anharmonicity (terms higher order in uij , vij), or long-distance interactions (higher partial derivatives) is straightforward. An ex- tension to boron nitride nanotubes, with different coef- ficients for each sublattice in the direct terms of Eq. 2, might prove useful to study their piezoelectricity. Appendix: Derivation of Eq. 3 The term c[u, v] must be invariant under the combina- tion of 2π/6 rotations and the exchange of fields u ↔ v. Adding reflection through the x axis (Fig. 1) then im- plies C3v invariance. There is also a field translation invariance: u(x) → u(x) + p, v(x) → v(x) + p. The objects ui, vj , uij , and vij can be combined pairwise only into tensors of rank two, three and four; thus c[u, v] decomposes into three parts. The first part has terms like uivj ; symmetry then implies the form α(u − v)2 with α > 0 to ensure an energy minimum. The sec- ond part has terms like uijvkl; the only admissible form is 2µuijvij + λu j . The third part contains only rank three terms such as uijvk contracted with a C3v invari- ant tensor eijk, giving eijku ijvk. By requiring invariance under 2π/6 rotations conjugated with sublattice switch- ing, and also the field translation invariance, we obtain the form eijku ij(uk − vk) + e∗ijkvij(vk − uk), where the star means a 2π/6 rotation. Since C3v invariance implies e∗ijk = −eijk we finally obtain the third row of Eq. 3. [1] S. Iijima, Nature (London) 354, 56 (1991). [2] R. Saito, G. Dresselhaus and M. S. Dresselhaus, Physical properties of Carbon Nanotubes (Imperial College Press, London 1998). [3] M. S. Dresselhaus, G. Dresselhaus and P. C. Eklund, Sci- ence of Fullerences and Carbon Nanotubes (Academic, New York, 1996). [4] S. J. Tans et al., Nature 386, 474 (1997). [5] C. L. Kane et al., Europhys. Lett. 41, 683 (1998). [6] A. Rochefort, P. Avouris, F. Lesage, D. R. Salahub, Phys. Rev. B 60, 13824 (1999). [7] L. Yang and J. Han, Phys. Rev. Lett. 85, 154 (2000) [8] S. Berber, Y-K. Kwon, and D. Tománek, Phys. Rev. Lett. 84, 4613 (2000). [9] H.-Y. Chiu et al., Phys. Rev. Lett. 95, 226101 (2005). [10] P. Kim, L. Shi, A. Majumdar and P. L. McEuen Phys. Rev. Lett. 87, 215502 (2001). [11] E. Richter and K. R. Subbaswamy Phys. Rev. Lett. 79, 2738 (1997). [12] R. Saito et al., Phys. Rev. B 64, 085312 (2001). [13] A. Jorio et al., Phys. Rev. Lett. 86, 1118 (2001). [14] A. Jorio, R. Saito, G. Dresselhaus and M. S. Dresselhaus, Phil. Trans. R. Soc. Lond. A 362, 2311 (2004). [15] K. S. Novoselov et al., Nature 438, 197 (2005). [16] S. Y. Zhou et al., Nature Phys. 2, 595 (2006). [17] G. D. Mahan, Phys. Rev. B 65, 235402 (2002). [18] H. Suzuura and T . Ando, Phys. Rev. B 65, 235412 (2002); A. Raichura et al., J. Appl. Phys. 94, 4060 (2003). S.V. Goupalov, Phys. Rev. B 71, 085420 (2005) [19] F. Comas et al., Phys. Rev. B 47, 7602 (1993). [20] L. Chico and R. Pérez-Álvarez Phys. Rev. B 69, 35419 (2004). L. Chico and R. Pérez-Álvarez, Phys. Rev. B 73, 075425 (2006) [21] Y. N . Gartstein et al., Phys. Rev. B 68, 115415 (2003) [22] M. Verissimo-Alves, B. Koiller, H. Chacham, and R. B. Capaz, Phys. Rev. B 67, 161401 (R) (2003). [23] H. Deresiewicz et al. The collected papers of Raymond D. Mindlin, Springer-Verlag New York (1989). [24] L. D. Landau, E. M. Lifshitz “Theory of Elasticity” Perg- amon Press Oxford (1986). The density of elastic energy for an isotropic system has the form f = µuijuij+ where λ, µ are the Lamé coefficients. Here we renormalize the coefficients to σg = 1. [25] B. A. Foreman, Phys. Rev. B 52, 12260 (1995). [26] C. Oshima et al., Solid State Commun. 65, 1601 (1988). [27] S. Siebentritt, R. Pues, K. H. Rieder, and A. M. Shikin, Phys. Rev. B 55, 7927 (1997). [28] The fit around Γ returns (in Km s−1) vL = 21, vT = 14, −2µ+ 2µ′ − λ+ λ′ + β2/α = 4.4, (µ− µ′) 15, 6 and ℓ ≡ β/α = 0.3 Å [28]. The fit to the full zone uses vL = 16.5 Km s −1, vT = 10.8 Km 2µ− 2µ′ + λ− λ′ − β2/α = 8.7 Km s−1, (µ− µ′) = 6, 6 Km s−1, ℓ ≡ β/α = 0.24 Å; an exten- sion to higher derivatives would improve the agreement. [29] The chiral vector of the tube breaks the hexagonal sym- metry and allows for new terms to be introduced in V as curvature corrections, which for simplicity we won’t consider here. Different problems will suggest different leading corrections. [30] E. Dobardžić et al., Phys. Rev. B 68, 045408 (2003). [31] J. Kürti et al., New J. Phys.5, 125 (2003). [32] t(l), t 1 are the absolute values of the hopping integrals for nearest and next nearest neighbors. We assume they depend only on distance, and thus dt = −τde/e, dt1 = −τ1da/a. [33] O. Dubay et al., Phys. Rev. Lett. 88, 235506 (2002). [34] R. E. Margine et al. submitted to Phys. Rev. Lett. [35] G. Chen, C. A. Furtado, U. J. Kim, and P. C. Eklund Phys. Rev. B 72 155406 (2005). [36] G. Chen, C. A. Furtado, S. Bandow, S. Iijima and P. C. Eklund Phys. Rev. B 71 045408 (2005).

0704.1307

Ke4 decays and Wigner cusp

Ke4 decays and Wigner cusp Contribution to the proceedings of HQL06, Munich, October 16th-20th 2006 Lucia Masetti1 Institut für Physik Universität Mainz D-55099 Mainz, GERMANY 1 Introduction The single-flavour quark condensate 〈0 |qq| 0〉 is a fundamental parameter of χPT , determining the relative size of mass and momentum terms in the expansion. Since it can not be predicted theoretically, its value must be determined experimentally, e.g. by measuring the ππ scattering lengths, whose values are predicted very precisely within the framework of χPT , assuming a big quark condensate [1], or of generalised χPT , where the quark condensate is a free parameter [2]. The K+−e4 decay is a very clean environment for the measurement of ππ scattering lengths, since the two pions are the only hadrons and they are produced close to threshold. The only theoretical uncertainty enters through the constraint [3] between the scattering lengths a20 and a 0. In the K ± → π0π0π± decay a cusp-like structure can be observed at M200 = 4m π+ , due to re-scattering from K ± → π+π−π±. The scattering lengths can be extracted from a fit of the M200 distribution around the discontinuity. 2 Experimental setup Simultaneous K+ and K− beams were produced by 400 GeV energy protons from the CERN SPS, impinging on a beryllium target. The kaons were deflected in a front- end achromat in order to select the momentum band of (60± 3) GeV/c and focused at the beginning of the detector, about 200 m downstream. For the measurements presented here, the most important detector components are the magnet spectrom- eter, consisting of two drift chambers before and two after a dipole magnet and the quasi-homogeneous liquid krypton electromagnetic calorimeter. The momentum of 1Present address: Physikalisches Institut, Universität Bonn, D-53012 Bonn, GERMANY the charged particles and the energy of the photons are measured with a relative uncertainty of 1% at 20 GeV. A detailed description of the NA48/2 detector can be found in Ref. [4]. 3 K± → π+π−e±νe The K+−e4 selection consisted of geometrical criteria, like the requirement of having three tracks within the detector acceptance and building a good vertex; particle iden- tification requirements, based mainly on the different fraction of energy deposited by pions and electrons in the electromagnetic calorimeter; kinematical cuts for back- ground rejection, like an elliptical cut in the (pT ,M3π) plane centered at (0,MK). In order to improve the pion rejection, the electron identification also included a Linear Discriminant Analysis combining the three quantities with the highest discriminating power. Two reconstruction strategies can be applied to the K+−e4 events: either im- posing the kaon mass and extracting the kaon momentum from a quadratic equation, or imposing the kaon momentum to be the mean beam momentum (60 GeV/c along the beam axis) and extracting the kaon mass from a linear equation (see Fig. 1). 50 52 54 56 58 60 62 64 66 68 70 pK(GeV/c) )2 (GeV/c 0.3 0.4 0.5 0.6 0.7 0.8 510 datae4K MCe4K Background Figure 1: Kaon momentum (left) and mass (right) of the K+−e4 events reconstructed with a quadratic or a linear equation, respectively. The data (crosses) are compared to signal MC (open histogram) plus background (yellow). Analysing part of the 2003 data, 3.7× 105 K+−e4 events were selected with a back- ground contamination below 1%. The background level was estimated from data, using the so-called “wrong sign” events, i.e. with the signature π±π±e∓νe, that, at the present statistical level, can only be background, since the corresponding kaon decay violates the ∆S = ∆Q rule and is therefore strongly suppressed [5]. The main background contributions are due to K± → π+π−π± events with π → eν or a pion mis-identified as an electron. The background estimate from data was cross-checked using Monte Carlo simulation (MC). 3.1 Form factors Figure 2: Topology of the Ke4 decay. The form factors of the K+−e4 decay are parametrised as a function of five kinematic variables [6] (see Fig. 2): the invariant masses Mππ and Meν and the angles θπ, θe and φ. The matrix element V ∗usu(pν)γµ(1− γ5)v(pe)(V µ − Aµ) contains a hadronic part, that can be described using two axial (F and G) and one vector (H) form factors [7]. After expanding them into partial waves and into a Taylor series in q2 = M2ππ/4m π+ − 1, the following parametrisation was used to determine the form factors from the experimental data [8, 9]: F = (fs + f 2 + f ′′s q 4)eiδ 2) + fp cos θπe iδ11(q G = (gp + g 2)eiδ H = hpe iδ11(q In a first step, ten independent five-parameter fits were performed for each bin in Mππ, comparing data and MC in four-dimensional histograms in Meν , cos θπ, cos θe and φ, with 1500 equal population bins each. The second step consisted in a fit of the distributions in Mππ (see Figs. 3,4), to extract the (constant) form factor parameters. The polynomial expansion in q2 was truncated according to the experimental sensitivity. The dependence on Meν and the D-wave were found to be negligible within the total uncertainty and the corresponding parameters were therefore set to zero. The δ = δ00 − δ11 distribution was fitted with a one-parameter function given by the numerical solution of the Roy equations [3], in order to determine a00, while a was constrained to lie on the centre of the universal band. The following preliminary result was obtained: f ′s/fs = 0.169± 0.009stat ± 0.034syst 0.28 0.3 0.32 0.34 0.36 0.38 0.4 NA48/2 Ke4 PRELIMINARY statistical errors only ’ ’ ’ 0.28 0.3 0.32 0.34 0.36 0.38 0.4 NA48/2 Ke4 PRELIMINARY statistical errors only -0.15 -0.05 0.28 0.3 0.32 0.34 0.36 0.38 0.4 NA48/2 Ke4 PRELIMINARY statistical errors only 0.28 0.3 0.32 0.34 0.36 0.38 0.4 NA48/2 Ke4 PRELIMINARY statistical errors only Figure 3: F , G and H dependence on Mππ. The points represent the results of the first-step fits, the lines are fitted in the second step. f ′′s /fs = −0.091± 0.009stat ± 0.031syst fp/fs = −0.047± 0.006stat ± 0.008syst gp/fs = 0.891± 0.019stat ± 0.020syst g′p/fs = 0.111± 0.031stat ± 0.032syst hp/fs = −0.411± 0.027stat ± 0.038syst a00 = 0.256± 0.008stat ± 0.007syst ± 0.018theor, where the systematic uncertainty was determined by comparing two independent analyses and taking into account the effect of reconstruction method, acceptance, fit method, uncertainty on background estimate, electron-ID efficiency, radiative correc- tions and bias due to the neglected Meν dependence. The form factors are measured relative to fs, which is related to the decay rate. The obtained value for a 0 is compat- ible with the χPT prediction a00 = 0.220±0.005 [10] and with previous measurements [11, 12]. 4 K± → π0π0e±νe About 10,000 K00e4 events were selected from the 2003 data and about 30,000 from the 2004 data with a background contamination of 3% and 2%, respectively. The 0.28 0.3 0.32 0.34 0.36 0.38 0.4 NA48/2 Ke4 PRELIMINARY Universal Band fit statistical errors only Figure 4: δ = δ00 − δ11 distribution as a function of Mππ. The points represent the results of the first-step fits, the line is fitted in the second step. selection criteria were similar to the ones used for the K+−e4 events, apart from the requirement of containing one track and 4 photons compatible with two π0s at the same vertex. The electron identification was based on the fraction of energy deposited in the electromagnetic calorimeter and on the width of the corresponding shower. The background level was estimated from data by reversing some of the selection criteria and was found to be mainly due to K± → π0π0π± events with a pion mis-identified as an electron (see Fig. 5). 2 in GeV/cKm 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Signal MC00e4K )γe3(K Figure 5: Invariant mass distribution in logarithmic scale of the K00e4 events selected from the 2003 data (crosses) compared to the signal MC (red) plus physical (yellow) and accidental (blue) background. The branching fraction was measured, as a preliminary result from the 2003 data only, normalised to K± → π0π0π±: BR(K00e4 ) = (2.587± 0.026stat ± 0.019syst ± 0.029ext)× 10 where the systematic uncertainty takes into account the effect of acceptance, trigger efficiency and energy measurement of the calorimeter, while the external uncertainty is due to the uncertainty on the K± → π0π0π± branching fraction. This result is about eight times more precise than the best previous measurement [13]. For the form factors the same formalism is used as in K+−e4 , but, due to the symmetry of the π0π0 system, the P -wave is missing and only two parameters are left: f ′s/fs and f s /fs. Using the full data sample, the following preliminary result was obtained: f ′s/fs = 0.129± 0.036stat ± 0.020syst f ′′s /fs = −0.040± 0.034stat ± 0.020syst, which is compatible with the K+−e4 result (see Fig. 6). s/fsf’ 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 -0.14 -0.12 -0.08 -0.06 -0.04 -0.02 σ 1e4 68%e4 σ 1e4 68%e4 Figure 6: Comparison of the f ′s/fs and f s /fs measurements in K e4 and K 5 K± → π0π0π± From 2003 data, about 23 million K± → π0π0π± events were selected, with negligible background. The squared invariant mass of the π0π0 system (M200) was computed imposing the mean vertex of the π0s, in order to improve its resolution close to threshold. At M200 = 4m π+ , the distribution shows evidence for a cusp-like structure (see Fig. 7, left) due to ππ re-scattering. x 10 2 0.08 0.09 0.1 0.11 0.12 0.13 25000 30000 35000 40000 45000 50000 0.076 0.077 0.078 0.079 0.08 x 10 2 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Figure 7: Left: M200 of the selection K ± → π0π0π± data events. The arrow indicates the position of the cusp. Right: angle between the π± and the π0 in the π0π0 centre of mass system. The points represent the data, the three curves, the MC distribution for different values of k′ Fitting the distribution with the theoretical model presented in Ref. [14] and using the unperturbed matrix element M0 = A0(1 + h′u2 + 1 k′v2), the following result was obtained [15], assuming k′ = 0 [16]: g0 = 0.645± 0.004stat ± 0.009syst h′ = −0.047± 0.012stat ± 0.011syst a2 = −0.041± 0.022stat ± 0.014syst a0 − a2 = 0.268± 0.010stat ± 0.004syst ± 0.013theor, where the a0 − a2 measurement is dominated by the uncertainty on the theoretical model. In a further analysis, the value of k′ was obtained from a fit above the cusp in the plane cos θ vs M200, where θ is the angle between the π + and the π0 in the π0π0 centre of mass system. Evidence was found for a non-zero value of k′ (see Fig. 7, right): k′ = 0.0097± 0.0003stat ± 0.0008syst, where the systematic uncertainty takes into account the effect of acceptance and trigger efficiency. Reweighting the MC with the obtained value of k′, the standard fit of the M200 distribution with the Cabibbo-Isidori model was performed to obtain the cusp parameters, that were found to be compatible with the published values. References [1] G. Colangelo AIP Conf. Proc. 756, 60 (2005). [2] M. Knecht et al. Nucl. Phys. B 457, 513 (1995). [3] B. Ananthanarayan et al. Phys. Rept. 353, 207 (2001). [4] J. R. Batley et al. Phys. Lett. B 634, 474 (2006). [5] P. Bloch et al. Phys. Lett. B 60, 393 (1976). [6] N. Cabibbo and A. Maksymowicz Phys. Rev. 137, B438 (1965); Ibid. 168, 1926 (1968). [7] J. Bijnens et al. 2nd DAΦNE Phisics Handbook, 315 (1995). [8] A. Pais and S. B. Treiman Phys. Rev. 168, 1858 (1968). [9] G. Amoros and J. Bijnens J. Phys. G 25, 1607 (1999). [10] G. Colangelo et al. Nucl. Phys. B 603, 125 (2001). [11] L. Rosselet et al. Phys. Rev. D 15, 574 (1977). [12] S. Pislak et al. Phys. Rev. D 67, 072004 (2003). [13] S. Shimizu et al. Phys. Rev. D 70, 037101 (2004). [14] N. Cabibbo and G. Isidori JHEP 0503, 021 (2005). [15] J. R. Batley et al. Phys. Lett. B 633, 173 (2006). [16] S. Eidelman et al. Phys. Lett. B 592, 1 (2004). Introduction Experimental setup K + - e e Form factors K 0 0 e e K 0 0

0704.1308

Antenna Combining for the MIMO Downlink Channel

Antenna Combining for the MIMO Downlink Channel Nihar Jindal University of Minnesota, Department of ECE Minneapolis, MN 55455, USA Email: [email protected] Abstract A multiple antenna downlink channel where limited channel feedback is available to the transmitter is considered. In a vector downlink channel (single antenna at each receiver), the transmit antenna array can be used to transmit separate data streams to multiple receivers only if the transmitter has very accurate channel knowledge, i.e., if there is high-rate channel feedback from each receiver. In this work it is shown that channel feedback requirements can be significantly reduced if each receiver has a small number of antennas and appropriately combines its antenna outputs. A combining method that minimizes channel quantization error at each receiver, and thereby minimizes multi-user interference, is proposed and analyzed. This technique is shown to outperform traditional techniques such as maximum-ratio combining because minimization of interference power is more critical than maximization of signal power in the multiple antenna downlink. Analysis is provided to quantify the feedback savings, and the technique is seen to work well with user selection and is also robust to receiver estimation error. I. INTRODUCTION Multi-user MIMO techniques such as zero-forcing beamforming allow for simultaneous transmission of multiple data streams even when each receiver (mobile) has only a single antenna, but very accurate channel state information (CSI) is generally required at the transmitter in order to utilize such techniques. In the practically motivated finite rate feedback model, each mobile feeds back a finite number of bits describing its channel realization at the beginning of each block or frame. In the vector downlink channel (multiple transmit antennas, single antenna at each receiver), the feedback bits are determined by quantizing the channel vector to one of 2B quantization vectors. While a relatively small number of feedback bits suffice to obtain near-perfect CSIT performance in a point-to- point vector/MISO (multiple-input, single-output) channel [1], considerably more feedback is required in a vector downlink channel. If zero-forcing beamforming (ZFBF) is used, the feedback rate must be scaled with the number of transmit antennas as well as SNR in order to achieve rates close to perfect CSIT systems [2]. In such a system the transmitter emits multiple beams and uses its channel knowledge to select beamforming vectors such that nulls are created at certain users. Inaccurate CSI leads to inaccurate nulling and thus translates directly into multi-user interference and reduced SINR/throughput. In this paper we consider the MIMO downlink channel, in which the transmitter and each mobile have multiple antennas (M transmit antennas, N antennas per mobile), in the same limited feedback setting. We propose a receive antenna combining technique, dubbed quantization-based combining (QBC), that converts the MIMO downlink into a vector downlink in such a way that the system is able to operate with reduced channel feedback. Each mobile linearly combines its N antenna outputs and thereby creates a single antenna channel. The resulting vector channel is quantized and fed back, and transmission is then performed as in a normal vector downlink channel. With QBC the combiner weights are chosen on the basis of both the channel and the vector quantization codebook to produce the effective single antenna channel that can be quantized most accurately. On the other hand, traditional combining techniques such as the maximum-ratio based technique that is optimal for point-to-point MIMO channels with limited channel feedback [3] or direct quantization of the maximum eigenmode are aimed towards maximization of received signal power but generally do not minimize channel quantization error. Since channel quantization error is so critical in the MIMO downlink channel, quantization-based combining leads to better performance by minimizing quantization error (i.e., interference power) possibly at the expense of channel (i.e., signal) power. One way to view the advantage of QBC is through its reduced feedback requirements relative to the vector downlink channel. In [2] it is shown that scaling (per mobile) feedback as B = M−1 PdB , where P represents the http://arXiv.org/abs/0704.1308v2 SNR, suffices to maintain a maximum gap of 3 dB (equivalent to 1 bps/Hz per mobile) between perfect CSIT and limited feedback performance in a vector downlink channel employing ZFBF. With QBC, our analysis shows that the same throughput (3 dB away from a vector downlink with perfect CSIT) can be achieved if feedback is scaled at the slower rate of B ≈ M−N PdB . In other words, QBC allows a MIMO downlink to mimic vector downlink performance with reduced channel feedback. Alternatively, QBC can be thought of as an effective method to utilize multiple receive antennas in a downlink channel in the presence of limited channel feedback. Although it is possible to send multiple streams to each mobile if receive combining is not performed, this requires even more feedback from each mobile than a single-stream approach. In addition, QBC has the advantage that the transmitter need not be aware of the number of receive antennas being used. The remainder of this paper is organized as follows: In Section II we introduce the system model and some preliminaries. In Section III we describe a simple antenna selection method that leads directly into Section IV where the much more powerful quantization-based combining technique is described in detail. In Section V we analyze the throughput and feedback requirements of QBC. In Section VI we compare QBC to alternative MIMO downlink techniques, and finally we conclude in Section VII. II. SYSTEM MODEL AND PRELIMINARIES We consider a K mobile (receiver) downlink channel in which the transmitter (access point) has M antennas, and each of the mobiles has N antennas. The received signal at the i-th antenna is given by: yi = h i x + ni, i = 1, . . . , NK (1) where h1,h2, . . . ,hKN are the channel vectors (with hi ∈ CM×1) describing the KN receive antennas, x ∈ CM×1 is the transmitted vector, and n1, . . . ,nNK are independent complex Gaussian noise terms with unit variance. The k-th mobile has access to y(k−1)N+1, . . . , yNk. The input must satisfy a power constraint of P , i.e. E[||x||2] ≤ P . We use Hk to denote the concatenation of the k-th mobile’s channels, i.e. Hk = [h(k−1)N+1 · · ·hNk]. We consider a block fading channel with iid Rayleigh fading from block to block, i.e., the channel coefficients are iid complex Gaussian with unit variance. Each of the mobiles is assumed to have perfect knowledge of its own channel Hi, although we analyze the effect of relaxing this assumption in Section V-C. In this work we study only the ergodic capacity, or the long-term average throughput. Furthermore, we only consider systems for which N < M because QBC is not very useful if N ≥ M ; this point is briefly discussed in Section IV. A. Finite Rate Feedback Model In the finite rate feedback model, each mobile quantizes its channel to B bits and feeds back the bits perfectly and instantaneously to the transmitter at the beginning of each block [3][4]. Vector quantization is performed using a codebook C of 2B M -dimensional unit norm vectors C , {w1, . . . ,w2B}, and each mobile quantizes its channel to the quantization vector that forms the minimum angle to it [3] [4]: ĥk = arg min w=w1,...,w2B sin2 (∠(hk,w)) . (2) For analytical tractability, we study systems using random vector quantization (RVQ) in which each of the 2B quantization vectors is independently chosen from the isotropic distribution on the M -dimensional unit sphere and where each mobile uses an independently generated codebook [5]. We analyze performance averaged over random codebooks; similar to Shannon’s random coding argument, there always exists at least one quantization codebook that performs as well as the ensemble average. B. Zero-Forcing Beamforming After receiving the quantization indices from each of the mobiles, the AP can use zero-forcing beamforming (ZFBF) to transmit data to up to M users. For simplicity let us consider the N = 1 scenario, where the channels are the vectors h1, . . . ,hM . When ZFBF is used, the transmitted signal is defined as x = k=1 xkvk, where each xk is a scalar (chosen complex Gaussian) intended for the k-th mobile, and vk ∈ CM is the k-th mobile’s BF vector. If there are M mobiles (randomly selected), the beamforming vectors v1, . . . ,vM are chosen as the normalized rows of the matrix [ĥ1 · · · ĥM ]−1, i.e., they satisfy ||vk|| = 1 for all k and ĥHk vj = 0 for all j 6= k. If all multi-user interference is treated as additional noise and equal power loading is used, the resulting SINR at the k-th receiver is given by: SINRk = |hHk vk|2 j 6=k . (3) The coefficient that determines the amount of interference received at mobile k from the beam intended for mobile j, |hH vj |2, is easily seen to be an increasing function of mobile k’s quantization error. In the above expression we have assumed that M mobiles are randomly selected for transmission and that equal power is allocated to each mobile. However, the throughput of zero-forcing based MIMO downlink channels can be significantly increased by transmitting to an intelligently selected subset of mobiles [6]. In order to maximize throughput, users with nearly orthogonal channels and with large channel magnitudes are selected, and waterfilling can be performed across the channels of the selected users. In [7] a low-complexity greedy algorithm that selects users and performs waterfilling is proposed. If this algorithm is used, a zero-forcing based system can come quite close to the true sum capacity of the MIMO downlink, even for a moderate number of users. C. MIMO Downlink with Single Antenna Mobiles In [2] the vector downlink channel (N = 1) is analyzed assuming that equal power ZFBF is performed without user selection on the basis of finite rate feedback (with RVQ). The basic result of [2] is that: RFB(P ) ≥ RCSIT (P ) − log2 1 + P · E ∠(ĥk,hk) where RFB(P ) and RCSIT (P ) are the ergodic per-user throughput with feedback and with perfect CSIT, respec- tively, and the quantity E ∠(ĥk,hk) is the expected quantization error. The expected quantization error can be accurately upper bounded by 2− M−1 and therefore the throughput loss due to limited feedback is upper bounded by log2 1 + P · 2− , which is an increasing function of the SNR P . If the number of feedback bits (per mobile) is scaled with P according to: B = (M − 1) log2 P ≈ M − 1 PdB , then the difference between RFB(P ) and RCSIT (P ) is upper bounded by 1 bps/Hz at all SNR’s, or equivalently the power gap is at most 3 dB. As the remainder of the paper shows, quantization-based combining significantly reduces the quantization error (more precisely, it increases the exponential rate at which quantization error goes to zero as B is increased) and therefore decreases the rate at which B must be increased as a function of SNR. III. ANTENNA SELECTION FOR REDUCED QUANTIZATION ERROR In this section we describe a simple antenna selection method that reduces channel quantization error. Description of this technique is primarily included for expository reasons, because the simple concept of antenna selection naturally extends to the more complex (and powerful) QBC technique. In point-to-point MIMO, antenna selection corresponds to choosing the receive antenna with the largest channel gain, while in the MIMO downlink the receive antenna that can be vector quantized with minimal angular error is selected. Mobile 1, which has channel matrix H1 = [h1 · · ·hN ] and a single quantization codebook consisting of 2B quantization vectors w1, . . . ,w2B , first individually quantizes each of its N vector channels h1, . . . ,hN ĝi = arg min w=w1,...,w2B sin2 (∠(hi,w)) i = 1, . . . , N, (5) and then selects the antenna with the minimum quantization error: j = arg min i=1,...,N sin2 (∠(hi, ĝi)) , (6) and feeds back the quantization index corresponding to ĝj . The mobile uses only antenna j for reception, and thus the system is effectively transformed into a vector downlink channel. 1,1γγγγ 2,1γγγγ effy1 1,2γγγγ 2,2γγγγ effy2 1,3γγγγ 2,3γγγγ effy3 Fig. 1. Effective Channel for M = K = 3, N = 2 System Due to the independence of the channel and quantization vectors, choosing the best of N channel quantizations is statistically equivalent to quantizing a single vector channel using a codebook of size N · 2B . Therefore, antenna selection effectively increases the quantization codebook size from 2B to N · 2B , and thus the system achieves the same throughput as a vector downlink with B + log2 N feedback bits. Although not negligible, this advantage is much smaller than that provided by quantization-based combining. IV. QUANTIZATION-BASED COMBINING In this section we describe the quantization-based combining (QBC) technique that reduces channel quantization error by appropriately combining receive antenna outputs. We consider a linear combiner at each mobile, which effectively converts each multiple antenna mobile into a single antenna receiver. The combiner structure for a 3 user channel with 3 transmit antennas (M = 3) and 2 antennas per mobile (N = 2) is shown in Fig. 1. Each mobile linearly combines its N outputs, using appropriately chosen combiner weights, to produce a scalar output (denoted by yeff ). The effective channel describing the channel from the transmit antenna array to the effective output of the k-th mobile (yeff ) is simply a linear combination of the N vectors describing the N receive antennas. After choosing combining weights the mobile quantizes the effective channel vector and feeds back the appropriate quantization index. Only the effective channel output is used to receive data, and thus each mobile effectively has only one antenna. The key to the technique is to choose combiner weights that produce an effective channel that can be quantized very accurately; such a choice must be made on the basis of both the channel vectors and the quantization codebook. This is quite different from maximum ratio combining, where the combiner weights and quantization vector are chosen such that received signal power is maximized but quantization error is generally not minimized. Note that antenna selection corresponds to choosing the effective channel from the N columns of Hk, while QBC allows for any linear combination of these N column vectors. A. General Description Let us consider the effective received signal at the first mobile for some choice of combiner weights, which we denote as γ1 = (γ1,1, . . . , γ1,N ). In order to maintain a noise variance of one, the combiner weights are constrained to have unit norm: ||γ1|| = 1. The (scalar) combiner output, denoted yeff1 , is: yeff1 = γH1,i(h i x + ni) = γH1,ih γH1,ink = (heff1 ) x + n, where n = i=1 γ 1,ini is unit variance complex Gaussian because |γ1| = 1. The effective channel vector heff1 is simply a linear combination of the vectors h1, . . . ,hN : h i=1 γ1,ihi = H1γ1. Since γ1 can be any unit norm vector, heff1 can be in any direction in the N -dimensional subspace spanned by h1, . . . ,hN , i.e., in span(H1). Because quantization error is so critical to performance, the objective is to choose combiner weights that yield an effective channel that can be quantized with minimal error. The error corresponding to effective channel heff1 is l=1,...,2B ∠(heff1 ,wl) . (7) 1By well known properties of iid Rayleigh fading, the matrix H1 is full rank with probability one [8]. Therefore, the optimal choice of the effective channel is the solution to: l=1,...,2B ∠(heff1 ,wl) , (8) where heff1 is allowed to be in any direction in span(H1). Once the optimal effective channel is determined, the combiner weights γ1 can be determined through a simple pseudo-inverse operation. Since the expression for the optimum effective channel given in (8) consists of two minimizations, without loss of optimality the order of the minimization can be switched to give: l=1,...,2B ∠(heff1 ,wl) , (9) For each quantization vector wl, the inner minimization finds the effective channel vector in span(H1) that forms the minimum angle with wl. By basic geometric principles, the minimizing h 1 is the projection of wl on span(H1). The solution to the inner minimization in (9) is therefore the sine squared of the angle between wl and its projection on span(H1), which is referred to as the angle between wl and the subspace 2. As a result, the best quantization vector, i.e., the solution of (9), is the vector that forms the smallest angle between itself and span(H1). The optimal effective channel is the (scaled) projection of this particular quantization vector onto span(H1). In order to perform quantization, the angle between each quantization vector and span(H1) must be computed. If q1, . . . ,qN form an orthonormal basis for span(H1) and Q1 , [q1 · · ·qN ], then sin2(∠(w, span(H1))) = 1 − ||QH1 w||2. Therefore, mobile 1’s quantized channel, denoted ĥ1, is: ĥ1 = arg min w=w1,...,w2B |∠(w, span(H1))| = arg max w=w1,...,w2B ||QH1 w||2. (10) Once the quantization vector has been selected, it only remains to choose the combiner weights. The projection of ĥ1 on span(H1), which is equal to Q1Q 1 ĥ1, is scaled by its norm to produce the unit norm vector s 1 . The direction specified by sproj1 has the minimum quantization error amongst all directions in span(H1), and therefore the effective channel should be chosen in this direction. First we find the vector u1 ∈ CN such that H1u1 = sproj1 , and then scale to get γ1. Since s 1 is in span(H1), u1 is uniquely determined by the pseudo-inverse of H1: 1 , (11) and the combiner weight vector γ1 is the normalized version of u1: γ = ||u1|| . The quantization procedure is illustrated for a N = 2 channel in Fig. 2. In the figure the span of the two channel vectors is shown along with the quantization vector h1, its projection on the channel subspace, and the effective channel. B. Algorithm Summary We now summarize the quantization-based combining procedure performed at the k-th mobile: 1) Find an orthonormal basis, denoted q1, . . . ,qN , for span(Hk) and define Qk , [q1 · · ·qN ]. 2) Find the quantization vector closest to the channel subspace: ĥk = arg max w=w1,...,w2B ||QHk w||2. (12) 3) Determine the direction of the effective channel by projecting ĥk onto span(Hk). ||QkQHk ĥk|| . (13) 4) Compute the combiner weight vector γk: . (14) 2If the number of mobile antennas is equal to the number of transmit antennas (N = M ), the channel vectors span CM with probability one. Therefore, each quantization vector has zero angle with the channel subspace and as a result the solution to the inner minimization in (9) is trivially zero for each wl. Thus, performing quantization with the sole objective of minimizing angular error (i.e., QBC) is not meaningful when N = M and is therefore not studied here. Fig. 2. Quantization procedure for a two antenna mobile Each mobile performs these steps, feeds back the index of its quantized channel ĥk, and then linearly combines its N received signals using vector γk to produce its effective channel output y = (heff )Hx+n with heff = Hkγk. Note that the transmitter need not be aware of the number of receive antennas or of the details of this procedure because the downlink channel appears to be a single receive antenna channel from the transmitter’s perspective; this clearly eases the implementation burden of QBC. V. THROUGHPUT ANALYSIS Quantization-based combining converts the MIMO downlink channel into a vector downlink with channel vectors heff1 , . . . ,h and channel quantizations ĥi · · · ĥK . We first derive the statistics of the effective vector channel, then analyze throughput for ZFBF with equal power loading and no user selection, and finally quantify the effect of receiver estimation error. A. Channel Statistics We first determine the distribution of the quantization error and the effective channel vectors with respect to both the random channels and random quantization codebooks. Lemma 1: The quantization error sin2(∠(ĥk,h )), is the minimum of 2B independent beta (M−N,N) random variables. Proof: If the columns of M×N matrix Qk form an orthonormal basis for span(Hk), then cos2 (∠(wl, span(Hk)) = ||QHk wl||2 for any quantization vector. Since the basis vectors and quantization vectors are isotropically chosen and are independent, this quantity is the squared norm of the projection of a random unit norm vector in CM onto a random N -dimensional subspace, which is described by the beta distribution with parameters N and M − N [9]. By the properties of the beta distribution, sin2 (∠(wl, span(Hk)) = 1−cos2 (∠(wl, span(Hk)) is beta (M −N,N). Finally, the independence of the quantization and channel vectors implies independence of the 2B random variables. Lemma 2: The normalized effective channels h ||heff1 || , . . . , ||heff || are iid isotropic vectors in C Proof: From the earlier description of QBC, note that ||heff || = s , which is the projection of the best quantization vector onto span(Hk). Since each quantization vector is chosen isotropically, its projection is isotropi- cally distributed within the subspace. Furthermore, the best quantization vector is chosen based solely on the angle between the quantization vector and its projection. Thus sproj is isotropically distributed in span(Hk). Since this subspace is also isotropically distributed, the vector sproj is isotropically distributed in CM . Finally, the independence of the quantization and channel vectors from mobile to mobile implies independence of the effective channel directions. Lemma 3: The quantity ||heff ||2 is χ2 2(M−N+1). Proof: Using the notation from Section IV-A, the norm of the effective channel is given by: ||heffk ||2 = ||Hkγk||2 = ||Hk ||uk|| ||2 = 1 ||uk||2 ||Hkuk||2 = ||sproj ||uk||2 ||uk||2 , (15) where we have used the definitions heff = Hkγk and γk = ||uk|| , and the fact that uk satisfies Hkuk = s Therefore, in order to characterize the norm of the effective channel it is sufficient to characterize 1||uk||2 . The N - dimensional vector uk is the set of coefficients that allows s , the normalized projection of the chosen quantization vector, to be expressed as a linear combination of the columns of Hk (i.e., the channel vectors). Because s is isotropically distributed in span(Hk) (Lemma 2), if we change coordinates to any (N -dimensional) basis for span(Hk) we can assume without loss of generality that the projection of the quantization vector is [1 0 · · · 0]T . Therefore, the distribution of 1||uk||2 is the same as the distribution of [(HHk Hk) . Since the N×N matrix HH is Wishart distributed with M degrees of freedom, this quantity is well-known to be χ2 2(M−N+1); see [10] for a proof. The norm of the effective channel has the same distribution as that of a (M −N +1)-dimensional random vector instead of a M -dimensional vector. An arbitrary linear combination (with unit norm) of the N channel vectors would result in another iid complex Gaussian M -dimensional vector, whose squared norm is χ22M , but the weights defining the effective channel are not arbitrary due to the inverse operation. B. Sum Rate Performance Relative to Perfect CSIT After receiving the quantization indices from each of the mobiles, a simple transmission option is to perform equal-power ZFBF based on the channel quantizations (as described in Section II-B). If K = M or K > M and M users are randomly selected, the resulting SINR at the k-th mobile is given by: SINRk = |(heff )Hvk|2 j 6=k |(heff )Hvj|2 . (16) The ergodic sum rate achieved by QBC, denoted RQBC(P ), is therefore given by: RQBC(P ) = EH,W |(heff )Hvk|2 j 6=k |(heff )Hvj |2 where the expectation is taken with respect to the fading and the random quantization codebooks. In order to study the benefit of QBC we compare RQBC(P ) to the sum rate achieved using zero-forcing beamforming on the basis of perfect CSIT in an M transmit antenna vector downlink channel (single receive antenna), denoted RZF−CSIT (P ). We use the vector downlink with perfect CSIT as the benchmark because QBC converts the system into a vector downlink, and the rates achieved by QBC cannot exceed RZF−CSIT (P ) (even as B → ∞). We later describe how this metric can easily be translated into a comparison between RQBC(P ) and the sum rate achievable with linear precoding (i.e., block diagonalization) in an N receive antenna MIMO downlink channel with CSIT. In a vector downlink with perfect CSIT, the BF vectors (denoted vZF,k) can be chosen perfectly orthogonal to all other channels. Thus, the SNR of each user is as given in (3) with zero interference terms in the denominator and the resulting average rate is: RZF−CSIT (P ) = EH |hHk vZF,k|2 Following the procedure in [2], the rate gap ∆R(P ) is defined as the difference between the per-user throughput achieved with perfect CSIT and with feedback-based QBC: ∆R(P ) , RZF−CSIT (P ) − RQBC(P ). (17) Similar to Theorem 1 of [2], we can upper bound this throughput loss: Theorem 1: The per-user throughput loss is upper bounded by: ∆R(P ) ≤ l=M−N+1 log2 e + log2 M− N+ 1 E[sin2(∠(ĥk,h k ))] Proof: See Appendix. The first term in the expression is the throughput loss due to the reduced norm (Lemma 3) of the effective channel, while the second (more significant) term, which is an increasing function of P , is due to quantization error. In order to quantify this rate gap, the expected quantization error needs to be bounded. By Lemma 1, the quantization error is the minimum of 2B iid beta(M − N,N) RV’s. Furthermore, a general result on ordered statistics applied to beta RV’s gives [9, Chapter 4.I.B]: E[sin2(∠(ĥk,h k ))] ≤ F where FX(x) is the inverse of the CDF of a beta (M − N,N) random variable, which is: FX(x) = N−1−i xM−N+i(1 − x)N−1+i ≈ xM−N , where the approximation is the result of keeping only the lowest order x term and dropping (1 − x) terms; this is valid for small values of x. Using this we get the following approximation: E[sin2(∠(ĥk,h k ))] ≈ 2 . (18) The accuracy of this approximation is later verified by our numerical results. Plugging this approximation into the upper bound in Theorem 1 we get: ∆R(P ) ≈ l=M−N+1 log2 e + log2 1+P · M−N+1 If B is fixed, quantization error causes the system to become interference-limited as the SNR is increased (see [2, Theorem 2] for a formal proof when N = 1). However, if B is scaled with the SNR P such that the quantization error decreases as 1 , the rate gap in (19) can be kept constant and the full multiplexing gain (M ) is achieved. In order to determine this scaling, we set the approximation of ∆R(P ) in (19) equal to a rate constant log2 b and solve for B as a function of P . Thus, a per-mobile rate loss of at most log2 b (relative to RZF−CSIT (P )) is maintained if B is scaled as: BN ≈ (M − N) log2 P − (M − N) log2 c − (M − N) log2 M−N+1 − log2 ≈ M − N PdB − (M − N) log2 c − (M − N) log2 M−N+1 − log2 , (20) where c = b · e−( l=M−N+1 ) − 1. Note that a per user rate gap of log2 b = 1 bps/Hz is equivalent to a 3 dB power gap in the sum rate curves. As discussed in Section II-C, scaling feedback in a single receive antenna downlink as B1 = PdB maintains a 3 dB gap from perfect CSIT throughput. Feedback must also be increased linearly if QBC is used, but the slope of this increase is M−1 when mobiles have only a single antenna compared to a slope of M−N for antenna combining. If we compute the difference between the N = 1 feedback load and the QBC feedback load, we can quantify how much less feedback is required to achieve the same throughput (3 dB away from a vector downlink channel with perfect CSIT) if QBC is used with N antennas/mobile: ∆QBC(N) = B1 − BN ≈ N − 1 PdB + log2 − (N − 1) log2 e. The sum rate of a 6 transmit antenna downlink channel (M = 6) is plotted in Fig. 3. The perfect CSIT zero- forcing curve is plotted along with the rates achieved using finite rate feedback with B scaled according to (20) for N = 1, 2 and 3. For N = 2 and N = 3 QBC is performed and the fact that the throughput curves are approximately 3 dB away from the perfect CSIT curve verify the accuracy of the approximations used to derive the feedback scaling expression in (20). In this system, the feedback savings at 20 dB are 7 and 12 bits, respectively, for 2 and 3 receive antennas. All numerical results in the paper are generated using the method described in Appendix II. It is also important to compare QBC throughput to the throughput of a MIMO downlink channel with N antennas per mobile. The most meaningful comparison is to the rate achievable with block diagonalization (BD) [11] without 0 5 10 15 20 25 SNR (dB) ZF−CSIT (N=1) (N=1,2,3) BD−CSIT (N=3) BD−CSIT (N=2) ~ 3 dB Fig. 3. Sum rate of M = K = 6 downlink channel user selection and with equal power loading. In this case, M mobiles are transmitted to (with N data streams per mobile). In [12] it is shown that the BD sum rate is ∆BD−ZF (N) = (log2 e) N − j larger than RZF−CSIT (P ) at asymptotically high SNR, and that this offset is accurate even for moderate SNR’s. This can be translated to a power offset by multiplying by 3 to give 3 log2 e dB, which equates to 2.16 dB and 3.61 dB for N = 2 and N = 3. Therefore, the rate offset between QBC and BD with CSIT is the sum of ∆R(P ) (equation 17) and ∆BD−ZF (N). In Fig. 3 the BD sum rate curves are plotted, and their shifts relative to ZF-CSIT are seen to follow the predicted power gaps. C. Effect of Receiver Estimation Error Although the analysis until now has assumed perfect CSI at the mobiles, a practical system always has some level of receiver error. We consider the scenario where a shared pilot sequence is used to train the mobiles. If βM downlink pilots are used (β ≥ 1 pilots per transmit antenna), channel estimation at the k-th mobile is performed on the basis of observation Gk = βPHk +nk. The MMSE estimate of Hk is Ĝk = Gk, and the true channel matrix can be written as the sum of the MMSE estimate and independent estimation error: Hk = Ĝk + ek, (21) where ek is white Gaussian noise, independent of the estimate Ĝk, with per-component variance (1+βP ) −1. After computing the channel estimate Ĝk, the mobile performs QBC on the basis of the estimate Ĝk to determine the combining vector γk. As a result, the quantization vector ĥk very accurately quantizes the vector Ĝkγk, which is the mobile’s estimate of the effective channel output, while the actual effective channel is given by heff = Hkγk. For simplicity we assume that coherent communication is possible, and therefore the long-term average throughput is again E[log2(1 + SINRk)] where the same expression for SINR given in (16) applies 3. The general throughput analysis in Section V still applies, and in particular, the rate gap upper bound given in Theorem 1 still holds if the expected quantization error takes into account the effect of receiver noise. As shown in Appendix III, the approximate rate loss with receiver error is: ∆R(P ) ≈ log2 e l=M−N+1 + log2 1+P · M−N+1 . (22) 3We have effectively assumed that each mobile can estimate the phase and SINR at the effective channel output. In practice this could be accomplished via a second round of pilots as described in [13]. 0 5 10 15 20 25 30 SNR (dB) Perfect CSIT & CSIR Combining & Perfect RX Error: Beta = 2 RX Error: Beta = 1 Fig. 4. Combining with Imperfect CSIR: M = 4, N = 2, K = 4, B scaled with SNR Comparing this expression to (19) we see that estimation error leads only to the introduction of an additional 1 term. If feedback is scaled according to (20) the rate loss is log2(b+β −1) rather than log2(b). In Figure 4 the throughput of a 4 mobile system with M = 4 and N = 2 is plotted for perfect CSIT/CSIR and for QBC performed on the basis of perfect CSIR (β = ∞) and imperfect CSIR for β = 1 and β = 2. Estimation error causes non-negligible degradation, but the loss decreases rather quickly with β (which can be increased at a reasonable resource cost because pilots are shared). VI. PERFORMANCE COMPARISONS In this section we compare the throughput of QBC to other receive combining techniques and to limited feedback- based block diagonalization4. For all results on receiving combining, the user selection algorithm of [7] is applied assuming limited feedback (B bits) regarding the direction of the effective channel and perfect knowledge of the effective channel norm5. We first describe these alternative approaches and then discuss some numerical results. A. Alternate Combining Techniques The optimal receive combining technique for a point-to-point MIMO channel in a limited feedback setting is to select the quantization vector that maximizes received power [3]: ĥk = arg max w=w1,...,w2B ||HHk w||2. (23) Because this method roughly corresponds to maximum ratio combining, it is referred to as MRC. If BF vector w is used by the transmitter, received power is maximized by choosing γ = w|| [3], which yields h = Hkγk = wk|| . When B is not very small, with high probability the quantization vector that maximizes ||H k w||2 is the vector that is closest to the eigenvector corresponding to the maximum eigenvalue of HkH . To see this, consider the maximization of ||HH w|| when w is constrained to have unit norm but need not be selected from a finite codebook. This corresponds to the classical definition of the matrix norm, and the optimizing w is in the direction of the maximum singular value of Hk. When B is not too small, the quantization error is very small and as a result the solution to (23) is extremely close to ||Hk||2. As a result, selecting the quantization vector according to the criteria in (23) is roughly equivalent to directly finding the quantization vector that is closest to the direction of the maximum singular value of Hk. 4 It should be noted that comparisons with block diagonalization are somewhat rough because systems that perform BD on the basis of limited feedback and that employ user/stream selection have not yet been extensively studied in the literature, to the best of our knowledge. As a result, it may be possible to improve upon the BD systems we use here as the point of comparison. 5Although the rate gap upper bound derived in Theorem 1 only rigorously applies to systems with equal power loading and random selection of M mobiles, the bound can be used to reasonably approximate the throughput degradation due to limited feedback even when user selection is performed. See [14] for a further discussion of the effect of limited feedback on systems employing user selection. 2 4 6 8 10 12 14 Feedback Bits (B) Approximation Approximation Ant. Selection Max Eig. Fig. 5. Quantization Error for Different Combining Techniques (M = 4, N = 2) Effective Channel Norm Quantization Error Single RX Antenna (N = 1) χ22M 2 −B/(M−1) Antenna Selection χ22M 2 −(B+log2 N)/(M−1) MRC ≈ max eigenvalue 2−B/(M−1) Max Eigenvector max eigenvalue 2−B/(M−1) QBC χ22(M−N+1) 2 −B/(M−N) TABLE I SUMMARY OF COMBINING TECHNIQUES The maximum singular value of Hk can be directly quantized if the mobile first selects the combiner weights γk such that the effective channel heff = Hkγk is in the direction of the maximum singular value, which corresponds to selecting γk equal to the eigenvector corresponding to the maximum eigenvalue of the N × N matrix HHk Hk, and then finds the quantization vector closest to heff . The effective channel norm satisfies ||heff ||2 = ||Hk||2, which can be reasonably approximated as a scaled version of a χ22MN random variable [15]. Therefore the norm of the effective channel is large, but notice that the quantization procedure reduces to standard vector quantization, for which the error is roughly 2− M−1 . In Figure 5, numerically computed values of the quantization error (log2(E[sin 2(∠(heff , ĥk))]) are shown for QBC, antenna selection, MRC (corresponding to equation 23), and direct quantization of the maximum eigenvector, along with approximation 2− M−1 as well as the approximation from (18), for a M = 4, N = 2 channel. Note that the error of QBC is very well approximated by (18), and the exponential rate of decrease of the other techniques are all well approximated by 2− M−1 . Each combining technique transforms the MIMO downlink into a vector downlink with a modified channel norm and quantization error. These techniques are summarized in Table I. The key point is that only QBC changes the exponent of the quantization error6, which determines the rate at which feedback increases with SNR. When comparing these techniques note that the complexity of QBC and MRC are essentially the same: QBC and MRC require computation of ||QHk w||2 and ||HHk w||2, respectively. B. Block Diagonalization An alternative manner in which multiple receive antennas can be used is to extend the linear precoding structure of ZFBF to allow for transmission of multiple data streams to each mobile. Block diagonalization (BD) selects 6An improvement over QBC is to choose the quantization vector and combining weights that maximize the expected received SINR (the true SINR depends on the BF vectors, which are unknown to the mobile). This extension of QBC, which will surely outperform QBC and MRC, has been under investigation by other researchers since the initial submission of this manuscript and the results will be published shortly [16]. 0 5 10 15 20 SNR (dB) BD (2 users) Ant. Selection Single Antenna Fig. 6. Different Combining Techniques: M = 4, N = 2, K = 4, B scaled with SNR precoding matrices such multi-user interference is eliminated at each receiver, similar to ZFBF. In order to select appropriate precoding matrices, the transmitter must know the N -dimensional subspace spanned by each mobile channel Hk. Thus an appropriate feedback strategy is to have each mobile quantize and feedback its channel subspace. The effect of limited feedback in this setting (assuming there are M mobiles and equal power loading across users and streams is performed) was studied in [17]. In order to achieve a bounded rate loss relative to a perfect CSIT (BD) system, feedback (per mobile) needs to scale approximately as N(M − N) log2 P . Thus, the aggregate feedback load summed over M mobiles is approximately M(M − N) log2 P , which is (approximately) the same as the aggregate feedback in a QBC system in which each of the M mobiles uses B ≈ (M −N) log2 P . Thus, there is a rough equivalence between QBC and BD in terms of feedback scaling, and this is later confirmed by our numerical results. It is also possible to perform user and stream selection when BD is used, and [18] presents an extension of the algorithm of [7] to the multiple receive antenna setting (referred to as maximum eigenmode transmission, or MET). In essence, MET treats each mobile’s N eigenmodes as a different single antenna receiver and selects eigenmodes in a greedy fashion using the approach of [7]. Thus, in a limited feedback setting a reasonable strategy is to have each user separately quantize the directions of its N eigenvectors and also feed back the corresponding eigenvalues. C. Numerical Results In Figures 6 and 7 throughput curves are shown for a 4 transmit antenna, 2 receive antenna (M = 4, N = 2) system with K = 4 mobiles. Sum rate is plotted for three different combining techniques (QBC, antenna selection, and MRC) and for a vector downlink channel (N = 1); the BD curves are discussed in later paragraphs. In Fig. 6, B (per mobile) is scaled according to (20), i.e., roughly as (M − N) log2 P , while in Fig. 7 each mobile uses 10 bits of feedback. As expected, the throughput of antenna selection, MRC, and the single antenna system all lag behind QBC in Fig. 6, particularly at high SNR. This is because the (M −N) log2 P scaling of feedback is simply not sufficient to maintain good performance if these techniques are used. To be more precise, the quantization error goes to zero slower than 1 which corresponds to interference power that increases with SNR, and thus a reduction in the slope (i.e., multiplexing gain) of these curves. In Fig. 7, MRC outperforms QBC for SNR less than approximately 12 dB because signal power is more important than quantization error (i.e., interference power), i.e., the system is not yet interference-limited. However, at higher SNR’s QBC outperforms MRC because of the increased importance of quantization error. Figures 6 and 7 also include plots of the throughput of a BD system. In this system, 2 of the 4 users are randomly selected to feedback subspace information, and equal power BD with no selection is used to send 2 streams to each of these mobiles, for a total of 4 streams. In order to equalize the aggregate feedback load, each of the 2 users is allocated double the feedback budget of the combining-based systems; this corresponds to using two times the scaling of (20) in Fig. 6 and 20 bits per mobile in Fig. 7. BD performs slightly better than QBC in both figures, but we later see that this advantage is lost for larger K. 0 5 10 15 20 SNR (dB) Single Antenna Ant. SelectionMRC BD (2 users) Fig. 7. Different Combining Techniques: M = 4, N = 2, K = 4, B = 10 Figures 8 displays throughput for a 4 transmit antenna, 2 receive antenna (M = 4, N = 2) system at 10 dB against K, the number of mobiles. Capacity refers to the sum capacity of the system (with CSIT), MET-CSIT is the throughput achieved using the MET algorithm on the basis of CSIT[18], and ZF-CSIT is the throughput of a vector downlink with CSIT and user selection [7]. Below these are four limited feedback curves for 10 bits of feedback per mobile. The first three, QBC, MRC, and antenna selection, correspond to different combining techniques, while MET-FB corresponds to performing MET on the basis of 5 bit quantization of each eigenmode (10 bits total feedback per mobile). QBC achieves significantly higher throughput than MRC or antenna selection, particularly for larger values of K. The ZF-CSIT curve is shown because it serves as an upper bound on the performance of QBC, and the gap between the two is quite reasonable even for B = 10. MET-FB is seen to perform extremely poorly: this is not too surprising because the MET algorithm is likely to only choose the strongest eigenmode of a few users [18], and thus half of the feedback is essentially wasted on quantization of each user’s weakest eigenmode. This motivates dedicating all 10 bits to quantization of the strongest eigenmode, but note that this essentially corresponds to MRC, which is outperformed by QBC. The huge gap between MET-CSIT and MET-FB indicates that MET has the potential to provide excellent performance, but extremely high levels of feedback may be necessary to realize MET’s potential. Finally, Figure 9 shows throughput versus number of users K for a 6 transmit antenna (M = 6) channel with either 1 or 2 receive antennas. Sum capacity for N = 1 and N = 2 is plotted, along with the sum rate of a perfect-CSIT TDMA system in which only the receiver with the largest point-to-point capacity is selected for transmission. The ZF and QBC curves correspond to systems with user selection and either single receive antennas or quantization-based combining, respectively, for feedback levels of 10, 15, and 20 bits per mobile. For each feedback level, an additional receive antenna with QBC provides a significant throughput gain relative to a single receive antenna system. Furthermore, QBC significantly outperforms TDMA (N = 2) for B = 15 or B = 20, and provides an advantage over TDMA for B = 10 when the number of users is sufficiently large. Note, however, that there is a significant gap between QBC and N = 2 capacity even when 20 bits of feedback are used; this indicates that there may be room for significant improvement beyond QBC. VII. CONCLUSION The performance of multi-user MIMO techniques such as zero-forcing beamforming critically depend on the accuracy of the channel state information provided to the transmitter. In this paper, we have shown that receive antenna combining can be used to reduce channel quantization error in limited feedback MIMO downlink chan- nels, and thus significantly reduce channel feedback requirements. Unlike traditional maximum-ratio combining techniques that maximize received signal power, the proposed quantization-based combining technique minimizes quantization error, which translates into minimization of multi-user interference power. Antenna combining is just one method by which multiple receive antennas can be used in the MIMO downlink. It is also possible to transmit multiple streams to each mobile, or to use receive antennas for interference cancellation 0 20 40 60 80 100 Users Capacity MET−CSIT ZF−CSIT (N=1) Ant. Selection MET− FB Fig. 8. Combining and User Selection: M = 4, N = 2, B = 10 0 20 40 60 80 100 120 Users Capacity (N=2) Capacity (N=1) QBC (N=2) B=10,15,20 ZF (N=1) B=10,15,20 N=1,2 Fig. 9. Combining and User Selection: M = 6, N = 1, 2 if the structure of the transmitted signal is known to the mobile. It remains to be seen which of these techniques is most beneficial in practical wireless systems when channel feedback resources and complexity requirements are carefully accounted for. APPENDIX I PROOF OF THEOREM 1 Plugging the rate expressions into the definition of ∆(P ), we have ∆(P ) = ∆a + ∆b where ∆a = EH 1 + ρ|hHk vZF,k|2 − EH,W log2 ρ|(heffk )Hvj|2 ∆b = EH,W log2 j 6=k ρ|(heffk )Hvj |2 where ρ , P . To upper bound ∆a, we define normalized vectors h̃k = hk/||hk|| and h̃effk = h /||heff ||, and note that the norm and directions of hk and of h are independent. Using this we have: log2 ρ|(heffk )Hvj |2  ≥ EH,W 1 + ρ|(heffk )Hvk|2 = EH,W 1 + ρ||heffk || 2|h̃eff 1 + ρXβ||hk||2|h̃k vZF,k|2 , (24) where Xβ is β(M −N + 1, N − 1). Since the BF vector vZF,k is chosen orthogonal to the (M − 1) other channel vectors {hj}j 6=k, each of which is an iid isotropic vector, it is isotropic and is independent of h̃k. By Lemma 2 the same is also true of vk and h̃ , and therefore we can substitute |h̃k vZF,k|2 for |(heffk )Hvk|2. Finally, note that the product Xβ||hk||2 is χ22(M−N+1) because ||hk|| 2 is χ22M , and therefore Xβ ||hk||2 and ||heffk ||2 have the same distribution. Using (24) we get: ∆a ≤ EH 1 + ρ||hk||2|h̃k vZF,k|2 1 + ρXβ||hk||2|h̃k vZF,k|2 ≤ −E [log2 (Xβ)] = log2 e l=M−N+1 where we have used log2 (Xβ) = log2 2(M−N+1) and results from [8] to to compute E [log2 (Xβ)]. Finally, we upper bound ∆b using Jensen’s inequality: ∆b ≤ log2 1 + E j 6=k ρ|(heffk )Hvj |2 = log2 1 + ρ(M − 1)E ||(heffk )||2 |(h̃effk)Hvj |2 = log2 1 + ρ(M − 1)(M − N + 1)E |(h̃effk)Hvj |2 = log2 1 + ρ(M − N + 1)E h̃effk,hk where the final step uses Lemma 2 of [2] to get E |(h̃effk)Hvj|2 h̃effk,hk APPENDIX II GENERATION OF NUMERICAL RESULTS Rather than performing brute force simulation of RVQ, which becomes infeasible for B larger than 15 or 20, the statistics of RVQ can be exploited to efficiently and exactly emulate the quantization process: 1) Draw a realization of the quantization error Z according to its known CDF (Lemma 1). 2) Draw a realization of the corresponding quantization vector according to: ĥk = 1 − Z where u is isotropic in span(Hk), s is isotropic in the nullspace of span(Hk), with u, s independent. These steps exactly emulate step 2 of QBC. The same procedure can also be used to emulate antenna selection, quantization of the maximum eigenvector, and no combining (N = 1). Because the CDF of the quantization error is not known for MRC, MRC results are generated using brute force RVQ. APPENDIX III RATE GAP WITH RECEIVER ESTIMATION ERROR We bound the rate gap using the technique of [13]. We first restate the result of Theorem 1 in terms of the interference terms E |(heff )Hvj |2 ∆R ≤ log2 e l=M−N+1 + log2 1 + P M − 1 |(heffk ) vj |2 . (25) Using the representation of the channel matrix given in (21), we can write the interference term as: (heffk ) vj = (Hkγk) Ĝkγk vj + (ekγk) The first term in the sum is statistically identical to the interference term when there is perfect CSIR, while the second term represents the additional interference due to the receiver estimation error. Because the noise and the channel estimate are each zero-mean and are independent we have: |(heffk ) vj |2 Ĝkγk ∣(ekγk) The first term comes from the perfect CSIR analysis and is equal to the product of 1 M−1 and the expected quantization error with perfect CSIR. Because γk and vj are each unit norm and ek is independent of these two vectors, the quantity (ekγk) vj is (zero-mean) complex Gaussian with variance (1 + βP ) −1, which is less than (1 + βP )−1. We finally reach (22) by using the approximation for quantization error from (18) and plugging into (25), and noting that (1 + βP )−1 ≈ (βP )−1. REFERENCES [1] D. Love, R. Heath, W. Santipach, and M. Honig, “What is the value of limited feedback for MIMO channels?” IEEE Communications Magazine, vol. 42, no. 10, pp. 54–59, Oct. 2004. [2] N. Jindal, “MIMO broadcast channels with finite rate feedback,” IEEE Trans. on Inform. Theory, vol. 52, no. 11, pp. 5045–5059, 2006. [3] D. Love, R. Heath, and T. Strohmer, “Grassmannian beamforming for multiple-input multiple-output wireless systems,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2735–2747, Oct. 2003. [4] K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, “On beamforming with finite rate feedback in multiple-antenna systems,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2562–2579, Oct. 2003. [5] W. Santipach and M. Honig, “Asymptotic capacity of beamforming with limited feedback,” in Proceedings of Int. Symp. Inform. Theory, July 2004, p. 290. [6] T. Yoo and A. Goldsmith, “On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming,” IEEE Journal on Selected Areas in Communications, vol. 24, no. 3, pp. 528–541, 2006. [7] G. Dimic and N. Sidiropoulos, “On downlink beamforming with greedy user selection: Performance analysis and simple new algorithm,” IEEE Trans. Sig. Proc., vol. 53, no. 10, pp. 3857–3868, October 2005. [8] A. Tulino and S. Verdu, “Random matrix theory and wireless communications,” Foundations and Trends in Communications and Information Theory, vol. 1, no. 1, 2004. [9] A. K. Gupta and S. Nadarajah, Handbook of Beta Distribution and Its Applications. CRC, 2004. [10] J. Winters, J. Salz, and R. Gitlin, “The impact of antenna diversity on the capacity of wireless communication systems,” IEEE Trans. on Communications, vol. 42, no. 234, pp. 1740–1751, 1994. [11] Q. Spencer, A. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Trans. Sig. Proc., vol. 52, no. 2, pp. 461–471, 2004. [12] J. Lee and N. Jindal, “High SNR analysis for MIMO broadcast channels: Dirty paper coding vs. linear precoding,” to appear in IEEE Trans. Inform. Theory, 2007. [13] G. Caire, N. Jindal, M. Kobayashi, and N. Ravindran, “Quantized vs. analog feedback for the MIMO downlink: A comparison between zero-forcing based achievable rates,” in Proceedings of Int. Symp. Inform. Theory, June 2007. [14] T. Yoo, N. Jindal, and A. Goldsmith, “Finite-rate feedback MIMO broadcast channels with a large number of users,” 2007, to appear in IEEE J. Sel. Areas on Commun. [15] A. Paulraj, D. Gore, and R. Nabar, Introduction to Space-Time Wireless Communications. Cambridge University Press, 2003. [16] M. Trivellato, H. Huang, and F. Boccardi, “Antenna combining and codebook design for MIMO broadcast channel with limited feedback,” in Proc. Asilomar Conf. on Sig. and Systems, Nov. 2007. [17] N. Ravindran and N. Jindal, “MIMO broadcast channels with block diagonalization and finite rate feedback,” in Proc. ICASSP, April 2007. [18] F. Boccardi and H. Huang, “A near-optimum technique using linear precoding for the MIMO broadcast channel,” in Proc. ICASSP, April 2007. Introduction System Model and Preliminaries Finite Rate Feedback Model Zero-Forcing Beamforming MIMO Downlink with Single Antenna Mobiles Antenna Selection for Reduced Quantization Error Quantization-Based Combining General Description Algorithm Summary Throughput Analysis Channel Statistics Sum Rate Performance Relative to Perfect CSIT Effect of Receiver Estimation Error Performance Comparisons Alternate Combining Techniques Block Diagonalization Numerical Results Conclusion Appendix I: Proof of Theorem ?? Appendix II: Generation of Numerical Results Appendix III: Rate Gap with Receiver Estimation Error References

0704.1309

Quantum State Transfer with Spin Chains

Quantum State Transfer with Spin Chains Daniel Klaus Burgarth A thesis submitted to the University of London for the degree of Do tor of Philosophy Department of Physi s and Astronomy University College London De ember 2006 http://arxiv.org/abs/0704.1309v1 De laration I, Daniel Klaus Burgarth, on�rm that the work presented in this thesis is my own. Where information has been derived from other sour es, I on�rm that this has been indi ated in the thesis. Abstra t In the last few de ades the idea ame up that by making use of the superposition prin iple from Quantum Me hani s, one an pro ess information in a new and mu h faster way. Hen e a new �eld of information te hnology, QIT (Quantum Information Te hnology), has emerged. From a physi s point of view it is important to �nd ways of implementing these new methods in real systems. One of the most basi tasks required for QIT is the ability to onne t di�erent omponents of a Quantum Computer by quantum wires that obey the superposition prin iple. Sin e superpositions an be very sensitive to noise this turns out to be already quite di� ult. Re ently, it was suggested to use hains of permanently oupled spin-1/2 parti les (quantum hains) for this purpose. They have the advantage that no external ontrol along the wire is required during the transport of information, whi h makes it possible to isolate the wire from sour es of noise. The purpose of this thesis is to develop and investigate advan ed s hemes for using quantum hains as wires. We �rst give an introdu tion to basi quantum state transfer and review existing advan ed s hemes by other authors. We then introdu e two new methods whi h were reated as a part of this thesis. First, we show how the �delity of transfer an be made perfe t by performing measurements at the re eiving end of the hain. Then we introdu e a s heme whi h is based on performing unitary operations at the end of the hain. We generalise both methods and dis uss them from the more fundamental point of view of mixing properties of a quantum hannel. Finally, we study the e�e ts of a non-Markovian environment on quantum state transfer. A knowledgements Most of all, I would like to thank my supervisor Sougato Bose for mu h inspira- tion and advi e. I am very grateful for many inspiring and fruitful dis ussions and ollaborations with Vittorio Giovannetti, and with Floor Paauw, Christoph Bruder, Jason Twamley, Andreas Bu hleitner and Vladimir Korepin. Furthermore I would like to thank all my tea hers and those who have guided and motivated me along my journey through physi s, in luding Heinz-Peter Breuer, Fran es o Petru ione, Lewis Ryder, John Strange, Werner Riegler, Carsten S huldt and Rolf Bussmann. I a knowl- edge �nan ial support by the UK Engineering and Physi al S ien es Resear h Coun il through the grant GR/S62796/01. Finally I would like to thank my parents for their loving support. Notation X,Y,Z Pauli matri es Xn, Yn, Zn Pauli matri es a ting on the Hilbert-spa e of qubit n |0〉, |1〉 Single qubit state in the anoni al basis |0〉 Quantum hain in the produ t state |0〉 ⊗ · · · ⊗ |0〉 |n〉 �Single ex itation� state Xn|0〉 TrX Partial tra e over subsystem X || . . . || Eu lidean ve tor norm || . . . ||1 Tra e norm || . . . ||2 Eu lidean matrix norm We also use the following graphi al representation: |n〉 ≡ |0〉 ≡ |0〉 ≡ |1〉 ≡ nth qubit |ψ〉 = α|0〉 + β|1〉 ≡ controlled region: quantum gates and measurements uncontrolled region coupling receiver ("Bob") sender ("Alice") Contents 1 Introdu tion 9 1.1 Quantum Computation and Quantum Information . . . . . . . . . . . 10 1.2 Quantum state transfer along short distan es . . . . . . . . . . . . . . 11 1.3 Implementations and experiments . . . . . . . . . . . . . . . . . . . . . 15 1.4 Basi ommuni ation proto ol . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.1 Initialisation and end-gates . . . . . . . . . . . . . . . . . . . . 17 1.4.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.3 Transfer fun tions . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.4.4 Heisenberg Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 20 1.4.5 Dynami and Dispersion . . . . . . . . . . . . . . . . . . . . . . 21 1.4.6 How high should p(t) be? . . . . . . . . . . . . . . . . . . . . . 27 1.5 Advan ed ommuni ation proto ols . . . . . . . . . . . . . . . . . . . . 29 1.5.1 Engineered Hamiltonians . . . . . . . . . . . . . . . . . . . . . 29 1.5.2 Weakly oupled sender and re eiver . . . . . . . . . . . . . . . . 29 1.5.3 En oding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.5.4 Time-dependent ontrol . . . . . . . . . . . . . . . . . . . . . . 31 1.6 Motivation and outline of this work . . . . . . . . . . . . . . . . . . . . 31 2 Dual Rail en oding 34 2.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2 S heme for on lusive transfer . . . . . . . . . . . . . . . . . . . . . . . 35 2.3 Arbitrarily perfe t state transfer . . . . . . . . . . . . . . . . . . . . . 38 2.4 Estimation of the time-s ale the transfer . . . . . . . . . . . . . . . . . 40 2.5 De oheren e and imperfe tions . . . . . . . . . . . . . . . . . . . . . . 42 2.6 Disordered hains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.7 Con lusive transfer in the presen e of disorder . . . . . . . . . . . . . . 45 2.8 Arbitrarily perfe t transfer in the presen e of disorder . . . . . . . . . 48 2.9 Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.10 Numeri al Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Contents 2.11 Coupled hains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.12 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Multi Rail en oding 58 3.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 E� ient en oding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4 Perfe t transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.5 Convergen e theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.6 Quantum hains with nearest-neighbour intera tions . . . . . . . . . . 69 3.7 Comparison with Dual Rail . . . . . . . . . . . . . . . . . . . . . . . . 70 3.8 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4 Ergodi ity and mixing 72 4.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2 Topologi al ba kground . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 Generalised Lyapunov Theorem . . . . . . . . . . . . . . . . . . . . . . 76 4.3.1 Topologi al spa es . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3.2 Metri spa es . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4 Quantum Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.4.1 Mixing riteria for Quantum Channels . . . . . . . . . . . . . . 84 4.4.2 Beyond the density matrix operator spa e: spe tral properties . 86 4.4.3 Ergodi hannels with pure �xed points . . . . . . . . . . . . . 88 4.5 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5 Read and write a ess by lo al ontrol 93 5.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.2 Proto ol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3 De omposition equations . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.4 Coding transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.5 Fidelities for reading and writing . . . . . . . . . . . . . . . . . . . . . 99 5.6 Appli ation to spin hain ommuni ation . . . . . . . . . . . . . . . . . 101 5.7 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6 A valve for probability amplitude 104 6.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2 Arbitrarily Perfe t State Transfer . . . . . . . . . . . . . . . . . . . . . 104 6.3 Pra ti al Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Contents 6.4 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7 External noise 110 7.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.4 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 8 Con lusion and outlook 120 List of Figures 121 List of Tables 126 Bibliography 127 Index 141 1 Introdu tion The Hilbert spa e that ontains the states of quantum me hani al obje ts is huge, s aling exponentially with the number of parti les des ribed. In 1982, Ri hard Feyn- man suggested to make use of this as a resour e for simulating quantum me hani s in a quantum omputer, i.e. a devi e where the physi al intera tion ould be �programmed� to yield a spe i� Hamiltonian. This has led to the new �elds of Quantum Computa- tion and Quantum Information. A quantum omputer an solve questions one ould never imagine to solve using an ordinary omputer. For example, it an fa torise large numbers into primes e� iently, a task of greatest importan e for ryptography. It may thus be a surprise that more than twenty years after the initial ideas, these devi es still haven't been built or only in ridi ulously small size. The largest quantum omputer so far an only solve problems that any hild ould solve within se onds. A loser look reveals that the main problem in the realisation of quantum omput- ers is the �programming�, i.e. the design of a spe i� (time-dependent) Hamiltonian, usually des ribed as a set of dis rete unitary gates. This turns out to be extremely di� ult be ause we need to onne t mi ros opi obje ts (those behaving quantum me hani ally) with ma ros opi devi es that ontrol the mi ros opi behaviour. Even if one manages to �nd a link between the mi ro- and the ma ros opi world, su h as laser pulses and ele tri or magneti �elds, then the onne tion introdu es not only ontrol but also noise (dissipation and de oheren e) to the mi ros opi system, and its quantum behaviour is diminished. The vision of this thesis is to develop theoreti al methods narrowing the gap between what is imagined theoreti ally and what an be done experimentally. As a method we onsider hains (or more general graphs) of permanently oupled quantum systems. This idea has been originally put forward by S. Bose for the spe i� task of quantum ommuni ation [1℄. Due to the permanent oupling, these devi es an in prin iple be built in su h a way that they don't require external ontrol to perform their tasks, just like a me hani al lo kwork. This also over omes the problem of de oheren e as they an be separated from any sour e of noise. Unfortunately, most s hemes that have been developed so far still require external ontrol, though mu h less than an �ordinary� quantum omputer. Furthermore, internal dispersion in these devi es is 1 Introdu tion leading to a de rease of their �delity. A third problem is, that for building these devi es the permanent ouplings still need to be realised, although only on e, and experimental onstraints su h as resolution and errors need to be onsidered. We are thus left with the following questions: whi h is the best way to perform quantum state transfer using a permanently oupled graph? How mu h ontrol do we need, and how di� ult will it be to implement the ouplings? How do errors and noise a�e t the s heme? All these points are highly related and it annot be expe ted to �nd an absolute, i.e. system independent answer. The purpose of this resear h is to develop advan ed s hemes for the transfer of quantum information, to improve and generalise existing ideas, to relate them to ea h other and to investigate their stability and e� ien y. 1.1 Quantum Computation and Quantum Information In this Se tion we review some of the basi on epts of Quantum Computation. We will be very brief and only fo us on those aspe ts that we require later on in the thesis. A more detailed introdu tion an be found in [2℄. In information s ien e, an algorithm is a list of instru tions that a omputer performs on a given input to a hieve a spe i� task. For instan e, a fa toring algorithm has an arbitrary integer as its input, and gives its prime fa tors as an output. A quantum fa toring algorithm an be thought of in a similar way, i.e. it has an integer as input, and its prime fa tors as an output. In-between however it en odes information in a quantum me hani al system. Due to the superposition prin iple, the information of a quantum system annot be represented as bits. The valid generalisation of the bit to the quantum ase is alled qubit. The possible states of a qubit are written as α|0〉 + β|1〉, (1.1) where α, β are normalised omplex oe� ients, and |0〉 and |1〉 are ve tors of a two- dimensional omplex ve tor spa e. Peter W. Shor has shown in a famous paper [3℄ that the detour of representing the intermediate part of a fa toring algorithm in a quantum system (as well as using quantum gates, see below) an be very bene� ial: it runs mu h faster. This is important, be ause many ryptographi methods rely on fa toring algorithms being slow. Shor's algorithm is de�nitely not the only reason why it would be very ni e to have a quantum omputer, i.e. a ma hine that represents information in a quantum way and an perform instru tions on it, and many more details an be found in the textbook mentioned above. 1 Introdu tion Algorithms on a omputer an be represented as list of logi al operations on bits. Likewise, a (standard) quantum algorithm an be represented as a list of quantum logi al operations, or quantum gates, a ting on qubits. The most general quantum algorithm is given by an arbitrary unitary operator. A universal set of gates is a set su h that any quantum algorithm (i.e. unitary operator) an be de omposed into a sequen e of gates belonging to this set. In the standard model of quantum omputation, one assumes that su h a set is available on the ma hine [4℄. Also the ability to perform measurements is assumed. We refer to this as the full ontrol ase. From a information theoreti point of view, qubits are not only useful obje ts to perform algorithms with, but also very interesting from a fundamental point of view. To give a (too simple) analogy onsider the following. If you read the word � ho olate�, you an asso iate a positive/negative or neutral feeling of whether you would like to eat some ho olate now. However, what was the state of your mind on erning ho olate before you read the word? Unless you were already raving for ho olate beforehand, or you have just eaten a lot, your mind was probably unde ided. Moreover, it would have been very di� ult - if not impossible - to des ribe to someone in plain language whi h opinion you had about the ho olate before you read the word. In a similar manner, the quantum information ontained in a single arbitrary and unknown qubit annot be des ribed by lassi al information. When it is measured, it behaves like a normal bit in the sense that the out ome is only 0 or 1, but when it is not measured, it behaves in some way as if it was unde ided between 0 and 1. Of ourse one has to be very areful with these analogies. But for the purpose of this thesis it is important to stress that quantum information annot be transported by any lassi al methods [5℄. This is why it is so important and also so di� ult to develop new wires, dubbed quantum wires, that are apable of doing this. 1.2 Quantum state transfer along short distan es In theory, additional devi es for the transfer of unknown quantum states are not required for building a quantum omputer, unless it is being used for typi al quantum ommuni ation purposes, su h as se ret key distribution [4℄. This is be ause the universal set of gates on the quantum omputer an be used to transfer quantum states by applying sequen es of two-qubit swap gates (Fig. 1.1). However in pra ti e it is ru ial to minimise the required number of quantum gates, as ea h gate typi ally introdu es errors. In this light it appears ostly to perform N−1 swap gates between nearest neighbours to just move a qubit state over a distan e of N sites. For example, Shor's algorithm on N qubits an be implemented by only 1 Introdu tion SN−1,N Figure 1.1: In areas of universal ontrol, quantum states an easily be transferred by sequen es of unitary swap gates Sj,k between nearest neighbours. logN quantum gating operations [6℄ if long distant qubit gates are available. These long distant gates ould onsist of lo al gates followed by a quantum state transfer. If however the quantum state transfer is implemented as a sequen e of lo al gates, then the number of operations blows up to the order of N gates. The quantum state transfer an even be thought of as the sour e of the power of quantum omputation, as any quantum ir uit with logN gates and lo al gates only an be e� iently simulated on a lassi al omputer [7, 8℄. A se ond reason to onsider devi es for quantum state transfer is related to s ala- bility . While small quantum omputers have already been built [9℄, it is very di� ult to build large arrays of fully ontrollable qubits. A bla k box that transports unknown quantum states ould be used to build larger quantum omputers out of small ompo- nents by onne ting them. Likewise, quantum state transfer an be used to onne t di�erent omponents of a quantum omputer, su h as the pro essor and the memory (see also Fig. 1.2). On larger distan es, �ying qubits su h as photons, ballisti ele trons and guided atoms/ions are onsidered for this purpose [10, 11℄. However, onverting ba k and forth between stationary qubits and mobile arriers of quantum information and interfa ing between di�erent physi al implementations of qubits is very di� ult and worthwhile only for short ommuni ation distan es. This is the typi al situation one has to fa e in solid state systems, where quantum information is usually ontained in the states of �xed obje ts su h as quantum dots or Josephson jun tions. In this ase permanently oupled quantum hains have re ently been proposed as prototypes of re- liable quantum ommuni ation lines [1,12℄. A quantum hain (also referred to as spin hain) is a one-dimensional array of qubits whi h are oupled by some Hamiltonian ( f. Fig. 1.3). These ouplings an transfer states without external lassi al ontrol. 1 Introdu tion In many ases, su h permanent ouplings are easy to build in solid state devi es (in fa t a lot of e�ort usually goes into suppressing them). The qubits an be of the same type as the other qubits in the devi e, so no interfa ing is required. Quantum processor Quantum memory Controller Input Output Figure 1.2: S hemati layout of a quantum omputer. The solid arrows represent the �ow of quantum information, and the dashed arrows the �ow of lassi al information. Figure 1.3: Permanently oupled quantum hains an transfer quantum states without ontrol along the line. Note that the ends still need to be ontrollable to initialise and read out quantum states. Another related motivation to onsider quantum hains is that they an simplify the layout of quantum devi es on wafers. A typi al hip an ontain millions of qubits, and the fabri ation of many qubits is in prin iple no more di� ult than the fabri ation of a single one. In the last ouple of years, remarkable progress was made in experiments with quantum dots [13, 14℄ and super- ondu ting qubits [15, 16℄. It should however be emphasised that for initialisation, ontrol and readout, those qubits have to be onne ted to the ma ros opi world (see Fig. 1.2). For example, in a typi al �ux qubit gate, mi rowave pulses are applied onto spe i� qubits of the sample. This requires many ( lassi al) wires on the hip, whi h is thus a ompound of quantum and lassi al omponents. The ma ros opi size of the lassi al ontrol is likely to be the bottlene k of the s alability as a whole. In this situation, quantum hains are useful in order to keep some distan e between the ontrolled quantum parts. A possible layout for su h 1 Introdu tion a quantum omputer is shown in Fig. 1.4. It is built out of blo ks of qubits, some of whi h are dedi ated to ommuni ation and therefore onne ted to another blo k through a quantum hain. Within ea h blo k, arbitrary unitary operations an be performed in a fast and reliable way (they may be de omposed into single and two- qubit operations). Su h blo ks do not urrently exist, but they are the fo us of mu h work in solid state quantum omputer ar hite ture. The distan e between the blo ks is determined by the length of the quantum hains between them. It should be large enough to allow for lassi al ontrol wiring of ea h blo k, but short enough so that the time-s ale of the quantum hain ommuni ation is well below the time-s ale of de oheren e in the system. Figure 1.4: Small blo ks (grey) of qubits (white ir les) onne ted by quantum hains. Ea h blo k onsists of (say) 13 qubits, 4 of whi h are onne ted to outgoing quantum hains (the thi k bla k lines denote their nearest-neighbour ouplings). The blo ks are onne ted to the ma ros opi world through lassi al wires (thin bla k lines with bla k ir les at their ends) through whi h arbitrary unitary operations an be triggered on the blo k qubits. The quantum hains require no external ontrol. Finally, an important reason to study quantum state transfer in quantum hains stems from a more fundamental point of view. Su h systems in prin iple allow tests of Bell-inequalities and non-lo ality in solid-state experiments well before the realisation of a quantum omputer. Although quantum transport is quite an established �eld, the quantum information point of view o�ers many new perspe tives. Here, one looks at the transport of information rather than ex itations, and at entanglement [17,18,19,20℄ rather than orrelation fun tions. It has re ently been shown that this sheds new light on well-known physi al phenomena su h as quantum phase transitions [21, 22, 1 Introdu tion 23,24℄, quantum haos [25, 26, 27, 28℄ and lo alisation [29, 30℄. Furthermore, quantum information takes on a more a tive attitude. The orrelations of the system are not just al ulated, but one also looks at how they may be hanged. 1.3 Implementations and experiments As we have seen above, the main advantage of state transfer with quantum hains is that the qubits an be of the same type as those used for the quantum omputation. Therefore, most systems that are thought of as possible realisations of a quantum omputer an also be used to build quantum hains. Of ourse there has to be some oupling between the qubits. This is typi ally easy to a hieve in solid state sys- tems, su h as Josephson jun tions with harge qubits [31, 32℄, �ux qubits [33, 34℄ (see also Fig. 1.5) or quantum dots dots using the ele trons [35, 36℄ or ex itons [37, 38℄. Other systems where quantum hain Hamiltonians an at least be simulated are NMR qubits [39,40,41℄ and opti al latti es [42℄. Su h a simulation is parti ularly useful in the latter ase, where lo al ontrol is extremely di� ult. Finally, qubits in avities [43,44℄ and oupled arrays of avities were onsidered [45, 46℄. Figure 1.5: A quantum hain onsisting of N = 20 �ux qubits [34℄ (pi ture and exper- iment by Floor Paauw, TU Delft). The hain is onne ted to four larger SQUIDS for readout and gating. For the more fundamental questions, su h as studies of entanglement transfer, non- lo ality and oherent transport, the quantum hains ould also be realised by systems whi h are not typi ally thought of as qubits, but whi h are natural spin hains. These an be mole ular systems [47℄ or quasi-1D solid state materials [48, 49℄. 1.4 Basi ommuni ation proto ol We now review the most basi transport proto ol for quantum state transfer, initially suggested in [1℄. For the sake of simpli ity, we on entrate on the linear hain setting, though more general graphs of qubits an be onsidered in the same way. The proto ol onsists of the following steps: 1 Introdu tion 1. Initialise the quantum hain in the ground state |G〉. (1.2) 2. Put an arbitrary and unknown qubit with (possibly mixed) state ρ at the sending end of the hain ρ⊗ Tr1 {|G〉〈G|} . (1.3) 3. Let the system evolve under its Hamiltonian H for a time t exp {−iHt} ρ⊗ Tr1 {|G〉〈G|} exp {iHt} . (1.4) 4. Pi k up the quantum state at the end of the hain σ ≡ Tr1,...,N−1 [exp {−iHt} ρ⊗ Tr1 {|G〉〈G|} exp {iHt}] . (1.5) Some pra ti al aspe ts how to realise these steps are dis ussed in the next se tion. For the moment, we will on entrate on the quality of quantum state transfer given that the above steps an be performed. From a quantum information perspe tive, the above equations des ribe a quantum hannel [5℄ τ that maps input states ρ at one end of the hain to output states τ(ρ) = σ on the other end. A very simple measure of the quality of su h a quantum hannel is the �delity [50, 51, 2℄ F (ρ, σ) ≡ ρ1/2σρ1/2 . (1.6) More advan ed measures of the quality of transfer will be dis ussed in Chapter 3. Note also that some authors de�ne the �delity without taking the square of the tra e. It is a real-valued, symmetri fun tion with range between 0 and 1, assuming unity if and only if ρ = σ. Sin e the transported state that is an unknown result of some quantum omputation, we are interested in the minimal �delity F0 ≡ minρF (ρ, τ(ρ)). (1.7) We remark that some authors also assume an equal distribution of input states and ompute the average �delity [1℄. Using the strong on avity of the �delity [2℄ and the linearity of τ we �nd that the minimum must be assumed on pure input states, F0 = minψ〈ψ|τ(ψ)|ψ〉. (1.8) 1 Introdu tion In the present ontext, F0 = F0(H, t) is a fun tion of of the Hamiltonian H of the quantum hain (through the spe i� role of the ground state in the proto ol and through the time evolution), and of the time interval t that the system is evolving in the third step of the proto ol. 1.4.1 Initialisation and end-gates There are two strong assumptions in the proto ol from the last se tion. The �rst one is that the hain an be initialised in the ground state |G〉. How an that be a hieved if there is no lo al ontrol along the hain? The answer appears to be quite easy: one just applies a strong global magneti �eld and strong ooling (su h as laser ooling or dilution refrigeration) and lets the system rea h its ground state by relaxation. The ooling needs to be done for the remaining parts of the quantum omputer anyway, so no extra devi es are required. However there is a problem with the time-s ale of the relaxation. If the system is brought to the ground state by ooling, it must be oupled to some environment. But during the quantum omputation, one learly does not want su h an environment. This is usually solved by having the time-s ale of the omputation mu h smaller (say mi rose onds) than the time-s ale of the ooling (say se onds or minutes). But if the quantum hain should be used multiple times during one omputation, then how is it reset between ea h usage? This is important to avoid memory e�e ts [52℄, and there are two solutions to this problem. Either the proto ol is su h that at the end the hain is automati ally in the ground state. Su h a proto ol usually orresponds to perfe t state transfer. The other way is to use the ontrol at the ends of the hain to bring it ba k to the ground state. A simple ooling proto ol is given by the following: one measures the state of the last qubit of the hain. If it is in |0〉, then one just lets the hain evolve again and repeats. If however it is found to be in |1〉, one applies the Pauli operator X to �ip it before evolving and repeating. It will be ome lear later on in the thesis that su h a proto ol typi ally onverges exponentially fast to the ground state of the hain. The se ond assumption in the last se tion is that the sender and re eiver are apable of swapping in and out the state mu h qui ker than the time-s ale of the intera tion of the hain. Alternatively, it is assumed that they an swit h on and o� the intera tion between the hain and their memory in su h time-s ale. It has re ently been shown [33℄ that this is not a fundamental problem, and that �nite swit hing times an even slightly improve the �delity if they are arefully in luded in the proto ol. But this requires to solve the full time-dependent S hrödinger equation, and introdu es further parameters to the model (i.e. the raise and fall time of the ouplings). For the sake of simpli ity, 1 Introdu tion we will therefore assume that the end gates are mu h faster then the time evolution of the hain (see also Se tion 6.3). 1.4.2 Symmetries The dimensionality of the Hilbert spa e H of a quantum hain of N qubits is 2N . This makes it quite hopeless in general to determine the minimal �delity Eq. (1.8) for long quantum hains. Most investigations on quantum state transfer with quantum hains up to date are therefore on entrating on Hamiltonians with additional sym- metries. With few ex eptions [34,21,22,53℄ Hamiltonians that onserve the number of ex itations are onsidered. In this ase the Hilbert spa e is a dire t sum of subspa es invariant under the time evolution, Hℓ, (1.9) with dimHℓ = , and where ℓ is the number of ex itations. These Hamiltonians are mu h easier to handle both analyti ally and numeri ally, and it is also easier to get an intuition of the dynami s. Furthermore, they o ur quite naturally as a oupling between qubits in the relevant systems. We stress though that there is no fundamental reason to restri t quantum hain ommuni ation to this ase. 1.4.3 Transfer fun tions The spa e H0 only ontains the state |0〉 whi h is thus always an eigenstate of H. We will assume here that it is also the ground state, |G〉 = |0〉. (1.10) This an be a hieved by applying a strong global magneti �eld (or equivalent) to the system. The spa e H1 is spanned by the ve tors {|k〉, k = 1, . . . , N} having exa tly one ex itation. The above proto ol be omes: 1. Initialise the quantum hain in the ground state |0〉 (1.11) 2. Put an arbitrary and unknown qubit in the pure state |ψ〉 = α|0〉 + β|1〉 at the 1 Introdu tion sending end of the hain α|0〉+ β|1〉 (1.12) 3. Let the system evolve for a time t α|0〉+ β exp {−iHt} |1〉 (1.13) 4. Pi k up the quantum state at the end of the hain (see [1℄) τ(ψ) = (1− p(t))|0〉〈0| + p(t)|ψ〉〈ψ|, (1.14) with the minimal �delity given by F0 = minψ〈ψ|τ(ψ)|ψ〉 (1.15) = p(t) + (1− p(t))minψ |〈0|ψ〉|2 = p(t). (1.16) The fun tion p(t) is the transition probability from the state |1〉 to |N 〉 given by p(t) = |〈N | exp {−iHt} |1〉|2 . (1.17) We see that in the ontext of quantum state transfer, a single parameter su� es to hara terise the properties of an ex itation onserving hain. The averaged �delity [1℄ is also easily omputed as . (1.18) Even more omplex measures of transfer su h as the quantum apa ity only depend on p(t) [54℄. It is also a physi ally intuitive quantity, namely a parti ular matrix element of the time evolution operator, fn,m(t) ≡ 〈n| exp {−iHt} |m〉 (1.19) e−iEkt〈n|Ek〉〈Ek|m〉, (1.20) where |Ek〉 and Ek are the eigenstates and energy levels of the Hamiltonian in H1. 1 Introdu tion 1.4.4 Heisenberg Hamiltonian The Hamiltonian hosen in [1℄ is a Heisenberg Hamiltonian H = −J (XnXn+1 + YnYn+1 + ZnZn+1)−B Zn + c, (1.21) with a onstant term J(N − 1) +NB (1.22) added to set the ground state energy to 0. For J > 0 it ful�ls all the assumptions dis ussed above, namely its ground state is given by |0〉 and it onserves the number of ex itations in the hain. The Heisenberg intera tion is very ommon and serves here as a typi al and analyti ally solvable model for quantum state transfer. In the �rst ex itation subspa e H1, the Heisenberg Hamiltonian Eq. (1.21) is ex- pressed in the basis {|n〉} as −1 2 −1 −1 2 . . . −1 2 −1 . (1.23) A more general study of su h tridiagonal matri es an be found in a series of arti les on oherent dynami s [55, 56, 57, 58℄. Some interesting analyti ally solvable models have also been identi�ed [59, 56, 57℄ (we shall ome ba k to that point later). For the present ase, the eigenstates of Eq (1.23) are [1℄ |Ek〉 = 1 + δk0 (2n− 1) |n〉 (k = 0, . . . , N − 1), (1.24) with the orresponding energies given by Ek = 2B + 2J 1− cos πk . (1.25) The parameter B has no relevan e for the �delity but determines the stability of the ground state (the energy of the �rst ex ited state is given by 2B). The minimal �delity 1 Introdu tion for a Heisenberg hain is given by p(t) = N−2 −2iJt(1− cos πk (−1)k 1 + cos . (1.26) As an example, Fig 1.6 shows p(t) for N = 50. 0.02 0.04 0.06 0.08 0.12 0.14 0 10 20 30 40 50 60 70 80 90 100 Time [1/J] Figure 1.6: Minimal �delity p(t) for a Heisenberg hain of length N = 50. 1.4.5 Dynami and Dispersion Already in [1℄ has been realised that the �delity for quantum state transfer along spin hains will in general not be perfe t. The reason for the imperfe t transfer is the dispersion [60℄ of the information along the hain. Initially the quantum information is lo alised at the sender, but as it travels through the hain it also spreads (see Fig. 1.7 and Fig. 1.8). This is not limited to the Heisenberg oupling onsidered here, but a very ommon quantum e�e t. Due to the dispersion, the probability amplitude peak that rea hes Bob is typi ally small, and be omes even smaller as the hains get longer. The �delity given Eq. (1.26) is shown in Fig. 1.6. We an see that a wave of quantum information is travelling a ross the hain. It rea hes the other end at a time of approximately (1.27) 1 Introdu tion 0 10 20 30 40 50 Position along chain Figure 1.7: Snapshots of the time evolution of a Heisenberg hain with N = 50. Shown is the distribution |fn,1(t)|2 of the wave-fun tion in spa e at di�erent times if initially lo alised at the �rst qubit. 0 20 40 60 80 100 Time [1/J] Relative mean Relative variance Figure 1.8: Mean and varian e of the state |1〉 as a fun tion of time. Shown is the ase N = 50 with the y-axis giving the value relative to the mean N/2 + 1 and varian e (N2 − 1)/12 of an equal distribution 1√ 1 Introdu tion As a rough estimate of the s aling of the �delity with respe t to the hain length around this peak we an use [1, 61℄ (see also Fig. 1.9) |fN,1(t)|2 ≈ |2JN ( )|2 ≈ | (N − 2t )]|2, (1.28) where JN (x) is a Bessel fun tion of �rst kind and ai(x) is the Airy fun tion. The airy fun tion ai(x) has a maximum of 0.54 at x = −1.02. Hen e we have ) = |fN,1( )|2 ≈ 1.82N−2/3. (1.29) It is however possible to �nd times where the �delity of the hain is mu h higher. The reason for this is that the wave-pa ket is re�e ted at the ends of the hain and starts interfering with itself (Fig 1.6). As the time goes on, the probability distribution be omes more and more random. Sometimes high peaks at the re eiving end o ur. From a theoreti al point of view, it is interesting to determine the maximal peak o urring, i.e. pM (T ) ≡ max 0<t<T p(t). (1.30) As we an see in Fig. 1.10 there is quite a potential to improve from the estimate Eq. (1.29). 0.02 0.04 0.06 0.08 0.12 0.14 20 22 24 26 28 30 Time [1/J] |fN,1(t)| |2 JN(2t)| |(16/N)1/3 ai[(2/N)1/3(N-2t)]|2 Figure 1.9: Approximation of the transfer amplitude for N = 50 around the �rst maximum by Bessel and Airy fun tions [1, 61℄. 1 Introdu tion 0.1 1 10 100 1000 10000 100000 1e+06 1e+07 Time [1/J] Figure 1.10: pM (T ) as a fun tion of T for di�erent hain lengths. The solid urve is given by 1.82(2T ) and orresponds to the �rst peak of the probability amplitude (Eq. 1.29) We will now show a perhaps surprising onne tion of the fun tion pM (T ) to number theory. Some spe ulations on the dependen e of the �delity on the hain length being divisible by 3 were already made in [1℄, but not rigorously studied. As it turns out, for hains with prime number length the maximum of the �delity is a tually onverging to unity (see Fig. 1.10). To show this, we �rst prove the following Lemma 1.1 Let N be an odd prime. Then the set (k = 0, 1, . . . , , N − 1 (1.31) is linear independent over the rationals Q. Proof Assume that λk cos = 0 (1.32) with λk ∈ Q. It follows that + exp = 0 (1.33) 1 Introdu tion and hen e λk exp λk exp i(N − k)π = 0. (1.34) Changing indexes on the se ond sum we get λk exp k=N+1 λN−k exp = 0. (1.35) and �nally λ̃k exp = 0, (1.36) where λ̃0 = 2λ0 (1.37) λ̃k = λk (k = 1, . . . , N − 1 ) (1.38) λ̃k = −λN−k (k = N + 1 , . . . , N − 1). (1.39) Sin e N is prime, the roots of unity in Eq. (1.36) are all primitive and therefore linearly independent over Q [62, Theorem 3.1, p. 313℄. Hen e λk = 0 for all k. � Theorem 1.1 (Half re urren e) Let N be an odd prime. For a Heisenberg hain of length N we have pM (T ) = lim 0<t<T = 1. (1.40) Proof The eigenfrequen ies of the Hamiltonian in the �rst ex itation se tor H1 are given by Ek = 2B + 2J 1− cos πk (k = 0, 1, . . . , , N − 1). (1.41) Using Krone ker's theorem [63℄ and Lemma 1.1, the equalities exp {itEk} = (−1)k e2(B+J)t (k = 0, 1, . . . , , N − 1 ) (1.42) 1 Introdu tion an be ful�lled arbitrarily well by hoosing an appropriate t. Sin e = − cos (N − k)π , (1.43) the equalities (1.42) are then also ful�lled arbitrarily well for k = 0, . . . , N − 1. This is known as as su� ient ondition for perfe t state transfer in mirror symmetri hains [64℄, where the eigenstates an be hosen su h that they are alternately sym- metri and antisymmetri . Roughly speaking, Eq. (1.42) introdu es the orre t phases (a sign hange for the antisymmetri eigenstates) to move the state |1〉 to |N〉 and hen e the theorem. � Remark 1.1 The time-s ale for �nding high valued peaks is however exponential in the hain length [63℄. Therefore the above theorem has little pra ti al use. For non- prime hain lengths, the eigenfrequen ies are not su� iently independent to guarantee a perfe t state transfer, with the algebrai dimensionality of the roots of unity for non- prime N given by the Euler totient fun tion φ(N) [62, Theorem 3.1, p. 313℄. We also remark that due to its asymptoti hara ter, the above result is not ontradi ting [65℄, where it was shown that hains longer than N ≥ 4 never have perfe t �delity. Having proved that there are many hains that an in prin iple perform arbitrarily well, it is important to �nd a ut-o� time for the optimisation Eq. (1.30). Faster transfer than linear in N using lo al Hamiltonians is impossible due to the Lieb-Robinson bound [66, 67℄, whi h is a �speed limit� in non-relativisti quantum me hani s giving rise to a well de�ned group velo ity. Transport faster than this group velo ity is exponentially suppressed. Going ba k to the motivation of quantum state transfer, a natural omparison [37℄ for the time-s ale of quantum state transfer is given by the time it would take to perform a sequen e of swap gates ( f. Fig 1.1) that are realised by a pairwise swit hable oupling Hamiltonian (XnXn+1 + YnYn+1). (1.44) This time is linear in the hain length: (N − 1)π . (1.45) Ideally one ould say that the time for quantum state transfer should not take mu h longer than this. However one may argue that there is a trade-o� between qui k transfer on one hand, and minimising ontrol on the other hand. A se ond ut-o� 1 Introdu tion time may be given by the de oheren e time of the spe i� implementation. But short de oheren e times ould always be ountera ted by in reasing the hain oupling J. A more general and implementation independent limit is given by the requirement that the peak width ∆t should not be too small with respe t to the total time. Otherwise it is di� ult to pi k up the state at the orre t time. For the �rst peak, we an estimate the width by using the full width at half height of the airy fun tion. From Eq. (1.28) we get an absolute peak width of ∆t ≈ 0.72N1/3/J and a relative width of ≈ 1.44N−2/3. (1.46) This is already quite demanding from an experimental perspe tive and we on lude that the transfer time should not be hosen mu h longer than those of the �rst peak. 1.4.6 How high should p(t) be? We have not dis ussed yet what the a tual value of p(t) should be to make su h a spin hain useful as a devi e for quantum state transfer. p(t) = 0 orresponds to no state transfer, p(t) = 1 to a perfe t state transfer. But what are the relevant s ales for intermediate p(t)? In pra ti e, the quantum transfer will su�er from additional external noise (Chapter 7) and also the quantum omputer itself is likely to be very noisy. From this point of view, requiring p(t) = 1 seems a bit too demanding. From a theoreti al perspe tive, it is interesting that for any p(t) > 0, one an al- ready do things whi h are impossible using lassi al hannels, namely entanglement transfer and distillation [2℄. The entanglement of formation between the sender (Al- i e) and the re eiver (Bob) is simply given by p(t) [1℄. This entanglement an be partially distilled [68℄ into singlets, whi h ould be used for state transfer using tele- portation [2℄. It is however not known how mu h, i.e. at whi h rate, entanglement an be distilled (we will develop lower bounds for the entanglement of distillation in Se tion 2.2 and Se tion 3.4). Also, entanglement distillation is a quite omplex pro- edure that requires lo al unitary operations and measurements, additional lassi al ommuni ation, and multiple hain usages; and few expli it proto ols are known. This is likely to preponderate the bene�ts of using a quantum hain. When the hain is used without en oding and further operation, the averaged �delity Eq. (1.18) be omes better than the lassi al averaged �delity [1℄ when p(t) > 3 − By � lassi al �delity�, we mean the �delity that an be a hieved by optimising the following pro- to ol: Ali e performs measurements on her state and sends Bob the out ome through a lassi al ommuni ation line. Bob then tries to rebuild the state that Ali e had before the measurement based on the information she sent. For qubits, the lassi al �delity is given by 2/3 [69℄. 1 Introdu tion 2. Following the on lusion from the last subse tion that the �rst peak is the most relevant one, this would mean that only hains with length until N = 33 perform better than the lassi al �delity. Finally, the quantum apa ity [54, 70℄ of the hannel be omes non-zero only when p(t) > 1/2, orresponding to hain lengths up to N = 6. Roughly speaking, it is a measure of the number of perfe tly transmitted qubits per hain usage that an be a hieved asymptoti ally using en oding and de oding operations on multiple hannel usages. The quantum apa ity onsidered here is not assumed to be assisted by a lassi al ommuni ation, and the threshold of p(t) > 0.5 to have a non-zero quantum apa ity is a result of the non- loning theorem [2℄. This is not ontradi ting the fa t that entanglement distillation is possible for any p(t) > 0, as the entanglement distillation proto ols require additional lassi al ommuni ation. All the above points are summarised in Fig. 1.11. We an see that only very short hains rea h reasonable values (say > 0.6) for the minimal �delity. 0 0.2 0.4 0.6 0.8 1 235610153380300 Corresponding chain length (first peak) Quantum capacity EOD (lower bound) Averaged fidelity Classical threshold Figure 1.11: Quantum apa ity, entanglement of formation (EOF), a lower bound for the entanglement of distillation (EOD) and the averaged �delity as a fun tion of p(t). We also show the orresponding hain length whi h rea hes this value as a �rst peak and the lassi al threshold 3− 2 2. The expli it expression for the quantum apa ity plotted here is given in [54℄, and the lower bound of the entanglement of distillation will be derived in Se tion 3.4. 1 Introdu tion 1.5 Advan ed ommuni ation proto ols We have seen in the last se tion that without mu h further e�ort, i.e. entanglement distillation, unmodulated Heisenberg hains are useful only when they are very short. Shortly after the initial proposal [1℄ it has been shown that there are ways to a hieve even perfe t state transfer with arbitrarily long hains. These advan ed proposals an roughly be grouped into four ategories, whi h we will now brie�y des ribe. 1.5.1 Engineered Hamiltonians The Heisenberg model hosen by Bose features many typi al aspe ts of oherent trans- port, i.e. the wave-like behaviour, the dispersion, and the almost-periodi ity of the �delity. These features do not depend so mu h on the spe i� hoi es of the parame- ters of the hain, su h as the ouplings strengths. There are however spe i� ouplings for quantum hains that show a quite di�erent time evolution, and it was suggested in [71℄ and independently in [72℄ to use these to a hieve a perfe t state transfer: H = −J n(N − n) (XnXn+1 + YnYn+1) (1.47) These values for engineered ouplings also appear in a di�erent ontext in [57,73℄. The time evolution under the Hamiltonian (1.47) features an additional mirror symmetry: the wave-pa ket disperses initially, but the dispersion is reversed after its entre has passed the middle of the hain (Fig. 1.12). This approa h has been extended by various authors [64, 74, 75, 76, 77, 78, 79, 80, 53, 81, 82, 83, 84, 65, 59, 19℄, and many other hoi es of parameters for perfe t or near perfe t state transfer in various settings were found [59, 83, 81℄. 1.5.2 Weakly oupled sender and re eiver A di�erent approa h of tuning the Hamiltonian was suggested in [85℄. There, only the �rst and the last ouplings j of the hain are engineered to be mu h weaker than the remaining ouplings J of the hain, whi h an be quite arbitrary. The �delity an be made arbitrarily high by making the edge oupling strengths smaller. It was shown [86, 87℄ that to a hieve a �delity of 1 − δ in a hain of odd length, it takes approximately a time of δ (1.48) 1 Introdu tion 0 10 20 30 40 50 Position along chain t=pi/8 t=pi/4 t=3pi/8 t=pi/2 Figure 1.12: Snapshots of the time evolution of a quantum hain with engineered ouplings (1.47) for N = 50. Shown is the distribution of the wave-fun tion in spa e at di�erent times if initially lo alised at the �rst qubit ( ompare Fig. 1.7). 1 Introdu tion and the oupling ratio has to be approximately j/J ≈ δ/N. Some spe i� types of quantum hains whi h show high �delity for similar reasons were also investigated [88, 89, 90, 91℄. 1.5.3 En oding We have seen in Subse . 1.4.6 that if p(t) < 1/2, the �delity annot be improved by using any en oding/de oding strategy (be ause the quantum apa ity is zero). However it is possible to hange the proto ol des ribed in Se . 1.4 slightly su h that the �delity is mu h higher. This an be thought of as a �hardware en oding�, and was suggested �rst in [60℄. There, it was assumed that the hain onsists of three se tions: one part of length ≈ 2N1/3 ontrolled by the sending party, one �free� part of length N and one part of length ≈ 2.8N1/3 ontrolled by the re eiving party. The sender en odes the qubit not only in a single qubit of the hain, but in a Gaussian- modulated superposition of his qubits. These Gaussian pa kets are known to have minimal dispersion. Likewise, the re eiver performs a de oding operation on all qubits he ontrols. Near-perfe t �delity an be rea hed. 1.5.4 Time-dependent ontrol Finally, a number of authors found ways of improving the �delity by time-dependent ontrol of some parameters of the Hamiltonian. In [92℄ it is shown that if the end ouplings an be ontrolled as arbitrary (in general omplex valued) smooth fun tions of time the en oding s heme [60℄ ould be simulated without the requirement of ad- ditional operations and qubits. Another possibility to a hieve perfe t state transfer is to have an Ising intera tion with additionally pulsed global rotations [93, 40, 94℄. Further related methods of manipulating the transfer by global �elds were reported in [95, 96, 25, 97, 98, 28℄. 1.6 Motivation and outline of this work While the advan ed transfer proto ols have shown that in prin iple high �delity an be a hieved with arbitrarily long hains, they have all ome at a ost. Engineering ea h oupling of the Hamiltonian puts extra demands on the experimental realisation, whi h is often already at its very limits just to ensure the oheren e of the system. Furthermore, the more a s heme relies on parti ular properties of the Hamiltonian, the more it will be a�e ted by imperfe tions in its implementation [99, 84℄. For example, simulating an engineered hain of length N = 50 with a (relative) disorder of 5%, we 1 Introdu tion get a �delity peak of 0.95±0.02. For a disorder of 10% we get 0.85±0.05. The weakly oupled system is very stable for o�-site disorder [85℄, but su�ers strongly from on-site disorder (i.e. magneti �elds in z−dire tion) at the ends of the hain. For example, for a hain of N = 50 with edge ouplings j = 0.01 and the remaining ouplings being J = 1, we �nd that already a magneti �eld of the order of 0.00001 lowers the �delity to 0.87 ± 0.12. For �elds of the order of 0.00005 we �nd 0.45 ± 0.32. This is be ause these �u tuations must be small with respe t to the small oupling, so there is a double s aling. Also, the time-s ale of the transfer is longer than in other s hemes (note though that this may sometimes even be useful for having enough time to pi k up the re eived state). On the other hand, en oding and time-dependent ontrol require additional resour es and gating operations. It is not possible to judge independently of the realisation whi h of the above s hemes is the �most pra ti al� one. We summarise the di�erent aspe ts that are important in the following �ve riteria for quantum state transfer: 1. High e� ien y: How does the �delity depend on the length of the hain? Whi h rate [100, 81, 74℄ an be a hieved? 2. Minimal ontrol : How many operations are required to a hieve a ertain �delity? Where is ontrol required? 3. Minimal resour es: What additional resour es are required? 4. Minimal design: How general is the oupling type ? What values of the oupling strengths are allowed? 5. Robustness: How is the �delity a�e ted by stati disorder, by time-dependent disorder, by gate and timing errors, and by external noise su h as de oheren e and dissipation? At the start of this resear h, only the engineering and en oding s hemes were available. The engineering s hemes are strong in the points 2 and 3, but quite weak in the points 4 and 5. The en oding s heme on the other hand has its weakness in points 2 and 3. It was hen e desirable to develop more balan ed s hemes. Sin e most experiments in Quantum Information are extremely sensitive and at the utting edge of their parameters (i.e. requiring extremely low temperatures, well tuned lasers, and so forth, For example, gates at the ends of the hain are always needed for write-in and read-out, and thus � heaper� than gates along the hain. Global ontrol along the whole hain is often easier than lo al ontrol. Often the oupling type is already �xed by the experiment 1 Introdu tion to maintain their quantum behaviour), we parti ularly wanted to �nd s hemes whi h are strong in the points 4 and 5. Also, from a more fundamental point of view, we were interested in seeing how mu h information on the state of a quantum hain ould be obtained by the re eiver in prin iple, and how the re eiver might even be able to prepare states on the whole hain. The main a hievements of this thesis are two s hemes for the transfer of quantum information using measurements (Chapter 2 and 3) or unitary operations (Chapter 5 and 6) at the re eiving end of the hain. Sin e both s hemes use onvergen e properties of quantum operations, it seemed natural to investigate these properties in a more abstra t way (Chapter 4). There, we found a new way of hara terising mixing maps, whi h has appli ations beyond quantum state transfer, and may well be relevant for other �elds su h as haos theory or statisti al physi s. Finally, in Chapter 7 we dis uss problems quantum state transfer in the presen e of external noise. The results in Chapters 3-6 were developed in ollaboration with Vittorio Giovannetti from S uola Normale, Pisa. Mu h of the material dis ussed in this thesis has been published or submitted for publi ation [101, 102, 103, 104, 105, 106, 107, 108, 109, 110℄. 2 Dual Rail en oding 2.1 Introdu tion The role of measurement in quantum information theory has be ome more a tive re ently. Measurements are not only useful to obtain information about some state or for preparation, but also, instead of gates, for quantum omputation [111℄. In the ontext of quantum state transfer, it seems �rst that measurements would spoil the oheren e and destroy the state. The �rst indi ation that measurements an a tually be used to transfer quantum information along anti-ferromagneti hains was given in [24℄. However there the measurements had to be performed along the whole hain. This may in some ases be easier than to perform swap gates, but still requires high lo al a essibility. We take a �hybrid� approa h here: along the hain, we let the system evolve oherently, but at the re eiving end, we try to help the transfer by measuring. The main disadvantage of the en oding used in the proto ols above is that on e the information dispersed, there is no way of �nding out where it is without destroying it. A dual rail en oding [112℄ as used in quantum opti s on the other hand allows us to perform parity type measurements that do not spoil the oheren e of the state that is sent. The out ome of the measurement tells us if the state has arrived at the end ( orresponding to a perfe t state transfer) or not. We all this on lusively perfe t state transfer. Moreover, by performing repetitive measurements, the probability of su ess an be made arbitrarily lose to unity. As an example of su h an amplitude delaying hannel, we show how two parallel Heisenberg spin hains an be used as quantum wires. Perfe t state transfer with a probability of failure lower than P in a Heisenberg hain of N qubits an be a hieved in a time-s ale of the order of 0.33J−1N1.7| lnP |. We demonstrate that our s heme is more robust to de oheren e and non-optimal timing than any s heme using single spin hains. We then generalise the dual rail en oding to disordered quantum hains. The s heme performs well for both spatially orrelated and un orrelated �u tuations if they are relatively weak (say 5%). Furthermore, we show that given a quite arbitrary pair of quantum hains, one an he k whether it is apable of perfe t transfer by only lo al operations at the ends of the hains, and the system in the middle being a bla k box. 2 Dual Rail en oding We argue that unless some spe i� symmetries are present in the system, it will be apable of perfe t transfer when used with dual rail en oding. Therefore our s heme puts minimal demand not only on the ontrol of the hains when using them, but also on the design when building them. This Chapter is organised as follows. In Se tion 2.2, we suggest a s heme for quan- tum ommuni ation using two parallel spin hains of the most natural type (namely those with onstant ouplings). We require modest en odings (or gates) and measure- ments only at the ends of the hains. The state transfer is on lusive, whi h means that it is possible to tell by the out ome of a quantum measurement, without destroy- ing the state, if the transfer took pla e or not. If it did, then the transfer was perfe t. The transmission time for on lusive transfer is not longer than for single spin hains. In Se tion 2.3, we demonstrate that our s heme o�ers even more: if the transfer was not su essful, then we an wait for some time and just repeat the measurement, without having to resend the state. By performing su� iently many measurements, the probability for perfe t transfer approa hes unity. Hen e the transfer is arbitrarily perfe t. We will show in Se tion 2.4 that the time needed to transfer a state with a given probability s ales in a reasonable way with the length of the hain. In Se tion 2.5 we show that en oding to parallel hains and the on lusiveness also makes our proto ol more robust to de oheren e (a hitherto unaddressed issue in the �eld of quan- tum ommuni ation through spin hains). In the last part of this hapter, we show how this s heme an be generalised to disordered hains (Se tions 2.6-2.10) and even oupled hains (Se tion 2.11). 2.2 S heme for on lusive transfer We intend to propose our s heme in a system-independent way with o asional refer- en es to systems where onditions required by our s heme are a hieved. We assume that our system onsists of two identi al un oupled spin-1/2- hains (1) and (2) of length N , des ribed by the Hamiltonian H = H(1) ⊗ I(2) + I(1) ⊗H(2) − EgI(1) ⊗ I(2). (2.1) The term identi al states that H(1) and H(2) are the same apart from the label of the Hilbert spa e they a t on. The requirement of parallel hains instead of just one is not a real problem, sin e in many experimental realisations of spin hains, it is mu h easier to produ e a whole bun h of parallel un oupled [48,49℄ hains than just a single 2 Dual Rail en oding Figure 2.1: Two quantum hains inter onne ting A and B. Control of the systems is only possible at the two qubits of either end. We assume that the ground state of ea h hain is |0〉i, i.e. a ferromagneti ground state, with H(i) |0〉i = Eg |0〉i , and that the subspa e onsisting of the single spin ex itations |n〉i is invariant under H(i). Let us assume that the state that Ali e wants to send is at the �rst qubit of the �rst hain, i.e. |ψA〉1 ≡ α |0〉1 + β |1〉1 , (2.2) and that the se ond hain is in the ground state |0〉2. The aim of our proto ol is to transfer quantum information from the 1st (�Ali e�) to the Nth (�Bob�) qubit of the �rst hain: |ψA〉1 → |ψB〉1 ≡ α |0〉1 + β |N 〉1 . (2.3) The �rst step (see also Fig. 2.2) is to en ode the input qubit in a dual rail [112℄ by applying a NOT gate on the �rst qubit of system (2) ontrolled by the �rst qubit of system (1) being zero, resulting in a superposition of ex itations in both systems, |s(0)〉 = α |0, 1〉+ β |1, 0〉 , (2.4) where we have introdu ed the short notation |n,m〉 ≡ |n〉1 ⊗ |m〉2. This is assumed to take pla e in a mu h shorter time-s ale than the system dynami s. Even though a 2-qubit gate in solid state systems is di� ult, su h a gate for harge qubits has been reported [15℄. For the same qubits, Josephson arrays have been proposed as single spin hains for quantum ommuni ation [31℄. For this system, both requisites of our s heme are thus available. In fa t, the demand that Ali e and Bob an do measurements and apply gates to their lo al qubits (i.e. the ends of the hains) will be naturally ful�lled in pra ti e sin e we are suggesting a s heme to transfer information between quantum omputers (as des ribed in Se tion 1.2). 2 Dual Rail en oding (1) ◦ spin chain (1) tℓ • |ψB〉 (2) ⊕ spin chain (2) tℓ ⊕ ✙ ❴❴❴❴❴❴❴❴ ✤✤✤✤✤✤✤ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ wait again if 0 success if 1 Alice Bob Figure 2.2: Quantum ir uit representation of on lusive and arbitrarily perfe t state transfer. The �rst gate at Ali e's qubits represents a NOT gate applied to the se ond qubit ontrolled by the �rst qubit being zero. The qubit |ψA〉1 on the left hand side represents an arbitrary input state at Ali e's site, and the qubit |ψB〉1 represents the same state, su essfully transferred to Bob's site. The tℓ-gate represents the unitary evolution of the spin hains for a time interval of tℓ. Under the system Hamiltonian, the ex itation in Eq. (2.4) will travel along the two systems. The state after the time t1 an be written as |φ(t1)〉 = fn,1(t1) |s(n)〉 , (2.5) where |s(n)〉 = α |0, n〉 + β |n, 0〉 and the omplex amplitudes fn,1(t1) are given by Eq. (1.19). We an de ode the qubit by applying a CNOT gate at Bob's site. Assuming that this happens on a time-s ale mu h shorter than the evolution of the hain, the resulting state is given by fn,1(t1) |s(n)〉+ fN,1(t1) |ψB〉1 ⊗ |N〉2 . (2.6) Bob an now perform a measurement on his qubit of system (2). If the out ome of this measurement is 1, he an on lude that the state |ψ〉(1)1 has been su essfully transferred to him. This happens with the probability |fN,1(t1)|2 . If the out ome is 0, the system is in the state P (1) fn,1(t1) |s(n)〉 , (2.7) where P (1) = 1− |fN,1(t1)|2 is the probability of failure for the �rst measurement. If the proto ol stopped here, and Bob would just assume his state as the transferred one, 2 Dual Rail en oding the hannel ould be des ribed as an amplitude damping hannel [54℄, with exa tly the same �delity as the single hain s heme dis ussed in [1℄. Note that here the en oding is symmetri with respe t to α and β, so the minimal �delity is the same as the averaged But su ess probability is more valuable than �delity: Bob has gained knowledge about his state, and may reje t it and ask Ali e to retransmit (this is known as a quantum erasure hannel [113℄). Of ourse in general the state that Ali e sends is the unknown result of some quantum omputation and annot be sent again easily. This an be over ome in the following way: Ali e sends one e-bit on the dual rail �rst. If Bob measures a su ess, he tells Ali e, and they both start to teleport the unknown state. If he measures a failure, they reset the hains and start again. Sin e the joint probability of failure onverges exponentially fast to zero this is quite e� ient. In fa t the on lusive transfer of entanglement is possible even on a single hain by using the same hain again instead of a se ond one [114℄. This an be seen as a very simple entanglement distillation pro edure, a hieving a rate of |fN,1(t)|2/2. However the hain needs to be reset between ea h transmission (see Se tion 1.4.1 for problems related to this), and Ali e and Bob require lassi al ommuni ation. We will show in the next se tion, that the reuse of the hain(s) is not ne essary, as arbitrarily perfe t state transfer an already a hieved in the �rst transmission. 2.3 Arbitrarily perfe t state transfer Be ause Bob's measurement has not revealed anything about the input state (the su ess probability is independent of the input state), the information is still residing in the hain. By letting the state (2.7) evolve for another time t2 and applying the CNOT gate again, Bob has another han e of re eiving the input state. The state before performing the se ond measurement is easily seen to be P (1) {fn,1(t2 + t1)− fn,N(t2)fN,1(t1)} |s(n)〉 . (2.8) Hen e the probability to re eive the qubit at Bobs site at the se ond measurement is P (1) |fN,1(t2 + t1)− fN,N (t2)fN,1(t1)|2 . (2.9) If the transfer was still unsu essful, this strategy an be repeated over and over. Ea h time Bob has a probability of failed state transfer that an be obtained from the 2 Dual Rail en oding generalisation of Eq. (2.8) to an arbitrary number of iterations. The joint probability that Bob fails to re eive the state all the time is just the produ t of these probabilities. We denote the joint probability of failure for having done l unsu essful measurements as P (ℓ). This probability depends on the time intervals tℓ between the (ℓ− 1)th and ℓth measurement, and we are interested in the ase where the tℓ are hosen su h that the transfer is fast. It is possible to write a simple algorithm that omputes P (ℓ) for any transition amplitude fr,s(t). Figure 2.3 shows some results for the Heisenberg Hamiltonian given by Eq. (1.21). 1e-06 1e-05 1e-04 0.001 0.01 0 5 10 15 20 25 Number of measurements N=150 N=100 N= 50 N= 20 N= 10 N= 5 Figure 2.3: Semilogarithmi plot of the joint probability of failure P (ℓ) as a fun tion of the number of measurements ℓ. Shown are Heisenberg spin-1/2- hains with di�erent lengths N . The times between measurements tℓ have been optimised numeri ally. An interesting question is whether the joint probability of failure an be made arbitrarily small with a large number of measurements. In fa t, the times tℓ an be hosen su h that the transfer be omes arbitrarily perfe t. We will prove this in the next Chapter, where a generalisation of the dual rail s heme and a mu h wider lass of Hamiltonians is onsidered. In the limit of large number of measurements, the spin hannel will not damp the initial amplitude, but only delay it. 2 Dual Rail en oding 2.4 Estimation of the time-s ale the transfer The a hievable �delity is an important, but not the only riterion of a state transfer proto ol. In this Se tion, we give an heuristi approa h to estimate the time that it needs to a hieve a ertain �delity in a Heisenberg spin hain. The omparison with numeri examples is on�rming this approa h. Let us �rst des ribe the dynami of the hain in a very qualitative way. On e Ali e has initialised the system, an ex itation wave pa ket will travel along the hain. As shown in Subse tion 1.4.5, it will rea h Bob at a time of the order of , (2.10) with an amplitude of ∣fN,1(t ≈ 1.82N−2/3. (2.11) It is then re�e ted and travels ba k and forth along the hain. Sin e the wave pa ket is also dispersing, it starts interfering with its tail, and after a ouple of re�e tions the dynami is be oming quite randomly. This e�e t be omes even stronger due to Bobs measurements, whi h hange the dynami s by proje ting away parts of the wave pa ket. We now assume that 2t (the time it takes for a wave pa ket to travel twi e along the hain) remains a good estimate of the time-s ale in whi h signi� ant probability amplitude peaks at Bobs site o ur, and that Eq. (2.11) remains a good estimate of the amplitude of these peaks . Therefore, the joint probability of failure is expe ted to s ale as P (ℓ) ≈ 1− 1.82N−2/3 (2.12) in a time of the order of t(ℓ) ≈ 2tmaxℓ = J−1Nℓ. (2.13) If we ombine Eq. (2.12) and (2.13) and solve for the time t(P ) needed to rea h a ertain probability of failure P , we get for N ≫ 1 t(P ) ≈ 0.55J−1N5/3 |lnP | . (2.14) We ompare this rough estimate with exa t numeri al results in Fig. 2.4. The best �t This is not a strong assumption. If the ex itation was fully randomly distributed, the probability would s ale as N−1. By sear hing for good arrival times, this an be slightly in reased to N−2/3. 2 Dual Rail en oding for the range shown in the �gure is given by t(P ) = 0.33J−1N5/3 |lnP | . (2.15) We an on lude that the transmission time for arbitrarily perfe t transfer is s aling not mu h worse with the length N of the hains than the single spin hain s hemes. Despite of the logarithmi dependen e on P, the time it takes to a hieve high �delity is still reasonable. For example, a system with N = 100 and J = 20K ∗ kB will take approximately 1.3ns to a hieve a �delity of 99%. In many systems, de oheren e is ompletely negligible within this time-s ale. For example, some Josephson jun tion systems [115℄ have a de oheren e time of Tφ ≈ 500ns, while trapped ions have even larger de oheren e times. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1000 1500 2000 2500 Transfer Time [1/J] Numerical data Chain length Joint probability of failure Transfer Time [1/J] Figure 2.4: Time t needed to transfer a state with a given joint probability of failure P a ross a hain of length N . The points denote exa t numeri al data, and the �t is given by Eq. (2.15). 2 Dual Rail en oding 2.5 De oheren e and imperfe tions If the oupling between the spins J is very small, or the hains are very long, the transmission time may no longer be negligible with respe t to the de oheren e time. It is interesting to note that the dual rail en oding then o�ers some signi� ant general advantages over single hain s hemes. Sin e we are suggesting a system-independent s heme, we will not study the e�e ts of spe i� environments on our proto ol, but just qualitatively point out its general advantages. At least theoreti ally, it is always possible to ool the system down or to apply a strong magneti �eld so that the environment is not ausing further ex itations. For example in �ux qubit systems, the system is ooled to ≈ 25mK to ensure that the energy splitting∆ ≫ kBT [116℄. Then, there are two remaining types of quantum noise that will o ur: phase noise and amplitude damping. Phase noise is a serious problem and arises here only when an environment an distinguish between spin �ips on the �rst hain and spin �ips on the se ond hain. It is therefore important that the environment annot resolve their di�eren e. In this ase, the environment will only ouple with the total z- omponent Z(1)n + Z n (2.16) of the spins of both hains at ea h position n. This has been dis ussed for spin-boson models in [117,118℄ but also holds for spin environments as long as the hains are lose enough. The qubit is en oded in a de oheren e-free subspa e [119℄ and the s heme is fully robust to phase noise. Even though this may not be true for all implementations of dual rail en oding, it is worthwhile noti ing it be ause su h an opportunity does not exist at all for single hain s hemes, where the oheren e between two states with di�erent total z- omponent of the spin has to be preserved. Having shown one way of avoiding phase noise, at least in some systems, we now pro eed to amplitude damping. The evolution of the system in presen e of amplitude damping of a rate Γ an be easily derived using a quantum-jump approa h [120℄. This is based on a quantum master equation approa h, whi h is valid in the Born-Markov approximation [121℄ (i.e. it holds for weakly oupled environments without memory e�e ts). Similarly to phase noise, it is ne essary that the environment a ts symmetri ally on the hains. The dynami s is then given by an e�e tive non-Hermitian Hamiltonian Heff = H + iΓ Z(1)n + Z n + 2 /2 (2.17) 2 Dual Rail en oding if no jump o urs. If a jump o urs, the system is ba k in the ground state |0〉. The state of the system before the �rst measurement onditioned on no jump is given by fn,1(t) |s(n)〉 , (2.18) and this happens with the probability of e−2Γt (the norm of the above state). If a jump o urs, the system will be in the ground state 1− e−2Γt |0, 0〉 . (2.19) The density matrix at the time t is given by a mixture of (2.18) and (2.19). In ase of (2.19), the quantum information is ompletely lost and Bob will always measure an unsu essful state transfer. If Bob however measures a su ess, it is lear that no jump has o urred and he has the perfe tly transferred state. Therefore the proto ol remains on lusive, but the su ess probability is lowered by e−2Γt. This result is still valid for multiple measurements, whi h leave the state (2.19) unaltered. The probability of a su essful transfer at ea h parti ular measurement ℓ will de rease by e−2Γt(ℓ), where t(ℓ) is the time at whi h the measurement takes pla e. After a ertain number of measurements, the joint probability of failure will no longer de rease. Thus the transfer will no longer be arbitrarily perfe t, but an still rea h a very high �delity. Some numeri al examples of the minimal joint probability of failure that an be a hieved, P (ℓ) ≈ 1− 1.35N−2/3e− (2.20) are given in Fig. 2.5. For J/Γ = 50K ns nearly perfe t transfer is still possible for hains up to a length of N ≈ 40. Even if the amplitude damping is not symmetri , its e�e t is weaker than in single spin s hemes. This is be ause it an be split in a symmetri and asymmetri part. The symmetri part an be over ome with the above strategies. For example, if the amplitude damping on the hains is Γ1 and Γ2 with Γ1 > Γ2, the state (2.18) will be fn,1(t) αe−Γ2t |0, n〉+ βe−Γ1t |n, 0〉 (2.21) ≈ e−Γ2t fn,1(t) |s(n)〉 (2.22) 2 Dual Rail en oding provided that t ≪ (Γ1 − Γ2)−1 . Using a hain of length N = 20 with J = 20K ∗ kB and Γ−11 = 4ns, Γ 2 = 4.2ns we would have to ful�l t ≪ 164ns. We ould perform approximately 10 measurements ( f. Eq. (2.13)) without deviating too mu h from the state (2.22). In this time, we an use our proto ol in the normal way. The resulting su ess probability given by the �nite version of Eq. (2.20) would be 75%. A similar reasoning is valid for phase noise, where the environment an be split into ommon and separate parts. If the hains are lose, the ommon part will dominate and the separate parts an be negle ted for short times. 0 20 40 60 80 100 120 140 160 180 200 Chain length 1 K ns 10 K ns 25 K ns 50 K ns 100 K ns 200 K ns Figure 2.5: The minimal joint probability of failure P (ℓ) for hains with length N in the presen e of amplitude damping. The parameter J/Γ of the urves is the oupling of the hain (in Kelvin) divided by the de ay rate (ns−1). 2.6 Disordered hains The main requirement for perfe t transfer with dual rail en oding in the above is that two identi al quantum hains have to be designed. While this is not so mu h a theoreti al problem, for possible experimental realizations of the s heme [31℄ the question arises naturally how to ope with slight asymmetries of the hannels. We are now going to demonstrate that in many ases, perfe t state transfer with dual rail en oding is possible for quantum hains with di�ering Hamiltonians. By doing so, we also o�er a solution to another and perhaps more general problem: 2 Dual Rail en oding if one implements any of the s hemes for quantum state transfer, the Hamiltonians will always be di�erent from the theoreti al ones by some random perturbation. This will lead to a de rease of �delity in parti ular where spe i� energy levels were assumed (see [99,84℄ for an analysis of �u tuations a�e ting the engineered hains des ribed in Subse tion 1.5.1). This problem an be avoided using the s heme des ribed below. In general, disorder an lead to a Anderson lo alisation [122,29,30℄ of the eigenstates (and therefore to low �delity transport of quantum information). In this se tion however this is not relevant, as we onsider only short hains (N < 100) and small disorder (≈ 10% of the oupling strength), and the lo alisation length is mu h longer then the length of the hain. We will show numeri ally that the dual rail s heme an still a hieve arbitrarily perfe t transfer for a uniformly oupled Heisenberg Hamiltonian with disordered oupling strengths (both for the ase of spatially orrelated and un- orrelated disorder). Moreover, for any two quantum hains, we show that Bob and Ali e an he k whether their system is apable of dual rail transfer without dire tly measuring their Hamiltonians or lo al properties of the system along the hains but by only measuring their part of the system. 2.7 Con lusive transfer in the presen e of disorder We onsider two un oupled quantum hains (1) and (2), as shown in Fig. 2.6. The hains are des ribed by the two Hamiltonians H(1) and H(2) with total Hamiltonian given by H = H(1) ⊗ I(2) + I(1) ⊗H(2), (2.23) and the time evolution operator fa torising as U(t) = exp −iH(1)t ⊗ exp −iH(2)t . (2.24) For the moment, we assume that both hains have equal length N , but it will be ome lear in Se tion 2.9 that this is not a requirement of our s heme. All other assumptions remain as in the �rst part of the hapter. Initially, Ali e en odes the state as α |0, 1〉+ β |1, 0〉 . (2.25) This is a superposition of an ex itation in the �rst qubit of the �rst hain and an 2 Dual Rail en oding Figure 2.6: Two disordered quantum hains inter onne ting A and B. Control of the systems is only possible at the two qubits of either end. ex itation in the �rst qubit of the se ond hain. The state will evolve into {αgn,1(t) |0, n〉+ βfn,1(t) |n, 0〉} , (2.26) fn,1(t) ≡ 〈n, 0 |U(t)| 1, 0〉 (2.27) gn,1(t) ≡ 〈0, n |U(t)| 0, 1〉 . (2.28) In Se tion 2.2, these fun tions were identi al. For di�ering hains this is no longer the ase. We may, however, �nd a time t1 su h that the modulus of their amplitudes at the last spins are the same (see Fig. 2.7), gN,1(t1) = e iφ1fN,1(t1). (2.29) At this time, the state (2.26) an be written as {αgn,1(t1) |0, n〉+ βfn,1(t1) |n, 0〉}+ fN,1(t1) eiφ1α |0,N 〉+ β |N, 0〉 . (2.30) Bob de odes the state by applying a CNOT gate on his two qubits, with the �rst qubit 2 Dual Rail en oding 0 5 10 15 20 25 Time [1/J] |fN,1(t)| |gN,1(t)| Figure 2.7: The absolute values of the transition amplitudes fN,1(t) and gN,1(t) for two Heisenberg hains of length N = 10. The ouplings strengths of both hains were hosen randomly from the interval [0.8J, 1.2J ] . The ir les show times where Bob an perform measurements without gaining information on α and β. as the ontrol bit. The state thereafter is {αgn,1(t1) |0, n〉+ βfn,1(t1) |n, 0〉}+ fN,1(t1) eiφ1α |0〉(1) + β |N 〉(1) ⊗ |N 〉(2) . (2.31) Bob then measures his se ond qubit. Depending on the out ome of this measurement, the systems will either be in the state {αgn,1(t1) |0, n〉+ βfn,1(t1) |n, 0〉} (2.32) or in eiφ1α |0〉(1) + β |N〉(1) ⊗ |N 〉(2) , (2.33) where p1 = 1−|fN,1(t1)|2 = 1−|gN,1(t1)|2 is the probability that Bob has not re eived the state. The state (2.33) orresponds to the orre tly transferred state with a known phase error (whi h an be orre ted by Bob using a simple phase gate). If Bob �nds the system in the state (2.32), the transfer has been unsu essful, but the information is 2 Dual Rail en oding still in the hain. We thus see that on lusive transfer is still possible with randomly oupled hains as long as the requirement (2.29) is met. This requirement will be further dis ussed and generalised in the next se tion. 2.8 Arbitrarily perfe t transfer in the presen e of disorder If the transfer was unsu essful, the state (2.32) will evolve further, o�ering Bob further opportunities to re eive Ali e's message. For identi al quantum hains, leads to a su ess for any reasonable Hamiltonian (Se tion 3.6). For di�ering hains, this is not ne essarily the ase, be ause measurements are only allowed at times where the probability amplitude at the end of the hains is equal, and there may be systems where this is never the ase. In this se tion, we will develop a riterion that generalises Eq. (2.29) and allows to he k numeri ally whether a given system is apable of arbitrarily perfe t state transfer. The quantity of interest for on lusive state transfer is the joint probability P (ℓ) that after having he ked l times, Bob still has not re eived the proper state at his end of the hains. Optimally, this should approa h zero if ℓ tends to in�nity. In order to derive an expression for P (ℓ), let us assume that the transfer has been unsu essful for ℓ− 1 times with time intervals tℓ between the the ℓth and the (ℓ− 1)th measurement, and al ulate the probability of failure at the ℓth measurement. In a similar manner, we assume that all the ℓ − 1 measurements have met the requirement of on lusive transfer (that is, Bob's measurements are unbiased with respe t to α and β) and derive the requirement for the ℓth measurement. To al ulate the probability of failure for the ℓth measurement, we need to take into a ount that Bob's measurements disturb the unitary dynami s of the hain. If the state before a measurement with the out ome �failure� is |ψ〉 , the state after the measurement will be Q |ψ〉 , (2.34) where Q is the proje tor Q = I − |0, N 〉 〈0, N | − |N, 0〉 〈N, 0| , (2.35) and pℓ is the probability of failure at the lth measurement. The dynami s of the hain is alternating between unitary and proje tive, su h that the state before the ℓth 2 Dual Rail en oding measurement is given by P (ℓ− 1) {U(tk)Q} {α |1, 0〉+ β |0, 1〉} , (2.36) where P (ℓ− 1) = pk. (2.37) Note that the operators in (2.36) do not ommute and that the time ordering of the produ t (the index k in reases from right to left) is important. The probability that there is an ex itation at the Nth site of either hain is given by P (ℓ− 1) |α|2 |F (ℓ)|2 + |β|2 |G(ℓ)|2 , (2.38) F (ℓ) ≡ 〈N, 0| {U(tk)Q} |1, 0〉 , (2.39) G(ℓ) ≡ 〈0, N | {U(tk)Q} |0, 1〉 . (2.40) Bob's measurements are therefore unbiased with respe t to α and β if and only if |F (ℓ)| = |G(ℓ)| ∀ℓ. (2.41) In this ase, the state an still be transferred on lusively (up to a known phase). The probability of failure at the ℓth measurement is given by pℓ = 1− |F (ℓ)|2 P (ℓ− 1) . (2.42) It is easy (but not very enlightening) to show [103℄ that the ondition (2.41) is equiv- alent to ∥ {U(tk)Q} |1, 0〉 {U(tk)Q} |0, 1〉 ∀ℓ, (2.43) and that the joint probability of failure - if at ea h measurement the above ondition 2 Dual Rail en oding is ful�lled - is simply given by P (ℓ) = {U(tk)Q} |1, 0〉 . (2.44) It may look as if Eq. (2.43) was a ompli ated multi-time ondition for the measuring times tℓ, that be omes in reasingly di� ult to ful�l with a growing number of mea- surements. This is not the ase. If proper measuring times have been found for the �rst ℓ − 1 measurements, a trivial time tℓ that ful�ls Eq. (2.43) is tℓ = 0. In this ase, Bob measures immediately after the (ℓ− 1)th measurement and the probability amplitudes on his ends of the hains will be equal - and zero (a useless measurement). But sin e the left and right hand side of Eq. (2.43) when seen as fun tions of tℓ are both almost-periodi fun tions with initial value zero, it is likely that they interse t many times, unless the system has some spe i� symmetry or the systems are om- pletely di�erent. Note that we do not laim at this point that any pair of hains will be apable of arbitrary perfe t transfer. We will dis uss in the next system how one an he k this for a given system by performing some simple experimental tests. 2.9 Tomography Suppose someone gives you two di�erent experimentally designed spin hains. It may seem from the above that knowledge of the full Hamiltonian of both hains is ne essary to he k how well the system an be used for state transfer. This would be a very di� ult task, be ause we would need a ess to all the spins along the hannel to measure all the parameters of the Hamiltonian. In fa t by expanding the proje tors in Eq. (2.43) one an easily see that the only matrix elements of the evolution operator whi h are relevant for on lusive transfer are fN,1(t) = 〈N, 0|U(t) |1, 0〉 (2.45) fN,N(t) = 〈N, 0|U(t) |N, 0〉 (2.46) gN,1(t) = 〈0, N |U(t) |0, 1〉 (2.47) gN,N (t) = 〈0, N |U(t) |0, N〉 . (2.48) Physi ally, this means that the only relevant properties of the system are the transition amplitudes to arrive at Bob's ends and to stay there. The modulus of fN,1(t) and fN,N (t) an be measured by initialising the system in the states |1, 0〉 and |N, 0〉 and then performing a redu ed density matrix tomography at Bob's site at di�erent times 2 Dual Rail en oding t, and the omplex phase of these fun tions is obtained by initialising the system in (|0, 0〉+ |1, 0〉) / 2 and (|0, 0〉+ |N, 0〉) / 2 instead. In the same way, gN,1(t) and gN,N (t) are obtained. All this an be done in the spirit of minimal ontrol at the sending and re eiving ends of the hain only, and needs to be done only on e. It is interesting to note that the dynami s in the middle part of the hain is not relevant at all. It is a bla k box (see Fig. 2.8) that may involve even ompletely di�erent intera tions, number of spins, et ., as long as the total number of ex itations is onserved. On e the transition amplitudes [Equations (2.45)-(2.48)℄ are known, one Figure 2.8: The relevant properties for on lusive transfer an be determined by mea- suring the response of the two systems at their ends only. an sear h numeri ally for optimised measurement times tℓ using Eq. (2.44) and the ondition from Eq. (2.43). One weakness of the s heme des ribed here is that the times at whi h Bob measures have to be very pre ise, be ause otherwise the measurements will not be unbiased with respe t to α and β. This demand an be relaxed by measuring at times where not only the probability amplitudes are similar, but also their slope (see Fig. 2.7). The omputation of these optimal timings for a given system may be ompli ated, but they only need to be done on e. 2.10 Numeri al Examples In this se tion, we show some numeri al examples for two hains with Heisenberg ouplings J whi h are �u tuating. The Hamiltonians of the hains i = 1, 2 are given 2 Dual Rail en oding H(i) = J(1 + δ(i)n ) X(i)n X n+1 + Y n+1 + Z , (2.49) where δ n are uniformly distributed random numbers from the interval [−∆,∆] . We have onsidered two di�erent ases: in the �rst ase, the δ n are ompletely un orre- lated (i.e. independent for both hains and all sites along the hain). In the se ond ase, we have taken into a ount a spa ial orrelation of the signs of the δ n along ea h of the hains, while still keeping the two hains un orrelated. For both ases, we �nd that arbitrarily perfe t transfer remains possible ex ept for some very rare realisations of the δ Be ause measurements must only be taken at times whi h ful�l the ondition (2.43), and these times usually do not oin ide with the optimal probability of �nding an ex itation at the ends of the hains, it is lear that the probability of failure at ea h measurement will in average be higher than for hains without �u tuations. Therefore, more measurements have to be performed in order to a hieve the same probability of su ess. The pri e for noisy ouplings is thus a longer transmission time and a higher number of gating operations at the re eiving end of the hains. Some averaged values are given in Table 2.1 for the Heisenberg hain with un orrelated oupling �u tuations. ∆ = 0 ∆ = 0.01 ∆ = 0.03 ∆ = 0.05 ∆ = 0.1 377 524 ± 27 694 ± 32 775± 40 1106 ± 248 M 28 43± 3 58± 3 65± 4 110 ± 25 Table 2.1: The total time t and the number of measurements M needed to a hieve a probability of su ess of 99% for di�erent �u tuation strengths ∆ (un orrelated ase). Given is the statisti al mean and the standard deviation. The length of the hain is N = 20 and the number of random samples is 10. For strong �u tuations ∆ = 0.1, we also found parti ular samples where the su ess probability ould not be a hieved within the time range sear hed by the algorithm. For the ase where the signs of the δ n are orrelated, we have used the same model as in [99℄, introdu ing the parameter c su h that δ(i)n δ n−1 > 0 with propability c, (2.50) δ(i)n δ n−1 < 0 with propability 1− c. (2.51) 2 Dual Rail en oding For c = 1 (c = 0) this orresponds to the ase where the signs of the ouplings are ompletely orrelated (anti- orrelated). For c = 0.5 one re overs the ase of un orrelated ouplings. We an see from the numeri al results in Table 2.2 that arbitrarily perfe t transfer is possible for the whole range of c. c = 0 c = 0.1 c = 0.3 c = 0.7 c = 0.9 c = 1 666± 20 725± 32 755± 41 797± 35 882± 83 714± 41 M 256± 2 62± 3 65± 4 67± 4 77± 7 60± 4 Table 2.2: The total time t and the number of measurements M needed to a hieve a probability of su ess of 99% for di�erent orrelations c between the ouplings [see Eq. (2.50) and Eq. (2.51)℄. Given is the statisti al mean and the standard deviation for a �u tuation strength of ∆ = 0.05. The length of the hain is N = 20 and the number of random samples is 20. For ∆ = 0, we know from Se tion 2.4 that the time to transfer a state with proba- bility of failure P s ales as t(P ) = 0.33J−1N1.6 |lnP | . (2.52) If we want to obtain a similar formula in the presen e of noise, we an perform a �t to the exa t numeri al data. For un orrelated �u tuations of ∆ = 0.05, this is shown in Fig. 2.9. The best �t is given by t(P ) = 0.2J−1N1.9 |lnP | . (2.53) We on lude that weak �u tuations (say up to 5%) in the oupling strengths do not deteriorate the performan e of our s heme mu h for the hain lengths onsidered. Both the transmission time and the number of measurements raise, but still in a reasonable way [ f. Table 2.1 and Fig. 2.9℄. For larger �u tuations, the s heme is still appli able in prin iple, but the amount of junk (i.e. hains not apable of arbitrary perfe t transfer) may get too large. Note that we have onsidered the ase where the �u tuations δin are onstant in time. This is a reasonable assumption if the dynami �u tuations (e.g. those arising from thermal noise) an be negle ted with respe t to the onstant �u tuations (e.g. those arising from manufa turing errors). If the �u tuations were varying with time, the tomography measurements in Se . 2.9 would involve a time-average, and Bob would not measure exa tly at the orre t times. The transferred state (2.33) would then be a�e ted by both phase and amplitude noise. 2 Dual Rail en oding 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1000 1500 2000 2500 3000 3500 4000 4500 5000 Transfer Time [1/J] Numerical data Chain length Joint probability of failure Transfer Time [1/J] Figure 2.9: Time t needed to transfer a state with a given joint probability of failure P a ross a hain of length N with un orrelated �u tuations of ∆ = 0.05. The points denote numeri al data averaged over 100 realisations, and the �t is given by Eq. (2.53). This �gure should be ompared with Fig. 2.4 where ∆ = 0. 2.11 Coupled hains Let us look at the ondition for on lusive transfer in the more general s enario indi- ated by Fig. 2.10: Ali e and Bob have a bla k box a ting as an amplitude damping hannel in the following way. It has two inputs and two outputs. If Ali e puts in state in the dual rail, |ψ〉 = α|01〉 + β|10〉, (2.54) where α and β are arbitrary and unknown normalised amplitudes, then the output at Bob is given by p|φ〉〈φ| + (1− p) |00〉〈00|, (2.55) with a normalised �su ess� state |φ〉 = 1√ αf |01〉 + βg|10〉 + αf̃ |10〉 + βg̃|01〉 . (2.56) 2 Dual Rail en oding This bla k box des ribes the behaviour of an arbitrarily oupled qubit system that onserves the number of ex itations and that is initialised in the all zero state, in luding parallel un oupled hains, and oupled hains. Figure 2.10: Most general setting for on lusive transfer: A bla k box with two inputs and two outputs, a ting as an amplitude damping hannel de�ned by Eqs. (2.54) and (2.55) From the normalisation of |φ〉 it follows that p = p(α, β) = |αf + βg̃|2 + ∣βg + αf̃ . (2.57) We are interested in on lusive transfer: by measuring the observable |00〉〈00| the Bob an proje t the output onto either the failure state |00〉 or |φ〉. This is learly possible, but the question is if the output |φ〉 and the input |ψ〉 are related by a unitary operation. If Bob is able to re over the full information that Ali e sent, then p(α, β) must be independent of α and β (otherwise, some information on these amplitudes ould be obtained by the measurement already, whi h ontradi ts the non- loning theorem [2℄). This implies that p(1, 0) = p(0, 1), i.e. |f |2 + = |g̃|2 + |g|2 . (2.58) Be ause |f + g̃|2 + 1 ∣g + f̃ (2.59) = p(1, 0) + Re f∗g̃ + gf̃∗ (2.60) it also implies that f∗g̃ + gf̃∗ = 0. (2.61) 2 Dual Rail en oding Using the same tri k for p we get that Im f∗g̃ + gf̃∗ = 0 and therefore f∗g̃ + gf̃∗ = 0. (2.62) If we write |ψ〉 = U |φ〉 we get , (2.63) whi h is a unitary operator if Eq. (2.58) and (2.62) hold. We thus ome to the on lusion that on lusive transfer with the bla k box de�ned above is possible if and only if the probability p is independent of α and β. It is interesting to note that a verti al mirror symmetry of the system does not guarantee this. A ounterexample is sket hed in Fig. 2.11: learly the initial (�dark�) state |01〉 − |10〉 does not evolve, whereas |01〉 + |10〉 does. Hen e the probability must depend on α and β. A trivial ase where on lusive transfer works is given by two un oupled hains, at times where |f |2 = |g|2. This was dis ussed in Se t. 2.8. A non-trivial example is given by the oupled system sket hed in Fig. 2.12. This an be seen by splitting the Hamiltonian in a horizontal and verti al omponent, H = Hv +Hz. (2.64) By applying HvHz and HzHv on single-ex itation states it is easily he ked that they ommute in the �rst ex itation se tor (this is not longer true in higher se tors). Sin e the probability is independent of α and β in the un oupled ase it must also be true in the oupled ase (a rotation in the subspa e {|01〉, |10〉} does not harm). Figure 2.11: A simple ounterexample for a verti ally symmetri system where dual rail en oding is not possible. The bla k lines represent ex hange ouplings. A �nal remark - as Ali e and Bob alway only deal with the states {|00〉, |10〉, |01〉} it is obvious that the en oding used in this hapter is really living on qutrits. In some sense it would be more natural to onsider permanently oupled systems of qutrits, 2 Dual Rail en oding Figure 2.12: An example for a verti ally symmetri system where dual rail en oding is possible. The bla k lines represent ex hange ouplings of equal strength. su h as SU(3) hains [123,102,124,125℄. The �rst level of the qutrit |0〉 is then used as a marker for �no information here�, whereas the information is en oded in the states |1〉 and |2〉. One would have to ensure that there is no transition between |0〉 and |1〉, |2〉, and that the system is initialised in the all zero state. 2.12 Con lusion In on lusion, we have presented a simple s heme for on lusive and arbitrarily per- fe t quantum state transfer. To a hieve this, two parallel spin hains (individually amplitude damping hannels) have been used as one amplitude delaying hannel. We have shown that our s heme is more robust to de oheren e and imperfe t timing than the single hain s hemes. We have also shown that the s heme is appli able to dis- ordered and oupled hains. The s heme an be used as a way of improving any of the other s hemes from the introdu tion. For instan e, one may try to engineer the ouplings to have a very high probability of su ess already at the �rst measurement, and use further measurements to ompensate the errors of implementing the orre t values for the ouplings. We remark that the dual rail proto ol is unrelated to error �ltration [126℄ where parallel hannels are used for �ltering out environmental e�e ts on �ying qubits, whereas the purpose of the dual rail proto ol is to ensure the ar- rival of the qubit. Indeed one ould ombine both proto ols to send a qubit on say four rails to ensure the arrival and �lter errors. Finally, we note that in some re ent work [80℄ it was shown that our en oding an be used to perform quantum gates while the state is transferred, and that it an in rease the onvergen e speed if one performs measurements at intermediate positions [110, 127℄. 3 Multi Rail en oding 3.1 Introdu tion In quantum information theory the rate R of transferred qubits per hannel is an important e� ien y parameter [70℄. Therefore one question that naturally arises is whether or not there is any spe ial meaning in the 1/2 value of R a hieved in the dual rail proto ol of the last hapter. We will show now that this is not the ase, be ause there is a way of bringing R arbitrarily lose to 1 by onsidering multi rail en odings. Furthermore, in Se tion 2.3 it was still left open for whi h Hamiltonians the probability of su ess an be made arbitrarily lose to 1. Here, we give a su� ient and easily attainable ondition for a hieving this goal. This hapter is organised as follows: the model and the notation are introdu ed in Se . 3.2. The e� ien y and the �delity of the proto ol are dis ussed in Se . 3.3 and in Se . 3.4, respe tively. Finally in Se . 3.5 we prove a theorem whi h provides us with a su� ient ondition for a hieving e� ient and perfe t state transfer in quantum hains. 3.2 The model Assume that the two ommuni ating parties operate on M independent (i.e. non intera ting) opies of the hain. This is quite a ommon attitude in quantum informa- tion theory [70℄ where su essive uses of a memoryless hannel are formally des ribed by introdu ing many parallel opies of the hannel (see [54℄ for a dis ussion on the possibility of applying this formal des ription to quantum hain models). Moreover for the ase at hand the assumption of Ali e and Bob dealing with �real� parallel hains seems reasonable also from a pra ti al point of view [48, 49℄. The idea is to use these opies to improve the overall �delity of the ommuni ation. As usual, we assume Ali e and Bob to ontrol respe tively the �rst and last qubit of ea h hain (see Fig. 3.1). By preparing any superposition of her spins Ali e an in prin iple transfer up toM logi al qubits. However, in order to improve the ommuni ation �delity the two parties will �nd it more onvenient to redundantly en ode only a small number (say Q(M) 6M) 3 Multi Rail en oding Length N Figure 3.1: S hemati of the system: Ali e and Bob operateM hains, ea h ontaining N spins. The spins belonging to the same hain intera t through the Hamiltonian H whi h a ounts for the transmission of the signal in the system. Spins of di�erent hains do not intera t. Ali e en odes the information in the �rst spins of the hains by applying unitary transformations to her qubits. Bob re overs the message in the last spins of the hains by performing joint measurements. of logi al qubits in the M spins. By adopting these strategies Ali e and Bob are ef- fe tively sa ri� ing the e� ien y R(M) = Q(M)/M of their ommuni ation line in order to in rease its �delity. This is typi al of any ommuni ation s heme and it is analogous to what happens in quantum error orre tion theory, where a single logi al qubit is stored in many physi al qubits. In the last hapter we have seen that for M = 2 it is possible to a hieve perfe t state transfer of a single logi al qubit with an e� ien y equal to 1/2. Here we will generalise su h result by proving that there exist an optimal en oding-de oding strategy whi h asymptoti ally allows to a hieve perfe t state transfer and optimal e� ien y, i.e. R(M) = 1 . (3.1) Our strategy requires Ali e to prepare superpositions of the M hains where ∼ M/2 of them have a single ex itation in the �rst lo ation while the remaining are in |0〉. Sin e in the limit M >> 1 the number of qubit transmitted is log ≈ M , this ar hite ture guarantees optimal e� ien y (3.1). On the other hand, our proto ol requires Bob to perform olle tive measurements on his spins to determine if all the ∼ M/2 ex itations Ali e is transmitting arrived at his lo ation. We will prove that 3 Multi Rail en oding by repeating these dete tions many times, Bob is able to re over the messages with asymptoti ally perfe t �delity. Before beginning the analysis let us introdu e some notation. The following de�- nitions look more ompli ated than they really are; unfortunately we need them to arefully de�ne the states that Ali e uses for en oding the information. In order to distinguish the M di�erent hains we introdu e the label m = 1, · · · ,M : in this for- malism |n〉m represents the state of m-th hain with a single ex itation in the n-th spin. In the following we will be interested in those on�gurations of the whole system where K hains have a single ex itation while the remaining M −K are in |0〉, as in the ase |1〉1 ⊗ |1〉2 · · · ⊗ |1〉K ⊗ |0〉K+1 · · · ⊗ |0〉M (3.2) where for instan e the �rst K hains have an ex itation in the �rst hain lo ation. Another more general example is given in Fig. 3.2. The omplete hara terisation of these ve tors is obtained by spe ifying i) whi h hains possess a single ex itation and ii) where these ex itations are lo ated horizontally along the hains. In answering to the point i) we introdu e the K-element subsets Sℓ, omposed by the labels of those hains that ontain an ex itation. Ea h of these subsets Sℓ orresponds to a subspa e of the Hilbert spa e H(Sℓ) with a dimension NK . The total number of su h subsets is equal to the binomial oe� ient , whi h ounts the number of possibilities in whi h K obje ts (ex itations) an be distributed among M parties (parallel hains). In parti ular for any ℓ = 1, · · · , the ℓ-th subset Sℓ will be spe i�ed by assigning its K elements, i.e. Sℓ ≡ {m(ℓ)1 , · · · ,m K } with m j ∈ {1, · · · ,M} and m j < m for all j = 1, · · · ,K. To hara terise the lo ation of the ex itations, point ii), we will introdu e instead theK-dimensional ve tors ~n ≡ (n1, · · · , nK) where nj ∈ {1, · · · , N}. We an then de�ne |~n; ℓ〉〉 ≡ |nj〉m(ℓ)j m′∈Sℓ |0〉m′ , (3.3) where Sℓ is the omplementary of Sℓ to the whole set of hains. The state (3.3) represents a on�guration where the j-th hain of the subset Sℓ is in |nj〉 while the hains that do not belong to Sℓ are in |0〉 (see Fig. 3.2 for an expli it example). The kets |~n; ℓ〉〉 are a natural generalisation of the states |n〉1 ⊗ |0〉2 and |0〉1⊗|n〉2 whi h were used for the dual rail en oding. They are useful for our purposes be ause they are mutually orthogonal, i.e. 〈〈~n; ℓ|~n′; ℓ′〉〉 = δℓℓ′ δ~n~n′ , (3.4) 3 Multi Rail en oding Length 6 Figure 3.2: Example of our notation for M = 5 hains of length N = 6 with K = 2 ex itations. The state above, given by |0〉1 ⊗ |3〉2 ⊗ |0〉3 ⊗ |1〉4 ⊗ |0〉5, has ex itations in the hains m1 = 2 and m2 = 4 at the horizontal position n1 = 3 and n2 = 1. It is in the Hilbert spa e H(S6) orresponding to the subset S6 = {2, 4} (assuming that the sets Sℓ are ordered in a anoni al way, i.e. S1 = {1, 2}, S2 = {1, 3} and so on) and will be written as |(3, 1); 6〉〉. There are = 10 di�erent sets Sℓ and the number of qubits one an transfer using these states is log2 10 ≈ 3. The e� ien y is thus given by R ≈ 3/5 whi h is already bigger than in the dual rail s heme. and their time evolution under the Hamiltonian does not depend on ℓ. Among the ve tors (3.3) those where all the K ex itations are lo ated at the beginning of the Sℓ hains play an important role in our analysis. Here ~n = ~1 ≡ (1, · · · , 1) and we an write |~1; ℓ〉〉 ≡ m′∈Sℓ |0〉m′ . (3.5) A ording to Eq. (3.4), for ℓ = 1, · · · , these states form orthonormal set of elements. Analogously by hoosing ~n = ~N ≡ (N, · · · , N) we obtain the orthonormal set of ve tors | ~N ; ℓ〉〉 ≡ |N 〉m m′∈Sℓ |0〉m′ , (3.6) where all the K ex itations are lo ated at the end of the hains. 3 Multi Rail en oding 3.3 E� ient en oding If all theM hains of the system are originally in |0〉, the ve tors (3.5) an be prepared by Ali e by lo ally operating on her spins. Moreover sin e these ve tors span a dimensional subspa e, Ali e an en ode in the hain Q(M,K) = log2 qubits of logi al information by preparing the superpositions, |Φ〉〉 = Aℓ |~1; ℓ〉〉 , (3.7) with Aℓ omplex oe� ients. The e� ien y of su h en oding is hen e R(M,K) = log2 ( whi h maximised with respe t to K gives, R(M) = for M even (M−1)/2 for M odd . (3.8) The Stirling approximation an then be used to prove that this en oding is asymptot- i ally e� ient (3.1) in the limit of large M , e.g. ≈ log2 (M/2)M =M. (3.9) Note that already for M = 5 the en oding is more e� ient ( f. Fig. 3.2) than in the dual rail en oding. In the remaining of the hapter we show that the en oding (3.7) provides perfe t state transfer by allowing Bob to perform joint measurements at his end of the hains. 3.4 Perfe t transfer Sin e the M hains do not intera t with ea h other and possess the same free Hamil- tonian H, the unitary evolution of the whole system is des ribed by U(t) ≡ ⊗mum(t), with um(t) being the operator a ting on the m-th hain. The time evolved of the input |~1; ℓ〉〉 of Eq. (3.5) is thus equal to U(t)|~1; ℓ〉〉 = F [~n,~1; t] |~n; ℓ〉〉 , (3.10) where the sum is performed for all nj = 1, · · · , N and F [~n, ~n′; t] ≡ fn1,n′1(t) · · · fnK ,n′K (t) , (3.11) 3 Multi Rail en oding is a quantity whi h does not depend on ℓ. In Eq. (3.10) the term ~n = ~N orresponds to having all the K ex itations in the last lo ations of the hains. We an thus write U(t)|~1; ℓ〉〉 = γ1(t)| ~N ; ℓ〉〉+ 1− |γ1(t)|2 |ξ(t); ℓ〉〉 , (3.12) where γ1(t) ≡ 〈〈 ~N ; ℓ|U(t)|~1; ℓ〉〉 = F [ ~N,~1; t] (3.13) is the probability amplitude that all the K ex itation of |~1; ℓ〉〉 arrive at the end of the hains, and |ξ(t); ℓ〉〉 ≡ ~n 6= ~N F1[~n,~1; t] |~n; ℓ〉〉 , (3.14) F1[~n,~1; t] ≡ F [~n,~1; t] 1− |γ1(t)|2 , (3.15) is a superposition of terms where the number of ex itations arrived to the end of the ommuni ation line is stri tly less then K. It is worth noti ing that Eq. (3.4) yields the following relations, 〈〈 ~N ; ℓ|ξ(t); ℓ′〉〉 = 0, 〈〈ξ(t); ℓ|ξ(t); ℓ′〉〉 = δℓℓ′ , (3.16) whi h shows that {||ξ(t); ℓ〉〉} is an orthonormal set of ve tors whi h spans a subspa e orthogonal to the states | ~N ; ℓ〉〉. The time evolution of the input state (3.7) follows by linearity from Eq. (3.12), i.e. |Φ(t)〉〉 = γ1(t) |Ψ〉〉+ 1− |γ1(t)|2 |Ψ(t)〉〉 , (3.17) |Ψ(t)〉〉 ≡ Aℓ |ξ(t); ℓ〉〉 , |Ψ〉〉 ≡ Aℓ | ~N ; ℓ〉〉 . (3.18) The ve tors |Ψ〉〉 and |Ψ(t)〉〉 are unitary transformations of the input message (3.7) where the orthonormal set {|~1; ℓ〉〉} has been rotated into {| ~N ; ℓ〉〉} and {|ξ(t); ℓ〉〉} respe tively. Moreover |Ψ〉〉 is the on�guration we need to have for perfe t state 3 Multi Rail en oding transfer at the end of the hain. In fa t it is obtained from the input message (3.7) by repla ing the omponents |1〉 (ex itation in the �rst spin) with |N 〉 (ex itation in the last spin). From Eq. (3.16) we know that |Ψ〉〉 and |Ψ(t)〉〉 are orthogonal. This property helps Bob to re over the message |Ψ〉〉 from |Φ(t)〉〉: he only needs to perform a olle tive measurement on the M spins he is ontrolling to establish if there are K or less ex itations in those lo ations. The above is learly a proje tive measurement that an be performed without destroying the quantum oheren e asso iated with the oe� ients Aℓ. Formally this an des ribed by introdu ing the observable Θ ≡ 1− | ~N ; ℓ〉〉〈〈 ~N ; ℓ| . (3.19) A single measurement of Θ on |Φ(t1)〉〉 yields the out ome 0 with probability p1 ≡ |γ1(t1)|2, and the out ome +1 with probability 1 − p1. In the �rst ase the system will be proje ted in |Ψ〉〉 and Bob will get the message. In the se ond ase instead the state of the system will be ome |Ψ(t1)〉〉. Already at this stage the two ommuni ating parties have a su ess probability equal to p1. Moreover, as in the dual rail proto ol, the hannels have been transformed into a quantum erasure hannel [113℄ where the re eiver knows if the transfer was su essful. Just like the dual rail en oding, this en oding an be used as a simple entanglement puri� ation method in quantum hain transfer (see end of Se tion 2.2). The rate of entanglement that an be distilled is given by ∣F [ ~N,~1; t] = R(M)p(t)⌊M/2⌋, (3.20) where we used Eq. (3.11) and p(t) ≡ |fN,1(t)|2 . As we an see, in reasing M on one hand in reases R(M), but on the other hand de reases the fa tor p(t)⌊M/2⌋. Its maximum with respe t toM gives us a lower bound of the entanglement of distillation for a single spin hain, as shown in Fig. 1.11. We an also see that it be omes worth en oding on more than three hains for on lusive transfer only when p(t) > 0.8. Consider now what happens when Bob fails to get the right answer from the mea- surement. The state on whi h the hains is proje ted is expli itly given by |Ψ(t1)〉〉 = ~n 6= ~N F1[~n,~1; t1] Aℓ|~n; ℓ〉〉 . (3.21) Let us now onsider the evolution of this state for another time interval t2. By repeat- ing the same analysis given above we obtain an expression similar to (3.17), i.e. |Φ(t2, t1)〉〉 = γ2 |Ψ〉〉+ 1− |γ2|2 |Ψ(t2, t1)〉〉 , (3.22) 3 Multi Rail en oding where now the probability amplitude of getting all ex itation in the N -th lo ations is des ribed by ~n 6= ~N F [ ~N,~n; t2] F1[~n,~1; t1]. (3.23) In this ase |Ψ(t)〉〉 is repla ed by |Ψ(t2, t1)〉〉 = Aℓ |ξ(t2, t1); ℓ〉〉 , (3.24) |ξ(t2, t1); ℓ〉〉 = ~n 6= ~N F2[~n,~1; t2, t1]|~n; ℓ〉〉, (3.25) and F2 de�ned as in Eq. (3.27) (see below). In other words, the state |Φ(t2, t1)〉〉 an be obtained from Eq. (3.17) by repla ing γ1 and F1 with γ2 and F2. Bob an hen e try to use the same strategy he used at time t1: i.e. he will he k whether or not his M qubits ontain K ex itations. With ( onditional) probability p2 ≡ |γ2|2 he will get a positive answer and his quantum register will be proje ted in the state |Ψ〉〉 of Eq. (3.18). Otherwise he will let the system evolve for another time interval t3 and repeat the proto ol. By reiterating the above analysis it is possible to give a re ursive expression for the onditional probability of su ess pq ≡ |γq|2 after q − 1 su essive unsu essful steps. The quantity γq is the analogue of γ2 and γ1 of Eqs. (3.13) and (3.22). It is given by ~n 6= ~N F [ ~N,~n; tq] Fq−1[~n,~1, tq−1, · · · , t1] , (3.26) where Fq−1[~n,~1; tq−1, · · · , t1] (3.27) ~n′ 6= ~N F [ ~N,~n′; tq−1] 1− |γq−1|2 Fq−2[~n ′,~1; tq−2, · · · , t1] and F1[~n,~1, t] is given by Eq. (3.15). In these equations tq, · · · , t1 are the time-intervals that o urred between the various proto ol steps. Analogously the onditional proba- bility of failure at the step q is equal to 1−pq. The probability of having j−1 failures and a su ess at the step j-th an thus be expressed as π(j) = pj(1− pj−1)(1 − pj−2) · · · (1− p1) , (3.28) 3 Multi Rail en oding while the total probability of su ess after q steps is obtained by the sum of π(j) for all j = 1, · · · , q, i.e. π(j) . (3.29) Sin e pj > 0, Eq. (3.29) is a monotoni fun tion of q. As a matter of fa t in the next se tion we prove that under a very general hypothesis on the system Hamiltonian, the probability of su ess Pq onverges to 1 in the limit of q → ∞. This means that by repeating many times the olle tive measure des ribed by Θ Bob is guaranteed to get, sooner or later, the answer 0 and hen e the message Ali e sent to him. In other words our proto ol allows perfe t state transfer in the limit of repetitive olle tive measures. Noti e that the above analysis applies for all lasses of subsets Sℓ. The only di�eren e between di�erent hoi es of K is in the velo ity of the onvergen e of Pq → 1. In any ase, by hoosing K ∼ M/2 Ali e and Bob an a hieve perfe t �delity and optimal e� ien y. 3.5 Convergen e theorem Theorem 3.1 (Arbitrarly perfe t transfer) If there is no eigenve tor |em〉 of the quantum hain Hamiltonian H whi h is orthogonal to |N 〉, then there is a hoi e of the times intervals tq, tq−1, · · · , t1 su h that the �delity onverges to 1 as q → ∞. Before proving this Theorem, let us give an intuitive reasoning for the onvergen e. The unitary evolution an be thought of of a rotation in some abstra t spa e, while the measurement orresponds to a proje tion. The dynami s of the system is then represented by alternating rotations and proje tions. In general this will de rease the norm of ea h ve tor to null, unless the rotation axis is the same as the proje tion axis. Proof The state of the system at a time interval of tq after the (q− 1)-th failure an be expressed in ompa t form as follows |Φ(tq, · · · , t1)〉〉 = U(tq)ΘU(tq−1)Θ · · ·U(t1)Θ|Φ〉〉 (1− pq−1) · · · (1− p1) (3.30) with U(t) the unitary time evolution generated by the system Hamiltonian, and with 3 Multi Rail en oding Θ the proje tion de�ned in Eq. (3.19). One an verify for instan e that for q = 2, the above equation oin ides with Eq. (3.22). [For q = 1 this is just (3.17) evaluated at time t1℄. By de�nition the onditional probability of su ess at step q-th is equal to pq ≡ |〈〈Ψ|Φ(tq, · · · , t1)〉〉|2. (3.31) Therefore, Eq. (3.28) yields π(q) = |〈〈Ψ|U(tq)ΘU(tq−1)Θ · · ·U(t1)Θ|Φ〉〉|2 (3.32) = |〈〈 ~N ; ℓ|U(tq)ΘU(tq−1)Θ · · ·U(t1)Θ|~1; ℓ〉〉|2 , where the se ond identity stems from the fa t that, a ording to Eq. (3.4), U(t)Θ preserves the orthogonality relation among states |~n; ℓ〉〉 with distin t values of ℓ. In analogy to the ases of Eqs. (3.11) and (3.13), the se ond identity of (3.32) establishes that π(q) an be omputed by onsidering the transfer of the input |~1; ℓ〉〉 for arbitrary ℓ. The expression (3.32) an be further simpli�ed by noti ing that for a given ℓ the hains of the subset Sℓ ontribute with a unitary fa tor to π(q) and an be thus negle ted (a ording to (3.5) they are prepared in |0〉 and do not evolve under U(t)Θ). Identify |~1〉〉ℓ and | ~N 〉〉ℓ with the omponents of |~1; ℓ〉〉 and | ~N ; ℓ〉〉 relative to the hains belonging to the subset Sℓ. In this notation we an rewrite Eq. (3.32) as π(q) = |ℓ〈〈 ~N |Uℓ(tq)Θℓ · · ·Uℓ(t1)Θℓ|~1〉〉ℓ|2 , (3.33) where Θℓ = 1 − | ~N 〉〉ℓ〈〈 ~N | and Uℓ(t) is the unitary operator ⊗m∈Sℓum(t) whi h de- s ribes the time evolution of the hains of Sℓ. To prove that there exist suitable hoi es of tℓ su h that the series (3.29) onverges to 1 it is su� ient to onsider the ase tℓ = t > 0 for all j = 1, · · · , q: this is equivalent to sele ting de oding proto- ols with onstant measuring intervals. By introdu ing the operator Tℓ ≡ Uℓ(t)Θℓ, Eq. (3.33) be omes thus π(q) = |ℓ〈〈 ~N | (Tℓ)q|~1〉〉ℓ|2 (3.34) =ℓ〈〈~1|(T †ℓ ) q| ~N 〉〉ℓ〈〈 ~N | (Tℓ)q|~1〉〉ℓ = w(q)− w(q + 1) , where w(j) ≡ℓ 〈〈~1|(T †ℓ ) j (Tℓ) j |~1〉〉ℓ = ‖(Tℓ)j |~1〉〉ℓ‖2 , (3.35) 3 Multi Rail en oding is the norm of the ve tor (Tℓ) j |~1〉〉ℓ. Substituting Eq. (3.34) in Eq. (3.29) yields [w(j) − w(j + 1)] = 1− w(q + 1) (3.36) where the property w(1) = ℓ〈〈~1|Θℓ|~1〉〉ℓ = 1 was employed. Proving the thesis is hen e equivalent to prove that for q → ∞ the su ession w(q) nulli�es. This last relation an be studied using properties of power bounded matri es [128℄. In fa t, by introdu ing the norm of the operator (Tℓ) we have, w(q) = ‖(Tℓ)q|~1〉〉ℓ‖2 6 ‖(Tℓ)q‖2 6 c 1 + ρ(Tℓ) (3.37) where c is a positive onstant whi h does not depend on q (if S is the similarity transformation that puts Tℓ into the Jordan anoni al form, i.e. J = S −1TℓS, then c is given expli itly by c = ‖S‖ ‖S−1‖) and where ρ(Tℓ) is the spe tral radius of Tℓ, i.e. the eigenvalue of Tℓ with maximum absolute value (N.B. even when Tℓ is not diagonalisable this is a well de�ned quantity). Equation (3.37) shows that ρ(Tℓ) < 1 is a su� ient ondition for w(q) → 0. In our ase we note that, given any normalised eigenve tor |λ〉〉ℓ of Tℓ with eigenvalue λ we have |λ| = ‖Tℓ|λ〉〉ℓ‖ = ‖Θℓ|λ〉〉ℓ‖ 6 1 , (3.38) where the inequality follows from the fa t that Θℓ is a proje tor. Noti e that in Eq. (3.38) the identity holds only if |λ〉〉 is also an eigenve tor of Θℓ with eigenvalue +1, i.e. only if |λ〉〉ℓ is orthogonal to | ~N 〉〉ℓ. By de�nition |λ〉〉ℓ is eigenve tor Tℓ = Uℓ(t)Θℓ: therefore the only possibility to have the equality in Eq. (3.38) is that i) |λ〉〉ℓ is an eigenve tor of Uℓ(t) (i.e. an eigenve tor of the Hamiltonian 1 Htotℓ of the hain subset Sℓ) and ii) it is orthogonal to | ~N 〉〉ℓ. By negating the above statement we get a su� ient ondition for the thesis. Namely, if all the eigenve tors | ~E〉〉ℓ of Htotℓ are not orthogonal to | ~N 〉〉ℓ than the absolute values of the eigenvalues λ of Tℓ are stri tly smaller than 1 whi h implies ρ(Tℓ) < 1 and hen e the thesis. Sin e the Sℓ hannels are identi al and do not intera t, the eigenve tors | ~E〉〉ℓ ≡ m∈Sℓ |em〉m are tensor produ t of eigenve tors |em〉 of the single hain Hamiltonians H. Therefore the Noti e that stri tly speaking the eigenve tors of the Hamiltonian are not the same as those of the time evolution operators. The latter still an have evolution times at whi h additional degenera y an in rease the set of eigenstates. A trivial example is given for t = 0 where all states be ome eigenstates. But it is always possible to �nd times t at whi h the eigenstates of U(t) oin ide with those of H . 3 Multi Rail en oding su� ient ondition be omes ℓ〈〈 ~E| ~N 〉〉ℓ = m〈N |em〉m 6= 0 , (3.39) whi h an be satis�ed only if 〈N |em〉 6= 0 for all eigenve tors |em〉 of the single hain Hamiltonian H. � Remark 3.1 While we have proven here that for equal time intervals the probability of su ess is onverging to unity, in pra ti e one may use optimal measuring time intervals ti for a faster transfer (see also Se tion 2.4). We also point out that timing errors may delay the transfer, but will not de rease its �delity. 3.6 Quantum hains with nearest-neighbour intera tions It is worth noti ing that Eq. (3.39) is a very weak ondition, be ause eigenstates of Hamiltonians are typi ally entangled. For instan e, it holds for open hains with nearest neighbour-intera tions: Theorem 3.2 (Multi rail proto ol) Let H be the Hamiltonian of an open nearest-neighbour quantum hain that onserves the number of ex itations. If there is a time t su h that f1,N (t) 6= 0 (i.e. the Hamiltonian is apable of transport be- tween Ali e and Bob) then the state transfer an be made arbitrarily perfe t by using the multi rail proto ol. Proof We show by ontradi tion that the riterion of Theorem 3.1 is ful�lled. As- sume there exists a normalised eigenve tor |e〉 of the single hain Hamiltonian H su h 〈N |e〉 = 0. (3.40) Be ause |e〉 is an eigenstate, we an on lude that also 〈e |H|N 〉 = 0. (3.41) If we a t with the Hamiltonian on the ket in Eq. (3.41) we may get some term propor- tional to 〈e|N 〉 ( orresponding to an Ising-like intera tion) and some part proportional to 〈e|N − 1〉 ( orresponding to a hopping term; if this term did not exist, then learly 3 Multi Rail en oding f1,N (t) = 0 for all times). We an thus on lude that 〈e|N − 1〉 = 0. (3.42) Note that for a losed hain, e.g. a ring, this need not be the ase, be ause then also a term proportional to 〈e|N + 1〉 = 〈e|1〉 would o ur. If we insert the Hamiltonian into Eq. (3.42) again, we an use the same reasoning to see that 〈e|N − 2〉 = · · · = 〈e|1〉 = 0 (3.43) and hen e |e〉 = 0, whi h is a ontradi tion to |e〉 being normalised. � 3.7 Comparison with Dual Rail As we have seen above, the Multi Rail proto ol allows us in prin iple to rea h in prin iple a rate arbitrarily lose to one. However for a fair omparison with the Dual Rail proto ol, we should also take into a ount the time-s ale of the transfer. For the on lusive transfer of entanglement, we have seen in Se tion 3.4 that only for hains whi h have a su ess probability higher than p(t) = 0.8 it is worth en oding on more than three rails. The reason is that if the probability of su ess for a single ex itation is p, then the probability of su ess for ⌊M/2⌋ ex itations on on M parallel hains is lowered to p⌊M/2⌋. The proto ol for three rails is always more e� ient than on two, as still only one ex itation is being used, but three omplex amplitudes an be transferred per usage. For arbitrarily perfe t transfer, the situation is slightly more ompli ated as the optimal hoi e of M also depends on the joint probability of failure that one plans to a hieve. Let us assume that at ea h step of the proto ol, the su ess probability on a single hain is p. Then the number of steps to a hieve a given probability of failure P using M hains is given by ℓ(P,M) = max ln(1− p⌊M/2⌋) . (3.44) If we assume that the total time-s ale of the transfer is proportional to the number of steps, then the number of qubits that an be transferred per time interval is given by v(P,M) ∝ R(M)/ℓ(P,M). (3.45) Optimising this rate with respe t to M we �nd three di�erent regimes of the joint 3 Multi Rail en oding probability of failure (see Fig. 3.3). If one is happy with a large P, then the Multi Rail proto ol be omes superior to the Dual Rail for medium p. For intermediate P, the threshold is omparable to the threshold of p = 0.8 for on lusive transfer of entangle- ment. Finally for very low P the Multi Rail only be omes useful for p very lose to one. In all three ases the threshold is higher than the p(t) that an usually a hieved with unmodulated Heisenberg hains. We an thus on lude that the Multi Rail proto ol only be omes useful for hains whi h already have a very good performan e. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Success probability on a single chain P=0.50 P=0.90 P=0.99 Figure 3.3: Optimal rates (maximisation of Eq. (3.45 with respe t toM) for the Multi Rail proto ol. Shown are three urves orresponding to di�erent values of the joint probability of failure P one plans to a hieve. 3.8 Con lusion We thus on lude that any nearest-neighbour Hamiltonian that an transfer quantum information with nonzero �delity (in luding the Heisenberg hains analysed above) is apable of e� ient and perfe t transfer when used in the ontext of parallel hains. Hamiltonians with non-nearest neighbour intera tions [89,81℄ an also be used as long as the riterion of Theorem 3.1 is ful�lled. 4 Ergodi ity and mixing 4.1 Introdu tion We have seen above that by applying measurements at the end of parallel hains, the state of the hain is onverging to the ground state, and the quantum information is transferred to the re eiver. Indeed, repetitive appli ation of the same transformation is the key ingredient of many ontrols te hniques. Beside quantum state transfer, they have been exploited to inhibit the de oheren e of a system by frequently perturbing its dynami al evolution [129,130,131,132,133℄ (Bang-Bang ontrol) or to improve the �delity of quantum gates [134℄ by means of frequent measurements (quantum Zeno- e�e t [135℄). Re ently analogous strategies have also been proposed in the ontext of state preparation [136, 137, 138, 139, 140, 141, 142℄. In Refs. [138, 139℄ for instan e, a homogenisation proto ol was presented whi h allows one to transform any input state of a qubit into a some pre-�xed target state by repetitively oupling it with an external bath. A similar thermalisation proto ol was dis ussed in Ref. [140℄ to study the e� ien y of simulating lassi al equilibration pro esses on a quantum omputer. In Refs. [141, 142℄ repetitive intera tions with an externally monitored environment were exploited instead to implement puri� ation s hemes whi h would allow one to extra t pure state omponents from arbitrary mixed inputs. ergodic mixing Figure 4.1: S hemati examples of the orbits of a ergodi and a mixing map. 4 Ergodi ity and mixing The ommon trait of the proposals [136,137,138,139,140,141,142℄ and the dual and multi rail proto ols is the requirement that repeated appli ations of a properly hosen quantum operation τ onverges to a �xed density matrix x∗ independently from the input state x of the system, i.e. τn(x) ≡ τ ◦ τ ◦ · · · ◦ τ ︸ ︷︷ ︸ −→ x∗ , (4.1) with �◦� representing the omposition of maps. Following the notation of Refs. [143, 144℄ we all Eq. (4.1) the mixing property of τ . It is related with another important property of maps, namely ergodi ity (see Fig. 4.1). The latter requires the existen e of a unique input state x0 whi h is left invariant under a single appli ation of the map τ(x) = x ⇐⇒ x = x0 . (4.2) Ergodi ity and the mixing property are of high interest not only in the ontext of the above quantum information s hemes. They also o ur on a more fundamental level in statisti al me hani s [147℄ and open quantum systems [121, 148℄, where one would like to study irreversibility and relaxation to thermal equilibrium. In the ase of quantum transformations one an show that mixing maps with on- vergen e point x∗ are also ergodi with �xed point x0 = x∗. The opposite impli ation however is not generally true sin e there are examples of ergodi quantum maps whi h are not mixing (see the following). Su� ient onditions for mixing have been dis ussed both in the spe i� ase of quantum hannel [140, 143, 146℄ and in the more abstra t ase of maps operating on topologi al spa es [147℄. In parti ular the Lyapunov dire t method [147℄ allows one to prove that an ergodi map τ is mixing if there exists a on- tinuous fun tional S whi h, for all points but the �xed one, is stri tly in reasing under τ . Here we strengthen this riterion by weakening the requirement on S: our gener- alised Lyapunov fun tions are requested only to have limiting values S(τn(x))|n→∞ whi h di�er from S(x) for all x 6= x0. It turns out that the existen e of su h S is not just a su� ient ondition but also a ne essary ondition for mixing. Exploiting this fa t one an easily generalise a previous result on stri tly ontra tive maps [143℄ De�nition (4.2) may sound unusual for readers who are familiar with a de�nition of ergodi ity from statisti al me hani s, where a map is ergodi if its invariant sets have measure 0 or 1. The notion of ergodi ity used here is ompletely di�erent, and was introdu ed in [143, 145, 146℄. The set X one should have in mind here is not a measurable spa e, but the ompa t onvex set of quantum states. A perhaps more intuitive de�nition of ergodi ity based on the time average of observables is given by Lemma 4.5). 4 Ergodi ity and mixing by showing that maps whi h are asymptoti deformations (see De�nition 4.14) are mixing. This has, unlike ontra tivity, the advantage of being a property independent of the hoi e of metri (see however [144℄ for methods of �nding �tight� norms). In some ases, the generalised Lyapunov method permits also to derive an optimal mix- ing ondition for quantum hannels based on the quantum relative entropy. Finally a slightly modi�ed version of our approa h whi h employs multi- entral Lyapunov fun tions yields a hara terisation of (not ne essarily mixing) maps whi h in the limit of in�nitely many appli ations move all points toward a proper subset (rather than a single point) of the input spa e. The introdu tion of a generalised Lyapunov method seems to be sound not only from a mathemati al point of view, but also from a physi al point of view. In e�e t, it often happens that the informations available on the dynami s of a system are only those related on its asymptoti behaviour (e.g. its thermalisation pro ess), its �nite time evolution being instead di� ult to hara terise. Sin e our method is expli itly onstru ted to exploit asymptoti features of the mapping, it provides a more e�e tive way to probe the mixing property of the pro ess. Presenting our results we will not restri t ourself to the ase of quantum operations. Instead, following [147℄ we will derive them in the more general ontext of ontinuous maps operating on topologi al spa es [149℄. This approa h makes our results stronger by allowing us to invoke only those hypotheses whi h, to our knowledge, are stri tly ne essary for the derivation. It is important to stress however that, as a parti ular instan e, all the Theorems and Lemmas presented in this hapter hold for any linear, ompletely positive, tra e preserving map (i.e. quantum hannels) operating on a ompa t subset of normed ve tors (i.e. the spa e of the density matri es of a �nite dimensional quantum system). Therefore readers who are not familiar with topologi al spa es an simply interpret our derivations as if they were just obtained for quantum hannels a ting on a �nite dimensional quantum system. This hapter is organised as follows. In Se . 4.3 the generalised Lyapunov method along with some minor results is presented in the ontext of topologi al spa es. Then quantum hannels are analysed in Se . 4.4 providing a omprehensive summary of the ne essary and su� ient onditions for the mixing property of these maps. Con lusions and remarks form the end of the hapter in Se . 4.5. 4.2 Topologi al ba kground Let us �rst introdu e some basi topologi al ba kground required for this hapter. A more detailed introdu tion is given in [149℄. Topologi al spa es are a very elegant way 4 Ergodi ity and mixing of de�ning ompa tness, onvergen e and ontinuity without requiring more than the following stru ture: De�nition 4.1 A topologi al spa e is a pair (X ,O) of a set X and a set O of subsets of X ( alled open sets) su h that 1. X and ∅ are open 2. Arbitrary unions of open sets are open 3. Interse tions of two open sets are open Example 4.1 If X is an arbitrary set, and O = {X , ∅}, then (X ,O) is a topologi al spa e. O is alled the trivial topology. De�nition 4.2 A topologi al spa e X is ompa t if any open over (i.e. a set of open sets su h that X is ontained in their union) ontains a �nite sub- over. De�nition 4.3 A sequen e xn ∈ X is onvergent with limit x∗ if ea h open neigh- bourhood O(x∗) (i.e. a set su h that x∗ ∈ O(x∗) ∈ O ontains all but �nitely many points of the sequen e. De�nition 4.4 A map on a topologi al spa e is ontinous if the preimage of any open set is open. This is already all we require to make useful statements about ergodi ity and mixing. However, there are some subtleties whi h we need to take are of: De�nition 4.5 A topologi al spa e is sequentially ompa t if every sequen e has a onvergent subsequen e. Sequentially ompa tness is in general not related to ompa tness! Another subtlety is that with the above de�nition, a sequen e an onverge to many di�erent points. For example, in the trivial topology, any sequen e onverges to any point. This motivates De�nition 4.6 A topologi al spa e is Hausdor� if any two distin t points an by separated by open neighbourhoods. A limit of a sequen e in a Hausdor� spa e is unique. All these problems disappear in metri al spa es: 4 Ergodi ity and mixing De�nition 4.7 Ametri spa e is a pair (X , d) of a set X and a fun tion d : X×X → R su h that 1. d(x, y) ≥ 0 and d(x, y) = 0 ⇔ x = y 2. d(x, y) = d(y, x) 3. d(x, z) ≤ d(x, y) + d(y, z) A metri spa e be omes a topologi al spa e with the anoni al topology De�nition 4.8 A subset O of a metri spa e X is open if ∀x ∈ O there is an ǫ > 0 su h that {y ∈ X |d(x, y) ≤ ǫ} ⊂ O. In a metri spa e with the anoni al topology, ompa tness and sequentially ompa t- ness be ome equivalent. Furthermore, it is automati ally Hausdor� (see Fig. 4.2). Compact spaces spaces Compact Hausdorff Topological spaces metric spaces spaces Sequentially compact Figure 4.2: Relations between topologi al spa es [149℄. The spa e of density matri es on whi h quantum hannels are de�ned, is a ompa t and onvex subset of a normed ve tors spa e (the spa e of linear operators of the system) whi h, in the above graphi al representation �ts within the set of ompa t metri spa es. 4.3 Generalised Lyapunov Theorem 4.3.1 Topologi al spa es In this se tion we introdu e the notation and derive our main result (the Generalised Lyapunov Theorem). 4 Ergodi ity and mixing De�nition 4.9 Let X be a topologi al spa e and let τ : X → X be a map. The sequen e xn ≡ τn(x), where τn is a short-hand notation for the n−fold omposition of τ, is alled the orbit of x. An element x∗ ∈ X is alled a �xed point of τ if and only τ(x∗) = x∗ . (4.3) τ is alled ergodi if and only if it has exa tly one �xed point. τ is alled mixing if and only if there exists a onvergen e point x∗ ∈ X su h that any orbit onverges to it, i.e. xn = x∗ ∀x ∈ X . (4.4) A dire t onne tion between ergodi ity and mixing an be established as follows. Lemma 4.1 Let τ : X → X be a ontinuous mixing map on a topologi al Hausdor� spa e X . Then τ is ergodi . Proof Let x∗ be the onvergen e point of τ and let x ∈ X arbitrary. Sin e τ is ontinuous we an perform the limit in the argument of τ, i.e. τ(x∗) = τ τn(x) = lim τn+1(x) = x∗, (4.5) whi h shows that x∗ is a �xed point of τ . To prove that it is unique assume by ontradi tion that τ possesses a se ond �xed point y∗ 6= x∗. Then limn→∞ τn(y∗) = y∗ 6= x∗, so τ ould not be mixing (sin e the limit is unique in a Hausdor� spa e � see Fig. 4.2). Hen e τ is ergodi . � Remark 4.1 The onverse is not true in general, i.e. not every ergodi map is mixing (not even in Hausdor� topologi al spa es). A simple ounterexample is given by τ : [−1, 1] → [−1, 1] with τ(x) ≡ −x and the usual topology of R, whi h is ergodi with �xed point 0, but not mixing sin e for x 6= 0, τn(x) = (−1)nx is alternating between two points. A similar ounterexample will be dis ussed in the quantum hannel se tion (see Example 4.2). A well known riterion for mixing is the existen e of a Lyapunov fun tion [147℄. De�nition 4.10 Let τ : X → X be a map on a topologi al spa e X . A ontinuous map S : X → R is alled a (stri t) Lyapunov fun tion for τ around x∗ ∈ X if and only S (τ(x)) > S(x) ∀x 6= x∗. (4.6) 4 Ergodi ity and mixing Remark 4.2 At this point is is neither assumed that x∗ is a �xed point, nor that τ is ergodi . Both follows from the theorem below. Theorem 4.1 (Lyapunov fun tion) Let τ : X → X be a ontinuous map on a sequentially ompa t topologi al spa e X . Let S : X → R be a Lyapunov fun tion for τ around x∗. Then τ is mixing with the �xed point x∗. The proof of this theorem is given in [147℄. We will not reprodu e it here, be ause we will provide a general theorem that in ludes this as a spe ial ase. In fa t, we will show that the requirement of the stri t monotoni ity an be mu h weakened, whi h motivates the following de�nition. De�nition 4.11 Let τ : X → X be a map on a topologi al spa e X . A ontinuous map S : X → R is alled a generalised Lyapunov fun tion for τ around x∗ ∈ X if and only if the sequen e S (τn(x)) is point-wise onvergent2 for any x ∈ X and S ful�ls S∗(x) ≡ lim S (τn(x)) 6= S(x) ∀x 6= x∗. (4.7) In general it may be di� ult to prove the point-wise onvergen e. However if S is monotoni under the a tion of τ and the spa e is ompa t, the situation be omes onsiderably simpler. This is summarised in the following Lemma. Lemma 4.2 Let τ : X → X be map on a ompa t topologi al spa e. A ontinuous map S : X → R whi h ful�ls S (τ(x)) > S(x) ∀x ∈ X , (4.8) S∗(x) ≡ lim S (τn(x)) > S(x) ∀x 6= x∗. (4.9) for some �xed x∗ ∈ X is a generalised Lyapunov fun tion for τ around x∗. Proof It only remains to show the (point-wise) onvergen e of S (τn(x)). Sin e S is a ontinuous fun tion on a ompa t spa e, it is bounded. By Eq. (4.8) the sequen e is monotoni . Any bounded monotoni sequen e onverges. � Corollary 4.1 Let τ : X → X be a map on a ompa t topologi al spa e. A ontinuous map S : X → R whi h ful�ls S (τ(x)) > S(x) ∀x ∈ X , (4.10) Point-wise onvergen e in this ontext means that for any �xed x the sequen e Sn ≡ S (τ n(x)) is onvergent. 4 Ergodi ity and mixing τN (x) > S(x) ∀x 6= x∗, (4.11) for some �xed N ∈ N and for some x∗ ∈ X is a generalised Lyapunov fun tion for τ around x∗. Remark 4.3 This implies that a stri t Lyapunov fun tion is a generalised Lyapunov fun tion (with N = 1). We an now state the main result of this se tion: Theorem 4.2 (Generalized Lyapunov fun tion) Let τ : X → X be a on- tinuous map on a sequentially ompa t topologi al spa e X . Let S : X → R be a generalised Lyapunov fun tion for τ around x∗. Then τ is mixing with �xed point Proof Consider the orbit xn ≡ τn(x) of a given x ∈ X . Be ause X is sequen- tially ompa t, the sequen e xn has a onvergent subsequen e (see Fig. 4.2), i.e. limk→∞ xnk ≡ x̃. Let us assume that x̃ 6= x∗ and show that this leads to a on- tradi tion. By Eq. (4.7) we know that there exists a �nite N ∈ N su h that τN (x̃) 6= S(x̃). (4.12) Sin e τN is ontinuous we an perform the limit in the argument, i.e. τN (xnk) = τ N (x̃). (4.13) Likewise, by ontinuity of S we have S (xnk) = S(x̃), (4.14) and on the other hand S (xN+nk) = lim τN (xnk) = S(τN x̃), (4.15) where the se ond equality stems from the ontinuity of the map S and τN . Be ause S is a generalised Lyapunov fun tion, the sequen e S (xn) is onvergent. Therefore the subsequen es (4.14) and (4.15) must have the same limit. We on lude that 4 Ergodi ity and mixing S(τN x̃) = S(x̃) whi h ontradi ts Eq. (4.12). Hen e x̃ = x∗. Sin e we have shown that any onvergent subsequen e of τn(x) onverges to the same limit x∗, it follows by Lemma 4.3 that τn(x) is onverging to x∗. Sin e that holds for arbitrary x, it follows that τ is mixing. � Lemma 4.3 Let xn be a sequen e in a sequentially ompa t topologi al spa e X su h that any onvergent subsequen e onverges to x∗. Then the sequen e onverges to x∗. Proof We prove by ontradi tion: assume that the sequen e does not onverge to x∗. Then there exists an open neighbourhood O(x∗) of x∗ su h that for all k ∈ N, there is a nk su h that xnk /∈ O(x∗). Thus the subsequen e xnk is in the losed spa e X\O(x∗), whi h is again sequentially ompa t. xnk has a onvergent subsequen e with a limit in X\O(x∗), in parti ular this limit is not equal to x∗. � There is an even more general way of de�ning Lyapunov fun tions whi h we state here for ompleteness. It requires the on ept of the quotient topology [149℄. De�nition 4.12 Let τ : X → X be a map on a topologi al spa e X . A ontinuous map S : X → R is alled a multi- entral Lyapunov fun tion for τ around F ⊆ X if and only if the sequen e S (τn(x)) is point-wise onvergent for any x ∈ X and if S and τ ful�l the following three onditions: S is onstant on F , τ(F) ⊆ F , and S∗(x) ≡ lim S (τn(x)) 6= S(x) ∀x /∈ F . (4.16) For these fun tions we annot hope that the orbit is mixing. We an however show that the orbit is � onverging� to the set F in the following sense: Theorem 4.3 (Multi- entral Lyapunov fun tion) Let τ : X → X be a ontinu- ous map on a sequentially ompa t topologi al spa e X . Let S : X → R be a multi- entral Lyapunov fun tion for τ around F . Let ϕ : X → X/F be the ontinuous mapping into the quotient spa e (i.e. ϕ(x) = [x] for x ∈ X\F and ϕ(x) = [F ] for x ∈ F). Then τ̃ : X/F → X/F given by τ̃([x]) = ϕ ϕ−1([x]) is mixing with �xed point [F ]. Proof First note that τ̃ is well de�ned be ause ϕ is invertible on X/F\[F ] and τ(F) ⊆ F , so that τ̃([F ]) = [F ]. Sin e X is sequentially ompa t, the quotient spa e X/F is also sequentially ompa t. Note that for O open, τ̃−1(O) = ϕ ϕ−1 (O) is the image of ϕ of an open set in X and therefore (by de�nition of the quotient topol- ogy) open in X/F . Hen e τ̃ is ontinuous. The fun tion S̃([x]) : X/F → X/F given by S̃([x]) = S(ϕ−1([x])) is ontinuous and easily seen to be a generalised Lyapunov fun tion around [F ]. By Theorem 4.2 it follows that τ̃ is mixing. � 4 Ergodi ity and mixing 4.3.2 Metri spa es We now show that for the parti ular lass of ompa t topologi al sets whi h posses a metri , the existen e of a generalised Lyapunov fun tion is also a ne essary ondition for mixing. Theorem 4.4 (Lyapunov riterion) Let τ : X → X be a ontinuous map on a ompa t metri spa e X . Then τ is mixing with �xed point x∗ if and only if a generalised Lyapunov fun tion around x∗ exists. Proof Firstly, in metri spa es ompa tness and sequential ompa tness are equiva- lent, so the requirements of Theorem 4.2 are met. Se ondly, for any mixing map τ with �xed point x∗, a generalised Lyapunov fun tion around x∗ is given by S(x) ≡ d(x∗, x). In fa t, it is ontinuous be ause of the ontinuity of the metri and satis�es S (τn(x)) = d(x∗, x∗) = 0 6 d(x∗, x) = S(x), (4.17) where the equality holds if and only x = x∗. We all d(x∗, x) the trivial generalised Lyapunov fun tion. � Remark 4.1 In the above Theorem we have not used all the properties of the metri . In fa t a ontinuous semi-metri (i.e. without the triangle inequality) would su� e. The trivial Lyapunov fun tion requires knowledge of the �xed point of the map. There is another way of hara terising mixing maps as those whi h bring elements loser to ea h other (rather than loser to the �xed point). De�nition 4.13 A map τ : X → X is on a metri spa e is alled a non-expansive map if and only if d(τ(x), τ(y)) 6 d(x, y) ∀x, y ∈ X , (4.18) a weak ontra tion if and only if d(τ(x), τ(y)) < d(x, y) ∀x, y ∈ X , x 6= y, (4.19) and a stri t ontra tion if and only if there exists a k < 1 su h that d(τ(x), τ(y)) 6 k d(x, y) ∀x, y ∈ X . (4.20) Remark 4.2 The notation adopted here is slightly di�erent from the de�nitions used by other Authors [143, 150, 5℄ who use ontra tion to indi ate our non-expansive maps. 4 Ergodi ity and mixing Our hoi e is motivated by the need to learly distinguish between non-expansive transformation and weak ontra tions. We an generalise the above de�nition in the following way: De�nition 4.14 A map τ : X → X on a metri spa e is alled an asymptoti defor- mation if and only if the sequen e d(τn(x), τn(y)) onverges point-wise for all x, y ∈ X d(τn(x), τn(y)) 6= d(x, y) ∀x, y ∈ X , x 6= y. (4.21) Lemma 4.4 Let τ : X → X be a non-expansive map on a metri spa e X , and let d(τN (x), τN (y)) < d(x, y) ∀x, y ∈ X , x 6= y (4.22) for some �xed N ∈ N. Then τ is an asymptoti deformation. Then τ is an asymptoti deformation. Proof The existen e of the limit limn→∞ d(τ n(x), τn(y)) follows from the monotoni - ity and the fa t the any metri is lower bounded. � Remark 4.4 Any weak ontra tion is an asymptoti deformation (with N = 1). Theorem 4.5 (Asymptoti deformations) Let τ : X → X be a ontinuous map on a ompa t metri spa e X with at least one �xed point. Then τ is mixing if and only if τ is an asymptoti deformation. Proof Firstly assume that τ is an asymptoti deformation. Let x∗ be a �xed point and de�ne S(x) = d(x∗, x). S(τn(x)) = lim d(x∗, τ n(x)) = lim d(τn(x∗), τ n(x)) 6= d(x∗, x) = S(x) ∀x 6= x∗, (4.23) hen e S(x) is a generalised Lyapunov fun tion. By Theorem 4.2 it follows that τ is mixing. Se ondly, if τ is mixing, then d(τn(x), τn(y)) = d(x∗, x∗) = 0 6= d(x, y) ∀x, y ∈ X , x 6= y, (4.24) so τ is an asymptoti deformation. � 4 Ergodi ity and mixing Remark 4.5 Note that the existen e of a �xed point is assured if τ is a weak ontra tion on a ompa t spa e [151℄, or if the metri spa e is onvex ompa t [152℄. As a spe ial ase, we get the following result: Corollary 4.2 Any weak ontra tion τ on a ompa t metri spa e is mixing. Proof Sin e the spa e is ompa t τ has at least one �xed point. Moreover from Lemma 4.4 we know that τ is an asymptoti deformation. Then Theorem 4.5 applies.� Remark 4.6 This result an be seen as an instan e of Bana h ontra tion prin iple on ompa t spa es. In the se ond part of the hapter we will present a ounterexample whi h shows that weak ontra tivity is only a su� ient riterion for mixing (see Ex- ample 4.3). In the ontext of quantum hannels an analogous riterion was suggested in [146, 143℄ whi h applied to stri t ontra tions. We also note that for weak and stri t ontra tions, the trivial generalised Lyapunov fun tion (Theorem 4.4) is a stri t Lyapunov fun tion. Lemma 4.5 states the ergodi theorem by Birkho� [153℄ whi h, in the ontext of normed ve tor spa es, shows the equivalen e between the de�nition of ergodi ity of Eq. (4.4) and the standard time average de�nition. Lemma 4.5 Let X be a onvex and ompa t subset of a normed ve tor spa e, and let τ : X → X be a ontinuous map. If τ is ergodi with �xed point x∗, then τ ℓ(x) = x∗ . (4.25) Proof De�ne the sequen e An ≡ 1n+1 ℓ=0 τ ℓ(x). Let then M be the upper bound for the norm of ve tors in X , i.e. M ≡ supx∈X ‖x‖ < ∞. whi h exists be ause X is ompa t. The sequen e An has a onvergent subsequen e Ank with limit Ã. Sin e τ is ontinuous one has limk→∞ τ(Ank) = τ(Ã). On the other hand, we have ‖τ(Ank)−Ank‖ = nk + 1 ‖τnk+1(x)− x‖ 6 ‖τ nk+1(x)‖+ ‖x‖ nk + 1 nk + 1 , (4.26) so the two sequen es must have the same limit, i.e. τ(Ã) = Ã. Sin e τ is ergodi , we have à = x∗ and limn→∞An = x∗ by Lemma 4.3. � Remark 4.7 Note that if τ has a se ond �xed point y∗ 6= x∗, then for all n one has ℓ=0 τ ℓ(y∗) = y∗, so Eq. (4.25) would not apply. 4 Ergodi ity and mixing 4.4 Quantum Channels In this Se tion we dis uss the mixing properties of quantum hannels [2℄ whi h a ount for the most general evolution a quantum system an undergo in luding measure- ments and oupling with external environments. In this ontext solving the mixing problem (4.1) is equivalent to determine if repetitive appli ation of a ertain physi al transformation will drive any input state of the system (i.e. its density matri es) into a unique output on�guration. The relationship between the di�erent mixing riteria one an obtain in this ase is summarised in Fig. 4.3. At a mathemati al level quantum hannels orrespond to linear maps a ting on the density operators ρ of the system and satisfying the requirement of being ompletely positive and tra e preserving (CPT). For a formal de�nition of these properties we refer the reader to [154, 5, 155℄: here we note only that a ne essary and su� ient ondition to being CPT is to allow Kraus de omposition [154℄ or, equivalently, Stine- spring dilation [156℄. Our results are appli able if the underlying Hilbert spa e is �nite-dimensional. In su h regime there is no ambiguity in de�ning the onvergen e of a sequen e sin e all operator norms are equivalent (i.e. given two norms one an onstru t an upper and a lower bound for the �rst one by properly s aling the se ond one). Also the set of bounded operators and the set of operators of Hilbert-S hmidt lass oin ide. For the sake of de�niteness, however, we will adopt the tra e-norm whi h, given the linear operator Θ : H → H, is de�ned as ‖Θ‖1 = Tr[ Θ†Θ] with Tr[· · · ] being the tra e over H and Θ† being the adjoint of Θ. This hoi e is in part motivated by the fa t [150℄ that any quantum hannel is non-expansive with respe t to the metri indu ed by ‖ ·‖1 (the same property does not ne essarily apply to other operator norms, e.g. the Hilbert-S hmidt norm, also when these are equivalent to ‖ · ‖1). We start by showing that the mixing riteria dis ussed in the �rst half of the hapter do apply to the ase of quantum hannel. Then we will analyse these maps by studying their linear extensions in the whole ve tor spa e formed by the linear operators of H. 4.4.1 Mixing riteria for Quantum Channels Let H be a �nite dimensional Hilbert spa e and let S(H) be the set of its density matri es ρ. The latter is a onvex and ompa t subset of the larger normed ve tor spa e L(H) omposed by the linear operators Θ : H → H of H. From this and from the fa t that CPT maps are ontinuous (indeed they are linear) it follows that for This is just the tra e distan e d(ρ, σ) = ‖ρ− σ‖1. 4 Ergodi ity and mixing asymptotic deformation mixing ergodic spectral gap generalized Lyapunov function exists ergodic with pure fixpoint strict Lyapunov function exists contraction Figure 4.3: Relations between the di�erent properties of a quantum hannel. a quantum hannel there always exists at least one density operator whi h is a �xed point [140℄. It also follows that all the results of the previous se tion apply to quantum hannels. In parti ular Lemma 4.1 holds, implying that any mixing quantum hannel must be ergodi . The following example shows, however, that it is possible to have ergodi quantum hannels whi h are not mixing. Example 4.2 Consider the qubit quantum hannel τ obtained by as ading a om- pletely de oherent hannel with a NOT gate. Expli itly τ is de�ned by the transfor- mations τ(|0〉〈0|) = |1〉〈1|, τ(|1〉〈1|) = |0〉〈0|, and τ(|0〉〈1|) = τ(|1〉〈0|) = 0 with |0〉, |1〉 being the omputational basis of the qubit. This map is ergodi with �xed point given by the ompletely mixed state (|0〉〈0| + |1〉〈1|)/2. However it is trivially not mixing sin e, for instan e, repetitive appli ation of τ on |0〉〈0| will os illate between |0〉〈0| and |1〉〈1|. Theorems 4.5 implies that a quantum hannel τ : S(H) → S(H) is mixing if and only if it is an asymptoti deformation. As already pointed out in the introdu tion, this property is metri independent (as opposed to ontra tivity). Alternatively, if the �xed point of a quantum hannel is known, then one may use the trivial generalised Lyapunov fun tion (Theorem 4.4) to he k if it is mixing. However both riteria depend on the metri distan e, whi h usually has no easy physi al interpretation. A more useful hoi e is the quantum relative entropy, whi h is de�ned as H(ρ, σ) ≡ Trρ(log ρ− log σ). (4.27) The quantum relative entropy is ontinuous in �nite dimension [157℄ and an be used as a measure of distan e (though it is not a metri ). It is �nite if the support of ρ is ontained in the support of σ. To ensure that it is a ontinuous fun tion on a ompa t spa e, we hoose σ to be faithful: Theorem 4.6 (Relative entropy riterion) A quantum hannel with faithful �xed point ρ∗ is mixing if and only if the quantum relative entropy with respe t to ρ∗ is a 4 Ergodi ity and mixing generalised Lyapunov fun tion. Proof Be ause of Theorem 4.2 we only need to prove the se ond part of the thesis, i.e. that mixing hannels admit the quantum relative entropy with respe t to the �xed point, S(ρ) ≡ H(ρ, ρ∗), as a generalised Lyapunov fun tion. Firstly noti e that the quantum relative entropy is monotoni under quantum hannels [158,159℄. Therefore the limit S∗(ρ) ≡ limn→∞ S (τn(ρ)) does exist and satis�es the ondition S∗(ρ) > S(ρ). Suppose now there exists a ρ su h that S∗(ρ) = S(ρ). Be ause τ is mixing and S is ontinuous we have S(ρ) = S∗(ρ) = lim S (τn(ρ)) = S(ρ∗) = 0, (4.28) and hen e H(ρ, ρ∗) = 0. Sin e H(ρ, σ) = 0 if and only if ρ = σ it follows that S is a Lyapunov fun tion around ρ∗. � Another important investigation tool is Corollary 4.2: weak ontra tivity of a quantum hannel is a su� ient ondition for mixing. As already mentioned in the previous se tion, unfortunately this not a ne essary ondition. Here we present an expli it ounterexample based on a quantum hannel introdu ed in Ref. [140℄. Example 4.3 Consider a three-level quantum system hara terised by the orthogo- nal ve tors |0〉, |1〉, |2〉 and the quantum hannel τ de�ned by the transformations τ(|2〉〈2|) = |1〉〈1|, τ(|1〉〈1|) = τ(|0〉〈0|) = |0〉〈0|, and τ(|i〉〈j|) = 0 for all i 6= j. Its easy to verify that after just two iterations any input state ρ will be transformed into the ve tor |0〉〈0|. Therefore the map is mixing. On the other hand it is expli itly not a weak ontra tion with respe t to the tra e norm sin e, for instan e, one has ‖ τ(|2〉〈2|) − τ(|0〉〈0|) ‖1 = ‖ |1〉〈1| − |0〉〈0| ‖1 = ‖ |2〉〈2| − |0〉〈0| ‖1 , (4.29) where in the last identity we used the invarian e of ‖ · ‖1 with respe t to unitary transformations. 4.4.2 Beyond the density matrix operator spa e: spe tral properties Exploiting linearity quantum hannels an be extended beyond the spa e S(H) of density operators to be ome maps de�ned on the full ve tor spa e L(H) of the linear operators of the system, in whi h basi linear algebra results hold. This allows one to simplify the analysis even though the mixing property (4.1) is still de�ned with respe t to the density operators of the system. 4 Ergodi ity and mixing Mixing onditions for quantum hannels an be obtained by onsidering the stru - ture of their eigenve tors in the extended spa e L(H). For example, it is easily shown that the spe tral radius [160℄ of any quantum hannel is equal to unity [140℄, so its eigenvalues are ontained in the unit ir le. The eigenvalues λ on the unit ir le (i.e. |λ| = 1) are referred to as peripheral eigenvalues. Also, as already mentioned, sin e S(H) is ompa t and onvex, CPT maps have always at least one �xed point whi h is a density matrix [140℄. Theorem 4.7 (Spe tral gap riterion) Let τ be a quantum hannel. τ is mixing if and only if its only peripheral eigenvalue is 1 and this eigenvalue is simple. Proof The �if� dire tion of the proof is a well known result from linear algebra (see for example [160, Lemma 8.2.7℄). Now let us assume τ is mixing towards ρ∗. Let Θ be a generi operator in L(H). Then Θ an be de omposed in a �nite set of non-orthogonal density operators , i.e. Θ = ℓ cℓρℓ, with ρℓ ∈ S(H) and cℓ omplex. Sin e Tr [ρℓ] = 1, we have have Tr [Θ] = ℓ cℓ. Moreover sin e τ is mixing we have limn→∞ τ n (ρℓ) = ρ∗ for all ℓ, with onvergen e with respe t to the tra e-norm. Be ause of linearity this implies τn (Θ) = cℓ ρ∗ = Tr [Θ] ρ∗ . (4.30) If there existed any other eigenve tor Θ∗ of τ with eigenvalue on the unit ir le, then limn→∞ τ n(Θ∗) would not satisfy Eq. (4.30). � The speed of onvergen e an also be estimated by [140℄ ‖τn (ρ)− ρ∗‖1 6 CN nN κn , (4.31) where N is the dimensionality of the underlying Hilbert spa e, κ is the se ond largest eigenvalue of τ , and CN is some onstant depending only on N and on the hosen norm. Hen e, for n ≫ N the onvergen e be omes exponentially fast. As mentioned in [143℄, the riterion of Theorem 4.7 is in general di� ult to he k. This is be ause one has to �nd all eigenvalues of the quantum hannel, whi h is hard espe ially in the high dimensional ase. Also, if one only wants to he k if a parti ular hannel To show that this is possible, onsider an arbitrary operator basis of L(H). If N is the �nite dimension of H the basis will ontain N2 elements. Ea h element of the basis an then be de omposed into two Hermitian operators, whi h themselves an be written as linear ombinations of at most N proje tors. Therefore there exists a generating set of at most 2N3 positive operators, whi h an be normalised su h that they are quantum states. There even exists a basis (i.e. a minimal generating set) onsisting of density operators, but in general it annot be orthogonalised. 4 Ergodi ity and mixing is mixing or not, then the amount of information obtained is mu h higher than the required amount. Example 4.4 As an appli ation onsider the non mixing CPT map of Example 4.2. One an verify that apart from the eigenvalue 1 asso iated with its �xed point (i.e. the ompletely mixed state), it possess another peripheral eigenvalue. This is λ = −1 whi h is asso iated with the Pauli operator |0〉〈0| − |1〉〈1|. Corollary 4.3 The onvergen e speed of any mixing quantum hannel is exponentially fast for su� iently high values of n. Proof From Theorem 4.7 mixing hannels have exa tly one peripheral eigenvalue, whi h is also simple. Therefore the derivation of Ref. [140℄ applies and Eq. (4.31) holds. � This result should be ompared with the ase of stri tly ontra tive quantum hannels whose onvergen e was shown to be exponentially fast along to whole traje tory [143, 146℄. 4.4.3 Ergodi hannels with pure �xed points An interesting lass of ergodi quantum hannel is formed by those CPT maps whose �xed point is a pure density matrix. Among them we �nd for instan e the maps employed in the ommuni ation proto ols dis ussed in this thesis or those of the pu- ri� ation s hemes of Refs. [142, 141℄. We will now show that within this parti ular lass, ergodi ity and mixing are indeed equivalent properties. We �rst need the following Lemma, whi h dis usses a useful property of quantum hannels (see also [161℄). Lemma 4.6 Let τ be a quantum hannel and Θ be an eigenve tor of τ with peripheral eigenvalue λ = eiϕ. Then, given g = Tr > 0, the density matri es ρ = ΘΘ†/g and σ = Θ†Θ/g are �xed points of τ . Proof Use the left polar de omposition to write Θ = g ρU where U is a unitary operator. The operator ρU is learly an eigenve tor of τ with eigenvalue eiϕ, i.e. τ(ρU) = λ ρU . (4.32) 4 Ergodi ity and mixing Hen e introdu ing a Kraus set {Kn}n of τ [154℄ and the spe tral de omposition of the density matrix ρ = j pj|ψj〉〈ψj | with pj > 0 being its positive eigenvalues, one gets λ = Tr[τ(ρU)U †] = j,ℓ,n pj〈φℓ|Kn|ψj〉〈ψj |UK†nU †|φℓ〉 , (4.33) where the tra e has been performed with respe t to an orthonormal basis {|φℓ〉}ℓ of H. Taking the absolute values of both terms gives |λ| = | j,ℓ,n pj〈φℓ|Kn|ψj〉〈ψj |UK†nU †|φℓ〉| j,ℓ,n pj〈φℓ|Kn|ψj〉〈ψj |K†n|φℓ〉 j,ℓ,n pj〈φℓ|UKnU †|ψj〉〈ψj |UK†nU †|φℓ〉 Tr[τ(ρ)] Tr[τ̃(ρ)] = 1, (4.34) where the inequality follows from the Cau hy-S hwartz inequality. The last identity instead is a onsequen e of the fa t that the transformation τ̃(ρ) = Uτ(U †ρU)U † is CPT and thus tra e preserving. Sin e |λ| = 1 it follows that the inequality must be repla ed by an identity. This happens if and only if there exist eiϑ su h that pj{〈φℓ|Kn|ψj〉}∗ = pj〈ψj |K†n|φℓ〉 = eiϑ pj〈ψj |UK†nU †|φℓ〉 , (4.35) for all j, ℓ and n. Sin e the |φℓ〉 form a basis of H, and pj > 0 this implies 〈ψj |K†n = eiϑ 〈ψj |UK†nU † ⇒ 〈ψj |UK†n = e−iϑ 〈ψj |K†nU , (4.36) for all n and for all the not null eigenve tors |ψj〉 of ρ. This yields τ(ρU) = Kn|ψj〉〈ψj |UK†n = e−iϑ Kn|ψj〉〈ψj |K†nU = e−iϑ τ(ρ)U (4.37) whi h, repla ed in (4.32) gives e−iϑ τ(ρ) = eiϕ ρ, whose only solution is e−iϑ = eiϕ. Therefore τ(ρ) = ρ and ρ is a �xed point of τ . The proof for σ goes along similar lines: simply onsider the right polar de omposition of Θ instead of the left polar de omposition. � Corollary 4.4 Let τ be an ergodi quantum hannel. It follows that its eigenve tors asso iated with peripheral eigenvalues are normal operators. 4 Ergodi ity and mixing Proof Let Θ be an eigenoperator with peripheral eigenvalue eiϕ su h that τ (Θ) = eiϕ Θ. By Lemma 4.6 we know that, given g = Tr the density matri es ΘΘ†/g and σ = Θ†Θ/g must be �xed points of τ . Sin e the map is ergodi we must have ρ = σ, i.e. ΘΘ† = Θ†Θ. � Theorem 4.8 (Purely ergodi maps) Let |ψ1〉〈ψ1| be the pure �xed point of an ergodi quantum hannel τ . It follows that τ is mixing. Proof We will use the spe tral gap riterion showing that |ψ1〉〈ψ1| is the only pe- ripheral eigenve tor of τ . Assume in fa t that Θ ∈ L(H) is a eigenve tor of τ with peripheral eigenvalue, i.e. τ (Θ) = eiϕΘ . (4.38) From Lemma 4.6 we know that the density matrix ΘΘ†/g, (4.39) with g = Tr > 0, must be a �xed point of τ . Sin e this is an ergodi map we must have ρ = |ψ1〉〈ψ1|. This implies Θ = g|ψ1〉〈ψ2|, with |ψ2〉 some normalised ve tor of H. Repla ing it into Eq. (4.38) and dividing both terms by g yields τ (|ψ1〉〈ψ2|) = eiϕ|ψ1〉〈ψ2| and |〈ψ1|τ(|ψ1〉〈ψ2|)|ψ2〉| = 1 . (4.40) Introdu ing a Kraus set {Kn}n of τ and employing Cau hy-S hwartz inequality one an then write 1 = |〈ψ1|τ(|ψ1〉〈ψ2|)|ψ2〉| = | 〈ψ1|Kn|ψ1〉〈ψ2|K†n|ψ2〉| (4.41) 〈ψ1|Kn|ψ1〉〈ψ1|K†n|ψ1〉 〈ψ2|Kn|ψ2〉〈ψ2|K†n|ψ2〉 〈ψ1|τ(|ψ1〉〈ψ1|)|ψ1〉 〈ψ2|τ(|ψ2〉〈ψ2|)|ψ2〉 = 〈ψ2|τ(|ψ2〉〈ψ2|)|ψ2〉 , where we used the fa t that |ψ1〉 is the �xed point of τ . Sin e τ is CPT the quantity 〈ψ2|τ(|ψ2〉〈ψ2|)|ψ2〉 is upper bounded by 1. Therefore in the above expression the 4 Ergodi ity and mixing inequality must be repla ed by an identity, i.e. 〈ψ2|τ(|ψ2〉〈ψ2|)|ψ2〉 = 1 ⇐⇒ τ(|ψ2〉〈ψ2|) = |ψ2〉〈ψ2| . (4.42) Sin e τ is ergodi , we must have |ψ2〉〈ψ2| = |ψ1〉〈ψ1|. Therefore Θ ∝ |ψ1〉〈ψ1| whi h shows that |ψ1〉〈ψ1| is the only eigenve tor of τ with peripheral eigenvalue of. � An appli ation of the previous Theorem is obtained as follows. Lemma 4.7 LetMAB =MA⊗1B+1A⊗MB be an observable of the omposite system HA ⊗HB and τ the CPT linear map on HA of Stinespring form [156℄ τ(ρ) = TrB U (ρ⊗ |φ〉B〈φ|)U † , (4.43) (here TrX [· · · ] is the partial tra e over the system X, and U is a unitary operator of HA⊗HB). Assume that [MAB , U ] = 0 and that |φ〉B is the eigenve tor orresponding to a non-degenerate maximal or minimal eigenvalue of MB. Then τ is mixing if and only if U has one and only one eigenstate that fa torises as |ν〉A ⊗ |φ〉B . Proof Let ρ be an arbitrary �xed point of τ (sin e τ is CPT it has always at least one), i.e. TrB U (ρ⊗ |φ〉B〈φ|)U † = ρ. Sin e MAB is onserved and TrA [MAρ] = TrA [MAτ(ρ)], the system B must remain in the maximal state, whi h we have assumed to be unique and pure, i.e. U (ρ⊗ |φ〉B〈φ|)U † = ρ⊗ |φ〉B〈φ| =⇒ [U, ρ⊗ |φ〉B〈φ|] = 0 . (4.44) Thus there exists a orthonormal basis {|uk〉}k ofHA⊗HB diagonalising simultaneously both U and ρ ⊗ |φ〉B〈φ|. We express the latter in this basis, i.e. ρ ⊗ |φ〉B〈φ| = k pk|uk〉〈uk| with pk > 0, and ompute the von Neumann entropy of subsystem B. This yields 0 = H(|φ〉B〈φ|) = H pk|uk〉〈uk| pk H (TrA [|uk〉〈uk|]) .(4.45) From the onvexity of the von Neumann entropy the above inequality leads to a ontradi tion unless TrA [|uk〉〈uk|] = |φ〉B〈φ| for all k. The |uk〉 must therefore be fa torising, |uk〉 = |νk〉A ⊗ |φ〉B . (4.46) If the fa torising eigenstate of U is unique, it must follow that ρ = |ν〉〈ν| for some |ν〉 and that τ is ergodi . By Theorem 4.8 it then follows that τ is also mixing. If on 4 Ergodi ity and mixing the other hand there exists more than one fa torising eigenstate, than all states of the form of Eq. (4.46) orrespond to a �xed point ρk = |νk〉〈νk| and τ is neither ergodi nor mixing. � Remark 4.3 An appli ation of this Lemma is the proto ol for read and write a ess by lo al ontrol dis ussed in the next hapter. 4.5 Con lusion In reviewing some known results on the mixing property of ontinuous maps, we obtained a stronger version of the dire t Lyapunov method. For ompa t metri spa es (in luding quantum hannels operating over density matri es) it provides a ne essary and su� ient ondition for mixing. Moreover it allows us to prove that asymptoti deformations with at least one �xed point must be mixing. In the spe i� ontext of quantum hannels we employed the generalised Lyapunov method to analyse the mixing properties. Here we also analysed di�erent mixing riteria. In parti ular we have shown that an ergodi quantum hannel with a pure �xed point is also mixing. 5 Read and write a ess by lo al ontrol 5.1 Introdu tion The unitarity of Quantum Me hani s implies that information is onserved. Whatever happens to a quantum system - as long as it is unitary, the original state an in prin iple be re overed by applying the inverse unitary transformation. However it is well known that in open quantum systems [121℄ the redu ed dynami s is no longer unitary. The redu ed dynami s is des ribed by a ompletely positive, tra e preserving maps, and we have seen in the last hapter that there are extreme examples, namely mixing maps, where all information about the initial state is eventually lost. Where has it gone? If the whole system evolves unitary, then this information must have been transferred in the orrelations between redu ed system and environment [162℄, and/or in the environment. We an see that this may be useful for quantum state transfer, in parti ular the ase where all information is transferred into the �environment�, whi h ould be another quantum system (the re eiver). A parti ularly useful ase is given by mixing maps with pure onvergen e points, be ause a pure state annot be orrelated, and be ause we have a simple onvergen e riterion in this ase (Subse tion 4.4.3). This is an example of homogenisation [138, 139℄. Furthermore, if the mixing property arises from some operations, we an expe t that by applying the inverse operations, information an also be transferred ba k to the system. This property was used in [137, 163℄ to generate arbitrary states of a avity �eld by sending atoms through the avity. The ru ial di�eren e is that in our system ontrol is only assumed to be available on a subsystem (su h as, for example, the ends of a quantum hain). Hen e we will show in this hapter how arbitrary quantum states an be written to (i.e. prepared on) a large system, and read from it, by lo al ontrol only. This is similar in spirit to universal quantum interfa es [164℄, but our di�erent approa h allows us to spe ify expli it proto ols and to give lower bounds for �delities. We also demonstrate how this an be used to signi� antly improve the quantum ommuni ation between two parties if the re eiver is allowed to store the re eived signals in a quantum memory 5 Read and write a ess by lo al ontrol before de oding them. In the limit of an in�nite memory, the transfer is perfe t. We prove that this s heme allows the transfer of arbitrary multi-partite states along Heisenberg hains of spin-1/2 parti les with random oupling strengths. Even though the onvergen e of a mixing map is essentially exponentially fast (Corollary 4.3), we still have to deal with in�nite limits. Looking at the environment this in turn would require to study states on an in�nite dimensional Hilbert spa e, and unfortunately this an introdu e many mathemati al di� ulties. We are mainly interested in bounds for the �nite ase: if the proto ol stops after �nitely many steps, what is the �delity of the reading/writing? Whi h en oding and de oding operations must be applied? By stressing on these questions, we an a tually avoid the in�nite dimensional ase, but the pri e we have to pay is that our onsiderations be ome a bit te hni ally involved. 5.2 Proto ol We onsider a tripartite �nite dimensional Hilbert spa e given by H = HC⊗HC̄⊗HM . We assume that full ontrol (the ability to prepare states and apply unitary transfor- mations) is possible on system C and M, but no ontrol is available on system C̄. However, we assume that C and C̄ are oupled by some time-independent Hamilto- nian H. We show here that under ertain assumptions, if the system CC̄ is initialised in some arbitrary state we an transfer (�read�) this state into the systemM by apply- ing some operations between M and C. Likewise, by initialising the system M in the orre t state, we an prepare (�write�) arbitrary states on the system CC̄. The system M fun tions as a quantum memory and must be at least as large as the system CC̄. As sket hed in Fig. 5.1 we an imagine it to be split into se tors Mℓ, I.e.. HMℓ (5.1) dimHMℓ = dimHC . (5.2) For the reading ase, we assume that the memory is initialised in the state |0〉M ≡ |0〉Mℓ (5.3) 5 Read and write a ess by lo al ontrol where |0〉 an stand for some generi state1. Like in the multi rail proto ols onsidered in Chapter 3, we let the system evolve for a while, perform an operation, let it evolve again and so forth, only that now the operation is not a measurement, but a unitary gate. More spe i� ally, at step ℓ of the proto ol we perform a unitary swap Sℓ between system C and systems Mℓ. After the Lth swap operation the proto ol stops. The proto ol for reading is thus represented by the unitary operator W ≡ SLUSL−1U · · ·SℓU · · ·S1U, (5.4) where U ∈ L(HCC̄) is the time-evolution operator U = exp {−iHt} for some �xed time interval t. As we will see in the next se tion, the redu ed evolution of the system C̄ under the proto ol an be expressed in terms of the CPT map τ(ρC̄) ≡ trC U (ρC̄ ⊗ |0〉C〈0|)U † , (5.5) where |0〉C is the state that is swapped in from the memory. Our main assumption now is that τ is ergodi with a pure �xed point (whi h we denote as |0〉C̄). By Theorem 4.8 this implies that τ is mixing, and therefore asymptoti ally all information is transferred into the memory. For writing states on the system, we just make use of the unitarity of W. Roughly speaking, we initialise the memory in the state that it would have ended up in after applying W if system CC̄ had started in the state we want to initialise. Then we apply the inverse of W given by W † = U †S1 · · ·U †Sℓ · · ·U †SL−1U †SL. (5.6) We will see in Se tion 5.4 how this gives rise to a unitary oding transformation on the memory system, su h that arbitrary and unknown states an be initialised on the system. The reader has probably noti ed that the inverse ofW is generally unphysi al in the sense that it requires ba kward time evolution, i.e. one has to wait negative time steps between the swaps. But we will see later how this an be �xed by a simple transformation. For the moment, we just assume that W † is physi al. Later on we will give an example where |0〉 represents a multi-qubit state with all qubits aligned, but here we don't need to assume this. 5 Read and write a ess by lo al ontrol Mℓ+1 Mℓ+2MℓMℓ−2 Mℓ−1 Writing Reading Figure 5.1: The system CC̄ an only be ontrolled by a ting on a (small) subsystem C. However system C is oupled to system C̄ by a unitary operator U = exp {−iHt} . This oupling an - in some ases - mediate the lo al ontrol on C to the full system CC̄. In our ase, system C is ontrolled by performing regular swap operations Sℓ between it and a quantum memory Mℓ. 5.3 De omposition equations In this se tion we give a de omposition of the state after applying the proto ol whi h will allow us to estimate the �delities for state transfer in terms of the mixing properties of the map τ. Let |ψ〉CC̄ ∈ HCC̄ be an arbitrary state. We noti e that the C omponent of W |ψ〉CC̄ |0〉M is always |0〉C . Therefore we an de ompose it as follows W |ψ〉CC̄ |0〉M = |0〉C ⊗ η|0〉C̄ |φ〉M + 1− η|∆〉C̄M (5.7) with |∆〉C̄M being a normalised ve tor of C̄ and M whi h satis�es the identity C̄〈0|∆〉C̄M = 0 . (5.8) 5 Read and write a ess by lo al ontrol It is worth stressing that in the above expression η, |φ〉M and |∆〉C̄M are depending on |ψ〉CC̄ . We de ompose W † a ting on the �rst term of Eq. (5.7) as W †|0〉CC̄ |φ〉M = η̃ |ψ〉CC̄ |0〉M + 1− η̃ |∆̃〉CC̄M , (5.9) where |∆̃〉CC̄M is the orthogonal omplement of |ψ〉CC̄ |0〉M , i.e. C̄C〈ψ|M 〈0|∆̃〉CC̄M = 0 . (5.10) Multiplying Eq. (5.9) from the left with CC̄〈ψ|M 〈0| and using the onjugate of Eq. (5.7) we �nd that η = η̃. An expression of η in terms of τ an be obtained by noti ing that for any ve tor |ψ〉C̄C the following identity applies τ(ρC̄) = trC U (ρC̄ ⊗ |0〉C〈0|)U † = trCM USℓ (|ψ〉C̄C〈ψ| ⊗ |0〉M 〈0|) SℓU † (5.11) with ρC̄ being the redu ed density matrix trC [|ψ〉C̄C〈ψ|]. Reiterating this expression one gets W (|ψ〉CC̄〈ψ| ⊗ |0〉M 〈0|)W † = τL−1 (5.12) with ρ′ = trC U (|ψ〉C̄C〈ψ|)U † . Therefore from Eq. (5.7) and the orthogonality relation (5.8) it follows that η = C̄〈0|τL−1 |0〉C̄ , (5.13) whi h, sin e τ is mixing, shows that η → 1 for L→ ∞. Moreover we an use Eq. (4.31) to laim that |η − 1| = |C̄〈0|τL−1 |0〉C̄ − 1| ≤ ‖τL−1 − |0〉C̄〈0|‖1 ≤ R (L− 1)dC̄ κL−1, (5.14) where R is a onstant whi h depends upon dC̄ ≡ dimHC̄ and where κ ∈]0, 1[ is the se ond largest eigenvalue of τ. 5.4 Coding transformation Here we derive the de oding/en oding transformation that relates states on the mem- ory M to the states that are on the system CC̄. We �rst apply the above de omposi- 5 Read and write a ess by lo al ontrol tions Eqs. (5.7) and (5.9) to a �xed orthonormal basis {|ψk〉CC̄} of HCC̄ , i.e. W |ψk〉CC̄ |0〉M = |0〉C ⊗ ηk|0〉C̄ |φk〉M + 1− ηk|∆k〉C̄M W †|0〉CC̄ |φk〉M = ηk |ψk〉CC̄ |0〉M + 1− ηk |∆̃k〉CC̄M . (5.15) De�ne a linear operator D on HM whi h performs the following transformation D|ψk〉M = |φk〉M . (5.16) Here |ψk〉M are orthonormal ve tors ofM whi h represent the states {|ψk〉CC̄} of HCC̄ (formally they are obtained by a partial isometry from C̄C to M). The ve tors |φk〉M are de�ned through Eq. (5.15) - typi ally they will not be orthogonal. We �rst show that for large L they be ome approximately orthogonal. From the unitarity ofW † and from Eq. (5.15) we an establish the following identity M 〈φk|φk′〉M = ηk ηk′ δkk′ + ηk (1− ηk′) C̄CM 〈ψk0|∆̃k′〉C̄CM (5.17) ηk′ (1− ηk) C̄CM 〈∆̃k|ψk′0〉C̄CM + (1− η̃k)(1 − η̃k′) CC̄M 〈∆̃k|∆̃k′〉CC̄M . De�ning η0 ≡ mink ηk it follows for k 6= k′ that |M 〈φk|φk′〉M | ≤ ηk (1− ηk′) |C̄CM 〈ψk0|∆̃k′〉C̄CM | (5.18) ηk′ (1− ηk) |C̄CM 〈∆̃k|ψk′0〉C̄CM | (1− η̃k)(1− η̃k′) |CC̄M 〈∆̃k|∆̃k′〉CC̄M | 1− η0 + (1− η0) ≤ 3 1− η0. (5.19) Therefore for all k, k′ the inequality |M 〈φk|φk′〉M − δk,k′ | ≤ 3 1− η0 (5.20) holds. It is worth noti ing that, sin e Eq. (5.14) applies for all input states |ψ〉C̄C , we |η0 − 1| ≤ C (L− 1)dC̄ κL−1 . (5.21) Eq. (5.20) allows us to make an estimation of the eigenvalues λk of D †D as |λk − 1| ≤ 3 dCC̄ 1− η0, (5.22) 5 Read and write a ess by lo al ontrol with dCC̄ ≡ dimHCC̄ . We now take a polar de omposition D = PV of D. V is the best unitary approximation to D [160, p 432℄ and we have ||D − V ||22 = λk − 1 |λk − 1| ≤ 3 d2 1− η0. (5.23) Therefore ||D − V ||2 ≤ 3 dCC̄ (1− η0)1/4, (5.24) whi h, thanks to Eq. (5.21), shows that D an be approximated arbitrary well by a unitary operator V for L→ ∞. 5.5 Fidelities for reading and writing In what follows we will use V † and V as our reading and writing transformation, respe tively. In parti ular, V † will be used to re over the input state |ψ〉CC̄ of the hain after we have (partially) transferred it into M through the unitary W (i.e. we �rst a t on |ψ〉CC̄ ⊗ |0〉M with W , and then we apply V † on M). Vi e-versa, in order to prepare a state |ψ〉CC̄ on CC̄ we �rst prepare M into |ψ〉M , then we apply to it the unitary transformation V and �nally we apply W †. We now give bounds on the �delities for both pro edures. The �delity for reading the state |ψ〉M is given by Fr(ψ) ≡ M 〈ψ|V † RM V |ψ〉M (5.25) where RM is the state of the memory after W , i.e. RM ≡ trCC̄ W (|ψ〉CC̄〈ψ| ⊗ |0〉M 〈0|)W † = η |φ〉M 〈φ|+ (1− η) σM . (5.26) In the above expression we used Eqs. (5.7) and (5.8) and de�ned σM = trC̄ [|∆〉C̄M 〈∆|]. Therefore by linearity we get Fr(ψ) = η |M 〈φ|V |ψ〉M |2 + (1− η) M 〈ψ|V † σM V |ψ〉M ≥ η |M 〈φ|V |ψ〉M |2 . (5.27) 5 Read and write a ess by lo al ontrol Noti e that |M 〈φ|V |ψ〉M | = |M 〈φ|V −D +D|ψ〉M | ≥ |M 〈φ|D|ψ〉M | − |M 〈φ|D − V |ψ〉M | .(5.28) Now we use the inequality (5.24) to write |M 〈φ|D − V |ψ〉M | ≤ ||D − V ||2 ≤ 3 dCC̄ (1− η0)1/4 . (5.29) If |ψ〉M was a basis state |ψk〉M , then |M 〈φ|D|ψ〉M | = 1 by the de�nition Eq. (5.16) of D. For generi |ψ〉M we an use the linearity to �nd after some algebra that η |M 〈φ|D|ψ〉M | ≥ η0 − 3 dCC̄ 1− η0 . (5.30) Therefore Eq. (5.28) gives η |M 〈φ|V |ψ〉M | > η0 − 5 dCC̄ (1− η0)1/4 . (5.31) By Eq. (5.27) it follows that Fr ≥ η0 − 10 dCC̄ (1− η0)1/4 . (5.32) The �delity for writing a state |ψ〉C̄C into C̄C is given by Fw(ψ) ≡ CC̄〈ψ|trM W †V (|ψ〉M 〈ψ| ⊗ |0〉C̄C〈0|) V †W |ψ〉CC̄ . (5.33) A lower bound for this quantity is obtained by repla ing the tra e over M with the expe tation value on |0〉M , i.e. Fw(ψ) ≥ CC̄〈ψ|M 〈0|W †V (|ψ〉M 〈ψ| ⊗ |0〉C̄C〈0|) V †W |0〉M |ψ〉CC̄ ∣CC̄〈0|M 〈ψ|V †W |0〉M |ψ〉CC̄ ∣M 〈ψ|V †|φ〉M = η |M 〈φ|V |ψ〉M |2 (5.34) where Eqs. (5.7) and the orthogonality relation (5.8) have been employed to derive the se ond identity. Noti e that the last term of the inequality (5.34) oin ides with the lower bound (5.27) of the reading �delity. Therefore, by applying the same derivation of the previous se tion we an write F ≥ η0 − 10 dCC̄ (1− η0)1/4, (5.35) 5 Read and write a ess by lo al ontrol NR NB S Figure 5.2: Ali e and Bob ontrol the spins NA and NB inter onne ted by the spins NR. At time jt Bob performs a swap Sj between his spins and the memory Mj . whi h shows that the reading and writing �delities onverge to 1 in the limit of large L. Note that this lower bound an probably be largely improved. 5.6 Appli ation to spin hain ommuni ation We now show how the above proto ol an be used to improve quantum state transfer on a spin hain. The main advantage of using su h a memory proto ol is that - opposed to all other s hemes - Ali e an send arbitrary multi-qubit states with a single usage of the hannel. She needs no en oding, all the work is done by Bob. The proto ol proposed here an be used to improve the performan es of any s heme mentioned in Se tion 1.5, and it works for a large lass of Hamiltonians, in luding Heisenberg and XY models with arbitrary (also randomly distributed) oupling strengths. Consider a hain of spin-1/2 parti les des ribed by a Hamiltonian H whi h onserves the number of ex itations. The hain is assumed to be divided in three portions A (Ali e), B (Bob) and R (the remainder of the hain, onne ting Ali e and Bob) ontaining respe tively the �rst NA spins of the hain, the last NB spins and the intermediate NR spins, and the total length of the hain is N = NA+NR+NB (see Fig 5.2). Bob has a ess also to a olle tion of quantum memories M1, · · · ,Mj · · · ,ML isomorphi with B, i.e. ea h having dimension equal to the dimension 2NB of B. We assume that Bob's memory is initialised in the zero ex itation state |0〉M . Ali e prepares an arbitrary and unknown state |ψ〉A on her NA qubits. By de�ning the (from Bob's perspe tive) ontrolled part of system C = B and the un ontrolled part C̄ = AR, we an apply the results of the last se tions and get the following Theorem 5.1 (Memory swapping) Let H be the Hamiltonian of an open nearest-neighbour quantum hain that onserves the number of ex itations. If there is a time t su h that f1,N (t) 6= 0 (i.e. the Hamiltonian is apable of transport be- tween Ali e and Bob) then the state transfer an be made arbitrarily perfe t by using the memory swapping proto ol. 5 Read and write a ess by lo al ontrol Proof We only have to show that the redu ed dynami s on the hain is mixing with a pure �xed point. Using the number of ex itations as a onserved additive observable, we an use the riterion of Lemma 4.7: If there exists exa tly one eigenstate |E〉 of fa torising form with |0〉B , i.e. ∃1 |λ〉AR : H|λ〉AR ⊗ |0〉B = E|λ〉AR ⊗ |0〉B , (5.36) then the redu ed dynami s is mixing toward |0〉AR. Assume by ontradi tion that has an eigenve tor |E〉AR 6= |0〉AR whi h falsi�es Eq. (5.36). Su h an eigenstate an be written as |E〉AR ⊗ |0〉B = a|µ〉AR ⊗ |0〉B + b|µ̄〉AR ⊗ |0〉B , (5.37) where a and b are omplex oe� ients and where the spin just before the se tion B (with position NA + NR) is in the state |0〉 for |µ〉AR and in the state |1〉 for |µ̄〉AR. Sin e the intera tion between this spin and the �rst spin of se tion B in ludes an ex hange term (otherwise f1,N (t)=0 for all t), then the a tion of H on the se ond term of (5.37) yields exa tly one state whi h ontains an ex itation in the se tor B. It annot be ompensated by the a tion of H on the �rst term of (5.37). But by assumption |E〉AR ⊗ |0〉B is an eigenstate of H, so we on lude that b = 0. This argument an be repeated for the se ond last spin of se tion R, the third last spin, and so on, to �nally yield |E〉AR = |0〉AR, as long as all the nearest neighbour intera tions ontain ex hange parts. � Remark 5.1 Theorem 5.1 should be ompared to Theorem 3.2 for the multi rail pro- to ol. They are indeed very similar. However the urrent theorem is mu h stronger, sin e it allows to send arbitrary multi-ex itation states, and also to write states ba k onto the hain. It is interesting to note that Lemma 4.7 and Theorem 5.1 indi ate a onne tion between the dynami al ontrollability of a system and its stati entangle- ment properties. It may be interesting to obtain a quantitative relation between the amount of entanglement and the onvergen e speed. Let us now ome ba k to the question raised in Se tion 5.2 about the operation W † being unphysi al. As mentioned before, this an be �xed using a simple transfor- mation: if the Hamiltonian H ful�ls the requirements of Lemma 4.7, then also the Hamiltonian −H ful�ls them. Now derive the oding transformation Ṽ as given in Se tion 5.4 for the Hamiltonian H̃ = −H. In this pi ture, the reading proto ol W is unphysi al, whereas the writing proto ol be omes physi al. In the more general ase 5 Read and write a ess by lo al ontrol where the ondition of Lemma 4.7 is not valid, but the map τ(ρC̄) ≡ trC U (ρC̄ ⊗ |0〉C〈0|)U † (5.38) is still ergodi with a pure �xed point, we then require the map τ̃(ρC̄) ≡ trC U † (ρC̄ ⊗ |0〉C〈0|)U (5.39) to be also ergodi with pure �xed point to be able to use this tri k. 5.7 Con lusion We have given an expli it proto ol for ontrolling a large permanently oupled system by a essing a small subsystem only. In the ontext of quantum hain ommuni ation this allows us to make use of the quantum memory of the re eiving party to improve the �delity to a value limited only by the size of the memory. We have shown that this s heme an be applied to a Heisenberg spin hain. The main advantage of this method is that arbitrary multi-ex itation states an be transferred. Also, our method an be applied to hains that do not onserve the number of ex itations in the system, as long as the redu ed dynami is ergodi with a pure �xed point. It remains an open question how mu h of our results remain valid if the hannel is mixing toward a mixed state. In this ase, a part of the quantum information will in general remain in the orrelations between the system and the memory, and it annot be expe ted that the �delity onverges to one. However, by on entrating only on the eigenstate of the �xed point density operator with the largest eigenvalue, it should be possible to derive some bounds of the amount of information that an be extra ted. 6 A valve for probability amplitude 6.1 Introdu tion We have mainly dis ussed two methods for quantum state transfer so far. In the �rst one, multiple hains where used, and in the se ond one, a single hain was used in ombination with a large quantum memory. Can we ombine the best of the two s hemes, i.e. is it possible to use only a single hain and a single memory qubit? In this hapter we will show that this is indeed the ase and that the �delity an be improved easily by applying in ertain time-intervals two-qubit gates at the re eiving end of the hain. These gates a t as a valve whi h takes probability amplitude out of the system without ever putting it ba k. The required sequen e is determined a priori by the Hamiltonian of the system. Su h a proto ol is optimal in terms of resour es, be ause two-qubit gates at the sending and re eiving end are required in order to onne t the hain to the blo ks in all above proto ols (though often not mentioned expli itly). At the same time, the engineering demands are not higher then for the memory swapping proto ol. Our s heme has some similarities with [92℄, but the gates used here are mu h simpler, and arbitrarily high �delity is guaranteed by a onvergen e theorem for arbitrary oupling strengths and all non-Ising oupling types that onserve the number of ex itations. Furthermore, we show numeri ally that our proto ol ould also be realised by a simple swit hable intera tion. 6.2 Arbitrarily Perfe t State Transfer We now show how the re eiver an improve the �delity to an arbitrarily high value by applying two-qubit gates between the end of the hain and a �target qubit� of the blo k. We label the qubits of the hain by 1, 2, · · · , N and the target qubit by N + 1 (see Fig. 6.1). The oupling of the hain is des ribed by a Hamiltonian H. We assume that the Hamiltonian H onserves the number of ex itations and that the target qubit N + 1 is un oupled, H|N + 1〉 = 0 (6.1) 6 A valve for probability amplitude and set the energy of the ground state |0〉 to zero. For what follows we restri t all operators to the N + 2 dimensional Hilbert spa e H = span {|n〉; n = 0, 1, 2, . . . , N + 1} . (6.2) Our �nal assumption about the Hamiltonian of the system is that there exists a time t su h that fN,t(t) ≡ 〈N | exp {−itH} |1〉 6= 0. (6.3) Physi ally this means that the Hamiltonian has the apability of transporting from the �rst to the last qubit of the hain. As mentioned in the introdu tion, the �delity of this transport may be very bad in pra ti e. 1 2 N N+1 Figure 6.1: A quantum hain (qubits 1, 2, · · · , N) and a target qubit (N + 1). By applying a sequen e of two-qubit unitary gates Vk on the last qubit of the hain and the target qubit, arbitrarily high �delity an be a hieved. We denote the unitary evolution operator for a given time tk as Uk ≡ exp {−itkH} and introdu e the proje tor P = 1− |0〉〈0| − |N 〉〈N | − |N + 1〉〈N + 1|. (6.4) A ru ial ingredient to our proto ol is the operator V (c, d) ≡ P + |0〉〈0|+ d|N 〉〈N |+ d∗|N + 1〉〈N + 1| +c∗|N + 1〉〈N | − c|N 〉〈N + 1|, (6.5) where c and d are omplex normalised amplitudes. It is easy to he k that V V † = V †V = 1, (6.6) so V is a unitary operator on H. V a ts as the identity on all but the last two qubits, and an hen e be realised by a lo al two-qubit gate on the qubits N and N +1. 6 A valve for probability amplitude Furthermore we have V P = P and V (c, d) [{c|N 〉+ d|N + 1〉}] = |N + 1〉. (6.7) The operator V (c, d) has the role of moving probability amplitude c from the Nth qubit to target qubit, without moving amplitude ba k into the system, and an be thought of as a valve. Of ourse as V (c, d) is unitary, there are also states su h that V (c, d) a ting on them would move ba k probability amplitude into the system, but these do not o ur in the proto ol dis ussed here. Using the time-evolution operator and two-qubit unitary gates on the qubits N and N +1 we will now develop a proto ol that transforms the state |1〉 into |N + 1〉. Let us �rst look at the a tion of U1 on |1〉. Using the proje tor P we an de ompose this time-evolved state as U1|1〉 = PU1|1〉+ |N 〉〈N |U1|1〉 ≡ PU1|1〉+ p1 {c1|N 〉+ d1|N + 1〉} , (6.8) where p1 = |〈N |U1|1〉|2 , c1 = 〈N |U1|1〉/ p1 and d1 = 0. Let us now onsider the a tion of V1 ≡ V (c1, d1) on the time-evolved state. By Eq. (6.7) it follows that V1U1|1〉 = PU1|1〉+ p1|N + 1〉. (6.9) Hen e with a probability of p1, the ex itation is now in the position N +1, where it is �frozen� (sin e that qubit is not oupled to the hain. We will now show that at the next step, this probability is in reased. Applying U2 to Eq. (6.9) we get U2V1U1|1〉 = PU2PU1|1〉+ 〈N |U2PU1|1〉|N 〉+ p1|N + 1〉 = PU2PU1|1〉+ p2 {c2|N 〉+ d2|N + 1〉} (6.10) with c2 = 〈N |U2PU1|1〉/ p2, d2 = p2 and p2 = p1 + |〈N |U2PU1|1〉|2 ≥ p1. (6.11) Applying V2 ≡ V (c2, d2) we get V2U2V1U1|1〉 = PU2PU1|1〉+ p2|N + 1〉. (6.12) 6 A valve for probability amplitude Repeating this strategy ℓ times we get |1〉 = |1〉+√pℓ|N + 1〉, (6.13) where the produ ts are arranged in the time-ordered way. Using the normalisation of the r.h.s. of Eq. (6.13) we get pℓ = 1− . (6.14) From Se tion 3.5 we know that there exists a t > 0 su h that for equal time intervals t1 = t2 = . . . = tk = t we have limℓ→∞ pℓ = 1. Therefore the limit of in�nite gate operations for Eq. (6.13) is given by |1〉 = |N + 1〉. (6.15) It is also easy to see that limk→∞ dℓ = 1, limk→∞ cℓ = 0 and hen e the gates Vk onverge to the identity operator. Furthermore, sin e VkUk|0〉 = |0〉 it also follows that arbitrary superpositions an be transferred. As dis ussed in Theorem 4.31, this onvergen e is asymptoti ally exponentially fast in the number of gate applied (a detailed analysis of the relevant s aling an be found in Chapter 2). Equation (6.15) is a surprising result, whi h shows that any non-perfe t transfer an be made arbitrarily perfe t by only applying two-qubit gates on one end of the quantum hain. It avoids restri ting the gate times to spe i� times (as opposed to the dual rail s heme) while requiring no additional memory qubit (as opposed to the memory swapping s heme). The sequen e Vk that needs to be applied to the end of the hain to perform the state transfer only depends on the Hamiltonian of the quantum hain. The relevant properties an in prin iple be determined a priori by pre eding measurements and tomography on the quantum hain (as dis ussed in Se t. 2.9). 6.3 Pra ti al Considerations Motivated by the above result we now investigate how the above proto ol may be implemented in pra ti e, well before the realisation of the quantum omputing blo ks from Fig. 1.4. The two-qubit gates Vk are essentially rotations in the {|01〉, |10〉} spa e of the qubits N and N +1. It is therefore to be expe ted that they an be realised (up 6 A valve for probability amplitude to a irrelevant phase) by a swit hable Heisenberg or XY type oupling between the Nth and the target qubit. However in the above, we have assumed that the gates Vk an be applied instantaneously, i.e. in a time-s ale mu h smaller than the time-s ale of the dynami s of the hain. This orresponds to a swit hable oupling that is mu h stronger than the oupling strength of the hain. 0 100 200 300 400 500 600 700 800 Time [1/J] switched magnetic field switched interaction Figure 6.2: Numeri al example for the onvergen e of the su ess probability. Sim- ulated is a quantum hain of length N = 20 with the Hamiltonian from Eq. (6.16) (dashed line) and Eq. (6.17) with B/J = 20 (solid line). Using the original proto- ol [1℄, the same hain would only rea h a su ess probability of 0.63 in the above time interval. Here, we numeri ally investigate if a onvergen e similar to the above results is still possible when this assumption is not valid. We do however assume that the swit hing of the intera tion is still des ribable by an instantaneous swit hing (i.e. the sudden approximation is valid). This assumption is mainly made to keep the numeri s simple. We do not expe t qualitative di�eren es when the swit hing times be ome �nite as long as the time-dependent Hamiltonian is still onserving the number of ex itations in the hain. In fa t it has re ently been shown that the �nite swit hing time an even improve the �delity [33℄. Intuitively, this happens be ause by gradually de reasing the oupling, he not only re eives the probability amplitude of the last qubit of the hain, but an also �swallow� a bit of the dispersed wave-pa ked (similar to the situation dis ussed in [92℄). We have investigated two types of swit hing. For the �rst type, the oupling itself 6 A valve for probability amplitude is swit hable, i.e. H(t) = J σ−n σ n+1 +∆(t)σ N+1 + h. ., (6.16) where ∆(t) an be 0 or 1. For the se ond type, the target qubit is permanently oupled to the remainder of the hain, but a strong magneti �eld on the last qubit an be swit hed, H(t) = J σ−n σ n+1 + h. .+B∆(t)σ N+1, (6.17) where again ∆(t) an be 0 or 1 and B ≫ 1. This suppresses the oupling between the Nth and N + 1th qubit due to an energy mismat h. In both ases, we �rst numeri ally optimise the times for unitary evolution tk over a �xed time interval su h that the probability amplitude at the Nth qubit is maximal. The algorithm then �nds the optimal time interval during whi h ∆(t) = 1 su h that the probability amplitude at the target qubit is in reased. In some ases the phases are not orre t, and swit hing on the intera tion would result in probability amplitude �oating ba k into the hain. In this situation, the target qubit is left de oupled and the hain is evolved to the next amplitude maximum at the Nth qubit. Surprisingly, even when the time-s ale of the gates is omparable to the dynami s, near-perfe t transfer remains possible (Fig 6.2). In the ase of the swit hed magneti �eld, the a hievable �delity depends on the strength of the applied �eld. This is be ause the magneti �eld does not fully suppress the oupling between the two last qubits. A small amount of probability amplitude is lost during ea h time evolution Uk, and when the gain by the gate is ompensated by this loss, the total su ess probability no longer in reases. 6.4 Con lusion We have seen that by having a simple swit hable intera tion a ting as a valve for probability amplitude, arbitrarily perfe t state transfer is possible on a single spin hain. In fa t, by using the inverse proto ol, arbitrary states in the �rst ex itation se tor an also be prepared on the hain. Furthermore, this proto ol an easily be adopted to arbitrary graphs onne ting multiple senders and re eivers (as dis ussed for weakly oupled systems in [86℄). Opposed to the method for state preparation developed in the last hapter this allows the reation of known states only (as the valve operations Vk depend expli itly on the state that one wants to prepare). 7 External noise 7.1 Introdu tion An important question that was left open so far is what happens to quantum state transfer in the presen e of external noise. It is well known from the theory of open quantum systems [121℄ that this an lead to dissipation and de oheren e, whi h also means that quantum information is lost. The evolution of a losed quantum system is des ribed by the S hrödinger equation ∂t|ψ〉 = −iH|ψ〉. (7.1) If a system is very strongly oupled to a environment, the dynami is ompletely in o- herent and des ribed by some simple rate equations for the o upation probabilities, ∂tPn = kn→mPn − km→nPm. (7.2) In the more general ase where the dynami onsists of oherent and in oherent parts, the evolution an sometimes be expressed as a Lindblad equation [121℄ ∂tρ = Lρ (7.3) for the redu ed density matrix. These three regimes are shown in Fig. 7.1. For quantum information theory, oheren e is essential [2℄, and one has to try to isolate the quantum hain as mu h as possible from the environment. In the partially oherent regime, typi ally the quantum behaviour de ays exponentially with a rate depending on the temperature of the environment. Not surprisingly, this has also been found in the ontext of quantum state transfer [165, 166, 167℄. From a theoreti al point of view it is perhaps more interesting to look at the low temperature and strong oupling regime, where the dynami s is often non-Markovian [121℄ and an no longer expressed as a simple Lindblad equation. This is also interesting from a pra ti al perspe tive, orresponding to e�e ts of the environment whi h annot be avoided by ooling. Here we onsider a model where the system is oupled to a spin environment 7 External noise coherent partially coherent incoherent System coupling Figure 7.1: Dominant regimes of dynami s depending on the relative strength of the system Hamiltonian and the environmental oupling [47℄. through an ex hange intera tion. This oupling o�ers the unique opportunity of an analyti solution of our problem without any approximations regarding the strength of system-environment oupling (in most treatments of the e�e t of an environment on the evolution of a quantum system, the system-environment oupling is assumed to be weak) and allows us to in lude inhomogeneous intera tions of the bath spins with the system. For su h oupling, de oheren e is possible for mixed (thermal) initial bath states [168, 169℄. However if the system and bath are both initially ooled to their ground states, is there still a non-trivial e�e t of the environment on the �delity? In this hapter we �nd that there are two important e�e ts: the spin transfer fun tions (Eq. 1.19) are slowed down by a fa tor of two, and destabilised by a modulation of |cosGt| , where G is the mean square oupling to the environment. This has both positive and negative impli ations for the use of strongly oupled spin systems as quantum ommuni ation hannels. The spin transfer fun tions also o ur in the harge and energy transfer dynami s in mole ular systems [47℄ and in ontinuous time random walks [170℄ to whi h our results equally apply. 7 External noise 7.2 Model We hoose to start with a spe i� spin system, i.e. an open spin hain of arbitrary length N, with a Hamiltonian given by HS = − Jℓ (XℓXℓ+1 + YℓYℓ+1) , (7.4) where Jℓ are some arbitrary ouplings and Xℓ and Yℓ are the Pauli-X and Y matri es for the ℓth spin. Toward the end of the se tion we will however show that our results hold for any system where the number of ex itations is onserved during dynami al evolution. In addition to the hain Hamiltonian, ea h spin ℓ of the hain intera ts with an independent bath of Mℓ environmental spins (see Fig 7.2) via an inhomogeneous Hamiltonian, I = − k + YℓY . (7.5) Figure 7.2: A spin hain of length N = 5 oupled to independent baths of spins. In the above expression, the Pauli matri es Xℓ and Yℓ a t on the ℓth spin of the hain, whereas X k and Y k a t on the kth environmental spin atta hed to the ℓth spin of the hain. We denote the total intera tion Hamiltonian by I . (7.6) The total Hamiltonian is given by H = HS +HI , where it is important to note that [HS,HI ] 6= 0.We assume that a homogeneous magneti �eld along the z-axis is applied. The ground state of the system is then given by the fully polarised state |0, 0〉, with all hain and bath spins aligned along the z-axis. The above Hamiltonian des ribes an extremely omplex and disordered system with a Hilbert spa e of dimension 2N+NM . In the ontext of state transfer however, only the dynami s of the �rst ex itation se tor is relevant. We pro eed by mapping this se tor to a mu h simpler system [171, 172, 7 External noise 173,174, 175℄. For ℓ = 1, 2, . . . , N we de�ne the states |ℓ, 0〉 ≡ Xℓ|0, 0〉 (7.7) |0, ℓ〉 ≡ 1 k |0, 0〉 (7.8) . (7.9) It is easily veri�ed that (setting J0 = JN = 0) HS|ℓ, 0〉 = −Jℓ−1|ℓ− 1, 0〉 − Jℓ|ℓ+ 1, 0〉 HS|0, ℓ〉 = 0, (7.10) HI |ℓ, 0〉 = −Gℓ|0, ℓ〉 (7.11) HI |0, ℓ〉 = −Gℓ|ℓ, 0〉. (7.12) Hen e these states de�ne a 2N−dimensional subspa e that is invariant under the a tion of H. This subspa e is equivalent to the �rst ex itation se tor of a system of 2N spin 1/2 parti les, oupled as it is shown in Fig 7.3. Figure 7.3: In the �rst ex itation se tor, the system an be mapped into an e�e tive spin model where the bath spins are repla ed by a single e�e tive spin, as indi ated here for N = 5. Our main assumption is that the bath ouplings are in e�e t the same, i.e. Gℓ = G for all ℓ. Note however that the individual number of bath spinsMℓ and bath ouplings may still depend on ℓ and k as long as their means square average is the same. Also, our analyti solution given in the next paragraph relies on this assumption, but numeri s show that our main result [Equation (7.28)℄ remains a good approximation if the Gℓ slightly vary and we take G ≡ 〈Gℓ〉 . Disorder in the verti al ouplings is treated exa tly in the sense that our results hold for any hoi e of ouplings Jℓ. 7 External noise 7.3 Results In this paragraph, we solve the S hrödinger equation for the model outlined above and dis uss the spin transfer fun tions. Firstly, let us denote the orthonormal eigenstates of HS alone by HS|ψk〉 = ǫk|ψk〉 (k = 1, 2 . . . , N) (7.13) |ψk〉 = akℓ|ℓ, 0〉. (7.14) For what follows, it is not important whether analyti expressions for the eigensystem ofHS an be found. Our result holds even for models that are not analyti ally solvable, su h as the randomly oupled hains onsidered in Se tion 2.6. We now make an ansatz for the eigenstates of the full Hamiltonian, motivated by the fa t that the states |φnℓ 〉 ≡ (|ℓ, 0〉 + (−1)n |0, ℓ〉) (n = 1, 2) (7.15) are eigenstates of H I with the orresponding eigenvalues ±G [this follows dire tly from Eqs. (7.11) and (7.12)℄. De�ne the ve tors |Ψnk〉 ≡ akℓ|φnℓ 〉 (7.16) with k = 1, 2, . . . , N and n = 0, 1. The |Ψnk〉 form an orthonormal basis in whi h we express the matrix elements of the Hamiltonian. We an easily see that HI |Ψnk〉 = − (−1) G|Ψnk〉 (7.17) HS|Ψnk〉 = akℓ|ℓ, 0〉 = |Ψ0k〉+ |Ψ1k〉 . (7.18) Therefore the matrix elements of the full Hamiltonian H = HS +HI are given by 〈Ψn′k′ |H|Ψnk〉 = δkk′ − (−1)nGδnn′ + . (7.19) The Hamiltonian is not diagonal in the states of Eq. (7.16). But H is now blo k diagonal onsisting of N blo ks of size 2, whi h an be easily diagonalised analyti ally. 7 External noise The orthonormal eigenstates of the Hamiltonian are given by |Enk 〉 = c−1kn ((−1)n∆k − 2G) |Ψ0k〉+ ǫk|Ψ1k〉 (7.20) with the eigenvalues Enk = (ǫk + (−1)n∆k) (7.21) and the normalisation ckn ≡ ((−1)n∆k − 2G)2 + ǫ2k, (7.22) where 4G2 + ǫ2k. (7.23) Note that the ansatz of Eq. (7.16) that put H in blo k diagonal form did not depend on the details of HS and H I . The methods presented here an be applied to a mu h larger lass of systems, in luding the generalised spin star systems (whi h in lude an intera tion within the bath) dis ussed in [175℄. After solving the S hrödinger equation, let us now turn to quantum state transfer. The relevant quantity [1, 92℄ is given by the transfer fun tion fN,1(t) ≡ 〈N, 0| exp {−iHt} |1, 0〉 exp {−iEnk t} 〈Enk |1, 0〉〈N, 0|Enk 〉. The modulus of fN,1(t) is between 0 (no transfer) and 1 (perfe t transfer) and fully determines the �delity of state transfer. Sin e 〈ℓ, 0|Enk 〉 = c−1kn ((−1)n∆k − 2G) 〈ℓ, 0|Ψ0k〉+ ǫk〈ℓ, 0|Ψ1k〉 c−1kn√ ((−1)n∆k − 2G+ ǫk) akℓ we get fN,1(t) = (7.24) (ǫk+(−1)n∆k) ((−1) ∆k − 2G+ ǫk)2 ((−1)n∆k − 2G)2 + ǫ2k Eq. (7.24) is the main result of this se tion, fully determining the transfer of quantum information and entanglement in the presen e of the environments. In the limit G→ 0, 7 External noise we have ∆k ≈ ǫk and fN,1(t) approa hes the usual result without an environment, f0N,1(t) ≡ exp {−itǫk} ak1a∗kN . (7.25) In fa t, a series expansion of Eq. (7.24) yields that the �rst modi� ation of the transfer fun tion is of the order of G2, exp {−itǫk} . (7.26) Hen e the e�e t is small for very weakly oupled baths. However, as the hains get longer, the lowest lying energy ǫ1 usually approa hes zero, so the hanges be ome more signi� ant (s aling as 1/ǫk). For intermediate G, we evaluated Eq. (7.24) numeri ally and found that the �rst peak of the transfer fun tion generally be omes slightly lower, and gets shifted to higher times (Figures 7.4 and 7.5). A numeri sear h in the oupling spa e {Jℓ, ℓ = 1, . . . , N − 1} however also revealed some rare examples where an environment an also slightly improve the peak of the transfer fun tion (Fig 7.6). 0 2 4 6 8 10 12 14 16 18 20 Time [1/J] Figure 7.4: The absolute value of the transport fun tion fN,1(t) of an uniform spin hain (i.e. Jℓ = 1) with length N = 10 for three di�erent values of the bath oupling G. The �lled grey urve is the envelope of the limiting fun tion for G≫ ǫk/2 given by |f0( t )|. We an see that Eq. (7.28) be omes a good approximation already at G = 4. In the strong oupling regimeG≫ ǫk/2, we an approximate Eq. (7.23) by∆k ≈ 2G. 7 External noise 0 2 4 6 8 10 12 14 16 18 20 Time [1/J] Figure 7.5: The same as Fig. 7.4, but now for an engineered spin hain [i.e. Jℓ =√ ℓ(N − ℓ)℄ as in Subse tion 1.5.1. For omparison, we have res aled the ouplings su h that ℓ Jℓ is the same as in the uniform oupling ase. Inserting it in Eq. (7.24) then be omes fN,1(t) ≈ e−iGt −itǫk −itǫk = cos(Gt)f0N,1( ). (7.27) This surprisingly simple result onsists of the normal transfer fun tion, slowed down by a fa tor of 1/2, and modulated by a qui kly os illating term (Figures 7.4 and 7.5). We all this e�e t destabilisation. Our derivation a tually did not depend on the indexes of f(t) and we get for the transfer from the nth to the mth spin of the hain fn,m(t) ≈ cos(Gt)f0n,m( ). (7.28) It may look surprising that the matrix fn,m is no longer unitary. This is be ause we are onsidering the dynami s of the hain only, whi h is an open quantum sys- tem [121℄. A heuristi interpretation of Eq. (7.28) is that the ex itation os illates ba k and forth between the hain and the bath (hen e the modulation), and spends half 7 External noise of the time trapped in the bath (hen e the slowing). If the time of the maximum of the transfer fun tion |f0n,m(t)| for G = 0 is a multiple of π/2G then this maximum is also rea hed in the presen e of the bath. We remark that this behaviour is strongly non-Markovian [121℄. Finally, we want to stress that Eq. (7.28) is universal for any spin Hamiltonian that onserves the number of ex itations, i.e. with [HS, ℓ Zℓ] = 0. Thus our restri tion to hain-like topology and ex hange ouplings for HS is not ne essary. In fa t the only di�eren e in the whole derivation of Eq. (7.28) for a more general Hamiltonian is that Eq. (7.10) is repla ed by HS |ℓ, 0〉 = hℓ′ |ℓ′, 0〉. (7.29) The Hamiltonian an still be formally diagonalised in the �rst ex itation se tor as in Eq. (7.14), and the states of Eq. (7.20) will still diagonalise the total Hamiltonian HS +HI . Also, rather than onsidering an ex hange Hamiltonian for the intera tion with the bath, we ould have onsidered a Heisenberg intera tion [176℄, but only for the spe ial ase where all bath ouplings g k are all the same [177℄. Up to some irrelevant phases, this leads to the same results as for the ex hange intera tion. 0 5 10 15 20 25 30 35 40 45 50 Time [1/J] G=0.066 Figure 7.6: A weakly oupled bath may even improve the transfer fun tion for some spe i� hoi es of the Jℓ. This plot shows the transfer fun tion |fN,1(t)| for N = 10. The ouplings Jℓ were found numeri ally. 7 External noise 7.4 Con lusion We found a surprisingly simple and universal s aling law for the spin transfer fun tions in the presen e of spin environments. In the ontext of quantum state transfer this result is double-edged: on one hand, it shows that even for very strongly oupled baths quantum state transfer is possible, with the same �delity and only reasonable slowing. On the other hand, it also shows that the �delity as a fun tion of time be omes destabilised with a qui kly os illating modulation fa tor. In pra ti e, this fa tor will restri t the time-s ale in whi h one has to be able to read the state from the system. The results here are very spe i� to the simple bath model and do not hold in more general models (su h as these dis ussed in [165, 167℄, where true de oheren e and dissipation takes pla e). What we intended to demonstrate is that even though a bath oupling need not introdu e de oheren e or dissipation to the system, it an ause other dynami al pro esses that an be problemati for quantum information pro essing. Be ause the e�e ts observed here annot be avoided by ooling the bath, they may be ome relevant in some systems as a low temperature limit. 8 Con lusion and outlook Our resear h on quantum state transfer with spin hains has taken us on a journey from a very pra ti al motivation to quite fundamental issues and ba k again. On one hand, our results are quite abstra t and fundamental, and have related state transfer to number theory, topology and quantum onvergen e. On the other hand, we have developed s hemes whi h are simple and pra ti al, taking into a ount experimental hurdles su h as disorder and restri ted ontrol. While the multi rail s heme and the memory swapping s heme will probably be ome useful only after mu h further progress in experimental QIT, the dual rail s heme and in parti ular the valve s heme have some good han es to be realised in the near future. State transfer with quantum hains has be ome an area of large interest, with more than seventy arti les on the subje t over the last three years. The most important goal now is an experiment that demonstrates oherent transfer on a short hain (say of length N ≥ 5). Su h an experiment is not only useful building a quantum om- puter, but also from a fundamental perspe tive. For instan e, the violation of a Bell- inequality between distant entangled solid state qubits would be a milestone in the �eld. Sin e this requires a very high transfer �delity, the design of su h an experiment would probably require system dependent theoreti al resear h on how to over ome spe i� types of noise and how to improve the �delity for spe i� Hamiltonians. List of Figures 1.1 In areas of universal ontrol, quantum states an easily be transferred by sequen es of unitary swap gates Sj,k between nearest neighbours. . 12 1.2 S hemati layout of a quantum omputer. The solid arrows represent the �ow of quantum information, and the dashed arrows the �ow of lassi al information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Permanently oupled quantum hains an transfer quantum states with- out ontrol along the line. Note that the ends still need to be ontrol- lable to initialise and read out quantum states. . . . . . . . . . . . . . 13 1.4 Small blo ks (grey) of qubits (white ir les) onne ted by quantum hains. Ea h blo k onsists of (say) 13 qubits, 4 of whi h are onne ted to outgoing quantum hains (the thi k bla k lines denote their nearest- neighbour ouplings). The blo ks are onne ted to the ma ros opi world through lassi al wires (thin bla k lines with bla k ir les at their ends) through whi h arbitrary unitary operations an be triggered on the blo k qubits. The quantum hains require no external ontrol. . . 14 1.5 A quantum hain onsisting of N = 20 �ux qubits [34℄ (pi ture and experiment by Floor Paauw, TU Delft). The hain is onne ted to four larger SQUIDS for readout and gating. . . . . . . . . . . . . . . . . . . 15 1.6 Minimal �delity p(t) for a Heisenberg hain of length N = 50. . . . . . 21 1.7 Snapshots of the time evolution of a Heisenberg hain with N = 50. Shown is the distribution |fn,1(t)|2 of the wave-fun tion in spa e at di�erent times if initially lo alised at the �rst qubit. . . . . . . . . . . 22 1.8 Mean and varian e of the state |1〉 as a fun tion of time. Shown is the ase N = 50 with the y-axis giving the value relative to the mean N/2 + 1 and varian e (N2 − 1)/12 of an equal distribution 1√ |n〉. 22 1.9 Approximation of the transfer amplitude for N = 50 around the �rst maximum by Bessel and Airy fun tions [1, 61℄. . . . . . . . . . . . . . 23 List of Figures 1.10 pM (T ) as a fun tion of T for di�erent hain lengths. The solid urve is given by 1.82(2T ) and orresponds to the �rst peak of the proba- bility amplitude (Eq. 1.29) . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.11 Quantum apa ity, entanglement of formation (EOF), a lower bound for the entanglement of distillation (EOD) and the averaged �delity as a fun tion of p(t). We also show the orresponding hain length whi h rea hes this value as a �rst peak and the lassi al threshold 3 − 2 The expli it expression for the quantum apa ity plotted here is given in [54℄, and the lower bound of the entanglement of distillation will be derived in Se tion 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.12 Snapshots of the time evolution of a quantum hain with engineered ouplings (1.47) for N = 50. Shown is the distribution of the wave- fun tion in spa e at di�erent times if initially lo alised at the �rst qubit ( ompare Fig. 1.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1 Two quantum hains inter onne ting A and B. Control of the systems is only possible at the two qubits of either end. . . . . . . . . . . . . . 36 2.2 Quantum ir uit representation of on lusive and arbitrarily perfe t state transfer. The �rst gate at Ali e's qubits represents a NOT gate applied to the se ond qubit ontrolled by the �rst qubit being zero. The qubit |ψA〉1 on the left hand side represents an arbitrary input state at Ali e's site, and the qubit |ψB〉1 represents the same state, su essfully transferred to Bob's site. The tℓ-gate represents the unitary evolution of the spin hains for a time interval of tℓ. . . . . . . . . . . . . . . . . 37 2.3 Semilogarithmi plot of the joint probability of failure P (ℓ) as a fun tion of the number of measurements ℓ. Shown are Heisenberg spin-1/2- hains with di�erent lengths N . The times between measurements tℓ have been optimised numeri ally. . . . . . . . . . . . . . . . . . . . . . 39 2.4 Time t needed to transfer a state with a given joint probability of failure P a ross a hain of length N . The points denote exa t numeri al data, and the �t is given by Eq. (2.15). . . . . . . . . . . . . . . . . . . . . . 41 2.5 The minimal joint probability of failure P (ℓ) for hains with length N in the presen e of amplitude damping. The parameter J/Γ of the urves is the oupling of the hain (in Kelvin) divided by the de ay rate (ns−1). 44 2.6 Two disordered quantum hains inter onne ting A and B. Control of the systems is only possible at the two qubits of either end. . . . . . . 46 List of Figures 2.7 The absolute values of the transition amplitudes fN,1(t) and gN,1(t) for two Heisenberg hains of length N = 10. The ouplings strengths of both hains were hosen randomly from the interval [0.8J, 1.2J ] . The ir les show times where Bob an perform measurements without gaining information on α and β. . . . . . . . . . . . . . . . . . . . . . . 47 2.8 The relevant properties for on lusive transfer an be determined by measuring the response of the two systems at their ends only. . . . . . 51 2.9 Time t needed to transfer a state with a given joint probability of failure P a ross a hain of length N with un orrelated �u tuations of∆ = 0.05. The points denote numeri al data averaged over 100 realisations, and the �t is given by Eq. (2.53). This �gure should be ompared with Fig. 2.4 where ∆ = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.10 Most general setting for on lusive transfer: A bla k box with two inputs and two outputs, a ting as an amplitude damping hannel de�ned by Eqs. (2.54) and (2.55) . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.11 A simple ounterexample for a verti ally symmetri system where dual rail en oding is not possible. The bla k lines represent ex hange ouplings. 56 2.12 An example for a verti ally symmetri system where dual rail en oding is possible. The bla k lines represent ex hange ouplings of equal strength. 57 3.1 S hemati of the system: Ali e and Bob operate M hains, ea h on- taining N spins. The spins belonging to the same hain intera t through the Hamiltonian H whi h a ounts for the transmission of the signal in the system. Spins of di�erent hains do not intera t. Ali e en odes the information in the �rst spins of the hains by applying unitary trans- formations to her qubits. Bob re overs the message in the last spins of the hains by performing joint measurements. . . . . . . . . . . . . . . 59 3.2 Example of our notation for M = 5 hains of length N = 6 with K = 2 ex itations. The state above, given by |0〉1⊗|3〉2⊗|0〉3⊗|1〉4⊗|0〉5, has ex itations in the hains m1 = 2 and m2 = 4 at the horizontal position n1 = 3 and n2 = 1. It is in the Hilbert spa e H(S6) orresponding to the subset S6 = {2, 4} (assuming that the sets Sℓ are ordered in a anoni al way, i.e. S1 = {1, 2}, S2 = {1, 3} and so on) and will be written as |(3, 1); 6〉〉. There are = 10 di�erent sets Sℓ and the number of qubits one an transfer using these states is log2 10 ≈ 3. The e� ien y is thus given by R ≈ 3/5 whi h is already bigger than in the dual rail s heme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 List of Figures 3.3 Optimal rates (maximisation of Eq. (3.45 with respe t to M) for the Multi Rail proto ol. Shown are three urves orresponding to di�erent values of the joint probability of failure P one plans to a hieve. . . . . 71 4.1 S hemati examples of the orbits of a ergodi and a mixing map. . . . 72 4.2 Relations between topologi al spa es [149℄. The spa e of density matri- es on whi h quantum hannels are de�ned, is a ompa t and onvex subset of a normed ve tors spa e (the spa e of linear operators of the system) whi h, in the above graphi al representation �ts within the set of ompa t metri spa es. . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3 Relations between the di�erent properties of a quantum hannel. . . . 85 5.1 The system CC̄ an only be ontrolled by a ting on a (small) subsystem C. However system C is oupled to system C̄ by a unitary operator U = exp {−iHt} . This oupling an - in some ases - mediate the lo al ontrol on C to the full system CC̄. In our ase, system C is ontrolled by performing regular swap operations Sℓ between it and a quantum memory Mℓ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.2 Ali e and Bob ontrol the spins NA and NB inter onne ted by the spins NR. At time jt Bob performs a swap Sj between his spins and the memory Mj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.1 A quantum hain (qubits 1, 2, · · · , N) and a target qubit (N + 1). By applying a sequen e of two-qubit unitary gates Vk on the last qubit of the hain and the target qubit, arbitrarily high �delity an be a hieved. 105 6.2 Numeri al example for the onvergen e of the su ess probability. Sim- ulated is a quantum hain of length N = 20 with the Hamiltonian from Eq. (6.16) (dashed line) and Eq. (6.17) with B/J = 20 (solid line). Us- ing the original proto ol [1℄, the same hain would only rea h a su ess probability of 0.63 in the above time interval. . . . . . . . . . . . . . . 108 7.1 Dominant regimes of dynami s depending on the relative strength of the system Hamiltonian and the environmental oupling [47℄. . . . . . 111 7.2 A spin hain of length N = 5 oupled to independent baths of spins. . 112 7.3 In the �rst ex itation se tor, the system an be mapped into an e�e tive spin model where the bath spins are repla ed by a single e�e tive spin, as indi ated here for N = 5. . . . . . . . . . . . . . . . . . . . . . . . . 113 List of Figures 7.4 The absolute value of the transport fun tion fN,1(t) of an uniform spin hain (i.e. Jℓ = 1) with length N = 10 for three di�erent values of the bath oupling G. The �lled grey urve is the envelope of the limiting fun tion for G ≫ ǫk/2 given by |f0( t2 )|. We an see that Eq. (7.28) be omes a good approximation already at G = 4. . . . . . . . . . . . . 116 7.5 The same as Fig. 7.4, but now for an engineered spin hain [i.e. Jℓ = ℓ(N − ℓ)℄ as in Subse tion 1.5.1. For omparison, we have res aled the ouplings su h that ℓ Jℓ is the same as in the uniform oupling ase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.6 A weakly oupled bath may even improve the transfer fun tion for some spe i� hoi es of the Jℓ. This plot shows the transfer fun tion |fN,1(t)| for N = 10. The ouplings Jℓ were found numeri ally. . . . . . . . . . 118 List of Tables 2.1 The total time t and the number of measurements M needed to a hieve a probability of su ess of 99% for di�erent �u tuation strengths ∆ (un orrelated ase). Given is the statisti al mean and the standard deviation. The length of the hain is N = 20 and the number of random samples is 10. 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Index amplitude damping, 41 amplitude delaying hannel, 33 Anderson lo alisation, 43 arbitrary and unknown qubit, 11 asymptoti deformation, 81 bla k box, 12, 49, 53 ho olate, 11 lassi al averaged �delity, 27 oding transformation, 96 on lusively perfe t state transfer., 33 ooling proto ol, 17 oupled hains, 53 CPT, 83 riteria for quantum state transfer, 31 de oheren e-free subspa e, 41 destabilisation, 116 dispersion, 21 distillation, 27 dual rail, 35 e� ien y, 58 engineered ouplings, 29 entanglement distillation, 37 entanglement of distillation, 27, 63 entanglement of formation, 27 entanglement transfer, 27 ergodi , 76 experiments, 15 �delity, 16 �x-point, 76 �ux qubits, 15 generalised Lyapunov fun tion, 77 Heisenberg Hamiltonian, 20 homogenisation, 92 Lindblad equation, 109 maximal peak, 23 minimal �delity, 16, 20 mixing, 76 multi rail, 68 non-expansive map, 80 non-Markovian, 117 peak width, 27 peripheral eigenvalues, 86 phase noise, 41 pure �x-points, 87 quantum apa ity, 28 quantum hain, 12 quantum hannel, 16, 83 quantum omputer, 10 quantum erasure hannel, 37 quantum gates, 11 quantum memory, 93 quantum relative entropy, 84 Index quantum-jump approa h, 41 qutrits, 55 reading and writing �delities, 100 s alability, 12 S hrödinger equation, 109 Shor's algorithm, 10 spe tral radius, 67 spin hain, 12 stri t ontra tion, 80 swap gates, 26 time-s ale, 38 tomography, 49 topologi al spa e, 74 transfer fun tions, 18 valve, 103 weak ontra tion, 80 Introduction Quantum Computation and Quantum Information Quantum state transfer along short distances Implementations and experiments Basic communication protocol Initialisation and end-gates Symmetries Transfer functions Heisenberg Hamiltonian Dynamic and Dispersion How high should p(t) be? Advanced communication protocols Engineered Hamiltonians Weakly coupled sender and receiver Encoding Time-dependent control Motivation and outline of this work Dual Rail encoding Introduction Scheme for conclusive transfer Arbitrarily perfect state transfer Estimation of the time-scale the transfer Decoherence and imperfections Disordered chains Conclusive transfer in the presence of disorder Arbitrarily perfect transfer in the presence of disorder Tomography Numerical Examples Coupled chains Conclusion Multi Rail encoding Introduction The model Efficient encoding Perfect transfer Convergence theorem Quantum chains with nearest-neighbour interactions Comparison with Dual Rail Conclusion Ergodicity and mixing Introduction Topological background Generalised Lyapunov Theorem Topological spaces Metric spaces Quantum Channels Mixing criteria for Quantum Channels Beyond the density matrix operator space: spectral properties Ergodic channels with pure fixed points Conclusion Read and write access by local control Introduction Protocol Decomposition equations Coding transformation Fidelities for reading and writing Application to spin chain communication Conclusion A valve for probability amplitude Introduction Arbitrarily Perfect State Transfer Practical Considerations Conclusion External noise Introduction Model Results Conclusion Conclusion and outlook List of Figures List of Tables Bibliography Index

0704.1310

Thistlethwaite's theorem for virtual links

THISTLETHWAITE’S THEOREM FOR VIRTUAL LINKS SERGEI CHMUTOV AND JEREMY VOLTZ Abstract. The celebrated Thistlethwaite theorem relates the Jones polynomial of a link with the Tutte polynomial of the corresponding planar graph. We give a generalization of this theorem to virtual links. In this case, the graph will be embedded into a (higher genus) surface. For such graphs we use the generalization of the Tutte polynomial discovered by B. Bollobás and O. Riordan. Introduction Regions of a link diagram can be colored black and white in a checkerboard pattern. Putting a vertex in each black region and connecting two vertices by an edge if the corresponding regions share a crossing yields a planar graph. In 1987 Thistlethwaite [Th] proved that the Jones poly- nomial of an alternating link can be obtained as a specialization of the Tutte polynomial of the corresponding planar graph. L. Kauffman [K2] generalized the theorem to arbitrary links using signed graphs and extending the Tutte polynomial to them. An expression for the Jones polyno- mial in terms of the Bollobás-Riordan polynomial, without signed graphs, was found in [DFKLS]. The idea to use the Bollobás-Riordan polynomial instead of the Tutte polynomial belongs to Igor Pak. It was first realized in [CP], where Thistlethwaite’s theorem was generalized to checkerboard colorable virtual links. Here we shall generalize this theorem to arbitrary virtual links. We recall the basic definitions of virtual links, their Jones polynomial through the Kauffman bracket, ribbon graphs, and the Bollobás-Riordan polynomial in Sections 1 and 2. The key construction of a ribbon graph from a digram of a virtual link is explained in Section 3. Our main theorem is formulated and proved in Section 4. The work has been done as part of the Summer 2006 VIGRE working group “Knots and Graphs” (http://www.math.ohio-state.edu/~chmutov/wor-gr-su06/wor-gr.htm) at the Ohio State University, funded by NSF grant DMS-0135308. We are grateful to O. Dasbach and N. Stoltzfus for useful and stimulating conversations. 1. Virtual links and the Kauffman bracket The theory of virtual links was discovered independently by L. Kauffman [K3] and M. Gous- sarov, M. Polyak, and O. Viro [GPV] around 1998. According to Kauffman’s approach, virtual links are represented by diagrams similar to ordinary knot diagrams, except some crossings are designated as virtual. Virtual crossings should be understood not as crossings but rather as defects of our two-dimensional pictures. They should be treated in the same way as the extra crossings appearing in planar pictures of non-planar graphs. Here are some examples. The virtual crossings in these pictures are circled to distinguish them from the classical ones. Key words and phrases. Virtual knots and links, ribbon graph, Kauffman bracket, Bollobás-Riordan polynomial. http://arxiv.org/abs/0704.1310v1 2 SERGEI CHMUTOV AND JEREMY VOLTZ Virtual link diagrams are considered up to the classical Reidemeister moves involving classical crossings: and the virtual Reidemeister moves: The Kauffman bracket and the Jones polynomial for virtual links are defined in the same way as for classical ones. Let L be a virtual link diagram. Consider two ways of resolving a classical crossing. The A-splitting, , is obtained by joining the two vertical angles swept out by the overcrossing arc when it is rotated counterclockwise toward the undercrossing arc. Similarly, the B-splitting, , is obtained by joining the other two vertical angles. A state S of a link diagram L is a choice of either an A- or B-splitting at each classical crossing of the diagram. Denote by S(L) the set of the states of L. Clearly, a diagram L with n crossings has |S(L)| = 2n different states. Denote by α(S) and ( S) the numbers of A-splittings and B-splittings in a state S, respectively. Also, denote by ( S) the number of components of the curve obtained from the link diagram L by splitting according to the state S ∈ S(L). Note that virtual crossings do not connect components. Definition 1.1. The Kauffman bracket of a diagram L is a polynomial in three variables A, B, d defined by the formula: [L](A,B, d) := S∈S(L) Aα(S) B Note that [L] is not a topological invariant of the link and in fact depends on the link diagram. However, it defines the Jones polynomial JL(t) by a simple substitution: JL(t) := (−1) w(L)t3w(L)/4[L](t−1/4, t1/4,−t1/2 − t−1/2) . Here w(L) denotes the writhe, determined by orienting L and taking the sum over the classical crossings of L of the following signs : The Jones polynomial is a classical topological invariant (see e.g. [K1]). Example 1.2. Consider the third virtual knot diagram L from the example above. It is shown on the left of the table below. It has one virtual and three classical crossings (one positive and two negative). So there are eight states and w(L) = −1. The curves obtained by the splittings and the corresponding parameters α(S), ( S), and ( S) are shown in the remaining columns of the THISTLETHWAITE’S THEOREM FOR VIRTUAL LINKS 3 table. (α, , (3, 0, 2) (2, 1, 1) (2, 1, 1) (1, 2, 2) (2, 1, 1) (1, 2, 1) (1, 2, 1) (0, 3, 2) We have [L] = A3d+ 3A2B + 2AB2 +AB2d+B3d ; JL(t) = t −2 − t−1 − t−1/2 + 1 + t1/2 . 2. Ribbon graphs and the Bollobás-Riordan polynomial Ribbon graphs are the objects of Topological Graph Theory. There are several books on this subject and its applications [GT, LZ, MT]. Our ribbon graphs are nothing else than the band decompositions from [GT, Section 3.2] with the interior of all 2-bands removed. We modify the definitions of [BR] to signed ribbon graphs. Definition 2.1. A (signed) ribbon graph G is a surface (possibly nonorientable) with boundary represented as the union of two sets of closed topological discs called vertices V (G) and edges E(G), satisfying the following conditions: • these vertices and edges intersect by disjoint line segments; • each such line segment lies on the boundary of precisely one vertex and precisely one edge; • every edge contains exactly two such line segments, together with a sign function ε : E(G) → {±1}. Here are a few examples (if the sign is omitted it is assumed to be +1). If we put a dot at the center of each vertex-disc and a line in each edge-disc we will get an ordinary graph Γ, the core graph, embedded into a surface of G. Conversely if we have a graph Γ embedded into a surface then it determines a ribbon graph structure on a small neighborhood of Γ inside the surface. To define the Bollobás-Riordan polynomial we need to introduce several parameters of a ribbon graph G. Let • v(G) := |V (G)| denote the number of vertices of G; • e(G) := |E(G)| denote the number of edges of G; • k(G) denote the number of connected components of G; • r(G) := v(G)− k(G) be the rank of G; • n(G) := e(G)− r(G) be the nullity of G; 4 SERGEI CHMUTOV AND JEREMY VOLTZ • bc(G) denote the number of connected components of the boundary of the surface of G. A spanning subgraph of a ribbon graph G is defined as a subgraph consisting of all the vertices of G and a subset of the edges of G. Let F(G) denote the set of spanning subgraphs of G. Clearly, |F(G)| = 2e(G). For a signed ribbon graph we need one more parameter of a spanning subgraph. Let e−(F ) be the number of negative edges in F . Denote the complement to F in G by F = G−F , i.e. the spanning subgraph of G with exactly those (signed) edges of G that do not belong to F . Finally, let s(F ) = e−(F )− e−(F ) Definition 2.2. The signed Bollobás-Riordan polynomial RG(x, y, z) is defined by RG(x, y, z) := F∈F(G) xr(G)−r(F )+s(F )yn(F )−s(F )zk(F )−bc(F )+n(F ) . In general this is a Laurent polynomial in x1/2, y1/2, and z. The signed version of the Bollobás-Riordan polynomial was introduced in [CP]. If all the edges are positive then it is obtained from the original Bollobás-Riordan polynomial [BR] by a simple substitution x + 1 for x. Note that the exponent k(F )− bc(F ) + n(F ) of the variable z is equal to 2k(F ) − χ(F̃ ), where χ(F̃ ) is the Euler characteristic of the surface F̃ obtained by gluing a disc to each boundary component of F . For orientable ribbon graphs it is twice the genus of F . In particular, for a planar ribbon graph G (i.e. when the surface G has genus zero) the Bollobás- Riordan polynomial RG does not depend on z. In this case it is essentially equal to the classical Tutte polynomial TΓ(x, y) of the core graph Γ of G: RG(x − 1, y − 1, z) = TΓ(x, y) if all edges are positive, and if not, to Kauffman’s signed Tutte polynomial for signed graphs. Similarly, a specialization z = 1 of the Bollobás-Riordan polynomial of an arbitrary ribbon graph G gives the (signed) Tutte polynomial of the core graph: RG(x− 1, y − 1, 1) = TΓ(x, y) . So one may think about the Bollobás-Riordan polynomial as a generalization of the Tutte poly- nomial to graphs embedded into a surface. Example 2.3. Consider the third ribbon graph G from our example above and shown on the left in the table below. The other columns show eight possible spanning subgraphs F and the corresponding values of k(F ), r(F ), n(F ), bc(F ), and s(F ). (k, r, n, bc, s) (1, 1, 1, 2, 1) (1, 1, 0, 1, 0) (1, 1, 0, 1, 0) (2, 0, 0, 2,−1) (1, 1, 2, 1, 1) (1, 1, 1, 1, 0) (1, 1, 1, 1, 0) (2, 0, 1, 2,−1) We have RG(x, y, z) = x+ 2 + y + xyz 2 + 2yz + y2z . THISTLETHWAITE’S THEOREM FOR VIRTUAL LINKS 5 3. Ribbon graphs associated with virtual links In this section we describe a construction of a ribbon graph starting with a virtual link diagram. Our construction is similar to the classical Seifert algorithm of a construction of the Seifert surface of a link. There are two differences. The first one is that we do not twist the bands in small neighborhoods of the crossings. The second one is that we do not care how our ribbon graph is embedded into the three space. Suppose our virtual link diagram L is oriented. Then there is a state where all the splittings preserve orientation. Following a suggestion of N. Stoltzfus, we will call it the Seifert state, because its state circles are the Seifert circles of the link diagram. Also we will call all splittings in the Seifert state Seifert splittings. The Seifert circles will be the boundary circles of the vertex-discs of the future ribbon graph. So we are going to glue in a disc to each Seifert circle. Before that, though, let us describe the edges of the ribbon graph. When we are doing a Seifert splitting in a vicinity of a crossing we place a small planar band connecting two branches of the splitting. These bands will be the edge-discs of our ribbon graph. If the Seifert splitting was an A-splitting we assign +1 to the corresponding edge-band, if it was a B-splitting then we assign −1. It is easy to see that this sign is equal to the local writhe of the crossing. So we get a sign function. Because of the presence of virtual crossings our Seifert circles may be twisted, i.e. they are actually immersed into the plane with double points at virtual crossings. In the next step of the construction we untwist all Seifert circles to resolve the double points. This may result in some twisting on the edge-bands. After that we pull all Seifert circles apart, which could lead to additional twisting of our edges. In the last step, we glue the vertex-discs into the circles. The signed ribbon graph produced is denoted by GL. The next example illustrates this procedure. Example 3.1. + + − Diagram Seifert state Attaching bands to Seifert circles Untwisting Seifert circles Pulling Seifert circles apart Glue in the vertex-discs Another way to explain the same construction is the following. Instead of attaching bands to the Seifert circles we only mark the places on the Seifert circles where the bands have to be attached and memorize the order in which the marks occur according to the orientation on the circles. Then we draw each Seifert circle separately on a plane as a perfect circle oriented counterclockwise. Now attach edge-bands according to the marks; it is easy to see that this will always result in a half-twist on each band. The sign function is defined as before. 4. Main Theorem Theorem 4.1. Let L be a virtual link diagram, GL be the corresponding signed ribbon graph, and n := n(GL), r := r(GL), k := k(GL). Then [L](A,B, d) = AnBrdk−1 RGL Proof. Let L be a virtual link diagram, GL be the corresponding signed ribbon graph, and denote the Seifert state of GL as S. There is a natural bijection between S(L), the set of states of L, and F(GL), the set of spanning subgraphs of GL. Namely, given a state S, associate to it a spanning subgraph FS by the following construction. If a crossing in S is split differently than it is in S, include its associated edge-band in the spanning subgraph FS . If a crossing is split the same way for both S and S, do not include the associated edge-band in FS . This gives 6 SERGEI CHMUTOV AND JEREMY VOLTZ the subgraph FS associated to S. (Certainly |S(L)| = |F(GL)|, because the number of classical crossings of L is equal to e(GL) by virtue of our construction above.) For example, consider the link L from example 1.2. We know from above that GL is the ribbon graph considered in example 2.3. For each state given in the table on page 3, we can associate to it a spanning subgraph from the table on page 4. Consider the first state S given in the table on page 3. The two rightmost crossings are split differently than they are in the Seifert state of L given above. Thus the spanning subgraph associated to this state S via the correspondence given above is the first subgraph in the table on page 4. In fact, each state in the table on page 3 corresponds correctly to its associated spanning subgraph in the table on page 4. Check that [L] computed in example 1.2 and RGL(x, y, z) computed in example 2.3 satisfy the theorem. Now, given that F ∈ F(GL) is associated to S ∈ S(L) as described above (for simplicity, we write F instead of FS), consider the term xr(GL)−r(F )+s(F )yn(F )−s(F )zk(F )−bc(F )+n(F ) . Substituting in x = Ad , y = Bd , and z = 1 and multiplying by the term AnBrdk−1 as in the theorem, we have AnBrdk−1(AdB−1)r−r(F )+s(F )(BdA−1)n(F )−s(F )d−k(F )+bc(F )−n(F ) = An+r−r(F )−n(F )+2s(F )Br(F )+n(F )−2s(F )dk+r−k(F )−r(F )+bc(F )−1 . Since r(G) := v(G)− k(G) and n(G) := e(G)− r(G) for any ribbon graph G, we can rewrite our term as e(GL)−e(F )+2s(F )B e(F )−2s(F ) v(GL)−v(F )+bc(F )−1 . And since v(GL) = v(F ) by the definition of a spanning subgraph, we have (1) Ae(GL)−e(F )+2s(F )Be(F )−2s(F )dbc(F )−1 . It suffices to show that this is equal to the Kauffman bracket term Aα(S) B , since our bijection described above will then imply the theorem. We first show that e(F )− 2s(F ) = ( S) by a counting argument. Using the definition of s(F ), we get (2) e(F )− 2s(F ) = e(F )− e−(F ) + e−(F ) . Consider the crossings of L and how they are split in S. Let m denote the number of crossings which are B-splittings in S. Let b denote the number of crossings which are B-splittings in S but are A-splittings in S. Let a denote the number of crossings which are A-splittings in S but are B-splittings in S. Now, since e(F ) is the number of edges included in F , e(F ) equals the number of crossings of L which are split differently between S and S. That is, e(F ) = a+ b. Recall that e−(F ) denotes the number of edges in F with sign −1. And since any such edge corresponds to a B-splitting in S, it is clear that e−(F ) = b. Since F is the complement of F , e−(F ) denotes the number of crossings which are B-splittings in S and also in S. So, we deduce that e−(F ) = m− b. Finally, we consider ( S), the number of crossings which are B-splittings in S. So clearly, ( S) = a+ e−(F ) = a+ (m− b). THISTLETHWAITE’S THEOREM FOR VIRTUAL LINKS 7 Thus, we have that e(F )− e−(F ) + e−(F ) = (a+ b)− b+ (m− b) = a+ (m− b) This with (2) gives the desired result. And the fact that e(GL)−e(F )+2s(F ) = α(S) is immediate, since we just showed that e(F )− 2s(F ) = ( S), and certainly e(GL) = α(S) + ( To finish the proof it remains to show that bc(F ) = ( S). For that let us trace simultaneously a circle of the state S and a boundary component of F . Suppose we are passing a place near a crossing. If this crossing is split in the same way as in S then we continue to go along the circle which is locally the same as in S. On the whole ribbon graph GL this means that we are passing a place where the corresponding edge-band is supposed to be attached. Or, in other words, we skip the edge and do not include it into the corresponding spanning subgraph. This is precisely how we obtained F . Now if the passed crossing in S is split differently compared with S, then we have to switch to another strand of S. This means that we should turn on the corresponding edge-band in GL, i.e. we should include this edge-band in the corresponding subgraph. Again this is precisely what we did with F . The next picture with the first state of example 1.2 (the table on page 3) illustrates this. Seifert state S GL Spanning subgraph F = FS S Therefore, the tracing of the state circles of S corresponds to the tracing of the boundary com- ponents of F , i.e. bc(F ) = ( So, we have shown that (1) is equal to the term of [L] corresponding to the state S, and thus theorem 4.1 is proved. � References [BR] B. Bollobás and O. Riordan, A polynomial of graphs on surfaces, Math. Ann. 323 (2002) 81–96. [CP] S. Chmutov, I. Pak, The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial., preprint arXiv:math.GT/0609012, to appear in the Moscow Mathematical Journal. [DFKLS] O. Dasbach, D. Futer, E. Kalfagianni, X.-S. Lin, N. Stoltzfus, The Jones polynomial and dessins d’enfant, Preprint math.GT/0605571. [GPV] M. Goussarov, M. Polyak and O. Viro, Finite type invariants of classical and virtual knots, Topology 39 (2000) 1045–1068. [GT] J. L. Gross and T. W. Tucker, Topological graph theory, Wiley, NY, 1987. [K1] L. H. Kauffman, New invariants in knot theory, Amer. Math. Monthly 95 (1988) 195–242. [K2] L. H. Kauffman, A Tutte polynomial for signed graphs, Discrete Appl. Math. 25 (1989) 105–127. [K3] L. H. Kauffman, Virtual knot theory, European J. of Combinatorics 20 (1999) 663–690. [LZ] S. K. Lando, A. K. Zvonkin, Graphs on surfaces and their applications, Springer, 2004. [MT] B. Mohar, C. Thomassen, Graphs on Surfaces, The Johns Hopkins University Press, 2001. [Th] M. Thistlethwaite, A spanning tree expansion for the Jones polynomial, Topology 26 (1987) 297–309. Sergei Chmutov Department of Mathematics The Ohio State University, Mansfield 1680 University Drive Mansfield, OH 44906 [email protected] Jeremy Voltz Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210 [email protected] http://arxiv.org/abs/math/0609012 http://arxiv.org/abs/math/0605571 Introduction 1. Virtual links and the Kauffman bracket 2. Ribbon graphs and the Bollobás-Riordan polynomial 3. Ribbon graphs associated with virtual links 4. Main Theorem References

0704.1312

Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise

Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise Robert C. Dalang1,4, Davar Khoshnevisan2,5, and Eulalia Nualart3 Abstract We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish upper and lower bounds on the one-point density of the solution u(t, x), and upper bounds of Gaussian-type on the two-point density of (u(s, y), u(t, x)). In particular, this estimate quantifies how this density degenerates as (s, y) → (t, x). From these results, we deduce upper and lower bounds on hitting probabilities of the process {u(t , x)}t∈R+,x∈[0,1], in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to show that points are polar when d ≥ 7 and are not polar when d ≤ 5. We also show that the Hausdorff dimension of the range of the process is 6 when d > 6, and give analogous results for the processes t 7→ u(t, x) and x 7→ u(t, x). Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes. AMS 2000 subject classifications: Primary: 60H15, 60J45; Secondary: 60H07, 60G60. Key words and phrases. Hitting probabilities, stochastic heat equation, space-time white noise, Malliavin calculus. Institut de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Station 8, CH-1015 Lausanne, Switzerland. [email protected] Department of Mathematics, The University of Utah, 155 S. 1400 E. Salt Lake City, UT 84112-0090, USA. [email protected] Institut Galilée, Université Paris 13, 93430 Villetaneuse, France. [email protected] Supported in part by the Swiss National Foundation for Scientific Research. Research supported in part by a grant from the US National Science Foundation. http://arxiv.org/abs/0704.1312v1 1 Introduction and main results Consider the following system of non-linear stochastic partial differential equations (spde’s) (t , x) = (t , x) + σi,j(u(t , x))Ẇ j(t , x) + bi(u(t , x)), (1.1) for 1 ≤ i ≤ d, t ∈ [0 , T ], and x ∈ [0 , 1], where u := (u1 , . . . , ud), with initial conditions u(0 , x) = 0 for all x ∈ [0 , 1], and Neumann boundary conditions (t , 0) = (t , 1) = 0, 0 ≤ t ≤ T. (1.2) Here, Ẇ := (Ẇ 1 , . . . , Ẇ d) is a vector of d independent space-time white noises on [0 , T ]× [0 , 1]. For all 1 ≤ i, j ≤ d, bi, σij : Rd → R are globally Lipschitz functions. We set b = (bi), σ = (σij). Equation (1.1) is formal: the rigorous formulation of Walsh [W86] will be recalled in Section 2. The objective of this paper is to develop a potential theory for the Rd-valued process u = (u(t, x), t ≥ 0, x ∈ (0, 1)). In particular, given A ⊂ Rd, we want to determine whether the process u visits (or hits) A with positive probability. The only potential-theoretic result that we are aware of for systems of non-linear spde’s with multiplicative noise (σ non-constant) is Dalang and Nualart [DN04], who study the case of the reduced hyperbolic spde on R2+ (essentially equivalent to the wave equation in spatial dimension 1): ∂2Xit ∂t1∂t2 σi,j(Xt) ∂t1∂t2 + bi(Xt), where t = (t1 , t2) ∈ R2+, and Xit = 0 if t1t2 = 0, for all 1 ≤ i ≤ d. There, Dalang and Nualart used Malliavin calculus to show that the solution (Xt) of this spde satisfies K−1Capd−4(A) ≤ P{∃t ∈ [a, b]2 : Xt ∈ A} ≤ KCapd−4(A), where Capβ denotes the capacity with respect to the Newtonian β-kernel Kβ(·) (see (1.6)). This result, particularly the upper bound, relies heavily on properties of the underlying two-parameter filtration and uses Cairoli’s maximal inequality for two-parameter processes. Hitting probabilities for systems of linear heat equations have been obtained inMueller and Tribe [MT03]. For systems of non-linear stochastic heat equations with additive noise, that is, σ in (1.1) is a constant matrix, so (1.1) becomes (t , x) = (t , x) + σi,j Ẇ j(t , x) + bi(u(t , x)), (1.3) estimates on hitting probabilities have been obtained in Dalang, Khoshnevisan and Nualart [DKN07]. That paper develops some general results that lead to upper and lower bounds on hitting probabilities for continuous two-parameter random fields, and then uses these, together with a careful analysis of the linear equation (b ≡ 0, σ ≡ Id, where Id denotes the d × d identity matrix) and Girsanov’s theorem, to deduce bounds on hitting probabilities for the solution to (1.3). In this paper, we make use of the general results of [DKN07], but then, in order to handle the solution of (1.1), we use a very different approach. Indeed, the results of [DKN07] require in particular information about the probability density function pt,x of the random vector u(t, x). In the case of multiplicative noise, estimates on pt,x can be obtained via Malliavin calculus. We refer in particular on the results of Bally and Pardoux [BP98], who used Malli- avin calculus in the case d = 1 to prove that for any t > 0, k ∈ N and 0 ≤ x1 < · · · < xk ≤ 1, the law of (u(t , x1), . . . , u(t , xk)) is absolutely continuous with respect to Lebesgue measure, with a smooth and strictly positive density on {σ 6= 0}k, provided σ and b are infinitely differentiable functions which are bounded together with their derivatives of all orders. A Gaussian-type lower bound for this density is established by Kohatsu-Higa [K03] under a uniform ellipticity condition. Morien [M98] showed that the density function is also Hölder-continuous as a function of (t , x). In this paper, we shall use techniques of Malliavin calculus to establish the following theorem. Let pt,x(z) denote the probability density function of the R d-valued random vector u(t , x) = (u1(t , x), . . . , ud(t , x)) and for (s, y) 6= (t, x), let ps,y; t,x(z1, z2) denote the joint density of the R2d-valued random vector (u(s , y), u(t , x)) = (u1(s , y), . . . , ud(s , y), u1(t , x), . . . , ud(t , x)) (1.4) (the existence of pt,x(·) is essentially a consequence of the result of Bally and Pardoux [BP98], see our Corollary 4.3; the existence of ps,y; t,x(·, ·) is a consequence of Theorems 3.1 and 6.3). Consider the following two hypotheses on the coefficients of the system (1.1): P1 The functions σij and bi are bounded and infinitely differentiable with bounded partial derivatives of all orders, for 1 ≤ i, j ≤ d. P2 The matrix σ is uniformly elliptic, that is, ‖σ(x)ξ‖2 ≥ ρ2 > 0 for some ρ > 0, for all x ∈ Rd, ξ ∈ Rd, ‖ξ‖ = 1 (‖ · ‖ denotes the Euclidean norm on Rd). Theorem 1.1. Assume P1 and P2. Fix T > 0 and let I ⊂ (0, T ] and J ⊂ (0, 1) be two compact nonrandom intervals. (a) The density pt,x(z) is uniformly bounded over z ∈ Rd, t ∈ I and x ∈ J . (b) There exists c > 0 such that for any t ∈ I, x ∈ J and z ∈ Rd, pt,x(z) ≥ ct−d/4 exp ct1/2 (c) For all η > 0, there exists c > 0 such that for any s, t ∈ I, x, y ∈ J , (s, y) 6= (t, x) and z1, z2 ∈ Rd, ps,y; t,x(z1, z2) ≤ c(|t− s|1/2 + |x− y|)−(d+η)/2 exp − ‖z1 − z2‖ c(|t− s|1/2 + |x− y|) . (1.5) (d) There exists c > 0 such that for any t ∈ I, x, y ∈ J , x 6= y and z1, z2 ∈ Rd, pt,y; t,x(z1, z2) ≤ c(|x− y|)−d/2 exp −‖z1 − z2‖ c|x− y| The main technical effort in this paper is to obtain the upper bound in (c). Indeed, it is not difficult to check that for fixed (s, y; t, x), (z1, z2) 7→ ps,y; t,x(z1, z2) behaves like a Gaussian density function. However, for (s, y) = (t, x), the R2d-valued random vector (u(s , y), u(t , x)) is concentrated on a d-dimensional subspace in R2d and therefore does not have a density with respect to Lebesgue measure in R2d. So the main effort is to estimate how this density blows up as (s, y) → (t, x). This is achieved by a detailed analysis of the behavior of the Malliavin matrix of (u(s , y), u(t , x)) as a function of (s, y; t, x), using a perturbation argument. The presence of η in statement (c) may be due to the method of proof. When t = s, it is possible to set η = 0 as in Theorem 1.1(d). This paper is organized as follows. After introducing some notation and stating our main results on hitting probabilities (Theorems 1.2 and 1.6), we assume Theorem 1.1 and use the theorems of [DKN07] to prove these results in Section 2. In Section 3, we recall some basic facts of Malliavin calculus and state and prove two results that are tailored to our needs (Propositions 3.4 and 3.5). In Section 4, we establish the existence, smoothness and uniform boundedness of the one-point density function pt,x, proving Theorem 1.1(a). In Section 5, we establish a lower bound on pt,x, which proves Theorem 1.1(b). This upper (respectively lower) bound is a fairly direct extension to d ≥ 1 of a result of Bally and Pardoux [BP98] (respectively Kohatsu-Higa [K03]) when d = 1. In Section 6, we establish Theorem 1.1(c) and (d). The main steps are as follows. The upper bound on the two-point density function ps,y; t,x involves a bloc-decomposition of the Malliavin matrix of the R2d-valued random vector (u(s, y), u(s, y) − u(t, x)). The entries of this matrix are of different orders of magnitude, depending on which bloc they are in: see Theorem 6.3. Assuming Theorem 6.3, we prove Theorem 1.1(c) and (d) in Section 6.3. The exponential factor in (1.5) is obtained from an exponential martingale inequality, while the factor (|t − s|1/2 + |x − y|)−(d+η)/2 comes from an estimate of the iterated Skorohod integrals that appear in Corollary 3.3 and from the block structure of the Malliavin matrix. The proof of Theorem 6.3 is presented in Section 6.4: this is the main technical effort in this paper. We need bounds on the inverse of the Malliavin matrix. Bounds on its cofactors are given in Proposition 6.5, while bounds on negative moments of its determinant are given in Proposition 6.6. The determinant is equal to the product of the 2d eigenvalues of the Malliavin matrix. It turns out that at least d of these eigenvalues are of order 1 (“large eigenvalues”) and do not contribute to the upper bound in (1.5), and at most d are of the same order as the smallest eigenvalue (“small eigenvalues”), that is, of order |t−s|1/2+|x−y|. If we did not distinguish between these two types of eigenvalues, but estimated all of them by the smallest eigenvalue, we would obtain a factor of (|t− s|1/2 + |x− y|)−d+η/2 in (1.5), which would not be the correct order. The estimates on the smallest eigenvalue are obtained by refining a technique that appears in [BP98]; indeed, we obtain a precise estimate on the density whereas they only showed existence. The study of the large eigenvalues does not seem to appear elsewhere in the literature. Coming back to potential theory, let us introduce some notation. For all Borel sets F ⊂ Rd, we define P(F ) to be the set of all probability measures with compact support contained in F . For all integers k ≥ 1 and µ ∈ P(Rk), we let Iβ(µ) denote the β-dimensional energy of µ, that is, Iβ(µ) := Kβ(‖x− y‖)µ(dx)µ(dy), where ‖x‖ denotes the Euclidian norm of x ∈ Rk, Kβ(r) := r−β if β > 0, log(N0/r) if β = 0, 1 if β < 0, (1.6) and N0 is a sufficiently large constant (see Dalang, Khoshnevisan, and Nualart [DKN07, (1.5)]. For all β ∈ R, integers k ≥ 1, and Borel sets F ⊂ Rk, Capβ(F ) denotes the β-dimensional capacity of F , that is, Capβ(F ) := µ∈P(F ) Iβ(µ) where 1/∞ := 0. Note that if β < 0, then Capβ(·) ≡ 1. Given β ≥ 0, the β-dimensional Hausdorff measure of F is defined by Hβ(F ) = lim (2ri) β : F ⊆ B(xi , ri), sup ri ≤ ǫ , (1.7) where B(x , r) denotes the open (Euclidean) ball of radius r > 0 centered at x ∈ Rd. When β < 0, we define Hβ(F ) to be infinite. Throughout, we consider the following parabolic metric: For all s, t ∈ [0 , T ] and x, y ∈ [0 , 1], ∆((t , x) ; (s , y)) := |t− s|1/2 + |x− y|. (1.8) Clearly, this is a metric on R2 which generates the usual Euclidean topology on R2. Then we obtain an energy form I∆β (µ) := Kβ(∆((t , x) ; (s , y)))µ(dt dx)µ(ds dy), and a corresponding capacity Cap∆β (F ) := µ∈P(F ) I∆β (µ) For the Hausdorff measure, we write β (F ) = lim (2ri) β : F ⊆ B∆((ti , xi) , ri), sup ri ≤ ǫ where B∆((t , x) , r) denotes the open ∆-ball of radius r > 0 centered at (t , x) ∈ [0 , T ] × [0 , 1]. Using Theorem 1.1 together with results fromDalang, Khoshnevisan, and Nualart [DKN07], we shall prove the following result. Let u(E) denote the (random) range of E under the map (t, x) 7→ u(t, x), where E is some Borel-measurable subset of R2. Theorem 1.2. Assume P1 and P2. Fix T > 0, M > 0, and η > 0. Let I ⊂ (0, T ] and J ⊂ (0, 1) be two fixed non-trivial compact intervals. (a) There exists c > 0 depending on M, I, J and η such that for all compact sets A ⊆ [−M,M ]d, c−1 Capd−6+η(A) ≤ P{u(I × J) ∩A 6= ∅} ≤ cHd−6−η(A). (b) For all t ∈ (0, T ], there exists c1 > 0 depending on T , M and J , and c2 > 0 depending on T , M , J and η > 0 such that for all compact sets A ⊆ [−M,M ]d, c1 Capd−2(A) ≤ P{u({t} × J) ∩A 6= ∅} ≤ c2 Hd−2−η(A). (c) For all x ∈ (0, 1), there exists c > 0 depending on M, I and η such that for all compact sets A ⊆ [−M,M ]d, c−1 Capd−4+η(A) ≤ P{u(I × {x}) ∩A 6= ∅} ≤ cHd−4−η(A). Remark 1.3. (i) Because of the inequalities between capacity and Hausdorff measure, the right-hand sides of Theorem 1.2 can be replaced by cCapd−6−η(A), cCapd−2−η(A) and cCapd−4−η(A) in (a), (b) and (c), respectively (cf. Kahane [K85, p. 133]). (ii) Theorem 1.2 also holds if we consider Dirichlet boundary conditions (i.e. ui(t, 0) = ui(t, 1) = 0, for t ∈ [0, T ]) instead of Neumann boundary conditions. (iii) In the upper bounds of Theorem 1.2, the condition in P1 that σ and b are bounded can be removed, but their derivatives of all orders must exist and be bounded. As a consequence of Theorem 1.2, we deduce the following result on the polarity of points. Recall that a Borel set A ⊆ Rd is called polar for u if P{u((0, T ] × (0, 1)) ∩ A 6= ∅} = 0; otherwise, A is called nonpolar. Corollary 1.4. Assume P1 and P2. (a) Singletons are nonpolar for (t, x) 7→ u(t, x) when d ≤ 5, and are polar when d ≥ 7 (the case d = 6 is open). (b) Fix t ∈ (0, T ]. Singletons are nonpolar for x 7→ u(t, x) when d = 1, and are polar when d ≥ 3 (the case d = 2 is open). (c) Fix x ∈ (0, 1). Singletons are not polar for t 7→ u(t, x) when d ≤ 3 and are polar when d ≥ 5 (the case d = 4 is open). Another consequence of Theorem 1.2 is the Hausdorff dimension of the range of the process u. Corollary 1.5. Assume P1 and P2. (a) If d > 6, then dim (u((0, T ] × (0, 1))) = 6 a.s. (b) Fix t ∈ R+. If d > 2, then dimH(u({t} × (0, 1))) = 2 a.s. (c) Fix x ∈ (0, 1). If d > 4, then dim (u(R+ × {x})) = 4 a.s. As in Dalang, Khoshnevisan, and Nualart [DKN07], it is also possible to use Theorem 1.1 to obtain results concerning level sets of u. Define L (z ;u) := {(t , x) ∈ I × J : u(t , x) = z} , T (z ;u) = {t ∈ I : u(t , x) = z for some x ∈ J} , X (z ;u) = {x ∈ J : u(t , x) = z for some t ∈ I} , Lx(z ;u) := {t ∈ I : u(t , x) = z} , t(z ;u) := {x ∈ J : u(t , x) = z} . We note that L (z ;u) is the level set of u at level z, T (z ;u) (resp. X (z ;u)) is the projection of L (z ;u) onto I (resp. J), and Lx(z ;u) (resp. L t(z ;u)) is the x-section (resp. t-section) of L (z ;u). Theorem 1.6. Assume P1 and P2. Then for all η > 0 and R > 0 there exists a positive and finite constant c such that the following holds for all compact sets E ⊂ (0, T ] × (0, 1), F ⊂ (0, T ], G ⊂ (0, 1), and for all z ∈ B(0 , R): (a) c−1 Cap∆(d+η)/2(E) ≤ P{L (z ;u) ∩E 6= ∅} ≤ cH ∆(d−η)/2(E); (b) c−1Cap(d−2+η)/4(F ) ≤ P{T (z ;u) ∩ F 6= ∅} ≤ cH(d−2−η)/4(F ); (c) c−1 Cap(d−4+η)/2(G) ≤ P{X (z ;u) ∩G 6= ∅} ≤ cH(d−4−η)/2(G); (d) for all x ∈ (0, 1), c−1 Cap(d+η)/4(F ) ≤ P{Lx(z ;u) ∩ F 6= ∅} ≤ cH(d−η)/4(F ); (e) for all t ∈ (0, T ], c−1 Capd/2(G) ≤ P{L t(z ;u) ∩G 6= ∅} ≤ cH(d−η)/2(G). Corollary 1.7. Assume P1 and P2. Choose and fix z ∈ Rd. (a) If 2 < d < 6, then dim T (z ;u) = 1 (6− d) a.s. on {T (z ;u) 6= ∅}. (b) If 4 < d < 6 (i.e. d = 5), then dim X (z ;u) = 1 (6− d) a.s. on {X (z ;u) 6= ∅}. (c) If 1 ≤ d < 4, then dim Lx(z ;u) = (4− d) a.s. on {Lx(z ;u) 6= ∅}. (d) If d = 1, then dim t(z ;u) = 1 (2− d) = 1 a.s. on {L t(z ;u) 6= ∅}. In addition, all four right-most events have positive probability. Remark 1.8. The results of the two theorems and corollaries above should be compared with those of Dalang, Khsohnevisan and Nualart [DKN07]. 2 Proof of Theorems 1.2, 1.6 and their corollaries (assuming Theorem 1.1) We first recall that equation (1.1) is formal: a rigorous formulation, following Walsh [W86], is as follows. Let W i = (W i(s, x))s∈R+, x∈[0,1], i = 1, ..., d, be independent Brownian sheets defined on a probability space (Ω,F ,P), and set W = (W 1, ...,W d). For t ≥ 0, let Ft = σ{W (s, x), s ∈ [0, t], x ∈ [0, 1]}. We say that a process u = {u(t, x), t ∈ [0, T ], x ∈ [0, 1]} is adapted to (Ft) if u(t, x) is Ft-measurable for each (t, x) ∈ [0, T ]× [0, 1]. We say that u is a solution of (1.1) if u is adapted to (Ft) and if for i ∈ {1, . . . , d}, ui(t, x) = Gt−r(x , v) σi,j(u(r , v))W j(dr , dv) Gt−r(x , v) bi(u(r , v)) drdv, (2.1) where Gt(x , y) denotes the Green kernel for the heat equation with Neumann boundary conditions (see Walsh [W86, Chap 3]), and the stochastic integral in (2.1) is interpreted as in [W86]. Adapting the results from [W86] to the case d ≥ 1, one can show that there exists a unique continuous process u = {u(t, x), t ∈ [0, T ], x ∈ [0, 1]} adapted to (Ft) that is a solution of (1.1). Moreover, it is shown in Bally, Millet, and Sanz-Solé [BMS95] that for any s, t ∈ [0, T ] with s ≤ t, x, y ∈ [0, 1], and p > 1, E[|u(t, x)− u(s, y)|p] ≤ CT,p(∆((t , x) ; (s , y)))p/2 , (2.2) where ∆ is the parabolic metric defined in (1.8). In particular, for any 0 < α < 1/2, u is a.s. α-Hölder continuous in x and α/2-Hölder continuous in t. Assuming Theorem 1.1, we now prove Theorems 1.2, 1.6 and their corollaries. Proof of Theorem 1.2. (a) In order to prove the upper bound we use Dalang, Khosh- nevisan, and Nualart [DKN07, Theorem 3.3]. Indeed, Theorem 1.1(a) and (2.2) imply that the hypotheses (i) and (ii), respectively, of this theorem, are satisfied, and so the conclusion (with β = d− η) is too. In order to prove the lower bound, we shall use of [DKN07, Theorem 2.1]. This requires checking hypotheses A1 and A2 in that paper. Hypothesis A1 is a lower bound on the one-point density function pt,x(z), which is an immediate consquence of Theorem 1.1(b). Hypothesis A2 is an upper bound on the two-point density function ps,y;t,x(z1, z2), which involves a parameter β; we take β = d + η. In this case, Hypothesis A2 is an immediate consequence of Theorem 1.1(c). Therefore, the lower bound in Theorem 1.2(a) follows from [DKN07, Theorem 2.1]. This proves (a). (b) For the upper bound, we again refer to [DKN07, Theorem 3.3] (see also [DKN07, Theorem 3.1]). For the lower bound, which involves Capd−2(A) instead of Capd−2+η(A), we refer to [DKN07, Remark 2.5] and observe that hypotheses A1t and A2t there are satisfied with β = d (by Theorem 1.1(d)). This proves (b). (c) As in (a), the upper bound follows from [DKN07, Theorem 3.3] with β = d − η (see also [DKN07, Theorem 3.1(3)]), and the lower bound follows from [DKN07, Theorem 2.1(3)], with β = d+ η. Theorem 1.2 is proved. Proof of Corollary 1.4. We first prove (a). Let z ∈ Rd. If d ≤ 5, then there is η > 0 such that d − 6 + η < 0, and thus Capd−6+η({z}) = 1. Hence, the lower bound of Theorem 1.2 (a) implies that {z} is not polar. On the other hand, if d > 6, then for small η > 0, d− 6− η > 0. Therefore, Hd−6−η({z}) = 0 and the upper bound of Theorem 1.2(a) implies that {z} is polar. This proves (a). One proves (b) and (c) exactly along the same lines using Theorem 1.2(b) and (c). Proof of Theorem 1.6. For the upper bounds in (a)-(e), we use Dalang, Khoshnevisan, and Nualart [DKN07, Theorem 3.3] whose assumptions we verified above with β = d−η; these upper bounds then follow immediately from [DKN07, Theorem 3.2]. For the lower bounds in (a)-(d), we use [DKN07, Theorem 2.4] since we have shown above that the assumptions of this theorem, with β = d+ η, are satisfied by Theorem 1.1. For the lower bound in (e), we refer to [DKN07, Remark 2.5] and note that by Theorem 1.1(d), Hypothesis A2t there is satisfied with β = d. This proves Theorem 1.6. Proof of Corollaries 1.5 and 1.7. The final positive-probability assertion in Corollary 1.7 is an immediate consequence of Theorem 1.6 and Taylor’s theorem Khoshnevisan [K02, Corollary 2.3.1 p. 523]. Let E be a random set. When it exists, the codimension of E is the real number β ∈ [0 , d] such that for all compact sets A ⊂ Rd, P{E ∩A 6= ∅} > 0 whenever dim (A) > β, = 0 whenever dim (A) < β. See Khoshnevisan [K02, Chap.11, Section 4]. When it is well defined, we write the said codimension as codim(E). Theorems 1.2 and 1.6 imply that for d ≥ 1: codim(u(R+ × (0, 1))) = (d − 6)+; codim(u({t} × (0, 1))) = (d − 2)+; codim(u(R+ × {x})) = (d − 4)+; codim(T (z)) = (d−2 )+; codim(X (z)) = (d−4 )+; codim(Lx(z)) = ; and codim(L t(z)) = . According to Theorem 4.7.1 of Khoshnevisan [K02, Chapter 11], given a random set E in Rn whose codimension is strictly between 0 and n, E + codim E = n a.s. on {E 6= ∅}. (2.3) This implies the statements of Corollaries 1.5 and 1.7. 3 Elements of Malliavin calculus In this section, we introduce, following Nualart [N95] (see also Sanz-Solé [S05]), some elements of Malliavin calculus. Let S denote the class of smooth random variables of the F = f(W (h1), ...,W (hn)), where n ≥ 1, f ∈ C∞P (Rn), the set of real-valued functions f such that f and all its partial derivatives have at most polynomial growth, hi ∈ H := L2([0, T ] × [0, 1],Rd), and W (hi) denotes the Wiener integral W (hi) = hi(t, x) ·W (dx, dt), 1 ≤ i ≤ n. Given F ∈ S , its derivative is defined to be the Rd-valued stochastic process DF = (Dt,xF = (D t,xF, ...,D t,xF ), (t, x) ∈ [0, T ]× [0, 1]) given by Dt,xF = (W (h1), ...,W (hn))hi(t, x). More generally, we can define the derivative DkF of order k of F by setting DkαF = i1,...,ik=1 · · · ∂ f(W (h1), ...,W (hn))hi1(α1)⊗ · · · ⊗ hik(αk), where α = (α1, ..., αk), and αi = (ti, xi), 1 ≤ i ≤ k. For p, k ≥ 1, the space Dk,p is the closure of S with respect to the seminorm ‖ · ‖p defined by = E[|F |p] + E[‖DjF‖p where ‖DjF‖2 i1,...,ij=1 dx1 · · · (t1,x1) · · ·D(ij) (tj ,xj) We set (D∞)d = ∩p≥1 ∩k≥1 Dk,p. The derivative operator D on L2(Ω) has an adjoint, termed the Skorohod integral and denoted by δ, which is an unbounded operator on L2(Ω,H ). Its domain, denoted by Dom δ, is the set of elements u ∈ L2(Ω,H ) such that there exists a constant c such that |E[〈DF, u〉H ]| ≤ c‖F‖0,2, for any F ∈ D1,2. If u ∈ Dom δ, then δ(u) is the element of L2(Ω) characterized by the following duality relation: E[Fδ(u)] = E t,xF uj(t, x) dtdx , for all F ∈ D1,2. A first application of Malliavin calculus to the study of probability laws is the following global criterion for smoothness of densities. Theorem 3.1. [N95, Thm.2.1.2 and Cor.2.1.2] or [S05, Thm.5.2] Let F = (F 1, ..., F d) be an Rd-valued random vector satisfying the following two conditions: (i) F ∈ (D∞)d; (ii) the Malliavin matrix of F defined by γF = (〈DF i,DF j〉H )1≤i,j≤d is invertible a.s. and (det γF ) −1 ∈ Lp(Ω) for all p ≥ 1. Then the probability law of F has an infinitely differentiable density function. A random vector F that satisfies conditions (i) and (ii) of Theorem 3.1 is said to be nondegenerate. For a nondegenerate random vector, the following integration by parts formula plays a key role. Proposition 3.2. [N98, Prop.3.2.1] or [S05, Prop.5.4] Let F = (F 1, ..., F d) ∈ (D∞)d be a nondegenerate random vector, let G ∈ D∞ and let g ∈ C∞P (Rd). Fix k ≥ 1. Then for any multi-index α = (α1, ..., αk) ∈ {1, . . . , d}k, there is an element Hα(F,G) ∈ D∞ such that E[(∂αg)(F )G] = E[g(F )Hα(F,G)]. In fact, the random variables Hα(F,G) are recursively given by Hα(F,G) = H(αk)(F,H(α1,...,αk−1)(F,G)), H(i)(F,G) = δ(G (γ−1F )i,j DF Proposition 3.2 with G = 1 and α = (1, ..., d) implies the following expression for the density of a nondegenerate random vector. Corollary 3.3. [N98, Corollary 3.2.1] Let F = (F 1, ..., F d) ∈ (D∞)d be a nondegenerate random vector and let pF (z) denote the density of F . Then for every subset σ of the set of indices {1, ..., d}, pF (z) = (−1)d−|σ|E[1{F i>zi,i∈σ, F i<zi,i 6∈σ}H(1,...,d)(F, 1)], where |σ| is the cardinality of σ, and, in agreement with Proposition 3.2, H(1,...,d)(F, 1) = δ((γ F DF ) dδ((γ−1F DF ) d−1δ(· · · δ((γ−1F DF ) 1) · · · ))). The next result gives a criterion for uniform boundedness of the density of a nondegen- erate random vector. Proposition 3.4. For all p > 1 and ℓ ≥ 1, let c1 = c1(p) > 0 and c2 = c2(ℓ, p) ≥ 0 be fixed. Let F ∈ (D∞)d be a nondegenerate random vector such that (a) E[(det γF ) −p] ≤ c1; (b) E[‖Dl(F i)‖p ] ≤ c2, i = 1, ..., d. Then the density of F is uniformly bounded, and the bound does not depend on F but only on the constants c1(p) and c2(ℓ, p). Proof. The proof of this result uses the same arguments as in the proof of Dalang and Nualart [DN04, Lemma 4.11]. Therefore, we will only give the main steps. Fix z ∈ Rd. Thanks to Corollary 3.3 and the Cauchy-Schwarz inequality we find that |pF (z)| ≤ ‖H(1,...,d)(F, 1)‖0,2. Using the continuity of the Skorohod integral δ (cf. Nualart [N95, Proposition 3.2.1] and Nualart [N98, (1.11) and p.131]) and Hölder’s inequality for Malliavin norms (cf. Watanabe [W84, Proposition 1.10, p.50]), we obtain ‖H(1,...,d)(F, 1)‖0,2 ≤ c‖H(1,...,d−1)(F, 1)‖1,4 ‖(γ−1F )d,j‖1,8 ‖D(F j)‖1,8. (3.1) In agreement with hypothesis (b), ‖D(F j)‖m,p ≤ c. In order to bound the second factor in (3.1), note that ‖(γ−1F )i,j‖m,p = E[|(γ−1F )i,j| E[‖Dk(γ−1F )i,j‖ . (3.2) For the first term in (3.2), we use Cramer’s formula to get that |(γ−1F )i,j| = |(det γF ) −1(AF )i,j |, where AF denotes the cofactor matrix of γF . By means of Cauchy-Schwarz inequality and hypotheses (a) and (b) we find that E[((γ−1F )i,j) p] ≤ cd,p{E[(det γF )−2p]}1/2 × {E[‖D(F )‖4p(d−1)H ]} ≤ cd,p, where none of the constants depend on F . For the second term on the right-hand side of (3.2), we iterate the equality (cf. Nualart [N95, Lemma 2.1.6]) D(γ−1F )i,j = − k,ℓ=1 (γ−1F )i,kD(γF )k,ℓ(γ F )ℓ,j, (3.3) in the same way as in the proof of Dalang and Nualart [DN04, Lemma 4.11]. Then, appealing again to hypotheses (a) and (b) and iterating the inequality (3.1) to bound the first factor on the right-hand side of (3.2), we obtain the uniform boundedness of pF (z). We finish this section with a result that will be used later on to bound negative moments of a random variable, as is needed to check hypothesis (a) of Proposition 3.4. Proposition 3.5. Suppose Z ≥ 0 is a random variable for which we can find ǫ0 ∈ (0, 1), processes {Yi,ǫ}ǫ∈(0,1) (i = 1, 2), and constants c > 0 and 0 ≤ α2 ≤ α1 with the property that Z ≥ min(cǫα1 − Y1,ǫ cǫα2 − Y2,ǫ) for all ǫ ∈ (0, ǫ0). Also suppose that we can find βi > αi (i = 1, 2), not depending on ǫ0, such that C(q) := sup 0<ǫ<1 E[|Y1,ǫ|q] E[|Y2,ǫ|q] <∞ for all q ≥ 1. Then for all p ≥ 1, there exists a constant c′ ∈ (0,∞), not depending on ǫ0, such that E[|Z|−p] ≤ c′ǫ−pα10 . Remark 3.6. This lemma is of interest mainly when β2 ≤ α1. Proof. Define k := (2/c)ǫ 0 . Suppose that y ≥ k, and let ǫ := (2/c)1/α1y−1/α1 . Then 0 < ǫ ≤ ǫ0, y−1 = (c/2)ǫα1 , and for all q ≥ 1, Z−1 > y Z < y−1 Y1,ǫ ≥ Y2,ǫ ≥ cǫα2 − ≤ C(q) 2qǫq(β1−α1) + ǫqβ2 ǫα2 − 1 The inequality ǫα2 − (1/2)ǫα1 ≥ (1/2)ǫα2 implies that Z−1 > y ≤ C(q) 2qǫq(β1−α1) + 2qǫq(β2−α2) ≤ ay−qb, where a and b are positive and finite constants that do not depend on y, ǫ0 or q. We apply this with q := (p/b) + 1 to find that for all p ≥ 1, |Z|−p yp−1P Z−1 > y ≤ kp + ap y−b−1 dy = kp + Because k ≥ (2/c) and b > 0, it follows that E[|Z|−p] ≤ (1 + c1(ap/b))kp, where c1 := (c/2)b+p. This is the desired result. 4 Existence, smoothness and uniform boundedness of the one-point density Let u = {u(t, x), t ∈ [0, T ], x ∈ [0, 1]} be the solution of equation (2.1). In this section, we prove the existence, smoothness and uniform boundedness of the density of the random vector u(t, x). In particular, this will prove Theorem 1.1(a). The first result concerns the Malliavin differentiability of u and the equations satisfied by its derivatives. We refer to Bally and Pardoux [BP98, Proposition 4.3, (4.16), (4.17)] for its proof in dimension one. As we work coordinate by coordinate, the following proposition follows in the same way and its proof is therefore omitted. Proposition 4.1. Assume P1. Then u(t, x) ∈ (D∞)d for any t ∈ [0, T ] and x ∈ [0, 1]. Moreover, its iterated derivative satisfies D(k1)r1,v1 · · ·D rn,vn (ui(t, x)) Gt−rl(x, vl) D(k1)r1,v1 · · ·D (kl−1) rl−1,vl−1D (kl+1) rl+1,vl+1 · · ·D(kn)rn,vn(σikl(u(rl, vl))) r1∨···∨rn Gt−θ(x, η) D(kl)rl,vl(σij(u(θ, η)))W j(dθ, dη) r1∨···∨rn Gt−θ(x, η) D(kl)rl,vl(bi(u(θ, η))) dθdη if t ≤ r1 ∨ · · · ∨ rn and D(k1)r1,v1 · · ·D rn,vn(ui(t, x)) = 0 otherwise. Finally, for any p > 1, sup(t,x)∈[0,T ]×[0,1]E ∥Dn(ui(t, x)) < +∞. (4.1) Note that, in particular, the first-order Malliavin derivative satisfies, for r < t, D(k)r,v (ui(t, x)) = Gt−r(x, v)σik(u(r, v)) + ai(k, r, v, t, x), (4.2) where ai(k, r, v, t, x) = Gt−θ(x, η)D r,v (σij(u(θ, η)))W j(dθ, dη) Gt−θ(x, η)D r,v (bi(u(θ, η))) dθdη, (4.3) and D r,v (ui(t, x)) = 0 when r > t. The next result proves property (a) in Proposition 3.4 when F is replaced by u(t, x). Proposition 4.2. Assume P1 and P2. Let I and J two compact intervals as in Theorem 1.1. Then, for any p ≥ 1, (det γu(t,x)) is uniformly bounded over (t, x) ∈ I × J . Proof. This proof follows Nualart [N98, Proof of (3.22)], where it is shown that for fixed (t, x), E[(detγu(t,x)) −p] < +∞. Our emphasis here is on the uniform bound over (t, x) ∈ I × J . Assume that I = [t1, t2] and J = [x1, x2], where 0 < t1 < t2 ≤ T , 0 < x1 < x2 < 1. Let (t, x) ∈ I × J be fixed. We write det γu(t,x) ≥ infξ∈Rd:‖ξ‖=1 ξ Tγu(t,x)ξ Let ξ = (ξ1, ..., ξd) ∈ Rd with ‖ξ‖ = 1 and fix ǫ ∈ (0, 1). Note the inequality (a+ b)2 ≥ 2 a2 − 2b2, (4.4) valid for all a, b ≥ 0. Using (4.2) and the fact that γu(t,x) is a matrix whose entries are inner-products, this implies that ξTγu(t,x)ξ = Dr,v(ui(t, x))ξi t(1−ǫ) Dr,v(ui(t, x))ξi ≥ I1 − I2, where t(1−ǫ) Gt−r(x, v)σik(u(r, v))ξi I2 =2 t(1−ǫ) ai(k, r, v, t, x)ξi and ai(k, r, v, t, x) is defined in (4.3). In accord with hypothesis P2 and thanks to Lemma I1 ≥ c(tǫ)1/2, (4.5) where c is uniform over (t, x) ∈ I × J . Next we apply the Cauchy-Schwarz inequality to find that, for any q ≥ 1, supξ∈Rd:‖ξ‖=1|I2|q ≤ c(E[|A1|q] + E[|A2|q]), where i,j,k=1 t(1−ǫ) Gt−θ(x, η)D r,v (σij(u(θ, η)))W j(dθ, dη) i,k=1 t(1−ǫ) Gt−θ(x, η)D r,v (bi(u(θ, η))) dθdη We bound the q-th moment of A1 and A2 separately. As regards A1, we use Burkholder’s inequality for martingales with values in a Hilbert space (Lemma 7.6) to obtain E[|A1|q] ≤ c k,i=1 t(1−ǫ) t(1−ǫ) , (4.6) where Θ := 1{θ>r}Gt−θ(x, η) D(k)r,v (σij(u(θ, η))) ≤ c1{θ>r}Gt−θ(x, η) D(k)r,v (ul(θ, η)) thanks to hypothesis P1. Hence, E[|A1|q] ≤ c t(1−ǫ) dη G2t−θ(x, η) ∫ t∧θ t(1−ǫ) where Ψ := r,v (ul(θ, η)). We now apply Hölder’s inequality with respect to the measure G2t−θ(x, η)dθdη to find that E[|A1|q] ≤ C t(1−ǫ) dη G2t−θ(x, η) t(1−ǫ) dη G2t−θ(x, η) t(1−ǫ) Lemmas 7.3 and 7.5 assure that E[|A1|q] ≤ CT (tǫ) 2 (tǫ)q/2 t(1−ǫ) G2t−θ(x, η) dθdη ≤ CT (tǫ)q, where CT is uniform over (t, x) ∈ I × J . We next derive a similar bound for A2. By the Cauchy–Schwarz inequality, E [|A2|q] ≤ c(tǫ)q i,k=1 t(1−ǫ) where Φ := Gt−θ(x, η)|D(k)r,v (bi(u(θ, η))) |. From here on, the q-th moment of A2 is estimated as that of A1 was; cf. (4.6), and this yields E[|A2|q] ≤ CT (tǫ)2q. Thus, we have proved that supξ∈Rd:‖ξ‖=1|I2|q ≤ CT (tǫ)q, (4.7) where the constant CT is clearly uniform over (t, x) ∈ I × J . Finally, we apply Proposition 3.5 with Z := inf‖ξ‖=1(ξ Tγu(t,x)ξ), Y1,ǫ = Y2,ǫ = sup‖ξ‖=1I2, ǫ0 = 1, α1 = α2 = 1/2 and β1 = β2 = 1, to get (detγu(t,x)) ≤ CT , where all the constants are clearly uniform over (t, x) ∈ I×J . This is the desired result. Corollary 4.3. Assume P1 and P2. Fix T > 0 and let I and J be a compact intervals as in Theorem 1.1. Then, for any (t, x) ∈ (0, T ]× (0, 1), u(t, x) is a nondegenerate random vector and its density funciton is infinitely differentiable and uniformly bounded over z ∈ Rd and (t, x) ∈ I × J . Proof of Theorem 1.1(a). This is a consequence of Propositions 4.1 and 4.2 together with Theorem 3.1 and Proposition 3.4. 5 The Gaussian-type lower bound on the one-point density The aim of this section is to prove the lower bound of Gaussian-type for the density of u stated in Theorem 1.1(b). The proof of this result was given in Kohatsu-Higa [K03, Theorem 10] for dimension 1, therefore we will only sketch the main steps. Proof of Theorem 1.1(b). We follow [K03] and we show that for each (t, x), F = u(t, x) is a d-dimensional uniformly elliptic random vector and then we apply [K03, Theorem 5]. Let F in = Gt−r(x, v) σij(u(r, v))W j(dr, dv) + Gt−r(x, v) bi(u(r, v)) drdv, 1 ≤ i ≤ d, where 0 = t0 < t1 < · · · < tN = t is a sufficiently fine partition of [0, t]. Note that Fn ∈ Ftn . Set g(s, y) = Gt−s(x, y). We shall need the following two lemmas. Lemma 5.1. [K03, Lemma 7] Assume P1 and P2. Then: (i) ‖F in‖k,p ≤ ck,p, 1 ≤ i ≤ d; (ii) ‖((γFn(tn−1))ij)−1‖p,tn−1 ≤ cp(∆n−1(g))−1 = cp(‖g‖2L2([tn−1,tn]×[0,1])) where γFn(tn−1) denotes the conditional Malliavin matrix of Fn given Ftn−1 and ‖ · ‖p,tn−1 denotes the conditional Lp-norm. We define un−1i (s1, y1) = ∫ tn−1 Gs1−s2(y1, y2) σij(u(s2, y2))W j(ds1, dy2) ∫ tn−1 Gs1−s2(y1, y2) bi(u(s2, y2)) ds2dy2, 1 ≤ i ≤ d. Note that un−1 ∈ Ftn−1 . As in [K03], the following holds. Lemma 5.2. [K03, Lemma 8] Under hypothesis P1, for s ∈ [tn−1, tn], ‖ui(s, y)− un−1i (s, y)‖n,p,tn−1 ≤ (s− tn−1) 1/8, 1 ≤ i ≤ d, where ‖ · ‖n,p,tn−1 denotes the conditional Malliavin norm given Ftn−1 . The rest of the proof of Theorem 1.1(b) follows along the same lines as in [K03] for d = 1. We only sketch the remaining main points where the fact that d > 1 is important. In order to obtain the expansion of F in−F in−1 as in [K03, Lemma 9], we proceed as follows. By the mean value theorem, F in − F in−1 = Gt−r(x, v) σij(u n−1(r, v))W j(dr, dv) Gt−r(x, v) bi(u(r, v)) drdv Gt−r(x, v) j,l=1 ∂lσij(u(r, v, λ))dλ)(ul(r, v) − un−1l (r, v))W j(dr, dv), where u(r, v, λ) = (1−λ)u(r, v)+λun−1(r, v). Using the terminology of [K03], the first term is a process of order 1 and the next two terms are residues of order 1 (as in [K03]). In the next step, we write the residues of order 1 as the sum of processes of order 2 and residues of order 2 and 3 as follows: Gt−r(x, v) bi(u(r, v)) drdv Gt−r(x, v) bi(u n−1(r, v)) drdv Gt−r(x, v) ∂lbi(u(r, v, λ))dλ)(ul(r, v) − un−1l (r, v)) drdv Gt−r(x, v) j,l=1 ∂lσij(u(r, v, λ))dλ)(ul(r, v) − un−1l (r, v))W j(dr, dv) Gt−r(x, v) j,l=1 ∂lσij(u n−1(r, v))(ul(r, v) − un−1l (r, v))W j(dr, dv) Gt−r(x, v) j,l,l′=1 ∂l∂l′σij(u(r, v, λ))dλ) × (ul(r, v) − un−1l (r, v))(ul′ (r, v) − u (r, v))W j(dr, dv). It is then clear that the remainder of the proof of [K03, Lemma 9] follows for d > 1 along the same lines as in [K03], working coordinate by coordinate. Finally, in order to complete the proof of the proposition, it suffices to verify the hy- potheses of [K03, Theorem 5]. Again the proof follows as in the proof of [K03, Theorem 10], working coordinate by coordinate. We will only sketch the proof of his (H2c), where hypothesis P2 is used: (∆n−1(g)) (Gt−r(x, v)) 2‖σ(un−1(r, v))ξ‖2 drdv ≥ ρ2(∆n−1(g))−1 (Gt−r(x, v)) 2 drdv = ρ2 > 0, by the definition of g. This concludes the proof of Theorem 1.1 (b). 6 The Gaussian-type upper bound on the two-point density Let ps,y; t,x(z1, z2) denote the joint density of the 2d-dimensional random vector (u1(s, y), ..., ud(s, y), u1(t, x), ..., ud(t, x)), for s, t ∈ (0, T ], x, y ∈ (0, 1), (s, y) 6= (t, x) and z1, z2 ∈ Rd (the existence of this joint density will be a consequence of Theorem 3.1, Proposition 4.1 and Theorem 6.3). The next subsections lead to the proofs of Theorem 1.1(c) and (d). 6.1 Bounds on the increments of the Malliavin derivatives In this subsection, we prove an upper bound for the Sobolev norm of the derivative of the increments of our process u. For this, we will need the following preliminary estimate. Lemma 6.1. For any s, t ∈ [0, T ], s ≤ t, and x, y ∈ [0, 1], (g(r, v))2 drdv ≤ CT (|t− s|1/2 + |x− y|), where g(r, v) := gt,x,s,y(r, v) = 1{r≤t}Gt−r(x, v) − 1{r≤s}Gs−r(y, v). Proof. Using Bally, Millet, and Sanz-Solé [BMS95, Lemma B.1] with α = 2, we see (g(r, v))2 drdv (Gt−r(x, v)) 2 drdv + 2 (Gt−r(x, v) −Gs−r(x, v))2 drdv (Gs−r(x, v)−Gs−r(y, v))2 drdv ≤ CT (|t− s|1/2 + |x− y|). Proposition 6.2. Assuming P1, for any s, t ∈ [0, T ], s ≤ t, x, y ∈ [0, 1], p > 1, m ≥ 1, ∥Dm(ui(t, x)− ui(s, y)) |t− s|1/2 + |x− y| , i = 1, ..., d. Proof. Let m = 1. Consider the function g(r, v) defined in Lemma 6.1. Using the integral equation (4.2) satisfied by the first-order Malliavin derivative, we find that ∥D(ui(t, x)− ui(s, y)) E[|I1|p/2] + E[|I2|p/2] + E[|I3|p/2] where dv (g(r, v)σik(u(r, v))) j,k=1 g(θ, η)D(k)r,v (σij(u(θ, η)))W j(dθ, dη) g(θ, η)D(k)r,v (bi(u(θ, η)))dθdη We bound the p/2-moments of I1, I2 and I3 separately. By hypothesis P1 and Lemma 6.1, E[|I1|p/2] ≤ CT (|t − s|1/2 + |x − y|)p/2. Using Burkholder’s inequality for Hilbert-space-valued martingales (Lemma 7.6) and hypothesis P1, we obtain E[|I2|p/2] ≤ C dη (g(θ, η))2 where Θ := r,v (ul(θ, η)). From Hölder’s inequality with respect to the measure (g(θ, η))2dθdη, we see that this is bounded above by (g(θ, η))2dθdη × sup(θ,η)∈[0,T ]×[0,1] dη(g(θ, η))2 ≤ CT (|t− s|1/2 + |x− y|)p/2, thanks to (4.1) and Lemma 6.1. We next derive a similar bound for I3. By the Cauchy–Schwarz inequality, E[|I3|p/2] ≤ CT dη (g(θ, η))2 From here on, the p/2-moment of I3 is estimated as was that of I2, and this yields E[|I3|p/2] ≤ CT (|t − s|1/2 + |x − y|)p/2. This proves the desired result for m = 1. The case m > 1 follows using the stochastic differential equation satisfied by the iterated Malliavin deriva- tives (Proposition 4.1), Hölder’s and Burkholder’s inequalities, hypothesis P1, (4.1) and Lemma 6.1 in the same way as we did for m = 1, to obtain the desired bound. 6.2 Study of the Malliavin matrix For s, t ∈ [0, T ], s ≤ t, and x, y ∈ [0, 1] consider the 2d-dimensional random vector Z := (u1(s, y), ..., ud(s, y), u1(t, x)− u1(s, y), ..., ud(t, x)− ud(s, y)). (6.1) Let γZ the Malliavin matrix of Z. Note that γZ = ((γZ)m,l)m,l=1,...,2d is a symmetric 2d×2d random matrix with four d× d blocs of the form ... γ · · · ... · · · ... γ where Z = (〈D(ui(s, y)),D(uj(s, y))〉H )i,j=1,...,d, Z = (〈D(ui(s, y)),D(uj(t, x)− uj(s, y))〉H )i,j=1,...,d, Z = (〈D(ui(t, x)− ui(s, y)),D(uj(s, y))〉H )i,j=1,...,d, Z = (〈D(ui(t, x)− ui(s, y)),D(uj(t, x)− uj(s, y))〉H )i,j=1,...,d. We let (1) denote the set of indices {1, ..., d}×{1, ..., d}, (2) the set {1, ..., d}×{d+1, ..., 2d}, (3) the set {d+ 1, ..., 2d} × {1, ..., d} and (4) the set {d+ 1, ..., 2d} × {d+ 1, ..., 2d}. The following theorem gives an estimate on the Sobolev norm of the entries of the inverse of the matrix γZ , which depends on the position of the entry in the matrix. Theorem 6.3. Fix η, T > 0. Assume P1 and P2. Let I and J be two compact intervals as in Theorem 1.1. (a) For any (s, y) ∈ I × J , (t, x) ∈ I × J , s ≤ t, (s, y) 6= (t, x), k ≥ 0, p > 1, ‖(γ−1Z )m,l‖k,p ≤ ck,p,η,T (|t− s|1/2 + |x− y|)−dη if (m, l) ∈ (1), ck,p,η,T (|t− s|1/2 + |x− y|)−1/2−dη if (m, l) ∈ (2) or (3), ck,p,η,T (|t− s|1/2 + |x− y|)−1−dη if (m, l) ∈ (4). (b) For any s = t ∈ (0, T ], (t, y) ∈ I × J , (t, x) ∈ I × J , x 6= y, k ≥ 0, p > 1, ‖(γ−1Z )m,l‖k,p ≤ ck,p,T if (m, l) ∈ (1) , ck,p,T |x− y|−1/2 if (m, l) ∈ (2) or (3), ck,p,T |x− y|−1 if (m, l) ∈ (4). (Note the slight improvements in the exponents in case (b) where s = t.) The proof of this theorem is deferred to Section 6.4. We assume it for the moment and complete the proof of Theorem 1.1(c) and (d). 6.3 Proof of Theorem 1.1(c) and (d) Fix two compact intervals I and J as in Theorem 1.1. Let (s, y), (t, x) ∈ I × J , s ≤ t, (s, y) 6= (t, x), and z1, z2 ∈ Rd. Let Z be as in (6.1) and let pZ be the density of Z. Then ps,y; t,x(z1, z2) = pZ(z1, z1 − z2). Apply Corollary 3.3 with σ = {i ∈ {1, ..., d} : zi1 − zi2 ≥ 0} and Hölder’s inequality to see pZ(z1, z1 − z2) ≤ |ui(t, x)− ui(s, y)| > |zi1 − zi2| × ‖H(1,...,2d)(Z, 1)‖0,2. Therefore, in order to prove the desired results (c) and (d) of Theorem 1.1, it suffices to prove that: |ui(t, x)− ui(s, y)| > |zi1 − zi2| ≤ c exp − ‖z1 − z2‖ cT (|t− s|1/2 + |x− y|) , (6.2) ‖H(1,...,2d)(Z, 1)‖0,2 ≤ cT (|t− s|1/2 + |x− y|)−(d+η)/2, (6.3) and if s = t, then ‖H(1,...,2d)(Z, 1)‖0,2 ≤ cT |x− y|−d/2. (6.4) Proof of (6.2). Let ũ denote the solution of (2.1) for b ≡ 0. Consider the continuous one- parameter martingale (Mu = (M u , ...,M u ), 0 ≤ u ≤ t) defined by M iu = (Gt−r(x, v)−Gs−r(y, v)) j=1 σij(ũ(r, v))W j(dr, dv) if 0 ≤ u ≤ s, (Gt−r(x, v) −Gs−r(y, v)) j=1 σij(ũ(r, v))W j(dr, dv) Gt−r(x, v) j=1 σij(ũ(r, v))W j(dr, dv) if s ≤ u ≤ t, for all i = 1, ..., d, with respect to the filtration (Fu, 0 ≤ u ≤ t). Notice that M0 = 0, Mt = ũ(t, x)− ũ(s, y). Moreover, by hypothesis P1 and Lemma 6.1, 〈M i〉t = (Gt−r(x, v) −Gs−r(y, v))2 (σij(ũ(r, v))) 2 drdv (Gt−r(x, v)) (σij(ũ(r, v))) 2 drdv (g(r, v))2 drdv ≤ CT (|t− s|1/2 + |x− y|). By the exponential martingale inequality Nualart [N95, A.5], |ũi(t, x)− ũi(s, y)| > |zi1 − zi2| ≤ 2 exp 1 − zi2|2 CT (|t− s|1/2 + |x− y|) . (6.5) We will now treat the case b 6≡ 0 using Girsanov’s theorem. Consider the random variable Lt = exp σ−1(u(r, v)) b(u(r, v)) ·W (dr, dv) ‖σ−1(u(r, v)) b(u(r, v))‖2 drdv The following Girsanov’s theorem holds. Theorem 6.4. [N94, Prop.1.6] E[Lt] = 1, and if P̃ denotes the probability measure on (Ω,F ) defined by (ω) = Lt(ω), then W̃ (t, x) = W (t, x) + σ−1(u(r, v)) b(u(r, v)) drdv is a standard Brownian sheet under P̃. Consequently, the law of u under P̃ coincides with the law of ũ under P. Consider now the random variable Jt = exp σ−1(ũ(r, v)) b(ũ(r, v)) ·W (dr, dv) ‖σ−1(ũ(r, v)) b(ũ(r, v))‖2 drdv Then, by Theorem 6.4, the Cauchy-Schwarz inequality and (6.5), |ui(t, x)− ui(s, y)| > |zi1 − zi2| 1{|ui(t,x)−ui(s,y)||zi1−zi2|} 1{|ũi(t,x)−ũi(s,y)||zi1−zi2|} |ũi(t, x)− ũi(s, y)||zi1 − zi2| })1/2( ≤ 2 exp 1 − zi2|2 CT (|t− s|1/2 + |x− y|) Now, hypothesis P1 and P2 give t ] ≤ EP 2σ−1(ũ(r, v)) b(ũ(r, v)) ·W (dr, dv) 4 ‖σ−1(ũ(r, v)) b(ũ(r, v))‖2 drdv × exp ‖σ−1(ũ(r, v)) b(ũ(r, v))‖2 drdv since the second exponential is bounded and the first is an exponential martingale. Therefore, we have proved that |ui(t, x)− ui(s, y)| > |zi1 − zi2| ≤ C exp 1 − zi2|2 CT (|t− s|1/2 + |x− y|) from which we conclude that |ui(t, x)− ui(s, y)| > |zi1 − zi2| ≤ C exp − ‖z1 − z2‖ CT (|t− s|1/2 + |x− y|) This proves (6.2). Proof of (6.3). As in (3.1), using the continuity of the Skorohod integral δ and Hölder’s inequality for Malliavin norms, we obtain ‖H(1,...,2d)(Z, 1)‖0,2 ≤ C‖H(1,...,2d−1)(Z, 1)‖1,4 ‖(γ−1Z )2d,j‖1,8 ‖D(Z j)‖1,8 j=d+1 ‖(γ−1Z )2d,j‖1,8 ‖D(Z j)‖1,8 Notice that the entries of γ−1Z that appear in this expression belong to sets (3) and (4) of indices, as defined before Theorem 6.3. From Theorem 6.3(a) and Propositions 4.1 and 6.2, we find that this is bounded above by CT ‖H(1,...,2d−1)(Z, 1)‖1,4 (|t−s|1/2+ |x−y|)− j=d+1 (|t−s|1/2+ |x−y|)−1−dη+ that is, by CT ‖H(1,...,2d−1)(Z, 1)‖1,4(|t− s|1/2 + |x− y|)−1/2−dη . Iterating this procedure d times (during which we only encounter coefficients (γ−1Z )m,l for (m, l) in blocs (3) and (4), cf. Theorem 6.3(a)), we get, for some integers m0, k0 > 0, ‖H(1,...,2d)(Z, 1)‖0,2 ≤ CT ‖H(1,...,d)(Z, 1)‖m0 ,k0(|t− s|1/2 + |x− y|)−d/2−d Again, using the continuity of δ and Hölder’s inequality for the Malliavin norms, we obtain ‖H(1,...,d)(Z, 1)‖m,k ≤ C‖H(1,...,d−1)(Z, 1)‖m1 ,k1 ‖(γ−1Z )d,j‖m2,k2 ‖D(Z j)‖m3,k3 j=d+1 ‖(γ−1Z )d,j‖m4,k4 ‖D(Z j)‖m5,k5 for some integers mi, ki > 0, i = 1, ..., 5. This time, the entries of γ Z that appear in this expression come from the sets (1) and (2) of indices. We appeal again to Theorem 6.3(a) and Propositions 4.1 and 6.2 to get ‖H(1,...,d)(Z, 1)‖m,k ≤ CT ‖H(1,...,d−1)(Z, 1)‖m1 ,k1(|t− s|1/2 + |x− y|)−dη. Finally, iterating this procedure d times (during which we encounter coefficients (γ−1Z )m,l for (m, l) in blocs (1) and (2) only, cf. Theorem 6.3(a)), and choosing η′ = 4d2η, we conclude ‖H(1,...,2d)(Z, 1)‖0,2 ≤ CT (|t− s|1/2 + |x− y|)−(d+η ′)/2, which proves (6.3) and concludes the proof of Theorem 1.1(c). Proof of (6.4). In order to prove (6.4), we proceed exactly along the same lines as in the proof of (6.3) but we appeal to Theorem 6.3(b). This concludes the proof of Theorem 1.1(d). 6.4 Proof of Theorem 6.3 Let Z as in (6.1). Since the inverse of the matrix γZ is the inverse of its determinant multiplied by its cofactor matrix, we examine these two factors separately. Proposition 6.5. Fix T > 0 and let I and J be compact intervals as in Theorem 1.1. Assuming P1, for any (s, y), (t, x) ∈ I × J , (s, y) 6= (t, x), p > 1, E[|(AZ)m,l|p]1/p ≤ cp,T (|t− s|1/2 + |x− y|)d if (m, l) ∈ (1), cp,T (|t− s|1/2 + |x− y|)d− 2 if (m, l) ∈ (2) or (3), cp,T (|t− s|1/2 + |x− y|)d−1 if (m, l) ∈ (4), where AZ denotes the cofactor matrix of γZ . Proof. We consider the four different cases. • If (m, l) ∈ (1), we claim that |(AZ)m,l| ≤ C ‖D(u(s, y))‖2k × ‖D(u(t, x)− u(s, y))‖2(d−1−k) × ‖D(u(s, y))‖2(d−1−k) × ‖D(u(t, x) − u(s, y))‖2(k+1) (6.6) Indeed, let A Z = (a m̄,l̄ )m̄,l̄=1,...,2d−1 be the (2d−1)× (2d−1)-matrix obtained by removing from γZ its row m and column l. Then (AZ)m,l = det ((AZ) m,l) = π permutation of (1,...,2d−1) 1,π(1) · · · am,l 2d−1,π(2d−1). Each term of this sum contains one entry from each row and column of A Z . If there are k entries taken from bloc (1) of γZ , these occupy k rows and columns of A Z . Therefore, d−1−k entries must come from the d−1 remaining rows of bloc (2), and the same number from the columns of bloc (3). Finally, there remain k+1 entries to be taken from bloc (4). Therefore, |(AZ)m,l| ≤ C (product of k entries from (1)) × (product of d− 1− k entries from (2)) × (product of d− 1− k entries from (3))× (product of k + 1 entries from (4)) Adding all the terms and using the particular form of these terms establishes (6.6). Regrouping the various factors in (6.6), applying the Cauchy-Schwarz inequality and using (4.1) and Proposition 6.2, we obtain |(AZ)m,l|p ‖D(u(s, y))‖2(d−1)p × ‖D(u(t, x) − u(s, y))‖2dp ≤ CT (|t− s|1/2 + |x− y|)dp. • If (m, l) ∈ (2) or (m, l) ∈ (3), then using the same arguments as above, we obtain |(AZ)m,l| ‖D(u(s, y))‖2(d−1−k) × ‖D(u(s, y))‖k × ‖D(u(t, x) − u(s, y))‖k × ‖D(u(s, y))‖k+1 × ‖D(u(t, x)− u(s, y))‖k+1 × ‖D(u(t, x)− u(s, y))‖2(d−1−k) ‖D(u(s, y))‖2d−1 × ‖D(u(t, x) − u(s, y))‖2d−1 from which we conclude, using (4.1) and Proposition 6.2, that |(AZ)m,l|p ≤ CT (|t− s|1/2 + |x− y|)(d− • If (i, j) ∈ (4), we obtain |(AZ)m,l| ≤ C ‖D(u(s, y))‖2(k+1) × ‖D(u(s, y))‖2(d−1−k) × ‖D(u(t, x) −D(u(s, y)))‖2(d−1−k) × ‖D(u(t, x) −D(u(s, y)))‖2k ‖D(u(s, y))‖2d × ‖D(u(t, x) − u(s, y))‖2d−2 from which we conclude that |(AZ)m,l|p ≤ CT (|t− s|1/2 + |x− y|)(d−1)p. This concludes the proof of the proposition. Proposition 6.6. Fix η, T > 0. Assume P1 and P2. Let I and J be compact intervals as in Theorem 1.1. (a) There exists C depending on T and η such that for any (s, y), (t, x) ∈ I × J , (s, y) 6= (t, x), p > 1, det γZ )−p]1/p ≤ C(|t− s|1/2 + |x− y|)−d(1+η). (6.7) (b) There exists C only depending on T such that for any s = t ∈ I, x, y ∈ J , x 6= y, p > 1, det γZ )−p]1/p ≤ C(|x− y|)−d. Assuming this proposition, we will be able to conclude the proof of Theorem 6.3, after establishing the following estimate on the derivative of the Malliavin matrix. Proposition 6.7. Fix T > 0. Let I and J be compact intervals as in Theorem 1.1. As- suming P1, for any (s, y), (t, x) ∈ I × J , (s, y) 6= (t, x), p > 1 and k ≥ 1, ‖Dk(γZ)m,l‖pH ⊗k ]1/p ≤ ck,p,T if (m, l) ∈ (1), ck,p,T (|t− s|1/2 + |x− y|)1/2 if (m, l) ∈ (2) or (3), ck,p,T (|t− s|1/2 + |x− y|) if (m, l) ∈ (4). Proof. We consider the four different blocs. • If (m, l) ∈ (4), proceeding as inDalang and Nualart [DN04, p.2131] and appealing to Proposition 6.2 twice, we obtain ∥Dk(γZ)m,l dv Dr,v(um(t, x)− um(s, y)) ·Dr,v(ul(t, x) − ul(s, y)) ≤ (k + 1)p−1 dv DjDr,v(um(t, x)− um(s, y)) ·Dk−jDr,v(ul(t, x)− ul(s, y)) ≤ C̃T (k + 1)p−1 ∥DjD(um(t, x)− um(s, y)) H ⊗(j+1) ])1/2 ∥Dk−jD(ul(t, x)− ul(s, y)) H ⊗(k−j+1) ])1/2} ≤ CT (|t− s|1/2 + |x− y|)p. • If (m, l) ∈ (2) or (m, l) ∈ (3), proceeding as above and appealing to (4.1) and Propo- sition 6.2, we get ∥Dk(γZ)m,l ≤ C̃T (k + 1)p−1 ∥DjD(um(t, x)− um(s, y)) H ⊗(j+1) ])1/2 ∥Dk−jD(ul(s, y)) H ⊗(k−j+1) ])1/2} ≤ CT (|t− s|1/2 + |x− y|)p/2. • If (m, l) ∈ (1), using (4.1), we obtain ∥Dk(γZ)m,l ≤ C̃T (k + 1)p−1 ∥DjD(um(s, y)) H ⊗(j+1) ])1/2 ∥Dk−jD(ul(s, y)) H ⊗(k−j+1) ])1/2} ≤ CT . Proof of Theorem 6.3. When k = 0, the result follows directly using the fact that the inverse of a matrix is the inverse of its determinant multiplied by its cofactor matrix and the estimates of Propositions 6.5 and 6.6. For k ≥ 1, we shall establish the following two properties. (a) For any (s, y), (t, x) ∈ I × J , (s, y) 6= (t, x), s ≤ t, k ≥ 1 and p > 1, E[‖Dk(γ−1Z )m,l‖ ]1/p ≤ ck,p,η,T (|t− s|1/2 + |x− y|)−dη if (m, l) ∈ (1) , ck,p,η,T (|t− s|1/2 + |x− y|)−1/2−dη if (m, l) ∈ (2), (3), ck,p,η,T (|t− s|1/2 + |x− y|)−1−dη if (m, l) ∈ (4). (b) For any s = t ∈ I, x, y ∈ J , x 6= y, k ≥ 1 and p > 1, E[‖Dk(γ−1Z )m,l‖ ]1/p ≤ ck,p,T if (m, l) ∈ (1) , ck,p,T |x− y|−1/2 if (m, l) ∈ (2) or (3), ck,p,T |x− y|−1 if (m, l) ∈ (4). Since ‖(γ−1Z )m,l‖k,p = E[|(γ−1Z )m,l| E[‖Dj(γ−1Z )m,l‖ (a) and (b) prove the theorem. We now prove (a) and (b). When k = 1, we will use (3.3) written as a matrix product: D(γ−1Z ) = γ Z D(γZ)γ Z . (6.8) Writing (6.8) in bloc product matrix notation with blocs (1), (2), (3) and (4), we get that D((γ−1Z ) (1)) = (γ−1Z ) (1)D(γ Z )(γ (1) + (γ−1Z ) (1)D(γ Z )(γ + (γ−1Z ) (2)D(γ Z )(γ (1) + (γ−1Z ) (2)D(γ Z )(γ D((γ−1Z ) (2)) = (γ−1Z ) (1)D(γ Z )(γ (2) + (γ−1Z ) (1)D(γ Z )(γ + (γ−1Z ) (2)D(γ Z )(γ (2) + (γ−1Z ) (2)D(γ Z )(γ D((γ−1Z ) (3)) = (γ−1Z ) (3)D(γ Z )(γ (1) + (γ−1Z ) (3)D(γ Z )(γ + (γ−1Z ) (4)D(γ Z )(γ (1) + (γ−1Z ) (4)D(γ Z )(γ D((γ−1Z ) (4)) = (γ−1Z ) (3)D(γ Z )(γ (2) + (γ−1Z ) (3)D(γ Z )(γ + (γ−1Z ) (4)D(γ Z )(γ (2) + (γ−1Z ) (4)D(γ Z )(γ It now suffices to apply Hölder’s inequality to each block and use the estimates of the case k = 0 and Proposition 6.7 to obtain the desired result for k = 1. For instance, for (m, l) ∈ (1), (γ−1Z ) (2)D(γ Z )(γ ≤ sup m1,l1 (γ−1Z ) m1,l1 ]1/(2p) m2,l2 m2,l2 ]1/(4p) × sup m3,l3 (γ−1Z ) m3,l3 ]1/(4p) ≤ c (|t− s|1/2 + |x− y|)− −dη+1− 1 −dη = c (|t− s|1/2 + |x− y|)−2dη . For k ≥ 1, in order to calculate Dk+1(γ(·)Z ), we will need to compute Dk(γ Z D(γZ)γ For bloc numbers i1, i2, i3 ∈ {1, 2, 3, 4} and k ≥ 1, we have (γ−1Z ) (i1)D(γ Z )(γ j1+j2+j3=k ji∈{0,...,k} j1 j2 j3 (γ−1Z ) (γ−1Z ) Note that by Proposition 6.7, the norms of the derivatives Dj2 Z ) of γ Z are of the same order for all j2. Hence, we appeal again to Hölder’s inequality and Proposition 6.7, and use a recursive argument in order to obtain the desired bounds. Proof of Proposition 6.6. The main idea for the proof of Proposition 6.6 is to use a pertur- bation argument. Indeed, for (t, x) close to (s, y), the matrix γZ is close to ... 0 · · · ... · · · ... 0 The matrix γ̂ has d eigenvectors of the form (λ̂1,0), ..., (λ̂d,0), where λ̂1, ..., λ̂d ∈ Rd are eigenvectors of γ Z = γu(s,y), and 0 = (0, ..., 0) ∈ Rd, and d other eigenvectors of the form (0, ei) where e1, ..., ed is a basis of Rd. These last eigenvectors of γ̂ are associated with the eigenvalue 0, while the former are associated with eigenvalues of order 1, as can be seen in the proof of Proposition 4.2. We now write det γZ = (ξi)T γZξ i, (6.9) where ξ = {ξ1, ..., ξ2d} is an orthonormal basis of R2d consisting of eigenvectors of γZ . We then expect that for (t, x) close to (s, y), there will be d eigenvectors close to the subspace generated by the (λ̂i,0), which will contribute a factor of order 1 to the product in (6.9), and d other eigenvectors, close to the subspace generated by the (0, ei), that will each contribute a factor of order (|t− s|1/2 + |x− y|)−1−η to the product. Note that if we do not distinguish between these two types of eigenvectors, but simply bound below the product by the smallest eigenvalue to the power 2d, following the approach used in the proof of Proposition 4.2, then we would obtain C(|t− s|1/2 + |x − y|)−2dp in the right-hand side of (6.7), which would not be the correct order. We now carry out this somewhat involved perturbation argument. Consider the spaces E1 = {(λ,0) : λ ∈ Rd,0 ∈ Rd} and E2 = {(0, µ) : µ ∈ Rd,0 ∈ Rd}. Note that every ξi can be written as ξi = (λi, µi) = αi(λ̃ i,0) + 1− α2i (0, µ̃ i), (6.10) where λi, µi ∈ Rd, (λ̃i,0) ∈ E1, (0, µ̃i) ∈ E2, with ‖λ̃i‖ = ‖µ̃i‖ = 1 and 0 ≤ αi ≤ 1. Note in particular that ‖ξi‖2 = ‖λi‖2 + ‖µi‖2 = 1 (norms of elements of Rd or R2d are Euclidean norms). Lemma 6.8. Given a sufficiently small α0 > 0, with probability one, there exist at least d of these vectors, say ξ1, ..., ξd, such that α1 ≥ α0, ..., αd ≥ α0. Proof. Observe that as ξ is an orthogonal family and for i 6= j, the Euclidean inner product of ξi and ξj is ξi · ξj = αiαj (λ̃i · λ̃j) + 1− α2i 1− α2j (µ̃ i · µ̃j) = 0. For α0 > 0, let D = {i ∈ {1, ..., 2d} : αi < α0}. Then, for i, j ∈ D, i 6= j, if α0 < 12 , then |µ̃i · µ̃j | = αiαj√ 1− α2i 1− α2j |λ̃i · λ̃j| ≤ α 1− α20 ‖λ̃i‖‖λ̃j‖ ≤ 1 Since the diagonal terms of the matrix (µ̃i · µ̃j)i,j∈D are all equal to 1, for α0 sufficiently small, it follows that det((µ̃i · µ̃j)i,j∈D) 6= 0. Therefore, {µ̃i, i ∈ D} is a linearly independent family, and, as (0, µ̃i) ∈ E2, for i = 1, ..., 2d, we conclude that a.s., card(D) ≤ dim(E2) = d. We can therefore assume that {1, ..., d} ⊂ Dc and so α1 ≥ α0,...,αd ≥ α0. By Lemma 6.8 and Cauchy-Schwarz inequality one can write det γZ )−p]1/p ≤ (ξi)T γZξ )−2p])1/(2p) ξ=(λ,µ)∈R2d : ‖λ‖2+‖µ‖2=1 ξTγZξ −2dp )1/(2p) With this, Propositions 6.9 and 6.15 below conclude the proof of Proposition 6.6. 6.4.1 Small Eigenvalues Let I and J two compact intervals as in Theorem 1.1. Proposition 6.9. Fix η, T > 0. Assume P1 and P2. (a) There exists C depending on η and T such that for all s, t ∈ I, 0 < t−s < 1, x, y ∈ J , x 6= y, and p > 1, ξ=(λ,µ)∈R2d : ‖λ‖2+‖µ‖2=1 ξTγZξ −2dp ≤ C(|t− s|1/2 + |x− y|)−2dp(1+η). (b) There exists C depending only on T such that for all s = t ∈ I, x, y ∈ J , x 6= y, and p > 1, ξ=(λ,µ)∈R2d: ‖λ‖2+‖µ‖2=1 ξTγZξ −2dp ≤ C(|x− y|)−2dp. Proof. We begin by proving (a). Since γZ is a matrix of inner products, we can write ξTγZξ = r,v (u(s, y)) + µi(D r,v (u(t, x)) −D(k)r,v (u(s, y))) Therefore, for ǫ ∈ (0, t− s), ξTγZξ ≥ J1 + J2, where J1 := (λi − µi) [Gs−r(y, v)σik(u(r, v)) + ai(k, r, v, s, y)] +W J2 := dvW 2, ai(k, r, v, s, y) is defined in (4.3) and [µiGt−r(x, v)σik(u(r, v)) + µiai(k, r, v, t, x)] . From here on, the proof is divided into two cases. Case 1. In the first case, we assume that |x− y|2 ≤ t− s. Choose and fix an ǫ ∈ (0, t − s). Then we may write ‖ξ‖=1 ξTγZξ ≥ min ‖ξ‖=1 ,‖µ‖≥ǫη/2 J2 , inf ‖ξ‖=1 ,‖µ‖≤ǫη/2 We are going to prove that ‖ξ‖=1 ,‖µ‖≥ǫη/2 J2 ≥ ǫ +η − Y1,ǫ, ‖ξ‖=1 ,‖µ‖≤ǫη/2 J1 ≥ ǫ1/2 − Y2,ǫ, (6.11) where, for all q ≥ 1, E [|Y1,ǫ|q] ≤ c1(q)ǫq and E [|Y2,ǫ|q] ≤ c2(q)ǫq( +η). (6.12) We assume these, for the time being, and finish the proof of the proposition in Case 1. Then we will return to proving (6.11) and (6.12). We can combine (6.11) and (6.12) with Proposition 3.5 to find that ‖ξ‖=1 ξTγZξ )−2pd ≤ c(t− s)−2pd( (t− s)1/2 + |x− y| ]−2pd(1+2η) whence follows the proposition in the case that |x− y|2 ≤ t− s. Now we complete our proof of Case 1 by deriving (6.11) and (6.12). Let us begin with the term that involves J2. Inequality (4.4) implies that ‖ξ‖=1 ,‖µ‖≥ǫη/2 J2 ≥ Ŷ1,ǫ − Y1,ǫ, where Ŷ1,ǫ := ‖µ‖≥ǫη/2 µiσik(u(r, v)) G2t−r(x, v), Y1,ǫ := 2 sup ‖µ‖≥ǫη/2 µiai(k, r, v, t, x) In agreement with hypothesis P2, and thanks to Lemma 7.2, Ŷ1,ǫ ≥ c inf ‖µ‖≥ǫη/2 ‖µ‖2ǫ1/2 ≥ cǫ Next we apply Lemma 6.11 below [with s := t] to find that E[|Y1,ǫ|q] ≤ cǫq. This proves the bounds in (6.11) and (6.12) that concern J2 and Y1,ǫ. In order to derive the second bound in (6.11), we appeal to (4.4) once more to find that ‖ξ‖=1 ,‖µ‖≤ǫη/2 J1 ≥ Ŷ2,ǫ − Y2,ǫ, where Ŷ2,ǫ := ‖µ‖≤ǫη/2 (λi − µi)σik(u(r, v)) G2s−r(y, v), Y2,ǫ := 2 (W1 +W2 +W3) , where W1 := sup ‖µ‖≤ǫη/2 µiGt−r(x, v)σik(u(r, v)) W2 := sup ‖ξ‖=1 (λi − µi)ai(k, r, v, s, y) W3 := sup ‖µ‖≤ǫη/2 µiai(k, r, v, t, x) Hypothesis P2 and Lemma 7.2 together yield Ŷ2,ǫ ≥ cǫ1/2. (6.13) Next, we apply the Cauchy–Schwarz inequality to find that E [|W1|q] ≤ sup ‖µ‖≤ǫη/2 ‖µ‖2q × E (σik(u(r, v))) G2t−r(x, v) ≤ cǫqη dv G2t−r(x, v) thanks to hypothesis P1. In light of this, Lemma 7.4 implies that E [|W1|q] ≤ cǫq( In order to bound the q-th moment of |W2|, we use the Cauchy–Schwarz inequality together with hypothesis P1, and write E [|W2|q] ≤ sup ‖µ‖≤ǫη/2 ‖λ− µ‖2q × E a2i (k, r, v, s, y) a2i (k, r, v, s, y) We apply Lemma 6.11 below [with s := t] to find that E [|W2|q] ≤ cǫq. Similarly, we find using Lemma 6.11 that E [|W3|q] ≤ sup ‖µ‖≤ǫη/2 ‖µ‖2q × E a2i (k, r, v, t, x) ≤ cǫqη (t− s+ ǫ)q/2ǫq/2 ≤ cǫq( The preceding bounds for W1, W2, and W3 prove, in conjunction, that E[|Y2,ǫ|q] ≤ c2(q)ǫ +η). This and (6.13) together prove the bounds in (6.11) and (6.12) that concern J1 and Y2,ǫ, whence follows the result in Case 1. Case 2. Now we work on the second case where |x − y|2 ≥ t − s ≥ 0. Let ǫ > 0 be such that (1+α)ǫ1/2 < 1 |x− y|, where α > 0 is large but fixed; its specific value will be decided on later. Then ξTγZξ ≥ I1 + I2 + I3, where I1 := dv (S1 + S2) I2 := dv (S1 + S2) I3 := (t−ǫ)∨s dvS 22 , S1 := (λi − µi) [Gs−r(y, v)σi,k(u(r, v)) + ai(k, r, v, s, y)] , S2 := µi [Gt−r(x, v)σik(u(r, v)) + ai(k, r, v, t, x)] . From here on, Case 2 is divided into two further sub-cases. Sub-Case A. Suppose, in addition, that ǫ ≥ t− s. In this case, we are going to prove that ‖ξ‖=1 ξTγZξ ≥ cǫ1/2 − Z1,ǫ, (6.14) where for all q ≥ 1, E [|Z1,ǫ|q] ≤ c(q)ǫ3q/4. (6.15) Apply (4.4) to find that Ã1 −B(1)1 −B where Ã1 := [(λi − µi)Gs−r(y, v) + µiGt−r(x, v)] σik(u(r, v)) 1 := 4‖λ− µ‖2 a2i (k, r, v, s, y), (6.16) 1 := 4‖µ‖ a2i (k, r, v, t, x). (6.17) Using the inequality (a− b)2 ≥ 2 a2 − 2ab, (6.18) we see that Ã1 ≥ A1 −B(3)1 , where A1 := (λi − µi)Gs−r(y, v)σik(u(r, v)) 1 := 2 (λi − µi)Gs−r(y, v)σik(u(r, v)) µiGt−r(x, v)σik(u(r, v)) We can combine terms to find that A1 −B(1)1 −B We proceed in like manner for I2, but obtain slightly sharper estimates as follows. Owing to (6.18), A2 −B(1)2 −B where A2 := µiGt−r(x, v)σik(u(r, v)) 2 := 2 µiGt−r(x, v)σik(u(r, v)) µiai(k, r, v, t, x) 2 := 2 µiGt−r(x, v)σik(u(r, v)) (λi − µi)ai(k, r, v, s, y) 2 := 2 µiGt−r(x, v)σik(u(r, v)) (λi − µi)Gs−r(y, v)σik(u(r, v)) Finally, we appeal to (4.4) to find that A3 −B3, where A3 := (t−ǫ)∨s µiGt−r(x, v)σik(u(r, v)) B3 := 2 (t−ǫ)∨s µiai(k, r, v, t, x) . (6.19) By hypothesis P2, A1 +A2 +A3 ≥ ρ2 ‖λ− µ‖2 dv G2s−r(y, v) + ‖µ‖2 dv G2t−r(x, v) + ‖µ‖2 dv G2t−r(x, v) Note that we have used the defining assumption of Sub-Case A, namely, that ǫ ≥ t − s. Next, we group the last two integrals and apply Lemma 7.2 to find that A1 +A2 +A3 ≥ c ‖λ− µ‖2ǫ1/2 + ‖µ‖2 dv G2t−r(x, v) ‖λ− µ‖2 + ‖µ‖2 ≥ cǫ1/2. (6.20) We are aiming for (6.14), and propose to bound the absolute moments of B 1 , B i = 1, 2, 3 and B3, separately. According to Lemma 6.11 below with s = t, ‖ξ‖=1 |B3|q ≤ c(q)ǫq. (6.21) Next we bound the absolute moments of B 1 , i = 1, 2, 3. Using hypothesis P1 and Lemma 6.11, with t = s, we find that for all q ≥ 1, ‖ξ‖=1 ≤ cǫq. (6.22) In the same way, we see that ‖ξ‖=1 ≤ c(t− s+ ǫ)q/2ǫq/2. (6.23) We are in the sub-case A where t− s ≤ ǫ. Therefrom, we obtain the following: ‖ξ‖=1 ≤ cǫq. (6.24) Finally, we turn to bounding the absolute moments of B 1 . Hypothesis P1 assures us dv Gs−r(y, v)Gt−r(x, v) dv Gs−r(y, v)Gt−r(x, v) dr Gt+s−2r(x, y), thanks to the semi-group property Walsh [W86, (3.6)] (see (6.44) below). This and Lemma 7.1 together prove that t− s+ 2u − |x− y| 2(t− s+ 2u) t− s+ 2u 2(t− s+ 2u) since |x−y| ≥ 2(1+α)ǫ1/2 ≥ αǫ1/2. Now we can change variables [z := 2(t−s+2u)/(α2ǫ)], and use the bounds 0 ≤ t− s ≤ ǫ to find that ∣ ≤ cǫ1/2Ψ(α), where Ψ(α) := α ∫ 6/α2 z−1/2e−1/z dz. (6.25) Following exactly in the same way, we see that ∣ ≤ cǫ1/2Ψ(α). (6.26) We can combine (6.22), (6.24) as follows: ‖ξ‖=1 ≤ c(q)ǫq. (6.27) On the other hand, we will see in Lemmas 6.13 and 6.14 below that ‖ξ‖=1 ≤ c(q)ǫ3q/4. (6.28) Now, by (6.20), (6.21), (6.25), (6.26), (6.27) and (6.28), ‖ξ‖=1 ξTγZξ A1 +A2 +A3 −B(3)1 −B 2 +B3 ≥ c1ǫ1/2 − c2Ψ(α)ǫ1/2 − Z1,ǫ, where Z1,ǫ := B 1 + B 1 + B 2 + B 2 + B3 satisfies E[|Z1,ǫ|q] ≤ c1(q)ǫ3q/4. Because limν→∞Ψ(ν) = 0, we can choose and fix α so large that c2Ψ(α) ≤ c1/4 for the c1 and c2 of the preceding displayed equation. This yields, ‖ξ‖=1 ξTγZξ ≥ cǫ1/2 − Z1,ǫ, (6.29) as in (6.14) and (6.15). Sub-Case B. In this final (sub-) case we suppose that ǫ ≤ t− s ≤ |x − y|2. Choose and fix 0 < ǫ < t− s. During the course of our proof of Case 1, we established the following: ‖ξ‖=1 ξTγZξ ≥ min +η − Y1,ǫ , cǫ1/2 − Y2,ǫ where E [|Y1,ǫ|q] ≤ c(q)ǫq and E [|Y2,ǫ|q] ≤ c(q)ǫq( See (6.11) and (6.12). Consider this in conjunction with (6.14) to find that for all 0 < ǫ < (1 + α)−2|x− y|2, ‖ξ‖=1 ξTγZξ ≥ min +η − Y1,ǫ , cǫ1/2 − Y2,ǫ − Z1,ǫ1{t−s<ǫ} Because of this and (6.15), Proposition 3.5 implies that ‖ξ‖=1 ξTγZξ )−2pd ≤ c|x− y|2(−2dp)( ≤ c(|t− s|1/2 + |x− y|)−2dp(1+2η). This concludes the proof of Proposition 6.9(a). If t = s, then Sub–Case B does not arise, and so we get directly from (6.29) and Proposition 3.5 that ‖ξ‖=1 ξTγZξ )−2pd ≤ c|x− y|−2dp. This proves (b) and concludes the proof of Proposition 6.9. Remark 6.10. If σ and b are constant, then ai = 0, so η can be taken to be 0. This gives the correct upper bound in the Gaussian case, which shows that the method of proof of Proposition 6.9 is rather tight. We finally prove three results that we have used in the proof of Proposition 6.9. Lemma 6.11. Assume P1. For all T > 0 and q ≥ 1, there exists a constant c = c(q, T ) ∈ (0,∞) such that for every 0 < ǫ ≤ s ≤ t ≤ T and x ∈ [0, 1], a2i (k, r, v, t, x) ≤ c(t− s+ ǫ)q/2ǫq/2. Proof. Define a2i (k, r, v, t, x). Use (4.3) to write E [|A|q] ≤ c (E [|A1|q] + E [|A2|q]) , where A1 := i,j,k=1 Gt−θ(x, η)D r,v (σij(u(θ, η))) W j(dθ, dη) A2 := i,k=1 dη Gt−θ(x, η)D r,v (bi(u(θ, η))) We bound the q-th moment of A1 and A2 separately. As regards A1, we apply the Burkholder inequality for Hilbert-space-valued martingales (Lemma 7.6) to find that E [|A1|q] ≤ c i,j,k=1 , (6.30) where Θ := 1{θ>r}Gt−θ(x, η) D(k)r,v (σij(u(θ, η))) ≤ c1{θ>r}Gt−θ(x, η) D(k)r,v (ul(θ, η)) thanks to hypothesis P1. Thus, E [|A1|q] ≤ c dη G2t−θ(x, η) ∫ s∧θ D(k)r,v (ul(θ, η)) We apply Hölder’s inequality with respect to the measure G2t−θ(x, η) dθ dη to find that E [|A1|q] ≤ c dη G2t−θ(x, η) dη G2t−θ(x, η) ∫ s∧θ (6.31) where Υ := r,v (ul(θ, η)). Lemma 7.3 assures us that dη G2t−θ(x, η) ≤ c(t− s+ ǫ)(q−1)/2. (6.32) On the other hand, Lemma 7.5 implies that ∫ s∧θ ≤ cǫq/2, where c ∈ (0,∞) does not depend on (θ, η, s, t, ǫ, x). Consequently, dη G2t−θ(x, η) ∫ s∧θ ≤ cǫq/2 dη G2t−θ(x, η) ≤ cǫq/2(t− s+ ǫ)1/2. (6.33) Equations (6.31), (6.32), and (6.33) together imply that E [|A1|q] ≤ c(t− s+ ǫ)q/2ǫq/2. (6.34) This is the desired bound for the q-th moment of A1. Next we derive a similar bound for A2. This will finish the proof. By the Cauchy–Schwarz inequality E [|A2|q] ≤ c(t− s+ ǫ)q i,k=1 where Φ := Gt−θ(x, η)|D(k)r,v (bi(u(θ, η))) |. From here on, the q-th moment of A2 is estimated as that of A1 was; cf. (6.30), and this yields E[|A2|q] ≤ c(t− s+ ǫ)3q/2ǫq/2. This completes the proof. Remark 6.12. It is possible to prove that E[|A1|] is at least a constant times (t−s+ǫ)1/2ǫ1/2. In this sense, the preceding result is not improvable. Lemma 6.13. Assume P1. Fix T > 0 and q ≥ 1. Then there exists c = c(q, T ) such that for all x ∈ [0, 1], 0 ≤ s ≤ t ≤ T , and ǫ ∈ (0, 1), µ∈Rd: ‖µ‖≤1 ≤ cǫ3q/4. Proof. Define 2 (k, i) := dv Gt−r(x, v) |ai(k, r, v, t, x)| . Then, by the Cauchy–Schwarz inequality, 2 (k, i) ≤ J1J2, (6.35) where J1 := dv G2t−r(x, v) J2 := E dv a2i (k, r, v, t, x) On one hand, according to Lemma 7.4, J1 ≤ c (t− s+ ǫ)q/4 . (6.36) On the other hand, Lemma 6.11 assures us that J2 ≤ c′(t− s+ ǫ)q/4ǫq/4. (6.37) By combining (6.35), (6.36), and (6.37), we find that E[|B̂(1)2 (k, i)|q ] ≤ cǫ3q/4 for a constant c ∈ (0,∞) that does not depend on ǫ. By hypothesis P1, µ∈Rd: ‖µ‖≤1 2 (k, i) ≤ cǫ3q/4, (6.38) as asserted. Lemma 6.14. Assume P1. Fix T > 0 and q ≥ 1. Then there exists c = c(q, T ) such that for any x ∈ [0, 1], 0 ≤ s ≤ t ≤ T , and ǫ ∈ (0, 1), ξ=(λ,µ)∈R2d : ‖ξ‖=1 ≤ cǫ3q/4. Proof. Define 2 (k, i) := dv Gt−r(x, v) |ai(k, r, v, s, y)| . Then, by the Cauchy–Schwarz inequality, 2 (k, i) ≤ J1J2, where J1 := dv G2t−r(x, v) J2 := E dv a2i (k, r, v, s, y) According to (6.36), J1 ≤ c (t− s+ ǫ)q/4 ≤ c′ǫq/4. On the other hand, Lemma 6.11, with t = s, assures us that J2 ≤ c′′ǫq/2. It follows that the q-th absolute moment of B̂ 2 (k, i) is at most cǫ 3q/4. An appeal to the triangle inequality finishes the proof; see (6.38) where a similar argument was worked out in detail. 6.4.2 Large Eigenvalues Proposition 6.15. Assume P1 and P2. Fix T > 0 and p > 1. Then there exists C = C(p, T ) such that for all 0 ≤ s < t ≤ T with t− s < 1 , x, y ∈ (0, 1), x 6= y, (ξi)TγZξ where ξ1, ..., ξd are the vectors from Lemma 6.8. Proof. Let ξ1, . . . , ξd, written as in (6.10), be such that α1 ≥ α0, . . . , αd ≥ α0 for some α0 > 0. In order to simplify the exposition, we assume that 0 < α = α1 = · · · = αd ≤ 1, since the general case follows along the same lines. Let 0 < ǫ < s ≤ t. As in the proof of Proposition 6.9, we note first that i=1(ξ i)TγZξ i is bounded below by αλ̃iGs−r(y, v) + µ̃i 1− α2 (Gt−r(x, v) −Gs−r(y, v)) σik(u(r, v)) + αλ̃iai(k, r, v, s, y) + µ̃i 1− α2 (ai(k, r, v, t, x) − ai(k, r, v, s, y)) s∨(t−ǫ) 1− α2 Gt−r(x, v)σik(u(r, v)) + µ̃i 1− α2 ai(k, r, v, t, x) (6.39) We intend to use Proposition 3.5 with ε0 > 0 fixed, so we seek lower bounds for this expression for 0 < ε < ε0. Case 1. t− s ≤ ǫ. Then, by (4.4), the expression in (6.39) is bounded below by (f1(s, t, ǫ, α, λ̃, µ̃, x, y) + f2(s, t, ǫ, α, λ̃, µ̃, x, y))− 2Iǫ, where, from hypothesis P2, f1 ≥ cρ2 αλ̃Gs−r(y, v) + 1− α2 µ̃(Gt−r(x, v) −Gs−r(y, v)) , (6.40) f2 ≥ cρ2 s∨(t−ǫ) 1− α2 Gt−r(x, v) , (6.41) and Iǫ = I1,ǫ + I2,ǫ + I3,ǫ, where I1,ǫ := αλ̃i − µ̃i 1− α2 ai(k, r, v, s, y) I2,ǫ := 1− α2 ai(k, r, v, t, x) I3,ǫ := 1− α2 ai(k, r, v, t, x) There are obvious similarities between the terms I1,ǫ and B 1 in (6.16). Thus, we apply the same method that was used to bound E[|B(1)1 |q] to deduce that E[|I1,ǫ|q] ≤ c(q)ǫq. Since I2,ǫ is similar to B 1 from (6.17) and t− s ≤ ǫ, we see using (6.24) that E[|I2,ǫ|q] ≤ c(q)ǫq. Finally, using the similarity between I3,ǫ and B3 in (6.19), we see that E[|I3,ǫ|q] ≤ c(q)ǫq. We claim that there exists α0 > 0, ǫ0 > 0 and c0 > 0 such that f1 + f2 ≥ c0 ǫ for all α ∈ [α0, 1], ǫ ∈ (0, ǫ0], s, t ∈ [1, 2], x, y ∈ [0, 1]. (6.42) This will imply in particular that for ǫ ≥ t− s, i ≥ c0ǫ1/2 − 2Iǫ, where E[|Iǫ|q] ≤ c(q)ǫq. In order to prove (6.42), first define pt(x, y) := (4πt) −1/2e−(x−y) 2/(4t). In addition, let g1(s, t, ǫ, α, λ̃, µ̃, x, y) and g2(s, t, ǫ, α, λ̃, µ̃, x, y) be defined by the same ex- pressions as the right-hand sides of (6.40) and (6.41), but with Gs−r(x, v) replaced by ps−r(x− v), and replaced by Observe that g1 ≥ 0, g2 ≥ 0, and if g1 = 0, then for all v ∈ Rd, ∥αps−r(y − v)λ̃+ 1− α2 (pt−r(x− v)− ps−r(y − v))µ̃ ∥ = 0. (6.43) If, in addition, λ̃ = µ̃, then we get that for all v ∈ Rd, 1− α2 ps−r(y − v) + 1− α2pt−r(x− v) = 0. We take Fourier transforms to deduce from this that for all ξ ∈ Rd, 1− α2 eiξy = − 1− α2eiξxe(s−t)ξ2 . If x = y, then it follows that s = t and α − 1− α2 = − 1− α2. Hence, if α 6= 0, x = y and λ̃ = µ̃, then g1 > 0. We shall make use of this observation shortly. Because ‖λ̃‖ = ‖µ̃‖ = 1, f1 is bounded below by α2G2s−r(y, v) + 1− α2 (Gt−r(x, v) −Gs−r(y, v))2 1− α2Gs−r(y, v)(Gt−r(x, v)−Gs−r(y, v))(λ̃ · µ̃) = cρ2 1− α2 G2s−r(y, v)) + 1− α2 G2t−r(x, v) 1− α2 1− α2Gs−r(y, v)Gt−r(x, v) 1− α2Gs−r(y, v)(Gt−r(x, v)−Gs−r(y, v))(λ̃ · µ̃− 1) Recall the semigroup property dv Gs−r(y, v)Gt−r(x, v) = Gs+t−2r(x, y) (6.44) (see Walsh [W86, (3.6)]). We set h := t− s and change variables [r̄ := s− r] to obtain the following bound: f1 ≥ cρ2 1− α2 G2r(y, y) + 1− α2 G2h+2r(x, x) 1− α2 1− α2Gh+2r(x, y) 1− α2(Gh+2r(x, y)−G2r(y, y)) λ̃ · µ̃− 1 Recall ([W86, p.318]), that Gt(x, y) = pt(x, y) +Ht(x, y), where Ht(x, y) is a continuous function that is uniformly bounded over (t, x, y) ∈ (0,∞)× (0, 1) × (0, 1). Therefore, f1 ≥ cρ2g̃1 − cǫ, where g̃1 := g̃1(h, ǫ, α, λ̃, µ̃, x, y) 1− α2 p2r(y, y) + 1− α2 p2h+2r(x, x) 1− α2 1− α2ph+2r(x, y) 1− α2 (ph+2r(x, y)− p2r(y, y)) λ̃ · µ̃− 1 We can recognize that ph+2r(x, y)− p2r(y, y) = exp(−(x− y)2/(4(h + 2r))) 4π(h + 2r) 4π(2r) Also, λ̃ · µ̃− 1 ≤ 0. Thus, g̃1 ≥ ĝ1, where ĝ1 := ĝ1(h, ǫ, α, x, y) 1− α2 p2r(y, y) + 1− α2 p2h+2r(x, x) 1− α2 1− α2ph+2r(x, y) Therefore, ĝ1 = 1− α2)2 1√ 1− α2 8π(h + r) + 2(α − 1− α2) 1− α2ph+2r(x, y) On the other hand, by (6.44) above, ∫ ǫ∧(t−s) 1− α2 G2r(y, y) ≥ g̃2 := ∫ ǫ∧h 1− α2 p2r(y, y)− Cǫ 1− α2 ǫ ∧ h− Cǫ. Finally, we conclude that f1 + f2 ≥ ĝ1 + g̃2 − 2Cǫ 1− α2 1− α2√ h+ ǫ− 1− α2 1− α2 dr ph+2r(x, y) 1− α2√ ǫ ∧ h− 2Cǫ. Now we consider two different cases. Case (i). Suppose α− 1− α2 ≥ 0, that is, α ≥ 2−1/2. Then ǫ−1/2 (ĝ1 + g̃2) ≥ φ1 − 2Cǫ1/2, where φ1(α , z) := 1− α2 1− α2 1 + z + 1− α2 1 ∧ z Clearly, α≥2−1/2 φ1(α, z) ≥ inf α>2−1/2 1− α2 1− α2 > φ0 > 0. Thus, α≥2−1/2, h≥0, 0<ǫ≤ǫ0 ǫ−1/2 (ĝ1 + g̃2) > 0. Case (ii). Now we consider the case where α − 1− α2 < 0, that is, α < 2−1/2. In this case, ǫ−1/2 (ĝ1 + g̃2) ≥ ψ1 − 2Cǫ1/2, where ψ1(α , z) := 1− α2 1− α2 1 + z + 1− α2 − α 1− α2 2 + z + 1− α2 1 ∧ z Note that ψ1(α, z) > 0 if α 6= 0. This corresponds to the observation made in the lines following (6.43). Moreover, for α ≥ α0 > 0, limz↓0 ψ1(α, z) ≥ (2π)−1/2α20, and ψ1(α , z) ≥ inf 1− α2)2 + 1− α2 Therefore, α∈[α0,2−1/2], z≥0 ψ1(α , z) > 0. This concludes the proof of the claim (6.42). Case 2. t− s > ǫ. In accord with (6.39), we are interested in 1≥α≥α0 i := min(E1,ǫ, E2,ǫ), where E1,ǫ := inf α0≤α≤ E2,ǫ := inf√ 1−ǫη≤α≤1 Clearly, E1,ǫ ≥ f2 − 2I3,ǫ. Since α ≤ 1− ǫη is equivalent to 1− α2 ≥ ǫη/2, we use hypothesis P2 to deduce that f2 ≥ cρ2ǫη dv G2t−r(x, v) ≥ cρ2ǫ Therefore, E1,ǫ ≥ cρ2ǫ +η − I3,ǫ, and we have seen that I3,ǫ has the desirable property E [|I3,ǫ|q] ≤ c(q)ǫq. In order to estimate E2,ǫ, we observe using (6.39) that E2,ǫ ≥ f̃1 − J̃1,ǫ − J̃2,ǫ − J̃3,ǫ − J̃4,ǫ, where f̃1 ≥ α2 λ̃iσik(u(r, v)) G2s−r(y, v), J̃1,ǫ = 2 1− α2 µ̃iσik(u(r, v)) G2t−r(x, v), J̃2,ǫ = 2 1− α2 µ̃iσik(u(r, v)) G2s−r(y, v), J̃3,ǫ = 2 αλ̃i − µ̃i 1− α2 ai(k, r, v, s, y) J̃4,ǫ = 2 1− α2 µ̃iai(k, r, v, t, x) Because α2 ≥ 1−ǫη and ǫ ≤ t−s ≤ 1 , hypothesis P2 and Lemma 7.2 imply that f̃1 ≥ cǫ1/2. On the other hand, since 1−α2 ≤ ǫη, we can use hypothesis P1 and Lemma 7.4 to see that J̃1,ǫ ≤ c(q)ǫqηǫq/2 = c(q)ǫ( +η)q, and similarly, using Lemma 7.3, E[|J̃2,ǫ|q] ≤ c(q)ǫ( +η)q. The term J̃3,ǫ is equal to 2I1,ε, so E[|J̃3,ǫ|q] ≤ cεq, and J̃4,ε is similar to B(2)1 from (6.17), so we find using (6.23) that J̃4,ǫ ≤ cǫqη(t− s+ ǫ)q/2ǫq/2 ≤ cǫ( +η)q. We conclude that when t − s > ǫ, then E2,ǫ ≥ cǫ1/2 − J̃ǫ, where E[|J̃ǫ|q] ≤ c(q)ǫ( +η)q. Therefore, when t− s > ǫ, 1≥α≥α0 i ≥ min +η − I3,ǫ , cǫ 2 − J̃ǫ Putting together the results of Case 1 and Case 2, we see that for 0 < ǫ ≤ 1 ‖ξ‖=1, 1≥α≥α0 i ≥ min +η − I3,ǫ, cǫ 2 − 2Iǫ1{ǫ≥t−s} − J̃ǫ1{ǫ<t−s} We take into account the bounds on moments of I3,ǫ, Iǫ and J̃ǫ, and then use Proposition 3.5 to conclude the proof of Proposition 6.15. 7 Appendix On several occasions, we have appealed to the following technical estimates on the Green kernel of the heat equation. Lemma 7.1. [BP98, (A.1)] There exists C > 0 such that for any 0 < s < t and x, y ∈ [0, 1], x 6= y, Gt−s(x, y) ≤ C 2π(t− s) −|x− y| 2(t− s) Lemma 7.2. [BP98, (A.3)] There exists C > 0 such that for any t ≥ ǫ > 0 and x ∈ [0, 1], G2t−s(x, y)dyds ≥ C Lemma 7.3. [BP98, (A.5)] There exists C > 0 such that for any ǫ > 0, q < 3 , t ≥ ǫ and x ∈ [0, 1], t−s(x, y)dyds ≤ Cǫ3/2−q. Lemma 7.4. There exists C > 0 such that for all 0 < a < b and x ∈ [0, 1], G2s(x, y) dyds ≤ C b− a√ Proof. Using Lemma 7.1 and the change of variables z = , we see that G2s(x, y) dyds ≤ C ds = 2C̃( which concludes the proof. The next result is a straightforward extension to d ≥ 1 of Morien [M98, Lemma 4.2] for d = 1. Lemma 7.5. Assume P1. For all q ≥ 1, T > 0 there exists C > 0 such that for all T ≥ t ≥ s ≥ ǫ > 0 and 0 ≤ y ≤ 1, D(k)r,v (ui(t, y))  ≤ Cǫq/2. The next result is Burkholder’s inequality for Hilbert-space-valued martingales. Lemma 7.6. [BP98, eq.(4.18)] Let Hs,t be a predictable L 2(([0, t]× [0, 1])m)-valued process, m ≥ 1. Then, for any p > 1, there exists C > 0 such that ([0,t]×[0,1])m Hs,y(α)W (dy, ds) ([0,t]×[0,1])m H2s,y(α)dα Acknowledgement. The authors thank V. Bally for several stimulating discussions. References [BMS95] Bally, V., Millet, A. and Sanz-Solé, M. (1995), Approximation and sup- port theorem in Hölder norm for parabolic stochastic partial differential equations, The Annals of Probability, 23, 178-222. [BP98] Bally, V. and Pardoux, E. (1998), Malliavin calculus for white noise driven parabolic SPDEs, Potential Analysis, 9, 27-64. [DN04] Dalang, R.C. and Nualart, E. (2004), Potential theory for hyperbolic SPDEs, The Annals of Probability, 32, 2099-2148. [DKN07] Dalang, R.C., Khoshnevisan, D. and Nualart, E. (2007), Hitting prob- abilities for systems of non-linear stochastic heat equations with additive noise, Submitted for publication. See http://arxiv.org/pdf/math.PR/0702710. [K85] Kahane, J.-P. (1985), Some random series of functions, Cambridge University Press. [K02] Khoshnevisan, D. (2002), Multiparameter processes. An introduction to random fields, Springer-Verlag. [K03] Kohatsu-Higa, A. (2003), Lower bound estimates for densities of uniformly elliptic random variables on Wiener space, Probab. Theory Related Fields, 126, 421-457. [M98] Morien, P.-L. (1998), The Hölder and the Besov regularity of the density for the solution of a parabolic stochastic partial differential equation, Bernouilli, 5, 275-298. http://arxiv.org/pdf/math.PR/0702710 [MT03] Mueller, C. and Tribe, R. (2002), Hitting properties of the random string, Electronic Journal of Probability, 7, 1-29. [N94] Nualart, D. and Pardoux, E. (1994), Markov field properties of solutions of white noise driven quasi-linear parabolic PDEs, Stochastics and Stochastics Reports, 48, 17-44. [N95] Nualart, D. (1995), The Malliavin calculus and related topics, Springer-Verlag. [N98] Nualart, D. (1998), Analysis on Wiener space and anticipating stochastic calcu- lus, Ecole d’Eté de Probabilités de Saint-Flour XXV, Lect. Notes in Math. 1690, Springer-Verlag, 123-227. [S05] Sanz-Solé, M. (2005), Malliavin calculus with applications to stochastic partial differential equations, EPFL Press. [W86] Walsh, J.B. (1986), An Introduction to Stochastic Partial Differential Equa- tions, Ecole d’Eté de Probabilités de Saint-Flour XIV, Lect. Notes in Math., 1180, Springer-Verlag, 266-437. [W84] Watanabe, S. (1984), Lectures on Stochastic Differential Equations and Malli- avin Calculus, Tata Institute of Fundamental Research Lectures on Math. and Physics, 73, Springer-Verlag, Berlin. Introduction and main results Proof of Theorems 1.2, 1.6 and their corollaries (assuming Theorem 1.1) Elements of Malliavin calculus Existence, smoothness and uniform boundedness of the one-point density The Gaussian-type lower bound on the one-point density The Gaussian-type upper bound on the two-point density Bounds on the increments of the Malliavin derivatives Study of the Malliavin matrix Proof of Theorem 1.1(c) and (d) Proof of Theorem 6.3 Small Eigenvalues Large Eigenvalues Appendix

0704.1313

Mutant knots and intersection graphs

Mutant knots and intersection graphs S. V. Chmutov∗, S. K. Lando † April 10, 2007 Abstract We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. The converse statement is easy and well known. We discuss relationship between our results and certain Lie algebra weight systems. 1 Introduction Below, we use standard notions of the theory of finite order, or Vassiliev, invariants of knots in 3-space; their definitions can be found, for example, in [6] or [14]. All knots are assumed to be oriented. Two knots are said to be mutant if they differ by a rotation/reflection of a tangle with four endpoints; if necessary, the orientation inside the tangle may be replaced by the opposite one. Here is a famous example of mutant knots, the Conway (11n34) knot C of genus 3, and Kinoshita–Terasaka (11n42) knot KT of genus 2 (see [1]). C = KT = Note that the change of the orientation of a knot can be achieved by a mutation in the complement to a trivial tangle. Most known knot invariants cannot distinguish mutant knots. Neither the (col- ored) Jones polynomial, nor the HOMFLY polynomial, nor the Kauffman two variable polynomial distinguish mutants. All Vassiliev invariants up to order 10 do not dis- tinguish mutants as well [17] (up to order 8 this fact was established by a direct computation [5, 6]). However, there is a Vassiliev invariant of order 11 distinguishing C and KT [16, 17]. It comes from the colored HOMFLY polynomial. ∗The Ohio State University, Mansfield. †Institute for System Research RAS and the Poncelet Laboratory, Independent University of Moscow, partly supported by the grant ACI-NIM-2004-243 (Noeuds et tresses), RFBR 05-01-01012- a, NWO-RFBR 047.011.2004.026 (RFBR 05-02-89000-NWOa), GIMP ANR-05-BLAN-0029-01. http://arxiv.org/abs/0704.1313v1 The main combinatorial objects of the Vassiliev theory of knot invariants are chord diagrams. To a chord diagram, its intersection graph (also called circle graph) is associated. The vertices of the graph correspond to chords of the diagram, and two vertices are connected by an edge if and only if the corresponding chords intersect. The value of a Vassiliev invariant of order n on a singular knot with n double points depends only on the chord diagram of the singular knot. Hence any such invariant determines a function, a weight system, on chord diagrams with n chords. Conversely, any weight system induces, in composition with the Kontsevich integral, which is the universal finite order invariant, a finite order invariant of knots. Such knot invariants are called canonical. Canonical invariants span the whole space of Vassiliev invariants. Direct calculations for small n show that the values of these functions are uniquely determined by the intersection graphs of the chord diagrams. This fact motivated the intersection graph conjecture in [5] (see also [6]) which states that any weight system depends on the intersection graph only. This conjecture happened to be false, because of the existence of a finite order invariant that distinguishes two mutant knots mentioned above and the following fact. The knot invariant induced by a weight system whose values depend only on the intersection graph of the chord diagrams cannot distinguish mutants. A justification of this statement, due to T. Le (unpublished), looks like follows (see details in [6]). If we have a knot (in general position) with a distinguished two- string tangle, then all the terms in the Kontsevich integral of the knot having chords connecting the tangle with its exterior vanish. Our goal is to prove the converse statement thus establishing an equivalence be- tween finite order knot invariants nondistinguishing mutants and weight systems de- pending on the intersection graphs of chord diagrams only. Theorem 1 If a finite order knot invariant does not distinguish mutants, then the corresponding weight system depends only on the intersection graphs of chord dia- grams. Together, the two statements can be combined as follows. A canonical knot invariant does not distinguish mutants if and only if its weight system depends on the intersection graphs of chord diagrams only. Recently, B. Mellor [15] extended the concept of intersection graph to string links. We do not know whether our Theorem 1 admits an appropriate generalization. Section 2 is devoted to the proof of Theorem 1. In Sec. 3, we discuss relationship between intersection graphs and the weight systems associated to the Lie algebra sl(2) and the Lie algebra gl(1|1). The paper was written during the second author’s visit to the Mathematical De- partment of the Ohio State University. He expresses his gratitude to this institution for warm hospitality and excellent working conditions. The authors are grateful to S. Duzhin, K. J. Supowit, and A. Vaintrob for useful discussions. 2 Proof 2.1 Representability of graphs as the intersection graphs of chord diagrams Not every graph can be represented as the intersection graph of a chord diagram. For example, the following graphs are not intersection graphs. A characterization of those graphs that can be realized as intersection graphs is given by an elegant theorem of A. Bouchet [4]. On the other hand, distinct diagrams may have coinciding intersection graphs. For example, next three diagrams have the same intersection graph : A combinatorial analog of the tangle in mutant knots is a share [5, 6]. Informally, a share of a chord diagram is a subset of chords whose endpoints are separated into at most two parts by the endpoints of the complementary chords. More formally, Definition 1 A share is a part of a chord diagram consisting of two arcs of the outer circle possessing the following property: each chord one of whose ends belongs to these arcs has both ends on these arcs. Here are some examples: A share Not a share Two shares The complement of a share also is a share. The whole chord diagram is its own share whose complement contains no chords. Definition 2 A mutation of a chord diagram is another chord diagram obtained by a rotation/reflection of a share. For example, three mutations of the share in the first chord diagram above produce the following chord diagrams: Obviously, mutations preserve the intersection graphs of chord diagrams. Theorem 2 Two chord diagrams have the same intersection graph if and only if they are related by a sequence of mutations. This theorem is contained implicitly in papers [3, 8, 11] where chord diagrams are written as double occurrence words, the language better suitable for describing algorithms than for topological explanation. Proof of Theorem 2. The proof of this theorem uses Cunningham’s theory of graph decompositions [9]. A split of a (simple) graph Γ is a disjoint bipartition {V1, V2} of its set of vertices V (Γ) such that each part contains at least 2 vertices, and there are subsets W1 ⊆ V1, W2 ⊆ V2 such that all the edges of Γ connecting V1 with V2 form the complete bipartite graph K(W1,W2) with the parts W1 and W2. Thus for a split {V1, V2} the whole graph Γ can be represented as a union of the induced subgraphs Γ(V1) and Γ(V2) linked by a complete bipartite graph. Another way to think about splits, which is sometimes more convenient and which we shall use in the pictures below, looks like follows. Consider two graphs Γ1 and Γ2 each having a distinguished vertex v1 ∈ V (Γ1) and v2 ∈ V (Γ2), respectively, called markers. Construct the new graph Γ = Γ1 ⊠(v1,v2) Γ2 whose set of vertices is V (Γ) = {V (Γ1)− v1} ⊔ {V (Γ2)− v2} and whose set of edges is E(Γ) = {(v′1, v 1) ∈ E(Γ1) : v 1 6= v1 6= v 1} ⊔ {(v 2) ∈ E(Γ2) : v 2 6= v2 6= v {(v′1, v 2) : (v 1, v1) ∈ E(Γ1) and (v2, v 2) ∈ E(Γ2)} . Representation of Γ as Γ1⊠(v1,v2)Γ2 is called a decomposition of Γ, Γ1 and Γ2 are called the components of the decomposition. The partition {V (Γ1)−v1, V (Γ2)−v2} is a split of Γ. Graphs Γ1 and Γ2 might be decomposed further giving a finer decomposition of the initial graph Γ. Pictorially, we represent a decomposition by pictures of its components where the corresponding markers are connected by a dashed edge. A prime graph is a graph with at least three vertices admitting no splits. A de- composition of a graph is said to be canonical if the following conditions are satisfied: (i) each component is either a prime graph, or a complete graph Kn, or a star Sn, which is the tree with a vertex, the center, adjacent to n other vertices; (ii) no two components that are complete graphs are neighbors, that is, their mark- ers are not connected by a dashed edge; (iii) the markers of two components that are star graphs connected by a dashed edge are either both centers or both not centers of their components. W. H. Cunningham proved [9, Theorem 3] that each graph with at least six vertices possesses a unique canonical decomposition. Let us illustrate the notions introduced above by two examples of canonical de- composition of the intersection graphs of chord diagrams. We number the chords and the corresponding vertices in our graphs, so that the unnumbered vertices are the markers of the components. The first example is our example from page 3: A chord diagram The intersection graph 1������������ 6 5 4 The canonical decomposition The second example represents the chord diagram of the double points in the plane diagram of the Conway knot C from page 1. The double points of the shaded tangle are represented by the chords 1,2,9,10,11. 8 6 9 10 Chord diagram Intersection graph Canonical decomposition The key observation in the proof of Theorem 2 is that components of the canoni- cal decomposition of any intersection graph admit a unique representation by chord diagrams. For a complete graph and star components, this is obvious. For a prime component, this was proved by A. Bouchet [3, Statement 4.4] (see also [11, Section 6] for an algorithm finding such a representation for a prime graph). Now to describe all chord diagrams with a given intersection graph, we start with a component of its canonical decomposition. There is only one way to realize the component by a chord diagram. We draw the chord corresponding to the marker as a dashed chord and call it the marked chord. This chord indicates the places where we must cut the circle removing the marked chord together with small arcs containing its endpoints. As a result we obtain a chord diagram on two arcs. Repeating the same procedure with a neighbor component of the canonical decomposition, we get another chord diagram on two arcs. We have to sew these two diagrams together by their arcs in an alternating order. There are four possibilities to do this, and they differ by mutations of the share corresponding to the second (or, alternatively, the first) component. This completes the proof of Theorem 2. � To illustrate the last stage of the proof consider our standard example and take the star 2-3-4 component first and then the triangle component. We get �� CUT Because of the symmetry, the four ways of sewing these diagrams produce only two distinct chord diagrams with a marked chord: repeating the same procedure with the marked chord for the last 1-6 component of the canonical decomposition, we get 1 ���� CUT Sewing this diagram into the previous two in all possible ways we get four mutant chord diagrams from page 3. As an enjoyable exercise we leave to the reader to work out our second example with the chord diagram of the diagram of the Conway knot and find the mutation producing the chord diagram of the plane diagram of the Kinoshita–Terasaka knot using the canonical decomposition. 2.2 Proof of Theorem 1 Suppose we have a Vassiliev knot invariant v of order at most n that does not dis- tinguish mutant knots. Let D1 and D2 be chord diagrams with n chords whose intersection graphs coincide. We are going to prove that the values of the weight system of v on D1 and D2 are equal. By Theorem 2, it is enough to consider the case when D1 and D2 differ by a single mutation in a share S. Let K1 be a singular knot with n double points whose chord diagram is D1. Consider the collection of double points of K1 corresponding to the chords occurring in the share S. By the definition of a share, K1 has two arcs containing all these double points and no others. By sliding the double points along one of these arcs and shrinking the other arc we may enclose these arcs into a ball whose interior does not intersect the rest of the knot. In other words, we may isotope the knot K1 to a singular knot so as to collect all the double points corresponding to S in a tangle TS. Performing an appropriate rotation of TS we obtain a singular knot K2 with the chord diagram D2. Since v does not distinguish mutants, its values on K1 and K2 are equal. Theorem 1 is proved. � To illustrate the proof, let D1 be the chord diagram from our standard example. Pick a singular knot representing D1, say K1 = 61 2 3 5 D1 = To perform a mutation in the share containing the chords 1,5,6, we must slide the double point 1 close to the double points 5 and 6, and then shrink the corresponding arcs: Sliding the double point 1 Shrinking the arcs Forming the tangle TS Now doing an appropriate rotation of the tangle TS we obtain a singular knot K2 representing the chord diagram D2. 3 Lie algebra weight systems and intersection graphs Kontsevich [12] generalized a construction of Bar-Natan [2] of weight systems defined by a Lie algebra and its representation to a universal weight system, with values in the universal enveloping algebra of the Lie algebra. In [18], Vaintrob extended this construction to Lie superalgebras. Our main goal in this section is to prove Theorem 3 The universal weight systems associated to the Lie algebra sl(2) and to the Lie superalgebra gl(1|1) depend on the intersection graphs of chord diagrams rather than on the diagrams themselves. It follows immediately that the canonical knot invariants corresponding to these two algebras do not distinguish mutants. The latter fact is already known, but we did not manage to find appropriate references; instead, we give a direct proof on the intersection graphs side. Note that for more complicated Lie algebras the statement of Theorem 3 is no longer true. For example, the universal sl(3) weight system distinguishes between the Conway and the Kinoshita–Terasaka knots. In fact, for each of the two algebras we prove more subtle statements. Theorem 4 The universal weight system associated to the Lie algebra sl(2) depends on the matroid of the intersection graph of a chord diagram rather than on the inter- section graph itself. This theorem inevitably leads to numerous questions concerning relationship be- tween weight systems and matroid theory, which specialists in this theory may find worth being investigated. Weight systems have a graph counterpart, so-called 4-invariants of graphs [13]. The knowledge that a weight system depends only on the intersection graphs does not guarantee, however, that it arises from a 4-invariant. In particular, we do not know, whether this is true for the universal sl(2) weight system. Either positive (with an explicit description) or negative answer to this question would be extremely interesting. For gl(1|1), the answer is positive. Theorem 5 The universal weight systems associated to the Lie superalgebra gl(1|1) is induced by a 4-invariant of graphs. In the first two subsections below, we recall the construction of universal weight systems associated to Lie algebras and the notion of 4-invariant of graphs. The next two subsections are devoted to separate treating of the Lie algebra sl(2) and the Lie superalgebra gl(1|1) universal weight systems. 3.1 Weight systems via Lie algebras Our approach follows that of Kontsevich in [12]. In order to construct a weight system, we need a complex Lie algebra endowed with a nondegenerate invariant bilinear form (·, ·). The invariance requirement means that (x, [y, z]) = ([x, y], z) for any three elements x, y, z in the Lie algebra. Pick an orthonormal basis a1, . . . , ad, (ai, aj) = δij , d being the dimension of the Lie algebra. Any chord diagram can be made into an arc diagram by cutting the circle at some point and further straightening it. For an arc diagram of n arcs, write on each arc an index i between 1 and d, and then write on both ends of the arc the letter ai. Reading all the letters left to right we obtain a word of length 2n in the alphabet a1, . . . , ad, which is an element of the universal enveloping algebra of our Lie algebra. The sum of all these words over all possible settings of the indices is the element of the universal enveloping algebra assigned to the chord diagram. This element is independent of the choice of the cutting point of the circle, as well as the orthonormal basis. It belongs to the center of the universal enveloping algebra and satisfies the 4-term relation, whence can be extended to a weight system. The latter is called the universal weight system associated to the Lie algebra and the bilinear form, and it can be specialized to specific representations of the Lie algebra as in the original Bar-Natan’s approach. Obviously, any universal weight system is multiplicative: its value on a product of chord diagrams coincides with the product of its values on the factors. The simplest noncommutative Lie algebra with a nondegenerate invariant bilinear form is sl(2). It is 3-dimensional, and the center of its universal enveloping algebra is the ring C[c] of polynomials in a single variable c, the Casimir element. The corresponding universal weight system was studied in detail in [7]. It attracts a lot of interest because of its equivalence to the colored Jones polynomials. In [18], Kontsevich’s construction was generalized to Lie superalgebras, and this construction was elaborated in [10] for the simplest non-commutative Lie superalgebra gl(1|1). The center of the universal enveloping algebra of this algebra is the ring of polynomials C[c, y] in two variables. The value of the corresponding universal weight system on a chord diagram with n chords is a quasihomogeneous polynomial in c and y, of degree n, where the weight of c is set to be 1, and the weight of y is set to be 2. 3.2 The 4-bialgebra of graphs By a graph, we mean a finite undirected graph without loops and multiple edges. Let Gn denote the vector space freely spanned over C by all graphs with n vertices, G0 = C being spanned by the empty graph. The direct sum G = G0 ⊕ G1 ⊕ G2 ⊕ . . . carries a natural structure of a commutative cocommutative graded Hopf algebra. The multiplication in this Hopf algebra is induced by the disjoint union of graphs, and the comultiplication is induced by the operation taking a graph G into the sum GU ⊗GŪ , where U is an arbitrary subset of vertices of G, Ū its complement, and GU denotes the subgraph of G induced by U . The 4-term relation for graphs is defined in the following way. By definition, the 4-term element in Gn determined by a graph G with n vertices and an ordered pair A,B of its vertices connected by an edge is the linear combination G−G′AB − G̃AB + G̃ where • G′AB is the graph obtained by deleting the edge AB in G; • G̃AB is the graph obtained by switching the adjacency to A of all the vertices adjacent to B in G; • G̃′AB is the graph obtained by deleting the edge AB in G AB (or, equivalently, by switching the adjacency to A of all the vertices adjacent to B in G′AB). All the four terms in a 4-term element have the same number n of vertices. The quotient of Gn modulo the span of all 4-term elements in Gn (defined by all graphs and all ordered pairs of adjacent vertices in each graph) is denoted by Fn. The direct F = F0 ⊕F1 ⊕F2 ⊕ . . . is the quotient Hopf algebra of graphs, called the 4-bialgebra. The mapping taking a chord diagram to its intersection graph extends to a graded Hopf algebra homomor- phism γ from the Hopf algebra of chord diagrams to F . Being commutative and cocommutative, the 4-bialgebra is isomorphic to the polynomial ring in its basic primitive elements, that is, it is the tensor product S(P1)⊗ S(P2)⊗ . . . of the symmetric algebras of its homogeneous primitive spaces. 3.3 The sl(2) weight system Our treatment of the universal weight system associated with the Lie algebra sl(2) is based on the recurrence formula for computing the value of this weight system on chord diagrams due to Chmutov and Varchenko [7]. The recurrence states that if a chord diagram contains a leaf, that is, a chord intersecting only one other chord, then the value of the sl(2) universal weight system on the diagram is (c − 1/2) times its value on the result of deleting the leaf, and, in addition, − − + = 2 − 2 meaning that the value of the weight system on the chord diagram on the left-hand side coincides with the linear combinations of its values on the chord diagrams indi- cated on the right. Now, in order to prove Theorem 3 for the universal sl(2) weight system, we must prove that mutations of a chord diagram preserve the values of this weight system. Take a chord diagram and a share in it. Apply the above reccurence formula to a chord and two its neighbors belonging to the chosen share. The recurrence relation does not affect the complementary share, while all the instances of the modified first share are simpler than the initial one (each of them contains either fewer chords or the same number of chords but with fewer intersections). Repeating this process, we can replace the original share by a linear combination of the simplest shares, chains, which are symmetric meaning that they remain unchanged under rotations. The sl(2) case of Theorem 3 is proved. � Now let us turn to the proof of Theorem 4. For elementary notions of matroid theory we refer the reader to any standard reference, say to [19]. Recall that a matroid can be associated to any graph. It is easy to check that the matroid associated to the disjoint union of two graphs coincides with that for the graph obtained by identifying a vertex in the first graph with a vertex in the second one. We call the result of gluing a vertex in a graph G1 to a vertex in a graph G2 a 1-product of G1 and G2. The converse operation is 1-deletion. Of course, the 1-product depends on the choice of the vertices in each of the factors, but the corresponding matroid is independent of this choice. Similarly, let G1, G2 be two graphs, and pick vertices u1, v1 in G1 and u2, v2 in G2. Then the matroid associated to the graph obtained by identifying u1 with u2 and v1 with v2 coincides with the one associated to the graph obtained by identifying u1 with v2 and u2 with v1. The operation taking the result of the first identification to that of the second one is called the Whitney twist on graphs. Both the 1-product and the Whitney twist have chord diagram analogs. For two chord diagrams with a distinguished chord in each of them, we define their 1-product as a chord diagram obtained by replacing the distinguished chords in the ordinary product of chord diagrams chosen so as to make them neighbors by a single chord connecting their other ends. The Whitney twist also is well defined because of the following statement. Lemma 1 Suppose the intersection graph of a chord diagram is the result of identi- fying two pairs of vertices in two graphs G1 and G2. Then both graphs G1 and G2 are intersection graphs, as well as the Whitney twist of the original graph. The assertion concerning the graphs G1 and G2 is obvious. In order to prove that the result of the Whitney twist also is an intersection graph, let c1, c2 denote the two chords in a chord diagram C such that deleting these chords makes C into an ordinary product of two chord diagrams C1, C2. By reflecting the diagram C2 and restoring the chords c1 and c2 we obtain a chord diagram whose intersection graph is the result of the desired Whitney twist. The lemma is proved. According to the Whitney theorem, two graphs have the same matroid iff they can be obtained from one another by a sequence of 1-products/deletions and Whitney twists. Therefore, Theorem 4 follows from Lemma 2 (i) The value of the universal sl(2) weight system on the 1-product of chord diagrams coincides with the product of its values on the factors divided by c. (ii) The value of the universal sl(2) weight system remains unchanged under the Whitney twist of the chord diagram. Statement (i) is proved in [7]. The proof of statement (ii) is similar to that of Theorem 3. Consider the part C2 participating in the Whitney twist and apply to it the recurrence relations. Note that the relations do not affect the complementary diagram C1. Simplifying the part C2 we reduce it to a linear combination of the simplest possible diagrams, chains, which are symmetric under reflection. Reflecting a chain preserves the chord diagram, whence the value of the sl(2) weight system. Theorem 4 is proved. � 3.4 The gl(1|1) weight system Define the (unframed) Conway graph invariant with values in the ring of polynomi- als C[y] in one variable y in the following way. We set it equal to (−y)n/2 on graphs with n vertices if the adjacency matrix of the graph is nondegenerate, and 0 otherwise. Recall that the adjacency matrix AG of a graph G with n vertices is an n× n-matrix with entries in Z2 obtained as follows. We choose an arbitrary numbering of the vertices of the graph, and the entry aij is 1 provided the i th and the j th vertices are adjacent and 0 otherwise (diagonal elements aii are 0). Note that for odd n, the adjacency matrix cannot be nondegenerate, hence the values indeed are in the ring of polynomials. The Conway graph invariant is multiplicative: its value on the disjoint union of graphs is the product of its values on the factors. Clearly, the Conway graph invariant is a 4-invariant. Moreover, it satisfies the 2- term relation, which is more restrictive than the 4-term one: its values on the graphsG and G̃AB coincide for any graph G and any pair of ordered vertices A,B in it. Indeed, consider the graph as a symmetric bilinear form on the Z2-vector space whose basis is the set of vertices of the graph, the adjacency matrix being the matrix of the bilinear form in this basis. In these terms, the transformation G 7→ G̃AB preserves the vector space and the bilinear form, but changes the basis A,B,C, · · · → A + B,B,C, . . . . Thus, it preserves the nondegeneracy property of the adjacency matrix. The subspace F1 is spanned by the graph p1 with a single vertex (whence no edges), which is a primitive element. Since F is the polynomial ring in its primitive elements, each homogeneous space Fn admits a decomposition into the direct sum of two subspaces, one of which is the subspace of polynomials in primitive elements of degree greater than 1, and the other one is the space of polynomials divisible by p1. We define the framed Conway graph invariant as the only multiplicative 4-invariant with values in the polynomial ring C[c, y] whose value on p1 is c, and on the projection of any graph to the subspace of p1-independent polynomials along the subspace of p1-divisible polynomials coincides with the Conway graph invariant of the graph. The values of the framed Conway graph invariant can be computed recursively. Take a graph G and consider its projection to the subspace of graphs divisible by p1. On this projection, the framed Conway graph invariant can be computed because of its multiplicativity. Now add to the result the value of the (unframed) Conway graph invariant on the graph. Now we can refine the statement of theorem 5. Theorem 6 The gl(1|1) universal weight system is the pullback of the framed Conway graph invariant to chord diagrams under the homomorphism γ. Proof. The proof follows from two statements in [10]. Theorem 3.6 there states that setting c = 0 in the value of the gl(1|1) universal weight system on a chord diagram we obtain the result of deframing this weight system. Theorem 4.4 asserts that this value is exactly the Conway invariant of the chord diagram. The latter coincides with the Conway graph invariant of the intersection graph of the chord diagrams defined above. Since the deframing for chord diagrams is a pullback of the deframing for graphs, we are done. � References [1] The knot Atlas, http://katlas.math.toronto.edu/wiki/Main_Page [2] D. Bar-Natan, On Vassiliev knot invariants, Topology, 34, 423–472 (1995) [3] A. Bouchet Reducing prime graphs and recognizing circle graphs, Combinatorica, 7, no. 3, 243–254 (1987) [4] A. Bouchet Circle graph obstructions, J. Combin. Theory Ser. B 60, no. 1, 107– 144 (1994) [5] S. V. Chmutov, S. V. Duzhin, S. K. Lando Vassiliev knot invariants I. Introduc- tion, Advances in Soviet Mathematics, vol. 21, 117-126 (1994) [6] S. Chmutov, S. Duzhin, J. Mostovoy, CDBooK. Introduction to Vassiliev Knot invariants. (a preliminary draft version of a book about Chord Diagrams.) http://www.math.ohio-state.edu/~chmutov/preprints/ [7] S. V. Chmutov, A. N. Varchenko, Remarks on the Vassiliev knot invariants com- ing from sl2, Topology 36, no. 1, 153–178 (1997) [8] B. Courcelle Circle graphs and Monadic Second-order logic, Preprint, June 2005, http://www.labri.fr/perso/courcell/ArticlesEnCours/CircleGraphsSubmitted.pdf [9] W. H. Cunningham, Decomposition of directed graphs, SlAM J. Algor. Discrete Math.,3, no. 2, 214–228 (1982) [10] J. M. Figueroa-O’Farrill, T. Kimura, A. Vaintrob, The universal Vassiliev in- variant for the Lie superalgebra gl(1|1), Comm. Math. Phys. 185, no. 1, 93– 127 (1997) [11] C. P. Gabor, K. J. Supowit, W.-L. Hsu, Recognizing circle graphs in polynomial time, Journal of the ACM (JACM) (3), 36, no. 3, 435–473 (1989) [12] M. Kontsevich, Vassiliev’s knot invariants, Adv. Soviet Math., 16, Part 2, AMS, Providence RI, 137–150 (1993) [13] S. K. Lando, On a Hopf algebra in graph theory, J. Comb. Theory Series B, 80, no. 1, 104–121 (2000) [14] S. K. Lando, A. K. Zvonkin, Graphs on surfaces and their applications, Springer (2004) [15] B. Mellor, Intersection graphs for string links, J. Knot Theory Ramif. 15, no. 1, 53-72 (2006). [16] H. R. Morton, P. R. Cromwell, Distinguishing mutants by knot polynomials, J. Knot Theory Ramif. 5 225–238 (1996) [17] J. Murakami, Finite type invariants detecting the mutant knots, Knot Thoery. A volume dedicated to Professor Kunio Murasugi for his 70th birthday. Edi- tors: M. Sakuma et al., Osaka University, March 2000. Preprint is available at http://www.f.waseda.jp/murakami/papers/finitetype.pdf [18] A. Vaintrob, Vassiliev knot invariants and Lie S-algebras, Math. Res. Lett. 1, no. 5, 579–595 (1994) [19] D. J. A. Welsh, Matroid theory, Academic Press, London (1976) Introduction Proof Representability of graphs as the intersection graphs of chord diagrams Proof of Theorem ?? Lie algebra weight systems and intersection graphs Weight systems via Lie algebras The 4-bialgebra of graphs The sl(2) weight system The gl(1|1) weight system

0704.1314

Acoustic resonances in microfluidic chips: full-image micro-PIV experiments and numerical simulations

Acoustic resonances in microfluidic chips: full-image micro-PIV experiments and numerical simulations S. M. Sundin, T. Glasdam Jensen, H. Bruus and J. P. Kutter MIC – Department of Micro and Nanotechnology, Technical University of Denmark DTU Bldg. 345 east, DK-2800 Kongens Lyngby, Denmark (Dated: 30 March 2007) We show that full-image micro-PIV analysis in combination with images of transient particle motion is a powerful tool for experimental studies of acoustic radiation forces and acoustic streaming in microfluidic chambers under piezo-actuation in the MHz range. The measured steady-state motion of both large 5 µm and small 1 µm particles can be understood in terms of the acoustic eigenmodes or standing ultra-sound waves in the given experimental microsystems. This interpretation is supported by numerical solutions of the corresponding acoustic wave equation. I. INTRODUCTION For the typical dimensions of microfluidic structures there are two acoustic effects of main importance: the acoustic radiation force [1, 2, 3], which moves suspended particles either towards or away from pressure nodes de- pending on their acoustic material properties, and acous- tic streaming [4, 5], which imparts movement onto the entire solvent. Both of these forces have been utilized, alone or in combination, for several lab-on-a-chip appli- cations. Yasuda et al. [6, 7], demonstrated concentration of particles by acoustic radiation forces, and separation of particles by acoustic forces in combination with elec- trostatic forces. One of the most attractive applications for acoustics in microfluidics is for mixing [8, 9, 10], as this process typically is diffusion limited in microscale devices. Valveless ultrasonic pumps, utilizing acoustic streaming, have also been presented [11, 12]. Numer- ous examples of microsystems where acoustics are ap- plied to handling and analysis of biological material have been suggested. Among others these include: trapping of microorganisms [13], bioassays [14], and separation and cleaning of blood [15, 16, 17]. Apart from on chip de- vices, acoustic forces have also been suggested for use in other µm-scale applications [18]. There are different imaging strategies and tools, which can be used in order to enhance the understanding, and to visualize the function of acoustic micro-devices during operation. For acoustic mixers the effect can be illus- trated and measured by partly filling the mixing cham- ber with a dye prior to piezo-actuation [9, 10]. However, this approach is mainly limited to determine the total, and not the local, mixing behavior within the cham- ber. A more refined method, which is not limited to the study of micromixers, is to apply streak- or stream- line analysis. This was shown by Lutz et al. [19, 20], who neatly demonstrated 3D steady micro streaming around a cylinder. Although streamline analysis can be employed to illustrate flow behaviour, it is not suitable in deter- mining local variations in velocity. For that purpose, the micron-resolution particle image velocimetry (micro- PIV) technique is the method of choice [21]. With this technique the motion of tracer particles, acquired from consecutive image frames, is utilized to obtain velocity vector fields. In a large chamber, local measurements of particle motion induced by acoustic radiation forces and acoustic streaming have been performed by Spen- gler et al. [22, 23], and further developed by Kuznetsova et al. [24]. Li and Kenny derived velocity profiles in a particle separating device utilizing the acoustic radiation force [17]. Jang et al. used confocal scanning microscopy to perform micro-PIV measurements on circulatory flows in a piezo-actuated fluidic chamber [25]. Furthermore, Manasseh et al. applied micro-PIV to measure stream- ing velocites around a bubble trapped in a microfluidic chamber [26]. As particles under the influence of acoustic fields do no longer function as true independent tracers in all sit- uations, and as several acoustic effects come into play at the same time, extra caution and consideration have to be taken when applying micro-PIV for microfluidic acoustic studies. These considerations will be discussed in more detail in section II C. The situation is further complicated by the coupling from the actuator to the structures and their acoustic resonances, which is a yet poorly understood mechanism. The resonances depend on the acoustic material parameters as well as the geom- etry of both the chip and the chamber. For substrate materials with low attenuation, such as silicon, the ac- tuation will result in strong resonances over the whole devices, whereas for substrate materials with high atten- uation, the effect will be mostly confined to the prox- imity of the actuator. Moreover, in a real system the coupling strengths vary for different resonances, and am- plitude fluctuations across the structures are often ob- served. Therefore, if investigations striving to yield a better understanding of acoustic resonances in low atten- uation microfluidic chips are to be performed, it is not sufficient only to study the acoustic phenomena locally. In this work, full-image micro-PIV analysis in combi- nation with images of transient particle motion is sug- gested as a tool for studying acoustic resonances in mi- crofluidic chambers under piezo-actuation. The acousto- fluidic phenomena mentioned above can be investigated by comparing these experimental images with plots of acoustic eigenmodes of the device structure obtained by http://arxiv.org/abs/0704.1314v1 FIG. 1: A top-view photograph of the silicon-glass chip (dark gray) containing a square chamber with straight inlet and outlet channels (light gray). numerical solution of the corresponding acoustic wave equation. II. MATERIALS AND EXPERIMENTAL METHODS A. Microchip fabrication In this study, two microfluidic chambers were inves- tigated, one of quadratic footprint with a side-length of 2 mm and one of circular shape with a diameter of 2 mm. The size was chosen to be a few times the acoustic wave- length of 2 MHz ultrasound waves in water, and the spe- cific shapes were employed to ensure simple patterns in the pressure field at the acoustic resonances. Both cham- bers were connected to 400 µm wide inlet and outlet channels, and the depth was 200 µm throughout. The microfluidic chips were fabricated in silicon via deep re- active ion etching (DRIE). The same technique was also applied on the backside of the chip to etch 300 µm diame- ter round fluidic inlets. Anodic bonding was used to seal the structures with a 500 µm thick pyrex glass lid on the channel side. Silicon rubber tubings were glued to the holes on the backside of the chip, for easy attachment of teflon tubing. A picture of one of our microfluidic chips is shown in Fig. 1, and a list of the geometrical parameters is given in Table I. B. Experimental setup and procedure The piezo-actuator (Pz27, Ferroperm) was pressed to the backside of the chip using an ultrasonic gel (ECO, Ceracarta) and biased by a 20 V ac tone generator (Model 195, Wavetek). Images were captured with a progressive scan interline CCD camera (Hisense MkII, Dantec Dy- namics), mounted with a 0.63x TV-adapter on an epiflu- orescent microscope (DMLB, Leica). The objective used was a Plan 5x with a numerical aperture NA of 0.12. For the given fluidic geometries, this combination allowed capture of full-image PIV vector fields, while utilizing the largest number of pixels on the CCD. A blue light emit- ting diode, LED, (Luxeon Star 3W, Lumileds) was used as illumination source in a front-lit configuration, which TABLE I: The geometrical parameters of the fabricated mi- crofluidic silicon-pyrex chip. chip length L0 49 mm silicon thickness hs 500 µm chip width w0 15 mm pyrex thickness hp 500 µm channel length Lc 26 mm chamber height h 200 µm channel width wc 400 µm chamber width w 2 mm is described elsewhere [27]. The LED was powered by an in-house built power supply controlled by a PIV timing system (Dantec Dynamics). Image acquisition was per- formed on a PC with Flowmanager software (Dantec Dy- namics). As tracer fluids solutions of 1 µm polystyrene micro-beads (Duke Scientific), 5 µm polyamide micro- beads (Danish Phantom Design), diluted milk, and fluo- rescein have been used. The investigations were performed by scanning the applied frequency from the tone generator and identi- fying those frequencies which led to a strong response, an acoustic resonance, in the microfluidic chamber. At the resonance frequencies, the behavior of the different tracer particle solutions was observed. Between succes- sive recordings the chip was flushed to assure homoge- neous seeding. Furthermore, to make sure that only par- ticle motion caused by acoustic forces were recorded, no external flow was applied during measurements. C. Micro-PIV considerations In micro-PIV tracer particles are chosen for their abil- ity to truthfully follow the motion of the flow that is to be investigated. Particles under the influence of an acoustic field do no longer fulfil this criterium in all situ- ations. Therefore, extra caution and considerations have to be taken regarding what movements are actually mea- sured when applying micro-PIV for these types of stud- ies. Given that particle motion caused by thermal or gravitational forces can be neglected, the main task is to determine if particle motion is caused by acoustic radi- ation forces, acoustic streaming or a combination of the two. In this study, this problem was tackled by applying three tracer solutions with different physical properties. Typically, the large polyamide particles are more strongly affected by the acoustic radiation forces than by the forces due to acoustic streaming of the surround- ing water. In contrast, since the acoustic radiation force scales with the volume of the particle, the small polystyrene particles will follow the motion of the water, if relatively strong acoustic streaming is present. How- ever, there is no simple relation between the two forces, and for an arbitrary frequency and geometry one can be strong whereas the other is not, and vice versa. There- fore, in order to determine whether particle motion is caused by acoustic radiation forces or acoustic streaming it is necessary to utilize the dependance of the acoustic radiation forces on the compressibility of the particle. TABLE II: The susceptibility to acoustic radiation forces for the particles used in this study, as well as for some other particles common to microfluidic applications. tracer type force direction beads (1 µm) weak nodes beads (5 µm) strong nodes red blood cells strong nodes milk particles weak anti-nodes large micelles strong anti-nodes fluorescein none - The polymer particles will move towards the pressure nodes since their compressibility is smaller than that of water. The opposite is true for the lipid particles in milk: their compressibility is larger than that of water, and consequently they will move towards pressure antinodes. Like the small polystyrene particles, the lipid particles we used were small enough to typically follow the net acoustic streaming flow of the water. Thus, if similar mo- tion is recorded with two types of tracers with different compressibilities compared to the medium, the acoustic radiation forces can be ruled out as cause of the motion. As an alternative or complementary technique to micro- PIV measurements, fluorescein can be used to investigate acoustic streaming. A summary of the acoustic behav- ior of the different particles used in this study, and some other bodies that are common in microfluidic applica- tions, is given in Table II. The speed of sound c in water has a significant de- pendence on temperature T given by the large deriva- tive ∂c/∂T ≃ 4 m s−1K−1. All tracer fluids were there- fore kept at room temperature, so that the temperature was not changed when the microchip was flushed dur- ing tracer particle exchange. The microchips used in this study are comparable in size and mode of actua- tion to those used for ultrasonic agitation in a study by Bengtsson and Laurell [28]. They performed sensitive temperature measurements on the reactor outlet, where no temperature increase caused by the acoustic power could be detected. In our study, the piezo-actuator was run at a moderate power-level and only for the short intervals during recordings (typically less than one sec- ond). Therefore, it can be ruled out that heating from the piezo-actuator would have any measurable impact on the measurements. One important factor, which needs to be accounted for when applying micro-PIV on systems affected by acoustic forces, is that the local seeding density will be distorted during actuation. This is normally not a problem when measuring on particle motion caused solely by acoustic streaming, as this motion generally will be of a circulat- ing nature. On the other hand, in the case of particle motion induced by acoustic radiation forces, it will typi- cally lead to total expulsion of particles from certain re- gions into others. If PIV-vector statistics is applied, only the first few image-pairs recorded after piezo-actuation TABLE III: The acoustic material parameters of the microsys- tem at 20 ◦C: sound velocities ci and densities ρi from the CRC Handbook of Chemistry and Physics. material speed of sound density water cw = 1483 m/s ρw = 998 kg/m silicon cs = 8490 m/s ρs = 2331 kg/m pyrex cp = 5640 m/s ρp = 2230 kg/m has been initiated can be used, and in this study, im- ages from a number of consecutively recorded sets have been used for averaging. Moreover, in the case of scan- ning, or mapping, techniques the expulsion of particles is especially problematic, as the seeding conditions in the device, or chamber, need to be restored for each measure- ment position. Also, the conditions may change during these lengthy recordings, leading to results that are dif- ficult to interpret. The acoustic resonances in low attenuation piezo- actuated microfluidic devices are formed over the whole devices, and they are also depending on the geometry of the whole device. As a consequence, there will typ- ically be amplitude fluctuations over the devices, due to unwanted artifacts, or deliberate designs. Therefore, when investigating acoustic resonances, and the influence caused by different modifications to the sample, it is im- portant to study the effects globally. If the acoustic ef- fects are only measured in a part of the device, this kind of information will not be yielded, independently on how detailed the flow is mapped within that region. There- fore, we suggest full-image micro-PIV for the investiga- tion of acoustic resonances in microfluidic devices. In this study, emphasis has been put on how to present the measured data in such a way that still images and PIV-vector plots give the best illustration of the tran- sient particle motion caused by the acoustic forces. To achieve this, we have chosen to superimpose the PIV- vector plots of the initial transient velocities on top of the pictures of the steady-state patterns of the particles obtained after a few seconds of actuation. After longer actuation times, secondary patterns will form, so images taken at this point can give a false impression of the parti- cle motion. This method of combining the transient PIV- vector plots and steady-state pictures has shown useful when comparing numerical simulations with micro-PIV measurements, especially for measuring amplitude fluc- tuations across the structures, and when discriminating between different numerical models. This will be demon- strated in Sec. IV. III. NUMERICAL SIMULATIONS In the experiments, the acoustic pressure field, which is superimposed on the ambient constant pressure, is driven by a harmonically oscillating piezo-actuator, i.e., the time-dependence can be described as cos(ωt). In this (a) 3D side-view (b) 3D side-view (c) 3D top-view (d) 3D top-view (e) 2D chip model (f) 2D chip model n, eigenmode no. 0 10 20 30 40 50✻ 3D model 2D chip model FIG. 2: Numerical simulations of the pressure eigenmodes pn(x, y, z) shown in gray-scale plots. (a) and (b) 3D model: side-view (xz-plane) of p1 and p31, respectively. (c) and (d) 3D model: top-view (xy-plane) of p1 and p31, respectively. (e) and (f) 2D chip model: top-view (xy-plane) of p1 and p31, respectively. (g) The eigenfrequencies ωn/2π versus mode number n for the 3D model and the 2D chip model. work, we focus on the acoustic resonances where the re- sponse of the bead solution is particularly strong. As the attenuation of the acoustic waves is relatively small, we can approximate the actual frequency-broadened acous- tic resonances of the driven system by the infinitely sharp eigenmodes of the isolated dissipationless chip. The pressure eigenmodes p (x, y, z) cos(ω t), labelled by an integer index n, and the angular eigenfrequencies or resonance frequencies ω are found as solutions to the Helmholtz eigenvalue equation ∇2p = −(ω2 where the index i is referring to the three material do- mains of silicon, water and glass in the chip. The bound- ary conditions at the outer edges of the system are given by the soft-wall condition p = 0 except at the bottom plane, where a hard-wall condition n ·∇p = 0 is chosen to mimic the piezo-actuator which fixes the velocity of the wall. At the internal interfaces between the different material regions the boundary conditions are continuity of the pressure p as well as of the wall-velocity. The lat- ter is ensured by continuity of the field (1/ρ . A list of the acoustic material parameters, i.e., sound velocities and densities ρ , is given in Table III. The Helmholtz equation was solved numerically using the COMSOL finite element method software. However, the large aspect ratio of the flat device made it impossible to simulate the actual device in 3D due to limited com- puter memory. We therefore investigated the possibility of making 2D simulations. The rationale for doing this is that the total height of the chip is only 1 mm. Given a weighted average speed of sound in the silicon-glass chip of 6900 m/s, the wavelength of a wave at the highest used frequency f = 2.5 MHz is 3 mm and thus three times the chip height. Similarly, at the same frequency the wave- length in water is 0.6 mm or three times the chamber height. Consequently, there is not room enough for even half a standing wave in the vertical direction neither in the water filled chamber nor in the silicon-glass chip. The first step towards a more rigorous justification for doing 2D simulations was to make a smaller 3D version of the system geometry. While keeping all the correct height measures as well as the chamber width as listed in Ta- ble I, we shrunk the planar dimensions of the surrounding chip to L = 8 mm, w = 6 mm and L = 6.8 mm. With this reduced geometry we could carry out the full 3D simulations, and the results thereof confirmed that the variations in the vertical z-direction of the 3D eigenmodes were modest, see the xz-plane plots of Figs. 2(a) and (b). A 2D simulation was then carried out for the horizontal xy center-plane of the chamber, i.e., a 2D water-filled area surrounded by a 2D silicon region. Comparing the 50 lowest 3D and 2D eigenmodes gave the following re- sults: (1) in the horizontal xy center-plane of the cham- ber the 3D eigenmodes agreed with the 2D eigenmodes, see Figs. 2(c–f); (2) due to the lack of the z-dependence in the Laplacian of the 2D Helmholtz equation, the 2D eigenfrequencies were systematically smaller than the 3D eigenfrequencies, see Fig. 2(g). It has thus been justified to simulate the experimentally observed resonances by 2D eigenmodes in the horizontal xy center-plane of the chamber. This we denote the 2D chip model. Due to the small acoustic impedance ratio ) = 0.08 between silicon and water, the simulations could be simplified even further. As demonstrated in Figs. 3(c) and (d), it suffices to find the eigenmodes of the chamber itself using hard-wall boundary conditions along its edges, except at the very ends of the inlet channels where soft-wall boundary conditions are employed to mimic in- and outlets. This we will refer to as the 2D chamber model. (c) (d) FIG. 3: Acoustic radiation force. (a) Experiments on 5 µm beads at the 1.936 MHz acoustic resonance. The white PIV- vectors indicate the initial bead velocities pointing away from pressure anti-nodes immediately after the piezo-actuation is applied. The picture underneath the PIV-vector plot shows the particles (black) gathered at the pressure nodal lines 3 seconds later. (b) As in panel (a) but now at 2.417 MHz. (c) and (d) Gray-scale plots of numerical simulations in the 2D chamber model of the corresponding acoustic pressure eigen- modes. Nodal lines are shown in black. IV. RESULTS AND DISCUSSION We have measured the flow response to the acoustic actuation in the frequency range from 0.5 to 2.5 MHz paying special attention to the strong responses corre- sponding to acoustic resonances. More than 30 of such resonances have been detected, but we present only a few, which we find to be representative for the method and the problems associated with acoustics in microfluidics. The most important results are the full-image micro- PIV analyses. For these, two types of experimental re- sults are presented. One type are the PIV-vector plots (white arrows) of the motion of the tracer particles, in most cases corresponding to the transient motion imme- diately after the onset of the acoustic piezo-actuation. The other type are micrographs of the microfluidic cham- ber with the steady-state particle patterns (often visible as narrow black bands) obtained after a few seconds of actuation. These two types of images are superimposed to illustrate the relation between the initial motion of the tracer beads and their final steady-state positions. The full-image micro-PIV analysis illustrations are also accompanied by the results of our numerical simulations in the form of gray-scale plots of the pressure eigenmodes (x, y, z). The pressure antinodes appear as white (pos- itive amplitude) and black (negative amplitude) regions. The pressure nodal lines are shown as thin black lines in the gray (small amplitude) regions. Additionally, we show a close up measurement of a streaming vortex, and provide a more in-depth compar- ison between the measured velocities and the calculated body force. A. Acoustic radiation force We first show results for the acoustic resonances at 1.936 and 2.417 MHz in the circular chamber containing large 5 µm tracer particles. In Figs. 3(a) and (b) are shown the measured tran- sient PIV-vector plots superimposed on the micrographs of the chamber with the static steady-state particle pat- terns. The fact that the particles accumulate in static patterns indicates that the dominant force on the tracer particles is the acoustic radiation force, an observation also expected from the relatively large size of the tracer particles. The matching numerically calculated acoustic eigenmodes of the 2D chamber model are shown in pan- els (c) and (d). It is noteworthy that even for the com- plicated resonance pattern of panels (b) and (d), the ob- served transient particle motion towards the steady-state positions, and the static steady-state patterns them- selves, are in good agreement with the numerically cal- culated pressure nodal lines. This demonstrates that even the simple 2D chamber model can predict what kind of fluidic behavior will be observed in the device. It also demonstrates that full-image micro-PIV analysis in combination with images of transient particle motion is effective in visualizing in-plane acoustic phenomena in micrometer-scale devices. FIG. 4: Acoustic streaming and radiation forces at the 2.17 MHz acoustic resonance. (a) Experiments on 5 µm beads similar to Fig. 3(a) showing that the acoustic radiation force dominates for large particles. (b) Experiments on 1 µm beads. Acoustic streaming dominates and the small beads act as trac- ers for the motion of the liquid. The resulting vortex structure in the flow-field prevents particle accumulation at the pres- sure nodes. (c) Gray-scale plot of numerical simulation in the 2D chamber model of the corresponding acoustic pressure eigenmode. Nodal lines are shown in black. B. Acoustic streaming To illustrate the difference between the acoustic radi- ation force and acoustic streaming, we now turn to the acoustic resonance at 2.17 MHz in the square chamber containing large 5 µm beads and small 1 µm beads as shown in Figs. 4(a) and (b), respectively. When micro-PIV is applied to investigate acoustic ef- fects in microfluidic chambers, the simultaneous presence of both acoustic radiation forces and acoustic streaming needs to be taken into account. For the large beads in Fig. 4(a) the acoustic radiation force dominates exactly as in Figs. 3(a) and (b), which results in particle accu- mulation at the pressure nodal lines. However, as shown in Fig. 4(b) reduction of the particle volume by a fac- tor of 125 leads to a qualitative change in the response. The motion of the smaller particles is dominated by the acoustic streaming of the water, and it manifests itself as a 6×6 pattern of vortices. The same 6×6 pattern was found by full-image micro-PIV when diluted milk was used as tracer solution, and also by optical inspec- tion with a fluorescein solution in the chamber (data not shown). All three experimental results strongly support the interpretation that the 6×6 vortex pattern is caused by acoustic streaming. In Fig. 4(b) is also seen a pronounced inhomogene- ity in the strength of the vortices across the microfluidic chamber. This effect cannot be ascribed to the geometry of the chamber, but is probably due to either a geomet- ric top-bottom asymmetry in the entire chip (similar to the left-right asymmetry discussed in Sec. IVC), or to an inhomogeneous coupling between the piezo-actuator and the silicon chip. If the frequency is shifted slightly in the vicinity of 2.17 MHz, the same vortex pattern will still be visible, but the strength distribution between the vortices will be altered. When investigating acoustic phe- nomena the advantage of full-image micro-PIV compared to partial-image micro-PIV is thus evident: partial-image micro-PIV employed locally in a part of the chamber would not have shown the symmetrical 6×6 vortex pat- tern, nor would it supply us with information of the in- homogeneity in strength for the same. Moreover, since the same inhomogeneity is not seen in the acoustic ra- diation force vector plot, this example shows that there is no direct relation between the strength of the acoustic streaming and the acoustic radiation force. Finally, we note that our measurements show that the acoustic radiation force on the large particles leads to a much larger particle velocity than the acoustic streaming velocities of the smaller particles. Turning to the numerical simulation in the 2D chamber model of the corresponding pressure eigenmode, shown in Fig. 4(c), we find good agreement with the experimental results. The calculated pressure nodal lines correspond well to the static steady-state particle patterns obtained with the large tracer particles dominated by the acous- tic radiation force. Moreover, the calculated 3×3 antin- ode pattern is also consistent with the observed period- doubled 6×6 vortex pattern of the small tracer particles dominated by acoustic streaming. The spatial period- doubling arises from the non-zero time-average of the non-linear term in the Navier–Stokes equation governing the attenuated acoustic flows leading to acoustic stream- ing [29]. C. Effects of geometric asymmetries For the results presented so far the simple 2D cham- ber model proved sufficient to interpret the experimental observations. However, as explained already in Sec. III the pressure eigenmodes are not confined to the chamber region but fill the entire chip. The acoustic resonances even propagate in all media (air and piezo-actuator) in contact with the chip. In the following we show one ex- ample of asymmetric resonance patterns that can only be explained by employing the more complete 2D chip model or by introducing asymmetries in the 2D chamber model. In Figs. 5(a) and (b) we consider the square cham- ber containing the large 5 µm beads at two nearby res- onance frequencies, 2.06 and 2.08 MHz. As before, the acoustic radiation force dominates and the beads accu- mulate at the pressure nodal lines. Note that the two patterns are similar, but that the first has a higher am- plitude on the left side, while the second has a higher amplitude on the right side. Both resonance patterns are similar to the acoustic pressure eigenmode shown in Fig. 5(c), which is found by numerical simulation using the 2D chamber model. However, since the chamber itself is left-right symmetric, the calculated eigenmode is also left-right symmetric, so to explain the observed asymme- try we have to break the left-right symmetry in the the- oretical model. We investigate two ways of doing this: first, in the 2D chip model by placing a symmetric cham- ber asymmetrically on the chip, and second, in the 2D chamber model by letting the inlet channel have a differ- ent length than the outlet channel. In Figs. 6(a-d) is shown the result of a numerical simu- lation in the 2D chip model where the left-right symme- try has been broken by displacing the chamber 1 mm left of the symmetry center of the chip. This displacement corresponds to the geometry of the actual chip used in the experiment. Panels (a) and (b) show the entire chip while panels (c) and (d) are the corresponding closeups of the chamber region. With this left-right asymmetric geometry, we do find asymmetric solutions at nearby fre- quencies that resemble the measured patterns: Figs. 6(c) and (d) correspond to Figs. 5(a) and (b), respectively. In the left-right symmetric case the left-right acoustic resonance is two-fold degenerate, i.e., two different res- onances have the same frequency. When the symme- try is broken the two resonances are affected differently: one gets a slightly higher eigenfrequency and the other a slightly lower, i.e., a splitting of the two-fold degenerate eigenfrequency into two non-degenerate nearly identical eigenfrequencies. The two closely spaced eigenmodes of the asymmetric 2D chamber model shown in Figs. 6(e) and (f) also resemble the measured patterns in Figs. 5(a) and (b). The calculated frequency splitting is 28 kHz, which is in fair agreement with the measured 20 kHz. Unquestionably, advanced models, like the chip model, are necessary for more complete theoretical investigations of how different factors contribute to the breaking of the FIG. 5: Splitting of a two-fold degenerate acoustic resonance due to geometrical asymmetry. (a) Acoustic radiation force as in Fig. 4(a) on 5 µm beads at the 2.06 MHz resonance. (b) The closely related 2.08 MHz resonance for the same sys- tem. (c) Gray-scale plot of numerical simulation in the left- right symmetric 2D chamber model of the corresponding two- fold degenerate, un-split, acoustic pressure eigenmode. Nodal lines are shown in black. symmetry of the simple chamber model. Experimentally this effect could be studied by measuring on a range of devices, with strictly controlled geometries of both struc- tures and substrates. We have only investigated two de- vices, and special concern was not taken as to the unifor- mity of the substrate. It is therefore not possible in the present study to determine whether the observed sym- metry breaking was due to geometric asymmetries in the chip, in the chip-actuator coupling, or in other parts of (c) (d) (e) (f) FIG. 6: (a) ad (b) Gray-scale plots of numerical simulations in the 2D chip model of two closely spaced acoustic pressure eigenmodes. The chamber is displaced 1 mm to the left of the symmetry center of the chip thereby breaking the left- right symmetry and splitting the two-fold eigenmode degen- eracy. The difference in eigenfrequency is only 1 kHz. (c) and (d) Closeups of the chamber region showing the asymmetric eigenmodes similar to the experimentally observed resonances seen in Fig. 5. (e) and (f) Gray-scale plots of numerically simulated pressure eigenmodes in the asymmetric 2D cham- ber model, where the left lead is 1 mm shorter than the right lead. The difference in eigenfrequency is 28 kHz, which is close to the observed difference of 20 kHz in Figs. 5(a) and (b). the system (such as air-bubbles trapped at the fluidic inlet and outlet). D. Validation of method Fig. 7 shows a micro-PIV vector plot of streaming mo- tion in the center of the square chamber at 2.17 MHz, recorded with a 20x microscope objective. With this kind of recording, detailed information of a section of the de- vice can be obtained, but it will not supply any informa- tion about the amplitude fluctuations over the device, nor does it reveal the 6x6 vortex pattern as seen in Fig. 4(b). Clearly, more detailed measurements of specific features are valuable, but for studies of resonances in low atten- uation microfluidic devices, full-image recordings are of 100 µm 500 µm/s FIG. 7: Micro-PIV velocity vector plot of streaming motion in the center of the square chamber at 2.17 MHz. Images were recorded with a 20x objective and a 0.63x Tv-adapter, and milk was used as tracer particles. most importance. By regarding the suspended particles as springs gov- erned by Hooke’s law, we can estimate the acoustic ra- diation force Fac, from the potential elastic energy. We find F = −τp , where τ is a constant parameter for each kind of particles, and p is the time-averaged second- order pressure field. As τ is an unknown positive constant for blood-cell like particles, the amplitude becomes a fit- ting parameter. Assuming the particles move in a quasi- stationary steady state, we can directly compare calcu- lated force patterns to measured velocity patterns. Such a comparison is seen in Fig. 8, where the calculated force pattern is compared with a scalar map of the velocity in y-direction, extruded from the measurement presented in Fig. 4(a). A comparison between the two is also seen in Fig. 9, where two vertical cross-sectional views, each lo- cated 330 µm away from the center of the chamber, are compared with the theoretical estimate. Both micro-PIV velocity plots show a good agreement with the calculated forces, and the fluctuations in amplitude over the device can be seen by comparing the two micro-PIV velocity plots with each other. V. CONCLUSION Using full-image micro-PIV we have made direct obser- vations of the acoustic resonances in piezo-actuated, flat microfluidic chambers containing various tracer particles. Depending on the size of the tracer particles either the acoustic radiation force or acoustic streaming of the solvent dominates their motion. Large particles are dom- FIG. 8: (a) The force in y-direction calculated with COMSOL finite element method software. (b) Scalar map of the velocity in y-direction, measured with micro-PIV. 0 10 20 30 40 50 -0.00025 -0.00020 -0.00015 -0.00010 -0.00005 0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 Point theory FIG. 9: Vertical cross-sectional plots of the velocity in the y-direction (PIV1) 330 µm left and (PIV2) 330 µm right of the center of the chamber. (theory) is the force in y-direction, calculated in COMSOL with the amplitude as the only fitting parameter. inated by the acoustic radiation force that pushes them to the static pressure nodal lines, while small particles are dominated by acoustic streaming and end up forming steady-state vortex patterns. However, for an arbitrary frequency and geometry one of the forces can be strong whereas the other is not, and it is therefore always nec- essary to apply more than one tracer solution in order to determine which forces are present. The observed acoustic resonances correspond to the pressure eigenmodes found by numerical simulation of 2D models of the system. The symmetric patterns can be explained by using the simple 2D chamber model, while asymmetric patterns can be explained by using the more complete 2D chip model taking into account the geomet- ric asymmetries of the surrounding chip, or in special cases, by an asymmetric 2D chamber model. We have demonstrated that full-image micro-PIV is a useful tool for complete characterization of the in-plane acoustically induced motion in piezo-actuated microfluidic chambers. Acknowledgement SMS was supported through Copenhagen Graduate School of Nanoscience and Nanotechnology, in a collab- oration between Dantec Dynamics A/S, and MIC, Tech- nical University of Denmark. [1] L.V. King, Proc. R. Soc. London, Ser. A, 1934, 147, 212. [2] K. Yosioka and Y. Kawasima, Acustica, 1955, 5, 167. [3] L.P. Gorkov, Sov. Phys. Doklady, 6, 1962, 773. [4] Lord Rayleigh, Proc. R. Soc. 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0704.1315

Observations towards early-type stars in the ESO-POP survey: II -- searches for intermediate and high velocity clouds

Mon. Not. R. Astron. Soc. 000, 000–000 () Printed 8 September 2021 (MN LaTEX style file v2.2) Observations towards early-type stars in the ESO-POP survey: II – searches for intermediate and high velocity clouds J. V. Smoker1⋆, I. Hunter1, P. M. W Kalberla2, F. P. Keenan1, R. Morras3, R. Hanuschik4, H. M. A. Thompson1, D. Silva5, E. Bajaja3, W. G. L Poppel3, M. Arnal3 1Astrophysics Research Centre, Department of Physics and Astronomy, Queen’s University Belfast, Belfast, BT7 1NN, U.K. 2Argelander-Institut für Astronomie, Universität †, Auf dem Hügel 71, 53121 Bonn, Germany 3Instituto Argentino de Radioastronomia, Casilla de correo 5, Villa Elisa, Argentina. 4European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748 Garching bei Mnchen, Germany 5TMT Observatory Scientist, AURA/Thirty Meter Telescope, 2636 East Washington Blvd., Pasadena, CA 91107, U.S.A. Accepted Received in original form ABSTRACT We present Ca ii K and Ti ii optical spectra of early-type stars taken mainly from the UVES Paranal Observatory Project, plus H i 21-cm spectra from the Vila-Elisa and Leiden-Dwingeloo surveys, which are employed to obtain distances to intermediate and high velocity clouds (IHVCs). H i emission at a velocity of –117 km s−1 towards the sightline HD30677 (l, b=190.2◦,–22.2◦) with column density ∼1.7×1019 cm−2 has no corresponding Ca ii K absorption in the UVES spectrum, which has a signal-to-noise (S/N) ratio of 610 per resolution element. The star has a spectroscopically determined distance of 2.7-kpc, and hence sets this as a firm lower distance limit towards Anti- Centre cloud ACII. Towards another sightline (HD 46185 with l, b=222.0◦, –10.1◦), H i at a velocity of +122 km s−1 and column density of 1.2×1019 cm−2 is seen. The corresponding Ca ii K spectrum has a S/N = 780, although no absorption is observed at the cloud velocity. This similarly places a firm lower distance limit of 2.9-kpc towards this parcel of gas that may be an intermediate velocity cloud. The lack of intermediate velocity (IV) Ca ii absorption towards HD196426 (l, b=45.8◦,–23.3◦) at a S/N = 500 reinforces a lower distance limit of ∼700-pc towards this part of Complex gp, where the H i column density is 1.1×1019 cm−2 and velocity is +78 km s−1. Additionally, no IV Ca ii is seen in absorption in the spectrum of HD19445, which is strong in H i with a column density of 8×1019 cm−2 at a velocity of ∼–42 km s−1, placing a firm although uninteresting lower distance limit of 39-pc to this part of IV South. Finally, no HV Ca ii K absorption is seen towards HD115363 (l, b=306.0◦,–1.0◦) at a S/N = 410, placing a lower distance of ∼3.2-kpc towards the HVC gas at velocity of ∼+224 km s−1 and H i column density of 5.2×1019 cm−2. This gas is in the same region of the sky as complex WE (Wakker 2001), but at higher velocities. The non-detection of Ca ii K absorption sets a lower distance of ∼3.2-kpc towards the HVC, which is unsurprising if this feature is indeed related to the Magellanic System. Key words: ISM: general – ISM: clouds – ISM: abundances – ISM: structure – stars: early-type ⋆ email: [email protected]. Based on observations taken at UT2, Kueyen, Cerro Paranal, c© RAS http://arxiv.org/abs/0704.1315v1 2 J. Smoker et al. 1 INTRODUCTION The Paranal Observatory Project (POP; Bagnulo et al. 2003)1 provides a wealth of high-resolution (R ∼80,000) op- tical spectra towards stars mainly in the Galactic disc that can be used to study subjects such as stellar properties, kinematics and the interstellar medium. In a previous paper (Hunter et al. 2006, hereafter Paper i) we used a sample of early-type stars in the POP survey in order to investigate the interstellar medium in the Na i UV, Ti ii and Ca ii K lines, using the stars as light sources to probe the material between the star and the Earth. Because these O- and B- type stars are often fast rotators with weak metal lines, they are ideal for probing narrow interstellar features. In the current paper, we use mainly Ca ii and Ti ii spec- tra in order to search for Intermediate and High Velocity Clouds (IHVCs) towards the sightlines investigated in Pa- per i, plus some additional sightlines for which high S/N data have now become available. Our aim is to improve the distances to these still enigmatic objects by searching for IHVCs in the Villa-Elisa Southern Sky 21-cm H i Survey (Bajaja et al. 2005) or Northern Hemisphere counterpart, the Leiden-Dwingeloo Survey (Hartmann & Burton 1997), and subsequently searching for corresponding absorption in the Ca ii or Ti ii optical spectra. Although the distance to many Intermediate Velocity Clouds (IVCs) is known (e.g. Kuntz & Danly 1996 and references therein), to date there are very few uncontroversial upper distance limits towards High Velocity Clouds (HVCs). Indeed towards the main complexes there are currently only three uncontroversial de- tections; towards Complex A (van Woerden et al. 1999), Complex M (Danly, Albert & Kuntz 1993) and Complex WB (Thom et al. 2006). Hence it is still unclear whether many of these objects are associated with the Galaxy, for example linked with a Galactic fountain with distances of ≈ 10 kpc (e.g. Quilis & Moore 2001), or are failed dwarf galaxies with distances of several hundred kpc (e.g. Braun & Burton 1999). Clearly, as the sample stars in the POP sur- vey were not chosen a−priori to intersect with known IHVC complexes, the majority of the sightlines do not cross known IHVCs. However, serendipitously a few of the sightlines in- tersect complexes and are studied in the current paper. In addition to the POP data, we include spectra from two re- cent high spectral resolution observing runs taken using the échelle spectrometer FEROS, plus further UVES observa- tions whose primary aim was to obtain spectra for a stellar library but that are also at high S/N and cover the Ca ii K line (Silva et al. 2007). The current work complements our previous studies in which we obtained improved distance limits towards IVC complexes gp and K and HVC complexes C, WA-WB, WE, and H (Smoker et al. 2004, 2006), by searching for absorp- tion in high-resolution spectra of mainly B-type stars taken Chile, ESO DDT programme 266.D-5655(A), UVES Paranal Observatory Project, with additional observations from 071.B- 0529(A), 072.B-0585(A), 073.B-0607(A), 074.B-0639(A), 076.D- 0018(A) and 077.D-0025(A). † Founded by merging of the Sternwarte, Radioastronomisches Institut, and Institut für Astrophysik und Extraterrestrische Forschung der Universität Bonn. 1 See also http://sc.eso.org/santiago/uvespop/ from the Edinburgh-Cape (Stobie et al. 1997) and Palomar- Green Surveys (Green, Schmidt & Liebert 1986). In partic- ular the current sightlines intersect IVC Complex K with previous distance limit of 0.7–6.8-kpc (de Boer & Savage 1983, Smoker 2006), the Anti-Centre clouds with previous distance limit of >0.4-kpc (Tamanaha 1996) and Complex WE with distance limit <12.8-kpc (Sembach et al. 1991). Finally, one of our current sightlines lies towards the M15 intermediate velocity cloud lying in the IVC complex gp. This cloud has been studied extensively, in the optical to determine variations in velocity and equivalent width varia- tions (Lehner et al. 1999, Meyer & Lauroesch 1999, Smoker et al. 2002), plus in the H i, infrared and Hα (Kennedy et al. 1998, Smoker et al. 2002). An improvement in the current distance limit of 0.8–4.3 kpc (Wakker 2001 and references therein) would be very useful to more accurately define the cloud parameters such as cloudlet sizes and densities and to provide clues to the high metalicity of this IVC (Little et al. 1994). Sect. 2 describes the sample, provides a table noting the cases where the current sightlines cross known IHVC complexes plus new observations not previously described in Paper I, and shows the optical and H i spectra. Sect. 3 gives the main results, including the cases where the cur- rent optical sightlines intersect IHVCs and an attempt to obtain improved distance limits towards these clouds. Sect. 4 discusses the most interesting lower limits to IHVCS and finally Sect. 5 gives a summary of the main findings. 2 THE SAMPLE, OBSERVATIONS AND DATA REDUCTION The list of sample stars is shown in Table 1. The table in- cludes all stars for which new observations were taken, plus sightlines that lie towards IHVC complexes that are dis- cussed in Sect. 3.2, but does not include the POP paper i objects that have no IHVC detection. Further information concerning the POP objects is given in Paper i. They are all O- and B-type stars with 2.3< mv <7.9 mag. For these POP optical spectroscopic data, we used the on-line versions of re- duced data from the Paranal Observatory Project (Bagnulo et al. 2003). These are spectra taken with the UVES échelle spectrometer mounted on the 8.2-m Kueyen telescope at the Very Large Telescope at a spectral resolution of 80,000 or 3.75 kms−1 and S/N pixel−1 ranging from 190–770. In this paper we concern ourselves with the Ca iiK (λair=3933.66Å) and Ti ii (λair=3383.76Å) species only. A further 9 stars were observed with FEROS on the ESO 2.2-m on La Silla during observing sessions in Oct. 2005 (FER1 in Table 1) and May 2006 (FER2 in Table 1). These stars are all B-type post- AGB stars or Planetary Nebulae and have fainter magni- tudes than the POP stars, with 9.4< mv < 13.3 mag. The S/N ratios pixel−1 at Ca ii K range from ∼40–120 and the resolution is R=48,000. The spectra shown in this paper are the quick-look pipeline products. As a check of their relia- bility, during each of the FEROS runs a bright B-type star from the POP survey was observed and the velocities and equivalent widths of some of the absorption lines were com- pared between the two datasets. Agreement was found to be excellent. Finally, 12 stars were taken from the dataset of Silva et al. (2007; S07 in Table 1). These are UVES spec- c© RAS, MNRAS 000, 000–000 http://sc.eso.org/santiago/uvespop/ IVCs and HVCs towards POP early-type stars 3 tra of early-type stars with 5.9 < mV < 11.3 mag., observed at a spectral resolution of ∼40,000 with S/N = 100–620 pixel−1, and were reduced using the ESO pipeline (MIDAS context) with calibrations taken the morning after the ob- servations. For the H i 21–cm spectra, we used either the Southern Villa-Elisa H i survey data (Bajaja et al. 2005), corrected for the effects of stray radiation or the Leiden- Dwingeloo survey for sightlines with Dec.>–20◦ (Hartman & Burton 1997). Both surveys have been merged to form the Leiden/Argentine/Bonn (LAB) H i line survey (Kalberla et al. 2005) which has a velocity resolution of 1 km s−1 and brightness temperature sensitivity of 0.07 K. In Table 1 the columns are as follows. Columns 1–5 give the star HD name, alternative name, Galactic coordi- nates and V -band magnitude taken from simbad. Columns 6–7 give the estimated stellar distance and z-height above or below the Galactic plane. These distances were primarily estimated using the method of spectroscopic parallax from the spectral type, apparent magnitude and reddening to- wards each star, estimated from the observed (B−V ) colour. Absolute magnitudes as a function of spectral type were taken from Schmidt-Kaler (1982) with colours from Wegner (1994). Details are given in Paper i. Excluding perhaps large systematic errors caused by the uncertainty in the absolute magnitude calibration of our sample, the distances have an uncertainty of ∼30 per cent. For a number of objects (in par- ticular the Wolf-Rayet stars, peculiar objects and Post-AGB stars), distances were taken from the reference given at the foot of the table. For example for HD179407 the distance is given as 76001 where the suffix refers to reference number 1 where the distance of 7600-pc was given. Column 8 gives the signal-to-noise (S/N) ratio pixel−1 in the Ca ii spectrum; to obtain the S/N per resolution element this needs to be multiplied by If the coordinate of the sightline lies within any of the figures of Wakker (2001) which display H i column densities towards IHVCs, this name is given in Column 9. We must stress that although more than 35 of our stars lie within the boundaries of these figures, often they are in regions where no IHVC is observed in H i, for example because the stars lie in holes in the H i distribution. Columns 10 and 11 give the minimum and maximum expected LSR velocity for gas orbiting the Galactic Centre, based on the direction of the sightline and the distance to the stellar target. To calculate the velocity range for ”normal” gas, we use the methodology of Wakker (1991), in that we assume a flat rotation curve with vrot = 220 km s −1 at r >0.5 kpc, decreasing linearly towards the Galactic Centre, together with equations from Mihalas & Binney (1981). A deviation velocity for interstel- lar cloud components which lie outside the expected velocity range is calculated, where the deviation velocity is defined as the difference between the velocity of the component and the nearest limit of the expected velocity range (Wakker 1991). We classify low velocity clouds (LVCs) as having absolute values of their deviation velocities below 30 km s−1, IVCs between 30 km s−1 and 90 km s−1, and HVCs greater than 90 km s−1. Finally, column 12 gives the source for the op- tical spectra (POP for stars from Paper i; FER1/FER2 for FEROS observations; S07 for stars from Silva et al. 2007), and H i data (LD for Leiden-Dwingeloo; VE for Villa-Elisa Survey sightlines). 3 RESULTS In this section we discuss those cases where the stellar sight- lines intersect with known IHVCs, and hence determine im- proved distance estimates towards a handful of objects. Fig. 1 shows the optical and H i spectra towards the sightlines where a distance limit has been determined to- wards an IHVC. Fig. 2 (available online) shows the remain- ing sightlines. Two plots are shown for each sightline in order to emphasise both weak and strong features. The majority of the optical spectra are in the Ca ii K line; where this was not available the Ti ii line is shown. The horizontal line at the top of the first of the optical plots shows the extent of the full width half maximum of the stellar line. In most cases, these lines are wide, hence there is no possibility that stellar lines could be misidentified as interstellar features, which tend to be much narrower. If the stellar lines have a FWHM exceeding ∼100 kms−1 they were removed in the normalisation process to facilitate visualisation of the inter- stellar lines in all cases apart from HD numbers 115363, 136239 and 142758 where too much overlap of stellar and interstellar components occur. 3.1 Methodology of estimating distances to IHVCs The method of estimating distances to IHVCs is discussed fully in Schwarz, Wakker & van Woerden (1995). For an up- per distance limit, detection of optical absorption, in associ- ation with an H i detection, is sufficient to provide an upper distance limit, being the distance of the stellar probe. Lower- distance limits are more problematic. A firm lower distance limit can only be set if no optical absorption is seen at a sufficient S/N ratio, the abundance of the optical element is known (generally from observations of the same part of the complex towards QSOs), and the H i column density is accu- rately defined using a pencil beam. For the current sample, the chemical abundance of the IHVC is often not known, and the observations in H i only have a spatial resolution of 0.5◦, which means that care must be taken in ascribing a lack of optical absorption as due to the stellar probe being closer than the IHVC. However, these factors are somewhat ameliorated by the fact that the optical spectra have high S/N, frequently being >500 per resolution element and with a median of 410 in the sightlines with a detected IHVC. 3.2 Distance limits towards individual complexes A number of the current sightlines either intersect with known IHVC complexes, or have gas present at IHVC ve- locities in the Villa-Elisa or Leiden-Dwingeloo H i spectra. These cases are discussed below, and lower distance esti- mates towards five IHVCs are determined. Table 2 summa- rizes these cases. Columns 1–6 gives the star name, stellar distance, previous IHVC distance limit, IHVC complex, ob- served H i velocity and corresponding log of the H i column density. Columns 7–8 give the previously-known abundance in Ca ii taken from Wakker (2001) and limiting 5σ Ca ii col- umn density estimated from the current spectra. This was derived using the observed S/N ratio and instrumental res- olution, assuming the the optically thin approximation. Fi- nally, column 9 gives the predicted Ca ii column density de- c© RAS, MNRAS 000, 000–000 4 J. Smoker et al. Table 1. The stellar subsample for new observations plus all sightlines which lie in the vicinity of IHVCs. The S/N ratios per pixel are for Ca ii K (3933Å). See text for details. Star Alt. l b mv d z S/N IHVC v Source Name (deg.) (deg.) (mag.) (pc) (pc) pixel−1 Opt./H i HD171432 BD-18 5008 14.62 -4.98 7.11 4014 -348 590 – 0.0 43.9 POP/VE EC20485-2420 21.76 -36.36 11.77 36005 -1200 40 gp 0.0 28.9 FER1/VE HD179407 BD-12 5308 24.02 -10.40 9.44 76001 -1400 120 gp 0.0 128.3 FER1/VE HD188294 57 Aql B 32.65 -17.77 6.44 212 -64 420 gp 0.0 2.3 POP/VE G169–28 HIP 82398 41.83 +36.06 11.26 1172 69 100 K 0.0 1.0 S07/LD HD196426 HR 7878 45.81 -23.32 6.21 7003 -280 360 gp 0.0 3.3 S07/LD HD344365 58.63 +3.41 10.8 103213 61 210 – 0.0 11.4 S07/LD HD2857 110.05 -67.64 9.95 7177 -663 270 IVS -0.9 0.0 S07/LD HD19445 157.48 -27.20 8.05 3912 -18 200 IVS, ACC -0.3 0.0 S07/LD HD30677 BD+08 775 190.18 -22.22 6.84 2707 -1023 430 ACII 0.0 8.1 POP/VE HD46185 BD-12 1520 221.97 -10.08 6.79 2937 -514 550 – 0.0 31.1 POP/VE BD-12 2669 239.12 +18.17 10.22 1588 49 250 IV Spur 0.0 1.6 S07/LD HD72067 HR 3356 262.08 -3.08 5.83 488 -26 450 – 0.0 2.0 POP/VE EC05229-6058 269.97 -34.08 11.4 22005 -2100 150 – 0.0 4.1 FER1/VE HD94910 HIP 53461 289.18 -0.69 7.09 60004 -72 430 – -12.2 3.4 POP/VE EC01483-6806 294.73 -48.36 11.1 26005 -2000 130 – -9.5 0.0 FER1/VE LB3193 297.32 -54.90 12.70 80006 1800 100 – -14.4 0.0 FER1/VE HD115363 HIP 64896 305.88 -0.97 7.82 3282 -55 290 WE -35.3 0.0 POP/VE ROA5701 309.24 +15.05 13.16 48007 1246 50 – -46.5 0.0 FER2/VE HD120908 312.25 +8.37 5.88 338 49 370 – -4.3 0.0 S07/VE HD480 319.45 -65.58 7.03 469 427 530 – -1.0 0.0 S07/VE HD142919 328.43 -0.76 6.10 268 -4 500 WE -3.2 0.0 S07/VE HD186837 335.85 -30.57 6.20 329 -167 620 WE -2.4 0.0 S07/VE IRAS17311 341.41 -9.04 11.4 11008 -174 55 – -9.5 0.0 FER1/VE HD163758 SAO 209560 355.36 -6.10 7.32 4103 -436 550 – -16.1 0.0 POP/VE HD163745 350.56 -8.79 6.50 2189 335 620 – -11.9 0.0 S07/VE BD+09 2860 353.04 +63.21 11.27 53310 475 250 – -0.4 0.0 S07/LD HD177566 355.55 -20.42 10.17 11009 -383 120 – -190.2 0.0 FER1/VE CD-41 13967 359.28 -33.50 9.5 350011 -1900 80 – -1.2 0.0 FER1/VE Reference codes: (1) Hoekzema, Lamers & van Genderen (1993), (1) Smartt, Dufton & Lennon (1997), (2) González et al. (2006) (3) Carney et al. (1994). (4) Hoekzema, Lamers & van Genderen (1993), (5) Smoker et al. (2003), (6) Quin & Lamers (1992) (7) Kinman et al. (2000), (8) Laird, Carney & Latham (1988), (9) Zsargó et al. (2003), (10) Beers et al. (2000), (11) McCarthy et al. (1991), (12) From parallax. (13) From RR-Lyrae calibration and magnitude. rived by subtracting the previously-known Ca ii abundance from the log of the observed H i column density. Where this predicted value is much higher than the limiting 5σ Ca ii column density a non-detection is interpreted as the cloud lying further away than the stellar probe. Individual com- plexes are discussed below. 3.2.1 Complex gp IVC Complex gp is a positive-velocity IVC lying in the direc- tion of the globular cluster M15, which has previously been studied in infrared, optical, Hα and H i by Smoker et al. (2002). The previously-existing distance limit was 0.8–4.3 kpc (Wakker 2001 and references therein) with an uncer- tain lower distance limit of 2.0-kpc (Smoker et al. 2006). The Complex has LSR velocities of ∼+60 to +90 kms−1. In our current sample, the star HD188294 lies towards this Complex, but only has a distance of 212-pc and no H i is detected for this sightline due to it being in a “hole” in the Complex. Additionally, HD196426 (l, b=45.81◦,–23.32◦) lies towards Complex gp, and weak H i is observed in emission in the Leiden-Dwingeloo spectrum, with a LSR velocity +78±1 kms−1, a FWHM of 24±2 km s−1, peak brightness tempera- ture TB=0.25±0.05 K and brightness temperature integral of 6.5±1.0 K km s−1, corresponding to an H i column density of 1.1±0.2×1019 cm−2. Although weak, this should have been detected in our UVES spectrum which has a S/N = 500 per resolution element. The star has a distance of 700-pc, which is similar to the distances for previous objects towards which there were non-detections. In Complex gp we also observed HD179407 (l, b=24.02◦,–10.4◦, distance=7600-pc) at a S/N pixel−1 of 120 in Ca ii K. At the current position, there are two weak H i velocity features, at v=+50±1 and v=+97±1 kms−1 with FWHM values of 26±2 and 42±4 kms−1 and brightness temperature integrals of 1.7±0.2 and 1.7±0.2 K kms−1 respectively, corresponding to column densities of ∼3×1018cm−2. There is obvious detection of Ca ii in the +50 kms−1 feature (as in the Ca ii spectrum of Sembach & Danks 1994), but no detection of the v= +97 kms−1 feature, perhaps due to clumpiness in the H i or ionisation issues; a higher S/N Ca ii spectrum would be useful. Given the weak nature of both H i features a higher spatial-resolution and sensitivity H i spectrum would be useful at this position al- though in any case the star lies at a distance exceeding the current upper limit of the cloud. Although HD 179407 was c© RAS, MNRAS 000, 000–000 IVCs and HVCs towards POP early-type stars 5 Figure 1. Optical Ca ii K and 21-cm H i spectra towards early-type stars for which a lower distance limit towards an IHVC has been determined. Two plots are shown per sightline in order to emphasise weak features. Further details are given in the text. Table 2. IHVC sightlines where H i is detected at intermediate or high velocities. Complex WEM is in the same part of the sky as complex WE of Wakker (2001), but at higher velocities. See Sect 3.2 for details. Star d d IHVC vIHVC(H i) log(NIHVC(H i)) A (Ca ii) log(Nlim(Ca ii)) log(Npred(Ca ii)) (pc) (pc) complex km s−1 (log(cm−2)) (log(cm−2)) (log(cm−2)) (log(cm−2)) HD196426 700 800-4300 gp +78 19.06 -7.42 10.20 11.65 HD179407 7600 ” gp +50 18.49 -7.42 10.60 11.07 ” ” ” gp +97 18.49 -7.42 10.60 11.07 HD19445 39 – IVS -45 19.49 -7.88 10.46 11.61 ” ” – IVS -40 19.71 -7.88 10.46 11.82 HD30677 2700 >400 ACII -117 19.24 <-8.39 9.82 – HD115363 3200 – WEM +224 19.71 – 9.99 – ” ” – WEM +240 19.30 – 9.99 – HD46185 2900 – Other +122 19.09 – 9.71 – also observed in the FUSE spectrum by Zsargo et al. (2003), the presence of a complex stellar continuum meant that no interstellar Ovi was observed. Finally, although EC 20485- 2420 lies in the general direction of this complex, no H i is obvious in the Villa-Elisa spectrum. 3.2.2 IV South IV South is a group of IVCs that extend over much of the southern sky, with velocities of ∼–85 to –45 kms−1. To- wards HD19445 (l, b=157.48◦,–27.20◦), no IV absorption is seen in the Ca ii spectrum at a S/N of 280 per resolution ele- ment, thus placing a rather uninteresting firm lower distance limit of 39-pc to this part of the IVC that has two compo- nents with v=–45±0.5 km s−1, –40.2±0.5 km s−1, FWHM values of 8±1 kms−1 and 22±2 kms−1, peak TB values of of 2.1±0.2 K and 1.2±0.2 K and brightness temperature in- tegral of 17±2 K kms−1 and 28±3 K kms−1. The combined H i column density in these two features is ∼8×1019 cm−2 which should have been easily detected in the current optical spectrum if the cloud were closer than the star. c© RAS, MNRAS 000, 000–000 6 J. Smoker et al. 3.2.3 Complex K Complex K is a Northern-Hemisphere cloud with LSR velocities ranging from –65 to –95 kms−1. Its previous distance bracket was ∼700–6800-pc (Smoker et al. 2006 and refs. therein). One of our sightlines towards G169- 28 (l, b=41.83◦,+36.06◦) lies in the general direction of Complex K, but no H i emission is visible in the Leiden- Dwingeloo spectrum and there are many stellar lines. Thus the current observations do not add anything to our knowl- edge of this IVC. 3.2.4 Anti-centre HVCs Seven of our sightlines lie in the region of the Anti-Centre HVC (Fig. 9 of Wakker 2001). No upper distance limit is available for this HVC and the previous lower-distance limit towards Cloud ACI is only 0.4-kpc (Tamanaha 1996). We only detect H i at high velocity towards one of the cur- rent sightlines which lies towards ACII, namely HD30677 at a velocity of –117±1 kms−1, peak brightness tempera- ture of 0.40±0.04 K, FWHM of 23±2 km s−1 and integrated brightness temperature of 9.5±1.0 K kms−1, corresponding to an HVC column density of 1.7±0.2×1019 cm−2. Assum- ing that the HVC has a similar abundance to the relation from Wakker & Mathis (2000), we would expect a corre- sponding column density log(Ca ii cm−2)=11.64. However, no corresponding optical absorption is detected in our Ca ii K spectrum, which has a S/N = 430 pixel−1 or 610 per res- olution element. Assuming that the cloud is optically thin in Ca, a 5σ detection, f = 0.634 for the Ca ii K transition and instrumental resolution of 0.05Å, the limiting column density observable with the current spectrum is log(Ca ii cm−2)=9.82, more than a factor 60 lower than predicted from the H i profile. Hence the current observations put a firm lower distance limit of 2.7-kpc towards complex ACII, assuming that the H i observed in the Villa-Elisa survey re- flects that in the pencil beam towards HD30677. 3.2.5 Complex WE/WEM HVC Complex WE is a group of small HVCs centred on (l, b)∼(320◦,0◦), first detected by Mathewson, Cleary & Murray (1974) and mapped in H i by Morras (1982). Parts of it lie in the same region of the sky as two large low- velocity H i shells in the direction of the Coalsack nebula described by McLure-Griffiths et al. (2001). At b ∼0◦ lat- itude the predicted values of Galactic rotation at l ∼320◦ are from ∼–120 to +70 km s−1, falling to ∼–100 to 0 km s−1 at b ∼–15◦. Towards HD156359 (l, b=328.68◦, –14.52◦), Sembach et al. (1991) found optical absorption at ∼+110 kms−1, putting an upper distance limit of 12.8 kpc. Eigh- teen of our sightlines lie within the general area of WE as defined in Fig. 11 of Wakker (2001). One of the sam- ple stars HD 115363 (l, b=306.0◦,–1.0◦ with spectroscopic distance=3.2-kpc) has HVC gas detected with two compo- nents at +224.5±3.0 kms−1, +240.0±5.0 km s−1, velocity widths 14.4±0.8 kms−1 and 19.2±2.4 km s−1, peak bright- ness temperatures of 1.8±0.06 K and 1.1±0.1 kms−1 and brightness temperature integrals of 28.3±1.0 K kms−1 and 11.1±0.8 K kms−1 which correspond to H i column densi- ties of 5.2± 0.2×1019 cm−2 and 2.0±0.1×1019 cm−2. There is no Ca ii K absorption present in the spectrum, which has a S/N = 410 per resolution element. This HVC is probably associated with the clouds defined by Putman (2000) as the Leading Arm: the counterpart of the Magellanic Stream, as projected on the sky, between the Magellanic Clouds and the Galactic Plane. These data hence set an unsurprising lower limit of 3.2-kpc towards this HVC that is probably re- lated to the Magellanic System, using our distance estimated spectroscopically . If we assume that HD115363 is a part of the Centaurus OB1 association, its distance is slightly closer at 2.5-kpc (McClure-Griffiths et al. 2001 and refs. therein). Finally we note that this HVC appears to be a different set of clouds to the lower-velocity and more negative galactic- latitude clouds described in Wakker (2001) and observed by Sembach et al. (1991), hence in the current paper it is named WEM due to its possible association with the Mag- ellanic system. 3.2.6 Other IVCs In the line-of-sight towards HD46185 (l, b=222.0◦,–10.1◦), H i emission is detected at +122±2 kms−1, with a peak brightness temperature of 0.35 K, FWHM of 17±3 kms−1 and brightness temperature integral of 6.7±0.7 K kms−1, corresponding to an H i column density of 1.2±0.1×1019 cm−2. Normal Galactic rotation predicts velocities of upto ∼+97 kms−1 in this part of the sky, so the deviation ve- locity is only ∼ 25 km s−1 and the cloud many not be an IVC. Assuming that the cloud has a similar abundance to the relation from Wakker & Mathis (2000), we would ex- pect a column density log(Ca ii K cm−2)=11.59. However, no corresponding optical absorption is detected in our Ca ii K spectrum, which has a S/N = 550 pixel−1 or 780 per res- olution element. The 5σ limiting column density observable with the current spectrum is log(Ca ii cm−2)=9.71, a fac- tor 75 lower than predicted from the H i profile. Hence the current observations put a firm lower distance limit of 2.9- kpc towards this parcel of gas that lies within ∼20◦ of the Anti-Centre Shell (Fig. 8 of Wakker 2001) but is at different velocities and probably unrelated. 3.3 IHVCs detected in Ca ii absorption A number of sightlines were already flagged in Paper i as having IHVC components detected in the optical spec- tra. These include the Wolf Rayet stars HD94910 and HD163758 and the sightline HD72067 which lies towards the Vela Supernova remnant. No H i is detected towards any of these sightlines. In the first two cases this implies the presence of circumstellar lines and in the latter case lines within the SN remnant. Similarly, towards HD171432 many IVCs are detected in the optical. This sightline lies towards the Scutum Supershell mapped in H i by Callaway et al. (2000) and with a distance of ∼3000-pc. Although to- wards HD171432 there is a dearth of H i detected in the Callaway maps, there is H i in our H i spectrum up to a velocity of ∼+90 kms−1, coincident with our detections of Ca ii. No H i is seen in our highest-velocity Ca ii component of ∼+120 kms−1, perhaps due to S/N limitations. The de- tections in Ca ii and H i are consistent with the supershell being closer than our stellar distance of ∼4000 pc and with c© RAS, MNRAS 000, 000–000 IVCs and HVCs towards POP early-type stars 7 the previous observations, but add nothing to the distance bracket. 4 DISCUSSION Table 3 gives a summary of the distance limits to IHVCs set by the current observations, plus existing limits to the clouds where available. Particularly interesting is the im- proved lower limit towards part of the Anti-Centre complex ACII which has firm lower-distance limit of >2.7-kpc. This compares with the indirect distance estimate of a part of the complex at l ∼60◦,b ∼–45◦ derived from morphological and kinematical arguments of ∼ 4-kpc (Peek et al. 2007), and an Hα estimated distance of between 8 and 20-kpc (Weiner et al. 2001) which is based upon the observed ionisation being caused by the Galactic radiation field. Although a big im- provement on the previous lower-distance limit of ∼0.4-kpc (Tamanaha 1996), the current observations cannot discrim- inate between the indirectly-estimated distances and clearly searches for more distant probe stars in this part of the sky would be useful. Other less interesting results are the con- solidation of the lower-distance limit towards complex gp and the first lower distance limit towards the WEM com- plex. The z-distance of the former IVC is now constrained to ∼300-1700-pc which compares to the H i scaleheight of <200-pc at Galactocentric radii of <10-kpc (Narayan, Saha & Jog 2005). Further progress on this sightline should in- volve performing obtaining a high-resolution spectrum of the star HD357657 and associated model atmosphere cal- culation and abundance analysis. Although Smoker et al. (2006) estimated a distance of ∼2.0-kpc for this object on the line of sight to Complex gp and found no associated Ca ii absorption, the distance of the star remains uncertain. If a firm lower distance limit of 2-kpc were confirmed, cloud parameters such as the cloudlet sizes, cloud electron den- sity, fractional H i to H ii ratios and ionizing radiation field could be better constrained (c.f. Smoker et al. 2002), and the position of the cloud relative to the H i disc of the Galaxy confirmed. Finally, the lower distance limit of 3.2-kpc towards HVC WEM is consistent with both a Magellanic origin as pro- posed for example by Putman (2000), or a ’classical’ high velocity cloud. H i synthesis mapping towards other HVCs in this part of the sky (e.g. Bekhti et al. 2006) have provided evidence from cloud structure and linewidths of distances of ∼10–60-kpc, consistent with a Magellanic origin, and the same observations could be performed for the present sight- line in order to obtain an indirect distance estimate, perhaps in conjunction with Hα mapping. However, in the absence of early-type stars present in the leading arm as present in the Magellanic Bridge (Rolleston et al. 1999), obtaining a firm upper distance limit will be difficult although perhaps possible due to the offset in velocity from the stellar and interstellar Ca ii K lines (c.f. Smoker et al. 2002). 5 SUMMARY We have correlated optical spectra in the Ca ii K and Ti ii lines observed towards early-type stars in the POP Survey, plus other optical data, with 21-cm H i spectra taken from Table 3. Distance limits and probe stars towards the IHVCs studied in this paper. IHVC (l, b) vIHVC(H i) Probes DIHVC (deg.) km s−1 (pc) gp 46,–23 +78 HD196426 800-43001,2 IVS 157,–27 –45, –40 HD19445 > 391 ACII 190,–22 –117 HD30677 > 27001 WEM 306,–1 +224, +240 HD115363 > 32001 Other 222,–10 +122 HD46185 > 29001 Reference codes: (1) This paper, (2) Little et al. (1994). the Villa-Elisa and Leiden-Dwingeloo Surveys, in order to determine the distances to Intermediate and High Velocity Clouds. The lack of Ca ii K absorption at –117 km s−1 to- wards HD30677 at a S/N ratio of ∼610 has set a firm lower distance limit towards Anti-Centre cloud ACII which previ- ously had a lower distance limit of 0.4-kpc. Likewise, towards HD46185 no Ca ii K absorption at +122 kms−1 is seen at a S/N ratio of ∼780, hence placing a lower distance limit of 2.9-kpc towards this gas that is perhaps an IVC. Towards Complex gp no Ca ii K absorption is seen in the spectrum of HD196426 at a S/N of ∼500, reinforcing the assertion that this IVC lies at a distance exceeding 0.7-kpc. Likewise, towards the nearby star HD19445 at 39-pc in the line of sight to IV South no Ca ii K absorption is seen setting a a firm but uninteresting distance limit towards this part of the complex. Finally, no HV Ca ii K absorption is seen in the stellar spectrum of HD115363 at a S/N = 410, placing a lower distance of ∼3.2-kpc towards the HVC gas at velocity of ∼+224 kms−1. This gas is in the same region of sky as the WE complex of Wakker (2001), but at higher velocities. If related to the Magellanic system (Putman 2000) then a distance limit of 3.2-kpc is not unexpected. A future paper will describe the use of new FEROS observations combined with UVES archive data to provide improved distance limits to complex EP, the Cohen Stream, IV South and the Anti-Centre shell. Concerning the POP data, future papers will investigate the neutral species of Ca i, Fe i, Na i D and K i as well as molecular line species CH, CH+ and CN in order to better understand the local interstellar medium. ACKNOWLEDGEMENTS We would like to thank the staff of the Very Large Telescope, Paranal for the large amount work involved in producing the POP Survey (ESO DDT programme ID 266.D-5655(A), http://www.eso.org/uvespop). Especially due thanks are S. Bagnulo, R. Cabanac, E. Jehin, C. Ledoux and C. Melo. In addition, we are grateful to the staffs of Dwingeloo/Leiden and the Villa-Elisa Telescope for producing the H i all sky surveys. JVS and HMAT thank the support staff at La Silla for their help with the FEROS observations. FPK is grate- ful to AWE Aldermaston for the award of a William Pen- ney Fellowship. This research has made use of the simbad Database, operated at CDS, Strasbourg, France. JVS ac- knowledges financial support from the Particle Physics and Astronomy Research Council with HMAT and IH thanking the Department of Education and Learning for Northern Ire- c© RAS, MNRAS 000, 000–000 http://www.eso.org/uvespop 8 J. Smoker et al. land. JVS thanks M. Garćıa Muñiz, L. 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D., 2003, ApJ, 586, 1019 c© RAS, MNRAS 000, 000–000 IVCs and HVCs towards POP early-type stars 9 Figure 2.Optical (Ca ii K or Ti ii) and 21-cm H i spectra towards early-type stars. Two plots are shown per sightline in order to emphasise weak features. In the cases where the FWHM of the stellar profile exceeded ∼100 kms−1 it has been removed in the normalisation process to emphasise the interstellar line features. This affects the stars with HD numbers 480, 2857, 2913, 49131, 61429, 74966, 76131, 90882, 100841, 115842, 122272. 145482, 156575, 163745, 186837, 188294, 344365, plus ROA 5701. For HD2857 some residuals are left by this process at ∼ –160 km s−1. c© RAS, MNRAS 000, 000–000 10 J. Smoker et al. Figure 2. ctd. c© RAS, MNRAS 000, 000–000 IVCs and HVCs towards POP early-type stars 11 Figure 2. ctd. c© RAS, MNRAS 000, 000–000 12 J. Smoker et al. Figure 2. ctd. c© RAS, MNRAS 000, 000–000 IVCs and HVCs towards POP early-type stars 13 Figure 2. ctd. c© RAS, MNRAS 000, 000–000 14 J. Smoker et al. Figure 2. ctd. c© RAS, MNRAS 000, 000–000 IVCs and HVCs towards POP early-type stars 15 Figure 2. ctd. c© RAS, MNRAS 000, 000–000 16 J. Smoker et al. Figure 2. ctd. c© RAS, MNRAS 000, 000–000 Introduction The sample, observations and data reduction Results Methodology of estimating distances to IHVCs Distance limits towards individual complexes IHVCs detected in Caii absorption Discussion Summary

0704.1316

Physisorption of Nucleobases on Graphene

Physisorption of Nucleobases on Graphene S. Gowtham1, Ralph H. Scheicher1,2, Rajeev Ahuja2,3, Ravindra Pandey1,∗ and Shashi P. Karna4 1Department of Physics and Multi-Scale Technologies Institute, Michigan Technological University, Houghton, Michigan 49931, USA 2Condensed Matter Theory Group, Department of Physics, Box 530, Uppsala University, S-751 21 Uppsala, Sweden 3Applied Materials Physics, Department of Materials and Engineering, Royal Institute of Technology (KTH), S-100 44 Stockholm, Sweden and 4US Army Research Laboratory, Weapons and Materials Research Directorate, ATTN: AMSRD-ARL-WM; Aberdeen Proving Ground, Maryland 21005-5069, USA (Dated: October 24, 2018) We report the results of our first-principles investigation on the interaction of the nucleobases adenine (A), cytosine (C), guanine (G), thymine (T), and uracil (U) with graphene, carried out within the density functional theory framework, with additional calculations utilizing Hartree–Fock plus second–order Møller–Plesset perturbation theory. The calculated binding energy of the nucle- obases shows the following hierarchy: G > T ≈ C ≈ A > U, with the equilibrium configuration being very similar for all five of them. Our results clearly demonstrate that the nucleobases exhibit significantly different interaction strengths when physisorbed on graphene. The stabilizing factor in the interaction between the base molecule and graphene sheet is dominated by the molecular polarizability that allows a weakly attractive dispersion force to be induced between them. The present study represents a significant step towards a first-principles understanding of how the base sequence of DNA can affect its interaction with carbon nanotubes, as observed experimentally. PACS numbers: 68.43.-h, 81.07.De, 82.37.Rs DNA-coated carbon nanotubes represent a hybrid sys- tem which unites the biological regime and the nanoma- terials world. They possess features which make them attractive for a broad range of applications, e.g., as an efficient method to separate carbon nanotubes (CNTs) according to their electronic properties [1, 2, 3], as highly specific nanosensors, or as an in vivo optical detector for ions. Potential applications of single-stranded DNA (ss- DNA) covered CNTs range from electron sensing of vari- ous odors [4], to probing conformational changes in DNA triggered by shifts in the surrounding ionic concentration [5], and detection of hybridization between complemen- tary strands of DNA [6, 7]. The interaction of DNA with CNT is not limited to the outer surface of the tube; it has also been experimentally demonstrated that ssDNA can be inserted into a CNT [8], further enhancing the potential applications of this nano-bio system. The details of the interaction of DNA with CNTs have not yet been fully understood, though it is gen- erally assumed to be mediated by the π-electron net- works of the base parts of DNA and the graphene-like surface of CNTs. One would like to obtain a better un- derstanding of the binding mechanism, and the relative strength of base-CNT binding as it is indicated experi- mentally from sequence-dependent interactions of DNA with CNTs [3, 4]. In this Letter, we present the results of our first-principles study of the interaction of nucle- obases with a graphene sheet as a significant step towards a deeper understanding of the interaction of ssDNA with ∗Corresponding authors. E-mail: [email protected], [email protected] CNTs. Previous theoretical studies focused on the adsorption of the nucleobase adenine on graphite [9]. In the present study, we have considered all five nucleobases of DNA and RNA, namely the two purine bases adenine (A) and guanine (G), and the three pyrimidine bases cytosine (C), thymine (T), and uracil (U). Our specific interest is to assess the subtle differences in the adsorption strength of these nucleobases on graphene, which in turn will allow us to draw conclusions for the interaction of DNA and RNA with CNTs as well. Calculations were performed using the plane-wave pseudopotential approach within the local density ap- proximation (LDA) [10, 11] of density functional theory (DFT) [16], as implemented in the Vienna Ab-initio Sim- ulation Package (vasp) [17]. The cutoff energy was set to 850 eV. For k-point sampling of the Brillouin zone we used the 1 × 1 × 1 Monkhorst-Pack grid [18], which we found from benchmark calculations to yield identical results as a 3× 3× 1 Monkhorst-Pack grid would. A 5×5 array of the graphene unit cell in the x-y plane and a separation of 15 Å between adjacent graphene sheets in the z-direction was found to be a suitable choice to represent the supercell. The base molecules were ter- minated at the cut bond to the sugar ring with a methyl group in order to generate an electronic environment in the nucleobase more closely resembling the situation in DNA and RNA rather than that of just individual iso- lated bases by themselves. This has the additional bene- fit that a small magnitude of steric hindrance can be ex- pected from the methyl group, quite similar to the case in which a nucleobase with attached sugar and phosphate group would interact with graphene. http://arxiv.org/abs/0704.1316v1 mailto:[email protected], [email protected] For each of the five nucleobases, an “initial force re- laxation” calculation step determined the preferred ori- entation and optimum height of the planar base molecule relative to the graphene sheet. A slice of the potential energy surface was then explored by translating the re- laxed base molecules in a fixed orientation parallel to the graphene plane in steps of 0.246 Å along the lattice vec- tors of graphene, covering its entire unit cell by a mesh of 10 × 10 scan points. The separation between base molecule and graphene sheet was held fixed at the opti- mum height determined previously. The determination of the minimum total energy configuration was then fol- lowed by a 360◦ rotation of the base molecules in steps of 5◦ to probe the dependence of the energy on the orienta- tion of the base molecules with respect to the underlying 2-D graphene sheet. The configuration yielding the mini- mum total energy was used in the final optimization step in which all atoms in the system were free to relax. We would like to emphasize here that for all five nucleobases, the eventually determined equilibrium configuration was characterized by a separation between base and graphene sheet that was equal to the optimum height chosen in the previous lateral potential energy surface scan. An additional set of calculations was performed us- ing the ab initio Hartree–Fock approach coupled with second–order Møller–Plesset perturbation theory (MP2) as implemented in the gaussian 03 suite of programs [19]. Due to the use of localized basis sets (rather than plane-wave), the system here consisted of the five nucle- obases on top of a patch of nanographene [20], i.e., a finite sheet containing 28 carbon atoms. The LDA optimized configuration and the 6-311++G(d,p) basis sets for C, H, N and O atoms were used for the MP2 calculations. The first optimization step involving the “initial force relaxation” led to a configuration of all five nucleobases in which their planes are likewise oriented almost exactly parallel to the graphene sheet with a separation of about 3.5 Å, characteristic for π–π stacked systems [21]. The in- teraction of the attached methyl group with the graphene sheet results in a very small tilt of the molecule, with an- gles less than 5◦. The base is translated 2.461 Å along both graphene lattice unit vectors respectively (maintaining a constant vertical distance of 3.5 Å from the sheet, as determined in the previous step), and rotated 360◦ in the equilibrium configuration with respect to the configuration obtained after the “initial force relaxation” step in the optimiza- tion procedure. From the optimization steps involving the translational scan of the energy surface, it is appar- ent that the energy barriers to lateral movement of a given base can range from 0.04 to 0.10 eV (Fig. 1), thus considerably affecting the mobility of the adsorbed nucle- obases on the graphene sheet at room temperature, and constricting their movement to certain directions. The rotational scans carried out by us found energy barriers of up to 0.10 eV, resulting in severe hindrance of changes in the orientation of the adsorbed nucleobase. In their equilibrium configuration, three of the five bases tend to position themselves on graphene in a con- figuration reminiscent of the Bernals AB stacking of two adjacent graphene layers in graphite (Fig. 2). Virtu- ally no changes in the interatomic structure of the nu- cleobases were found in their equilibrium configurations with respect to the corresponding gas-phase geometries, as it could be expected for a weakly interacting system such as the one studied here. A notable exception is the RC−O in guanine which shows a 10% contraction upon physisorption of the molecule on graphene. The stacking arrangement shown in Fig. 2 can be un- derstood from the tendency of the π–orbitals of the nucle- obases and graphene to minimize their overlap, in order to lower the repulsive interaction. The geometry deviates from the perfect AB base-stacking as, unlike graphene, the six- and five-membered rings of the bases possess a heterogeneous electronic structure due to the presence of both nitrogen and carbon in the ring systems. In ad- dition, there exist different side groups containing CH3, NH2, or O, all of which contribute to the deviation from the perfect AB base-stacking as well. Adenine, thymine and uracil display the least deviation from AB stacking (Fig. 2) out of the five nucleobases. For guanine and cy- tosine on the other hand, there is almost no resemblance to the AB stacking configuration recognizable (Fig. 2). We calculated the binding energy for all five nucle- obases. The binding energy of the system consisting of the nucleobase and the graphene sheet is taken as the energy of the equilibrium configuration with reference to the asymptotic limit obtained by varying the distance be- tween the base and the graphene sheet in the z-direction (Table I). Within LDA, we found adenine, cytosine and thymine to all possess nearly identical binding energies of about 0.49 eV, while guanine with 0.61 eV is bound more strongly, and uracil with 0.44 eV somewhat more weakly. It is somewhat surprising that guanine and adenine would possess such different physisorption energies, de- spite both containing a five- and a six-membered ring and featuring relatively similar molecular structures. A closer analysis of the various contributions to the total energy (Fig. 3) reveals that the Kohn-Sham kinetic en- ergy displays a slightly more pronounced minimum for guanine than for adenine, and that the position of that minimum is shifted by about 0.25 Å towards the graphene sheet. The exchange-correlation energy drops off some- what more rapidly in the case of adenine; however, the difference to the case for guanine is only very small. Table I also includes the polarizabilities of the nucle- obases calculated at the MP2 level of theory. The polar- izability of the nucleobase [22], which represents the de- formability of the electronic charge distribution, is known to arise from the regions associated with the aromatic rings, lone pairs of nitrogen and oxygen atoms. Accord- ingly, the purine base guanine appears to have the largest value, whereas the pyrimidine base uracil has the small- est value among the five nucleobases. Our calculations confirm this behavior. A remarkable correlation is found when the molecu- lar polarizabilities of the base molecules are compared with the binding energies, in particular when the latter are also determined at the MP2 level of theory (Table I). Clearly, the polarizability of a nucleobase is the key factor which governs the strength of interaction with the graphene sheet. This behavior is expected for a system that draws its stabilization from van der Waals (vdW) dispersion forces, since the vdW energy is proportional to the polarizabilities of the interacting entities. The observed correlation thus strongly suggests that vdW in- teraction is indeed the dominant source of attraction be- tween graphene and the nucleobases. The MP2 binding energies are systematically larger than those calculated within the LDA approximation (Table I). This is due to the well established fact that MP2 provides a more accurate treatment of the vdW interaction than LDA. We note that the adsystem con- sisting of the base and the sheet is not bound at the Hartree-Fock level of theory, which underscores the im- portance of electron correlation in describing the weak vdW interactions in this system. In the equilibrium configuration, a redistribution of the total charge density within a given base seems to appear. From an analysis of the Mulliken charges for the MP2 calculations, we also find a negligible charge transfer (< 0.02 e) between any of the five nucleobases and patch of nanographene in the equilibrium configuration. Elec- trostatic interactions in the adsystem are therefore very unlikely to contribute to the interaction energy. In summary, we investigated the physisorption of the five DNA/RNA nucleobases on a planar sheet of graphene. Our first-principles results clearly demon- strate that the nucleobases exhibit significantly differ- ent interaction strengths when physisorbed on graphene. This finding represents an important step towards a bet- ter understanding of experimentally observed sequence- dependent interaction of DNA with CNTs [3, 4]. The calculated trend in the binding energies strongly sug- gests that the polarizability of the base molecules de- termines the interaction strength of the nucleobases with graphene. As graphene can be regarded as a model sys- tem for CNTs with very small surface curvature, our con- clusions should therefore also hold for the physisorption of nucleobases on large-diameter CNTs. Further stud- ies involving the investigation of nucleobases interacting with small-diameter CNTs are currently underway. The authors acknowledge helpful discussions with Prof. Roberto Orlando of the University of Turin, Italy, and with Dr. Takeru Okada and Prof. Rikizo Hatakeyama of Tohoku University, Japan. S.G., R.H.S., and R.P. would like to thank DARPA for funding. R.H.S. and R.A. are grateful to the Swedish National Infrastructure (SNIC) for computing time. R.H.S. acknowledges support from EXC!TiNG (EU Research and Training Network) under contract HPRN-CT-2002-00317. The research reported in this document was performed in connection with con- tract DAAD17-03-C-0115 with the U.S. Army Research Laboratory. [1] N. Nakashima et al., Chem. Lett. 32, 456 (2003). [2] M. Zheng et al., Nature Mater. 2, 338 (2003). [3] M. Zheng et al., Science 302, 1545 (2003). [4] C. Staii et al., Nano Lett. 5, 1774 (2005). [5] D. A. Heller et al., Science 311, 508 (2006). [6] A. Star et al., Proc. Natl. Acad. Sci. U.S.A. 103, 921 (2006). [7] E. S. Jeng et al., Nano Lett. 6, 371 (2006). [8] T. Okada et al., Chem. Phys. Lett. 417, 288 (2006). [9] F. Ortmann, W. G. Schmidt, and F. Bechstedt, Phys. Rev. Lett. 95, 186101 (2005). [10] J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). [11] LDA appears to give a reliable description of disper- sive interactions, unlike the generalized gradient approx- imation (GGA) [12] for which binding is basically non- existent for van der Waals bound systems [13, 14]. The adsorption of adenine on graphite was recently investi- gated [9] using a modified version of the London dis- persion formula [15] in combination with GGA. The re- sults, however, clearly indicate that LDA, while under- binding the system, does in fact yield a potential energy surface for adenine on graphite which is almost indis- tinguishable in its structure from the one obtained via the GGA+vdW approach (cf. Figs. 1a and 1b of Ref. [9]). LDA yields almost the same equilibrium distance of adenine to graphene as GGA+vdW. The source of the attraction is identified as the exchange-correlation en- ergy, either calculated within LDA or calculated within GGA+vdW. [12] J. P. Perdew et al., Phys. Rev. B 46, 6671 (1992). [13] M. Simeoni et al., J. Chem. Phys. 122, 214710 (2005). [14] F. Tournus, S. Latil, M. I. Heggie, and J. C. Charlier, Phys. Rev. B. 72, 075431 (2005). [15] F. London, Z. Phys. 63, 245 (1930); Z. Phys. Chem., Abt. B 11, 222 (1930). [16] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964); W. Kohn and L. J. Sham, ibid. 140, A1133 (1965). [17] G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996); G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). [18] H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976). [19] Gaussian 03, Revision C.02, M. J. Frisch et al., Gaussian, Inc., Wallingford CT (2004). [20] The dangling bonds at the edge of the nanographene patch have been saturated by hydrogen atoms. For this and other types of nanographene, see Y. Shibayama, H. Sato, T. Enoki, and M. Endo, Phys. Rev. Lett. 84, 1744 (2000), and K. Harigaya and T. Enoki, Chem. Phys. Lett. 351, 128 (2002). [21] Klein, Cornelis and Cornelius S. Hurlbut, Jr., Manual of Mineralogy: after Dana, 20th ed. (1985); Watson JD, Baker TA, Bell SP, Gann A, Levine M, Losick R., Molec- ular Biology of the Gene. 5th ed. Pearson Benjamin Cum- mings: CSHL Press (2004). [22] Frank Seela, Anup M. Jawalekar, and Ingo Münster, Helevetica Chimica Acta 88, 751 (2005). base Eb(LDA) [eV] Eb(MP2) [eV] α [e G 0.61 1.07 131.2 A 0.49 0.94 123.7 T 0.49 0.83 111.4 C 0.49 0.80 108.5 U 0.44 0.74 97.6 TABLE I: Binding energy Eb of the DNA/RNA nucleobases with graphene as calculated within LDA are compared with binding energy and polarizability α from MP2 calculations. FIG. 1: Potential energy surface (PES) plot (in eV) for gua- nine on graphene. Qualitatively similar PES plots were ob- tained for the other four nucleobases. Approximately a 3 × 3 repetition of the unit cell is shown. The energy range between peak and valley is approximately 0.1 eV. Energy barriers of only about 0.04 eV separate adjacent global minima. FIG. 2: Equilibrium geometry of nucleobases on top of graphene: (a) guanine, (b) adenine, (c) thymine, (d) cyto- sine, and (e) uracil. -1.0 -0.5 0.0 0.5 1.0 displacement [A] -1.0 -0.5 0.0 0.5 1.0 displacement [A] FIG. 3: Plot of the relative total (black), exchange-correlation (blue) and kinetic energy (red) of guanine (top) and adenine (bottom) adsorbed on graphene calculated as a function of the displacement from their respective equilibrium position.

0704.1317

Low Density Lattice Codes

Low Density Lattice Codes Naftali Sommer, Senior Member, IEEE, Meir Feder, Fellow, IEEE, and Ofir Shalvi, Member, IEEE Abstract— Low density lattice codes (LDLC) are novel lattice codes that can be decoded efficiently and approach the capacity of the additive white Gaussian noise (AWGN) channel. In LDLC a codeword x is generated directly at the n-dimensional Euclidean space as a linear transformation of a corresponding integer message vector b, i.e., x = Gb, where H = G−1 is restricted to be sparse. The fact that H is sparse is utilized to develop a linear- time iterative decoding scheme which attains, as demonstrated by simulations, good error performance within ∼ 0.5dB from capacity at block length of n = 100, 000 symbols. The paper also discusses convergence results and implementation considerations. Index Terms— Lattices, lattice codes, iterative decoding, LDPC. I. INTRODUCTION If we take a look at the evolution of codes for binary or finite alphabet channels, it was first shown [1] that channel capacity can be achieved with long random codewords. Then, it was found out [2] that capacity can be achieved via a simpler structure of linear codes. Then, specific families of linear codes were found that are practical and have good minimum Hamming distance (e.g. convolutional codes, cyclic block codes, specific cyclic codes such as BCH and Reed-Solomon codes [4]). Later, capacity achieving schemes were found, which have special structures that allow efficient iterative decoding, such as low-density parity-check (LDPC) codes [5] or turbo codes [6]. If we now take a similar look at continuous alphabet codes for the additive white Gaussian noise (AWGN) channel, it was first shown [3] that codes with long random Gaussian code- words can achieve capacity. Later, it was shown that lattice codes can also achieve capacity ([7] – [12]). Lattice codes are clearly the Euclidean space analogue of linear codes. Similarly to binary codes, we could expect that specific practical lattice codes will then be developed. However, there was almost no further progress from that point. Specific lattice codes that were found were based on fixed dimensional classical lattices [19] or based on algebraic error correcting codes [13][14], but no significant effort was made in designing lattice codes directly in the Euclidean space or in finding specific capacity achieving lattice codes. Practical coding schemes for the AWGN channel were based on finite alphabet codes. In [15], “signal codes” were introduced. These are lattice codes, designed directly in the Euclidean space, where the information sequence of integers in, n = 1, 2, ... is encoded by convolving it with a fixed signal pattern gn, n = 1, 2, ...d. Signal codes are clearly analogous to convolutional codes, and The material in this paper was presented in part in the IEEE International Symposium on Information Theory, Seattle, July 2006, and in part in the Inauguration of the UCSD Information Theory and Applications Center, San Diego, Feb. 2006. in particular can work at the AWGN channel cutoff rate with simple sequential decoders. In [16] it is also demonstrated that signal codes can work near the AWGN channel capacity with more elaborated bi-directional decoders. Thus, signal codes provided the first step toward finding effective lattice codes with practical decoders. Inspired by LDPC codes and in the quest of finding practical capacity achieving lattice codes, we propose in this work “Low Density Lattice Codes” (LDLC). We show that these codes can approach the AWGN channel capacity with iterative decoders whose complexity is linear in block length. In recent years several schemes were proposed for using LDPC over continuous valued channels by either multilevel coding [18] or by non-binary alphabet (e.g. [17]). Unlike these LDPC based schemes, in LDLC both the encoder and the channel use the same real algebra which is natural for the continuous-valued AWGN channel. This feature also simplifies the convergence analysis of the iterative decoder. The outline of this paper is as follows. Low density lattice codes are first defined in Section II. The iterative decoder is then presented in Section III, followed by convergence analysis of the decoder in Section IV. Then, Section V describes how to choose the LDLC code parameters, and Section VI discusses implementation considerations. The computational complexity of the decoder is then discussed in Section VII, followed by a brief description of encoding and shaping in Section VIII. Simulation results are finally presented in Section IX. II. BASIC DEFINITIONS AND PROPERTIES A. Lattices and Lattice Codes An n dimensional lattice in Rm is defined as the set of all linear combinations of a given basis of n linearly independent vectors in Rm with integer coefficients. The matrix G, whose columns are the basis vectors, is called a generator matrix of the lattice. Every lattice point is therefore of the form x = Gb, where b is an n-dimensional vector of integers. The Voronoi cell of a lattice point is defined as the set of all points that are closer to this point than to any other lattice point. The Voronoi cells of all lattice points are congruent, and for square G the volume of the Voronoi cell is equal to det(G). In the sequel G will be used to denote both the lattice and its generator matrix. A lattice code of dimension n is defined by a (possibly shifted) lattice G in Rm and a shaping region B ⊂ Rm, where the codewords are all the lattice points that lie within the shaping region B. Denote the number of these codewords by N . The average transmitted power (per channel use, or per symbol) is the average energy of all codewords, divided by the codeword length m. The information rate (in bits/symbol) is log2(N)/m. When using a lattice code for the AWGN channel with power limit P and noise variance σ2, the maximal information rate is limited by the capacity 1 log2(1 + ). Poltyrev [20] considered the AWGN channel without restrictions. If there is no power restriction, code rate is a meaningless measure, since it can be increased without limit. Instead, it was suggested in [20] to use the measure of constellation density, leading to a generalized definition of the capacity as the maximal possible codeword density that can be recovered reliably. When applied to lattices, the generalized capacity implies that there exists a lattice G of high enough dimension n that enables transmis- sion with arbitrary small error probability, if and only if σ2 < |det(G)|2 . A lattice that achieves the generalized capacity of the AWGN channel without restrictions, also achieves the channel capacity of the power constrained AWGN channel, with a properly chosen spherical shaping region (see also [12]). In the rest of this work we shall concentrate on the lattice design and the lattice decoding algorithm, and not on the shaping region or shaping algorithms. We shall use lattices with det(G) = 1, where analysis and simulations will be carried for the AWGN channel without restrictions. A capacity achieving lattice will have small error probability for noise variance σ2 which is close to the theoretical limit 1 B. Syndrome and Parity Check Matrix for Lattice Codes A binary (n, k) error correcting code is defined by its n×k binary generator matrix G. A binary information vector b with dimension k is encoded by x = Gb, where calculations are performed in the finite field GF(2). The parity check matrix H is an (n−k)×n matrix such that x is a codeword if and only if Hx = 0. The input to the decoder is the noisy codeword y = x+ e, where e is the error sequence and addition is done in the finite field. The decoder typically starts by calculating the syndrome s = Hy = H(x+e) = He which depends only on the noise sequence and not on the transmitted codeword. We would now like to extend the definitions of the parity check matrix and the syndrome to lattice codes. An n- dimensional lattice code is defined by its n×n lattice generator matrix G (throughout this paper we assume that G is square, but the results are easily extended to the non-square case). Every codeword is of the form x = Gb, where b is a vector of integers. Therefore, G−1x is a vector of integers for every codeword x. We define the parity check matrix for the lattice code as H = G−1. Given a noisy codeword y = x+w (where w is the additive noise vector, e.g. AWGN, added by real arithmetic), we can then define the syndrome as = frac{Hy}, where frac{x} is the fractional part of x, defined as frac{x} = x−bxe, where bxe denotes the nearest integer to x. The syndrome s will be zero if and only if y is a lattice point, since Hy will then be a vector of integers with zero fractional part. For a noisy codeword, the syndrome will equal s = frac{Hy} = frac{H(x + w)} = frac{Hw} and therefore will depend only on the noise sequence and not on the transmitted codeword, as desired. Note that the above definitions of the syndrome and parity check matrix for lattice codes are consistent with the defini- tions of the dual lattice and the dual code[19]: the dual lattice of a lattice G is defined as the lattice with generator matrix H = G−1, where for binary codes, the dual code of G is defined as the code whose generator matrix is H , the parity check matrix of G. C. Low Density Lattice Codes We shall now turn to the definition of the codes proposed in this paper - low density lattice codes (LDLC). Definition 1 (LDLC): An n dimensional LDLC is an n- dimensional lattice code with a non-singular lattice generator matrix G satisfying |det(G)| = 1, for which the parity check matrix H = G−1 is sparse. The i’th row degree ri, i = 1, 2, ...n is defined as the number of nonzero elements in row i of H , and the i’th column degree ci, i = 1, 2, ...n is defined as the number of nonzero elements in column i of H . Note that in binary LDPC codes, the code is completely defined by the locations of the nonzero elements of H . In LDLC there is another degree of freedom since we also have to choose the values of the nonzero elements of H . Definition 2 (regular LDLC): An n dimensional LDLC is regular if all the row degrees and column degrees of the parity check matrix are equal to a common degree d. Definition 3 (magic square LDLC): An n dimensional reg- ular LDLC with degree d is called “magic square LDLC” if every row and column of the parity check matrix H has the same d nonzero values, except for a possible change of order and random signs. The sorted sequence of these d values h1 ≥ h2 ≥ ... ≥ hd > 0 will be referred to as the generating sequence of the magic square LDLC. For example, the matrix  0 −0.8 0 −0.5 1 0 0.8 0 0 1 0 −0.5 0 0.5 1 0 0.8 0 0 0 −0.5 −0.8 0 1 1 0 0 0 0.5 0.8 0.5 −1 −0.8 0 0 0  is a parity check matrix of a magic square LDLC with lattice dimension n = 6, degree d = 3 and generating sequence {1, 0.8, 0.5}. This H should be further normalized by the constant n |det(H)| in order to have |det(H)| = |det(G)| = 1, as required by Definition 1. The bipartite graph of an LDLC is defined similarly to LDPC codes: it is a graph with variable nodes at one side and check nodes at the other side. Each variable node corresponds to a single element of the codeword x = Gb. Each check node corresponds to a check equation (a row of H). A check equation is of the form k hkxik = integer, where ik de- notes the locations of the nonzero elements at the appropriate row of H , hk are the values of H at these locations and the integer at the right hand side is unknown. An edge connects check node i and variable node j if and only if Hi,j 6= 0. This edge is assigned the value Hi,j . Figure 1 illustrates the bi-partite graph of a magic square LDLC with degree 3. In the figure, every variable node xk is also associated with its noisy channel observation yk. Finally, a k-loop is defined as a loop in the bipartite graph that consists of k edges. A bipartite graph, in general, can only Fig. 1. The bi-partite graph of an LDLC contain loops with even length. Also, a 2-loop, which consists of two parallel edges that originate from the same variable node to the same check node, is not possible by the definition of the graph. However, longer loops are certainly possible. For example, a 4-loop exists when two variable nodes are both connected to the same pair of check nodes. III. ITERATIVE DECODING FOR THE AWGN CHANNEL Assume that the codeword x = Gb was transmitted, where b is a vector of integers. We observe the noisy codeword y = x + w, where w is a vector of i.i.d Gaussian noise samples with common variance σ2, and we need to estimate the integer valued vector b. The maximum likelihood (ML) estimator is then b̂ = arg min ||y −Gb||2. Our decoder will not estimate directly the integer vector b. Instead, it will estimate the probability density function (PDF) of the codeword vector x. Furthermore, instead of calculating the n-dimensional PDF of the whole vector x, we shall calculate the n one-dimensional PDF’s for each of the components xk of this vector (conditioned on the whole observation vector y). In appendix I it is shown that fxk|y(xk|y) is a weighted sum of Dirac delta functions: fxk|y(xk|y) = C · l∈G∩B δ(xk − lk) · e−d 2(l,y)/2σ2 (1) where l is a lattice point (vector), lk is its k-th component, C is a constant independent of xk and d(l, y) is the Euclidean distance between l and y. Direct evaluation of (1) is not X2X3X4X5X6X7 Tier1 (6 nodes) Tier 2 (24 nodes) X11 X10 X9 X8 Fig. 2. Tier diagram practical, so our decoder will try to estimate fxk|y(xk|y) (or at least approximate it) in an iterative manner. Our decoder will decode to the infinite lattice, thus ignoring the shaping region boundaries. This approximate decoding method is no longer exact maximum likelihood decoding, and is usually denoted “lattice decoding” [12]. The calculation of fxk|y(xk|y) is involved since the com- ponents xk are not independent random variables (RV’s), because x is restricted to be a lattice point. Following [5] we use a “trick” - we assume that the xk’s are independent, but add a condition that assures that x is a lattice point. Specifically, define s = H · x. Restricting x to be a lattice point is equivalent to restricting s ∈ Zn. Therefore, instead of calculating fxk|y(xk|y) under the assumption that x is a lattice point, we can calculate fxk|y(xk|y, s ∈ Z n) and assume that the xk are independent and identically distributed (i.i.d) with a continuous PDF (that does not include Dirac delta functions). It still remains to set fxk(xk), the PDF of xk. Under the i.i.d assumption, the PDF of the codeword x is fx(x) = k=1 fxk(xk). As shown in Appendix II, the value of fx(x) is not important at values of x which are not lattice points, but at a lattice point it should be proportional to the probability of using this lattice point. Since we assume that all lattice points are used equally likely, fx(x) must have the same value at all lattice points. A reasonable choice for fxk(xk) is then to use a uniform distribution such that x will be uniformly distributed in an n-dimensional cube. For an exact ML decoder (that takes into account the boundaries of the shaping region), it is enough to choose the range of fxk(xk) such that this cube will contain the shaping region. For our decoder, that performs lattice decoding, we should set the range of fxk(xk) large enough such that the resulting cube will include all the lattice points which are likely to be decoded. The derivation of the iterative decoder shows that this range can be set as large as needed without affecting the complexity of the decoder. The derivation in [5] further imposed the tree assumption. In order to understand the tree assumption, it is useful to define the tier diagram, which is shown in Figure 2 for a regular LDLC with degree 3. Each vertical line corresponds to a check equation. The tier 1 nodes of x1 are all the elements xk that take place in a check equation with x1. The tier 2 nodes of x1 are all the elements that take place in check equations with the tier 1 elements of x1, and so on. The tree assumption assumes that all the tree elements are distinct (i.e. no element appears in different tiers or twice in the same tier). This assumption simplifies the derivation, but in general, does not hold in practice, so our iterative algorithm is not guaranteed to converge to the exact solution (1) (see Section IV). The detailed derivation of the iterative decoder (using the above “trick” and the tree assumption) is given in Appendix III. In Section III-A below we present the final resulting algorithm. This iterative algorithm can also be explained by intuitive arguments, described after the algorithm specification. A. The Iterative Decoding Algorithm The iterative algorithm is most conveniently represented by using a message passing scheme over the bipartite graph of the code, similarly to LDPC codes. The basic difference is that in LDPC codes the messages are scalar values (e.g. the log likelihood ratio of a bit), where for LDLC the messages are real functions over the interval (−∞,∞). As in LDPC, in each iteration the check nodes send messages to the variable nodes along the edges of the bipartite graph and vice versa. The messages sent by the check nodes are periodic extensions of PDF’s. The messages sent by the variable nodes are PDF’s. LDLC iterative decoding algorithm: Denote the variable nodes by x1, x2, ..., xn and the check nodes by c1, c2, ...cn. • Initialization: each variable node xk sends to all its check nodes the message f (0)k (x) = − (yk−x) 2σ2 . • Basic iteration - check node message: Each check node sends a (different) message to each of the variable nodes that are connected to it. For a specific check node denote (without loss of generality) the appropriate check equa- tion by l=1 hlxml = integer, where xml , l = 1, 2...r are the variable nodes that are connected to this check node (and r is the appropriate row degree of H). Denote by fl(x), l = 1, 2...r, the message that was sent to this check node by variable node xml in the previous half- iteration. The message that the check node transmits back to variable node xmj is calculated in three basic steps. 1) The convolution step - all messages, except fj(x), are convolved (after expanding each fl(x) by hl): p̃j(x) = f1 ~ · · · fj−1 ~fj+1 ~ · · · · · ·~ fr 2) The stretching step - The result is stretched by (−hj) to pj(x) = p̃j(−hjx) 3) The periodic extension step - The result is extended to a periodic function with period 1/|hj |: Qj(x) = The function Qj(x) is the message that is finally sent to variable node xmj . • Basic iteration - variable node message: Each variable node sends a (different) message to each of the check nodes that are connected to it. For a specific variable node xk, assume that it is connected to check nodes channel PDF check node message #1 check node message #2 check node message #3 check node message #4 Final variable node message Fig. 3. Signals at variable node cm1 , cm2 , ...cme , where e is the appropriate column de- gree of H . Denote by Ql(x), l = 1, 2, ...e, the message that was sent from check node cml to this variable node in the previous half-iteration. The message that is sent back to check node cmj is calculated in two basic steps: 1) The product step: f̃j(x) = e − (yk−x) l 6=j Ql(x) 2) The normalization step: fj(x) = f̃j(x)R∞ −∞ f̃j(x)dx This basic iteration is repeated for the desired number of iterations. • Final decision: After finishing the iterations, we want to estimate the integer information vector b. First, we esti- mate the final PDF’s of the codeword elements xk, k = 1, 2, ...n, by calculating the variable node messages at the last iteration without omitting any check node message in the product step: f̃ (k)final(x) = e − (yk−x) l=1Ql(x). Then, we estimate each xk by finding the peak of its PDF: x̂k = argmaxx f̃ final(x). Finally, we estimate b as b̂ = bHx̂e. The operation of the iterative algorithm can be intuitively explained as follows. The check node operation is equivalent to calculating the PDF of xmj from the PDF’s of xmi , i = 1, 2, ..., j − 1, j + 1, ...r, given that l=1 hlxml = integer, and assuming that xmi are independent. Extracting xmj from the check equation, we get xmj = (integer− l 6=j hlxml). Since the PDF of a sum of independent random variables is the convolution of the corresponding PDF’s, equation (2) and the stretching step that follows it simply calculate the PDF of xmj , assuming that the integer at the right hand side of the check equation is zero. The result is then periodically extended such that a properly shifted copy exists for every possible value of this (unknown) integer. The variable node gets such a message from all the check equations that involve the corresponding variable. The check node messages and the channel PDF are treated as independent sources of information on the variable, so they are multiplied all together. Note that the periodic extension step at the check nodes is equivalent to a convolution with an infinite impulse train. With this observation, the operation of the variable nodes is completely analogous to that of the check nodes: the variable nodes multiply the incoming messages by the channel PDF, where the check nodes convolve the incoming messages with an impulse train, which can be regarded as a generalized “integer PDF”. In the above formulation, the integer information vector b is recovered from the PDF’s of the codeword elements xk. An alternative approach is to calculate the PDF of each integer element bm directly as the PDF of the left hand side of the appropriate check equation. Using the tree assumption, this can be done by simply calculating the convolution p̃(x) as in (2), but this time without omitting any PDF, i.e. all the received variable node messages are convolved. Then, the integer bm is determined by b̂m = argmaxj∈Z p̃(j). Figure 3 shows an example for a regular LDLC with degree d = 5. The figure shows all the signals that are involved in generating a variable node message for a certain variable node. The top signal is the channel Gaussian, centered around the noisy observation of the variable. The next 4 signals are the periodically extended PDF’s that arrived from the check nodes, and the bottom signal is the product of all the 5 signals. It can be seen that each periodic signal has a different period, according to the relevant coefficient of H . Also, the signals with larger period have larger variance. This diversity resolves all the ambiguities such that the multiplication result (bottom plot) remains with a single peak. We expect the iterative algorithm to converge to a solution where a single peak will remain at each PDF, located at the desired value and narrow enough to estimate the information. IV. CONVERGENCE A. The Gaussian Mixture Model Interestingly, for LDLC we can come up with a convergence analysis that in many respects is more specific than the similar analysis for LDPC. We start by introducing basic claims about Gaussian PDF’s. Denote Gm,V (x) = 1√ (x−m)2 Claim 1 (convolution of Gaussians): The convolution of n Gaussians with mean values m1,m2, ...,mn and variances V1, V2, ..., Vn, respectively, is a Gaussian with mean m1 + m2 + ...+mn and variance V1 + V2 + ...+ Vn. Proof: See [21]. Claim 2 (product of n Gaussians): Let Gm1,V1(x), Gm2,V2(x),...,Gmn,Vn(x) be n Gaussians with mean values m1,m2, ...,mn and variances V1, V2, ..., Vn respectively. Then, the product of these Gaussians is a scaled Gaussian: i=1Gmi,Vi(x) =  · Gm̂,V̂ (x), where 1 , m̂ = i=1miV i=1 V , and  = 1√ (2π)n−1V̂ −1 k=1 Vk j=i+1 (mi−mj) Vi·Vj . Proof: By straightforward mathematical manipulations. The reason that we are interested in the properties of Gaussian PDF’s lies in the following lemma. Lemma 1: Each message that is exchanged between the check nodes and variable nodes in the LDLC decoding al- gorithm (i.e. Qj(x) and fj(x)), at every iteration, can be expressed as a Gaussian mixture of the form M(x) =∑∞ j=1AjGmj ,Vj (x). Proof: By induction: The initial messages are Gaussians, and the basic operations of the iterative decoder preserve the Gaussian mixture nature of Gaussian mixture inputs (convolu- tion and multiplication preserve the Gaussian nature according to claims 1 and 2, stretching, expanding and shifting preserve it by the definition of a Gaussian, and periodic extension transforms a single Gaussian to a mixture and a mixture to a mixture). Convergence analysis should therefore analyze the conver- gence of the variances, mean values and amplitudes of the Gaussians in each mixture. B. Convergence of the Variances We shall now analyze the behavior of the variances, and start with the following lemma. Lemma 2: For both variable node messages and check node messages, all the Gaussians that take place in the same mixture have the same variance. Proof: By induction. The initial variable node messages are single element mixtures so the claim obviously holds. As- sume now that all the variable node messages at iteration t are mixtures where all the Gaussians that take place in the same mixture have the same variance. In the convolution step (2), each variable node message is first expanded. All Gaussians in the expanded mixture will still have the same variance, since the whole mixture is expanded together. Then, d − 1 expanded Gaussian mixtures are convolved. In the resulting mixture, each Gaussian will be the result of convolving d− 1 single Gaussians, one from each mixture. According to claim 1, all the Gaussians in the convolution result will have the same variance, which will equal the sum of the d−1 variances of the expanded messages. The stretching and periodic extension (3) do not change the equal variance property, so it holds for the final check node messages. The variable nodes multiply d− 1 check node messages. Each Gaussian in the resulting mixture is a product of d−1 single Gaussians, one from each mixture, and the channel noise Gaussian. According to claim 2, they will all have the same variance. The final normalization step does not change the variances so the equal variance property is kept for the final variable node messages at iteration t+ 1. Until this point we did not impose any restrictions on the LDLC. From now on, we shall restrict ourselves to magic square regular LDLC (see Definition 3). The basic iterative equations that relate the variances at iteration t + 1 to the variances at iteration t are summarized in the following two lemmas. Lemma 3: For magic square LDLC, variable node mes- sages that are sent at the same iteration along edges with the same absolute value have the same variance. Proof: See Appendix IV. Lemma 4: For magic square LDLC with degree d, denote the variance of the messages that are sent at iteration t along edges with weight ±hl by V l . The variance values 1 , V 2 , ..., V d obey the following recursion: (t+1) m 6=i h2m∑d j 6=m for i = 1, 2, ...d, with initial conditions V (0)1 = V 2 = ... = d = σ Proof: See Appendix IV. For illustration, the recursion for the case d = 3 is: (t+1) 1 + h 1 + h (t+1) 2 + h 1 + h (t+1) 2 + h 1 + h The lemmas above are used to prove the following theorem regarding the convergence of the variances. Theorem 1: For a magic square LDLC with degree d and generating sequence h1 ≥ h2 ≥ ... ≥ hd > 0, define α i=2 h . Assume that α < 1. Then: 1) The first variance approaches a constant value of σ2(1− α), where σ2 is the channel noise variance: = lim 1 = σ 2(1− α). 2) The other variances approach zero: = lim i = 0 for i = 2, 3..d. 3) The asymptotic convergence rate of all variances is exponential: 0 < lim ∣∣∣∣∣V i − V ∣∣∣∣∣ <∞ for i = 1, 2..d. 4) The zero approaching variances are upper bounded by the decaying exponential σ2αt: i ≤ σ for i = 2, 3..d and t ≥ 0. Proof: See Appendix IV. If α ≥ 1, the variances may still converge, but convergence rate may be as slow as o(1/t), as illustrated in Appendix IV. Convergence of the variances to zero implies that the Gaussians approach impulses. This is a desired property of the decoder, since the exact PDF that we want to calculate is indeed a weighted sum of impulses (see (1)). It can be seen that by designing a code with α < 1, i.e. h21 > i=2 h i , one variance approaches a constant (and not zero). However, all the other variances approach zero, where all variances converge in an exponential rate. This will be the preferred mode because the information can be recovered even if a single variance does not decay to zero, where exponential convergence is certainly preferred over slow 1/t convergence. Therefore, from now on we shall restrict our analysis to magic square LDLC with α < 1. Theorem 1 shows that every iteration, each variable node will generate d−1 messages with variances that approach zero, and a single message with variance that approaches a constant. The message with nonzero variance will be transmitted along the edge with largest weight (i.e. h1). However, from the derivation of Appendix IV it can be seen that the opposite happens for the check nodes: each check node will generate d − 1 messages with variances that approach a constant, and a single message with variance that approaches zero. The check node message with zero approaching variance will be transmitted along the edge with largest weight. C. Convergence of the Mean Values The reason that the messages are mixtures and not single Gaussians lies in the periodic extension step (3) at the check nodes, and every Gaussian at the output of this step can be related to a single index of the infinite sum. Therefore, we can label each Gaussian at iteration t with a list of all the indices that were used in (3) during its creation process in iterations 1, 2, ...t. Definition 4 (label of a Gaussian): The label of a Gaussian consists of a sequence of triplets of the form {t, c, i}, where t is an iteration index, c is a check node index and i is an integer. The labels are initialized to the empty sequence. Then, the labels are updated along each iteration according to the following update rules: 1) In the periodic extension step (3), each Gaussian in the output periodic mixture is assigned the label of the specific Gaussian of pj(x) that generated it, concate- nated with a single triplet {t, c, i}, where t is the current iteration index, c is the check node index and i is the index in the infinite sum of (3) that corresponds to this Gaussian. 2) In the convolution step and the product step, each Gaussian in the output mixture is assigned a label that equals the concatenation of all the labels of the specific Gaussians in the input messages that formed this Gaussian. 3) The stretching and normalization steps do not alter the label of each Gaussian: Each Gaussian in the stretched/normalized mixture inherits the label of the appropriate Gaussian in the original mixture. Definition 5 (a consistent Gaussian): A Gaussian in a mix- ture is called “[ta, tb] consistent” if its label contains no contradictions for iterations ta to tb, i.e. for every pair of triplets {t1, c1, i1}, {t2, c2, i2} such that ta ≤ t1, t2 ≤ tb, if c1 = c2 then i1 = i2. A [0, ∞] consistent Gaussian will be simply called a consistent Gaussian. We can relate every consistent Gaussian to a unique integer vector b ∈ Zn, which holds the n integers used in the n check nodes. Since in the periodic extension step (3) the sum is taken over all integers, a consistent Gaussian exists in each variable node message for every possible integer valued vector b ∈ Zn. We shall see later that this consistent Gaussian corresponds to the lattice point Gb. According to Theorem 1, if we choose the nonzero values of H such that α < 1, every variable node generates d− 1 messages with variances approaching zero and a single message with variance that approaches a constant. We shall refer to these messages as “narrow” messages and “wide” messages, respectively. For a given integer valued vector b, we shall concentrate on the consistent Gaussians that relate to b in all the nd variable node messages that are generated in each iteration (a single Gaussian in each message). The following lemmas summarize the asymptotic behavior of the mean values of these consistent Gaussians for the narrow messages. Lemma 5: For a magic square LDLC with degree d and α < 1, consider the d− 1 narrow messages that are sent from a specific variable node. Consider further a single Gaussian in each message, which is the consistent Gaussian that relates to a given integer vector b. Asymptotically, the mean values of these d− 1 Gaussians become equal. Proof: See Appendix V. Lemma 6: For a magic square LDLC with dimension n, degree d and α < 1, consider only consistent Gaussians that relate to a given integer vector b and belong to narrow messages. Denote the common mean value of the d− 1 such Gaussians that are sent from variable node i at iteration t by i , and arrange all these mean values in a column vector m(t) of dimension n. Define the error vector e(t) = m(t)− x, where x = Gb is the lattice point that corresponds to b. Then, for large t, e(t) satisfies: e(t+1) ≈ −H̃ · e(t) (6) where H̃ is derived from H by permuting the rows such that the ±h1 elements will be placed on the diagonal, dividing each row by the appropriate diagonal element (h1 or −h1), and then nullifying the diagonal. Proof: See Appendix V. We can now state the following theorem, which describes the conditions for convergence and the steady state value of the mean values of the consistent Gaussians of the narrow variable node messages. Theorem 2: For a magic square LDLC with α < 1, the mean values of the consistent Gaussians of the narrow variable node messages that relate to a given integer vector b are assured to converge if and only if all the eigenvalues of H̃ have magnitude less than 1, where H̃ is defined in Lemma 6. When this condition is fulfilled, the mean values converge to the coordinates of the appropriate lattice point: m(∞) = G · b. Proof: Immediate from Lemma 6. Note that without adding random signs to the LDLC nonzero values, the all-ones vector will be an eigenvector of H̃ with eigenvalue i=2 hi , which may exceed 1. Interestingly, recursion (6) is also obeyed by the error of the Jacobi method for solving systems of sparse linear equations [22] (see also Section VIII-A), when it is used to solve Hm = b (with solution m = Gb). Therefore, the LDLC decoder can be viewed as a superposition of Jacobi solvers, one for each possible value of the integer valued vector b. We shall now turn to the convergence of the mean values of the wide messages. The asymptotic behavior is summarized in the following lemma. Lemma 7: For a magic square LDLC with dimension n and α < 1, consider only consistent Gaussians that relate to a given integer vector b and belong to wide messages. Denote the mean value of such a Gaussian that is sent from variable node i at iteration t by m(t)i , and arrange all these mean values in a column vector m(t) of dimension n. Define the error vector = m(t) −Gb. Then, for large t, e(t) satisfies: e(t+1) ≈ −F · e(t) + (1− α)(y −Gb) (7) where y is the noisy codeword and F is an n × n matrix defined by: Fk,l = if k 6= l and there exist a row r of H for which |Hr,l| = h1 and Hr,k 6= 0 0 otherwise Proof: See Appendix V, where an alternative way to construct F from H is also presented. The conditions for convergence and steady state solution for the wide messages are described in the following theorem. Theorem 3: For a magic square LDLC with α < 1, the mean values of the consistent Gaussians of the wide variable node messages that relate to a given integer vector b are assured to converge if and only if all the eigenvalues of F have magnitude less than 1, where F is defined in Lemma 7. When this condition is fulfilled, the steady state solution is m(∞) = G · b+ (1− α)(I + F )−1(y −G · b). Proof: Immediate from Lemma 7. Unlike the narrow messages, the mean values of the wide messages do not converge to the appropriate lattice point coordinates. The steady state error depends on the difference between the noisy observation and the lattice point, as well as on α, and it decreases to zero as α → 1. Note that the final PDF of a variable is generated by multiplying all the d check node messages that arrive to the appropriate variable node. d−1 of these messages are wide, and therefore their mean values have a steady state error. One message is narrow, so it converges to an impulse at the lattice point coordinate. Therefore, the final product will be an impulse at the correct location, where the wide messages will only affect the magnitude of this impulse. As long as the mean values errors are not too large (relative to the width of the wide messages), this should not cause an impulse that corresponds to a wrong lattice point to have larger amplitude than the correct one. However, for large noise, these steady state errors may cause the decoder to deviate from the ML solution (As explained in Section IV-D). To summarize the results for the mean values, we considered the mean values of all the consistent Gaussians that correspond to a given integer vector b. A single Gaussian of this form exists in each of the nd variable node messages that are generated in each iteration. For each variable node, d − 1 messages are narrow (have variance that approaches zero) and a single message is wide (variance approaches a constant). Under certain conditions on H , the mean values of all the narrow messages converge to the appropriate coordinate of the lattice point Gb. Under additional conditions on H , the mean values of the wide messages converge, but the steady state values contain an error term. We analyzed the behavior of consistent Gaussian. It should be noted that there are many more non-consistent Gaussians. Furthermore non-consistent Gaussians are generated in each iteration for any existing consistent Gaussian. We conjecture that unless a Gaussian is consistent, or becomes consistent along the iterations, it fades out, at least at noise condi- tions where the algorithm converges. The reason is that non- consistency in the integer values leads to mismatch in the corresponding PDF’s, and so the amplitude of that Gaussian is attenuated. We considered consistent Gaussians which correspond to a specific integer vector b, but such a set of Gaussians exists for every possible choice of b, i.e. for every lattice point. Therefore, the narrow messages will converge to a solution that has an impulse at the appropriate coordinate of every lattice point. This resembles the exact solution (1), so the key for proper convergence lies in the amplitudes: we would like the consistent Gaussians of the ML lattice point to have the largest amplitude for each message. D. Convergence of the Amplitudes We shall now analyze the behavior of the amplitudes of consistent Gaussians (as discussed later, this is not enough for complete convergence analysis, but it certainly gives insight to the nature of the convergence process and its properties). The behavior of the amplitudes of consistent Gaussians is summarized in the following lemma. Lemma 8: For a magic square LDLC with dimension n, degree d and α < 1, consider the nd consistent Gaussians that relate to a given integer vector b in the variable node messages that are sent at iteration t (one consistent Gaussian per message). Denote the amplitudes of these Gaussians by i , i = 1, 2, ...nd, and define the log-amplitude as l log p(t)i . Arrange these nd log-amplitudes in a column vector l(t), such that element (k−1)d+i corresponds to the message that is sent from variable node k along an edge with weight ±hi. Assume further that the bipartite graph of the LDLC contains no 4-loops. Then, the log-amplitudes satisfy the following recursion: l(t+1) = A · l(t) − a(t) − c(t) (9) with initialization l(0) = 0. A is an nd× nd matrix which is all zeros except exactly (d − 1)2 ’1’s in each row and each column. The element of the excitation vector a(t) at location (k − 1)d+ i (where k = 1, 2, ...n and i = 1, 2, ...d) equals: (k−1)d+i = (10)  d∑ l 6=i j=l+1 j 6=i k,l − m̃ k,l · Ṽ l 6=i k,l − yk σ2 · Ṽ (t)k,l where m̃(t)k,l and Ṽ k,l denote the mean value and variance of the consistent Gaussian that relates to the integer vector b in the check node message that arrives to variable node k at iteration t along an edge with weight ±hl. yk is the noisy channel observation of variable node k, and V̂ (t)k,i l 6=i . Finally, c(t) is a constant excitation term that is independent of the integer vector b (i.e. is the same for all consistent Gaussians). Note that an iteration is defined as sending variable node messages, followed by sending check node messages. The first iteration (where the variable nodes send the initialization PDF) is regarded as iteration 0. Proof: At the check node, the amplitude of a Gaussian at the convolution output is the product of the amplitudes of the corresponding Gaussians in the appropriate variable node messages. At the variable node, the amplitude of a Gaussian at the product output is the product of the amplitudes of the corresponding Gaussians in the appropriate check node messages, multiplied by the Gaussian scaling term, according to claim 2. Since we assume that the bipartite graph of the LDLC contains no 4-loops, an amplitude of a variable node message at iteration t will therefore equal the product of (d − 1)2 amplitudes of Gaussians of variable node messages from iteration t− 1, multiplied by the Gaussian scaling term. This proves (9) and shows that A has (d−1)2 ’1’s in every row. However, since each variable node message affects exactly (d− 1)2 variable node messages of the next iteration, A must also have (d− 1)2 ’1’s in every column. The total excitation term −a(t)−c(t) corresponds to the logarithm of the Gaussian scaling term. Each element of this scaling term results from the product of d − 1 check node Gaussians and the channel Gaussian, according to claim 2. This scaling term sums over all the pairs of Gaussians, and in (10) the sum is separated to pairs that include the channel Gaussian and pairs that do not. The total excitation is divided between (10), which depends on the choice of the integer vector b, and c(t), which includes all the constant terms that are independent on b (including the normalization operation which is performed at the variable node). Since there are exactly (d − 1)2 ’1’s in each column of the matrix A, it is easy to see that the all-ones vector is an eigenvector of A, with eigenvalue (d − 1)2. If d > 2, this eigenvalue is larger than 1, meaning that the recursion (9) is non-stable. It can be seen that the excitation term a(t) has two compo- nents. The first term sums the squared differences between the mean values of all the possible pairs of received check node messages (weighted by the inverse product of the appropriate variances). It therefore measures the mismatch between the incoming messages. This mismatch will be small if the mean values of the consistent Gaussians converge to the coordinates of a lattice point (any lattice point). The second term sums the squared differences between the mean values of the incoming messages and the noisy channel output yk. This term measures the mismatch between the incoming messages and the channel measurement. It will be smallest if the mean values of the consistent Gaussians converge to the coordinates of the ML lattice point. The following lemma summarizes some properties of the excitation term a(t). Lemma 9: For a magic square LDLC with dimension n, degree d, α < 1 and no 4-loops, consider the consistent Gaussians that correspond to a given integer vector b. Accord- ing to Lemma 8, their amplitudes satisfy recursion (9). The excitation term a(t) of (9), which is defined by (10), satisfies the following properties: 1) a(t)i , the i’th element of a (t), is non-negative, finite and bounded for every i and every t. Moreover, a(t)i converges to a finite non-negative steady state value as 2) limt→∞ i=1 a (Gb− y)TW (Gb− y), where y is the noisy received codeword and W is a positive definite matrix defined by: = (d+ 1− α)I − 2(1− α)(I + F )−1+ (11) +(1− α)(I + F )−1 (d− 1)2I − F TF (I + F )−1 where F is defined in Lemma 7. 3) For an LDLC with degree d > 2, the weighted infinite i=1 a (d−1)2j+2 converges to a finite value. Proof: See Appendix VI. The following theorem addresses the question of which con- sistent Gaussian will have the maximal asymptotic amplitude. We shall first consider the case of an LDLC with degree d > 2, and then consider the special case of d = 2 in a separate theorem. Theorem 4: For a magic square LDLC with dimension n, degree d > 2, α < 1 and no 4-loops, consider the nd consistent Gaussians that relate to a given integer vector b in the variable node messages that are sent at iteration t (one consistent Gaussian per message). Denote the amplitudes of these Gaussians by p(t)i , i = 1, 2, ...nd, and define the product-of-amplitudes as P (t) i=1 p i . Define further i=1 a (d−1)2j+2 , where a i is defined by (10) (S is well defined according to Lemma 9). Then: 1) The integer vector b for which the consistent Gaussians will have the largest asymptotic product-of-amplitudes limt→∞ P (t) is the one for which S is minimized. 2) The product-of-amplitudes for the consistent Gaussians that correspond to all other integer vectors will decay to zero in a super-exponential rate. Proof: As in Lemma 8, define the log-amplitudes l(t)i log p(t)i . Define further s (t) ∆= i=1 l i . Taking the element- wise sum of (9), we get: s(t+1) = (d− 1)2s(t) − i (12) with initialization s(0) = 0. Note that we ignored the term∑nd i=1 c i . As shown below, we are looking for the vector b that maximizes s(t). Since (12) is a linear difference equation, and the term i=1 c i is independent of b, its effect on s is common to all b and is therefore not interesting. Define now s̃(t) (d−1)2t . Substituting in (12), we get: s̃(t+1) = s̃(t) − (d− 1)2t+2 i (13) with initialization s̃(0) = 0, which can be solved to get: s̃(t) = − i=1 a (d− 1)2j+2 We would now like to compare the amplitudes of consistent Gaussians with various values of the corresponding integer vector b in order to find the lattice point whose consistent Gaussians will have largest product-of-amplitudes. From the definitions of s(t) and s̃(t) we then have: P (t) = es = e(d−1) 2t·s̃(t) (15) Consider two integer vectors b that relate to two lattice points. Denote the corresponding product-of-amplitudes by P (t)0 and 1 , respectively, and assume that for these two vectors S converges to the values S0 and S1, respectively. Then, taking into account that limt→∞ s̃(t) = −S, the asymptotic ratio of the product-of-amplitudes for these lattice points will be: e−(d−1) 2t·S1 e−(d−1) 2t·S0 = e(d−1) 2t·(S0−S1) (16) It can be seen that if S0 < S1, the ratio decreases to zero in a super exponential rate. This shows that the lattice point for which S is minimized will have the largest product-of- amplitudes, where for all other lattice points, the product- of-amplitudes will decay to zero in a super-exponential rate (recall that the normalization operation at the variable node keeps the sum of all amplitudes in a message to be 1). This completes the proof of the theorem. We now have to find which integer valued vector b mini- mizes S. The analysis is difficult because the weighting factor inside the sum of (14) performs exponential weighting of the excitation terms, where the dominant terms are those of the first iterations. Therefore, we can not use the asymptotic results of Lemma 9, but have to analyze the transient behavior. However, the analysis is simpler for the case of an LDLC with row and column degree of d = 2, so we shall first turn to this simple case (note that for this case, both the convolution in the check nodes and the product at the variable nodes involve only a single message). Theorem 5: For a magic square LDLC with dimension n, degree d = 2, α < 1 and no 4-loops, consider the 2n consistent Gaussians that relate to a given integer vector b in the variable node messages that are sent at iteration t (one consistent Gaussian per message). Denote the amplitudes of these Gaussians by p(t)i , i = 1, 2, ...2n, and define the product- of-amplitudes as P (t) i=1 p i . Then: 1) The integer vector b for which the consistent Gaussians will have the largest asymptotic product-of-amplitudes limt→∞ P (t) is the one for which (Gb−y)TW (Gb−y) is minimized, where W is defined by (11) and y is the noisy received codeword. 2) The product-of-amplitudes for the consistent Gaussians that correspond to all other integer vectors will decay to zero in an exponential rate. Proof: For d = 2 (12) becomes: s(t+1) = s(t) − i (17) With solution: s(t) = − i (18) Denote Sa = limj→∞ i=1 a i . Sa is well defined according to Lemma 9. For large t, we then have s(t) ≈ −t · Sa. There- fore, for two lattice points with excitation sum terms which approach Sa0, Sa1, respectively, the ratio of the corresponding product-of-amplitudes will approach e−Sa1·t e−Sa0·t = e(Sa0−Sa1)·t (19) If Sa0 < Sa1, the ratio decreases to zero exponentially (unlike the case of d > 2 where the rate was super-exponential, as in (16)). This shows that the lattice point for which Sa is minimized will have the largest product-of-amplitudes, where for all other lattice points, the product-of-amplitudes will decay to zero in an exponential rate (recall that the normalization operation at the variable node keeps the sum of all amplitudes in a message to be 1). This completes the proof of the second part of the theorem. We still have to find the vector b that minimizes Sa. The basic difference between the case of d = 2 and the case of d > 2 is that for d > 2 we need to analyze the transient behavior of the excitation terms, where for d = 2 we only need to analyze the asymptotic behavior, which is much easier to handle. According to Lemma 9, we have: = lim (Gb− y)TW (Gb− y) (20) where W is defined by (11) and y is the noisy received codeword. Therefore, for d = 2, the lattice points whose consistent Gaussians will have largest product-of-amplitudes is the point for which (Gb − y)TW (Gb − y) is minimized. This completes the proof of the theorem. For d = 2 we could find an explicit expression for the “winning” lattice point. As discussed above, we could not find an explicit expression for d > 2, since the result depends on the transient behavior of the excitation sum term, and not only on the steady state value. However, a reasonable conjecture is to assume that b that maximizes the steady state excitation will also maximize the term that depends on the transient behavior. This means that a reasonable conjecture is to assume that the “winning” lattice point for d > 2 will also minimize an expression of the form (20). Note that for d > 2 we can still show that for “weak” noise, the ML point will have the minimal S. To see that, it comes out from (10) that for zero noise, the ML lattice point will have a(t)i = 0 for every t and i, where all other lattice points will have a(t)i > 0 for at least some i and t. Therefore, the ML point will have a minimal excitation term along the transient behavior so it will surely have the minimal S and the best product-of-amplitudes. As the noise increases, it is difficult to analyze the transient behavior of a(t)i , as discussed above. Note that the ML solution minimizes (Gb − y)T (Gb − y), where the above analysis yields minimization of (Gb − y)TW (Gb − y). Obviously, for zero noise (i.e. y = G · b) both minimizations will give the correct solution with zero score. As the noise increases, the solutions may deviate from one another. Therefore, both minimizations will give the same solution for “weak” noise but may give different solutions for “strong” noise. An example for another decoder that performs this form of minimization is the linear detector, which calculates b̂ =⌊ H · y (where bxe denotes the nearest integer to x). This is equivalent to minimizing (Gb − y)TW (Gb − y) with W = HTH = G−1 G−1. The linear detector fails to yield the ML solution if the noise is too strong, due to its inherent noise amplification. For the LDLC iterative decoder, we would like that the deviation from the ML decoder due to the W matrix would be negligible in the expected range of noise variance. Experi- mental results (see Section IX) show that the iterative decoder indeed converges to the ML solution for noise variance values that approach channel capacity. However, for quantization or shaping applications (see Section VIII-B), where the effective noise is uniformly distributed along the Voronoi cell of the lattice (and is much stronger than the noise variance at channel capacity) the iterative decoder fails, and this can be explained by the influence of the W matrix on the minimization, as described above. Note from (11) that as α→ 1, W approaches a scaled identity matrix, which means that the minimization criterion approaches the ML criterion. However, the variances converge as αt, so as α → 1 convergence time approaches infinity. Until this point, we concentrated only on consistent Gaus- sians, and checked what lattice point maximizes the product- of-amplitudes of all the corresponding consistent Gaussians. However, this approach does not necessarily lead to the lattice point that will be finally chosen by the decoder, due to 3 main reasons: 1) It comes out experimentally that the strongest Gaussian in each message is not necessarily a consistent Gaussian, but a Gaussian that started as non-consistent and became consistent at a certain iteration. Such a Gaussian will finally converge to the appropriate lattice point, since the convergence of the mean values is independent of initial conditions. The non-consistency at the first several iterations, where the mean values are still very noisy, allows these Gaussians to accumulate stronger amplitudes than the consistent Gaussians (recall that the exponential weighting in (14) for d > 2 results in strong dependency on the behavior at the first iterations). 2) There is an exponential number of Gaussians that start as non-consistent and become consistent (with the same integer vector b) at a certain iteration, and the final am- plitude of the Gaussians at the lattice point coordinates will be determined by the sum of all these Gaussians. 3) We ignored non-consistent Gaussians that endlessly re- main non-consistent. We have not shown it analytically, but it is reasonable to assume that the excitation terms for such Gaussians will be weaker than for Gaussians that become consistent at some point, so their amplitude will fade away to zero. However, non-consistent Gaus- sians are born every iteration, even at steady state. The “newly-born” non-consistent Gaussians may appear as sidelobes to the main impulse, since it may take several iterations until they are attenuated. Proper choice of the coefficients of H may minimize this effect, as discussed in Sections III-A and V-A. However, these Gaussians may be a problem for small d (e.g. d = 2) where the product step at the variable node does not include enough messages to suppress them. Note that the first two issues are not a problem for d = 2, where the winning lattice point depends only on the asymptotic behavior. The amplitude of a sum of Gaussians that converged to the same coordinates will still be governed by (18) and the winning lattice point will still minimize (20). The third issue is a problem for small d, but less problematic for large d, as described above. As a result, we can not regard the convergence analysis of the consistent Gaussians’ amplitudes as a complete conver- gence analysis. However, it can certainly be used as a qual- itative analysis that gives certain insights to the convergence process. Two main observations are: 1) The narrow variable node messages tend to converge to single impulses at the coordinates of a single lattice point. This results from (16), (19), which show that the “non-winning” consistent Gaussians will have am- plitudes that decrease to zero relative to the amplitude of the “winning” consistent Gaussian. This result remains valid for the sum of non-consistent Gaussians that be- came consistent at a certain point, because it results from the non-stable nature of the recursion (9), which makes strong Gaussians stronger in an exponential manner. The single impulse might be accompanied by weak “sidelobes” due to newly-born non-consistent Gaussians. Interestingly, this form of solution is different from the exact solution (1), where every lattice point is represented by an impulse at the appropriate coordinate, with amplitude that depends on the Euclidean distance of the lattice point from the observation. The iterative decoder’s solution has a single impulse that corresponds to a single lattice point, where all other impulses have amplitudes that decay to zero. This should not be a problem, as long as the ML point is the remaining point (see discussion above). 2) We have shown that for d = 2 the strongest consistent Gaussians relate to b that minimizes an expression of the form (Gb− y)TW (Gb− y). We proposed a conjecture that this is also true for d > 2. We can further widen the conjecture to say that the finally decoded b (and not only the b that relates to strongest consistent Gaussians) mini- mizes such an expression. Such a conjecture can explain why the iterative decoder works well for decoding near channel capacity, but fails for quantization or shaping, where the effective noise variance is much larger. E. Summary of Convergence Results To summarize the convergence analysis, it was first shown that the variable node messages are Gaussian mixtures. There- fore, it is sufficient to analyze the sequences of variances, mean values and relative amplitudes of the Gaussians in each mixture. Starting with the variances, it was shown that with proper choice of the magic square LDLC generating sequence, each variable node generates d−1 “narrow” messages, whose variance decreases exponentially to zero, and a single “wide” message, whose variance reaches a finite value. Consistent Gaussians were then defined as Gaussians that their generation process always involved the same integer at the same check node. Consistent Gaussians can then be related to an integer vector b or equivalently to the lattice point Gb. It was then shown that under appropriate conditions on H , the mean values of consistent Gaussians that belong to narrow messages converge to the coordinates of the appropriate lattice point. The mean values of wide messages also converge to these coordinates, but with a steady state error. Then, the amplitudes of consistent Gaussians were analyzed. For d = 2 it was shown that the consistent Gaussians with maximal product- of-amplitudes (over all messages) are those that correspond to an integer vector b than minimizes (Gb− y)TW (Gb− y), where W is a positive definite matrix that depends only on H . The product-of-amplitudes for all other consistent Gaussians decays to zero. For d > 2 the analysis is complex and depends on the transient behavior of the mean values and variances (and not only on their steady state values), but a reasonable conjecture is to assume that a same form of criterion is also minimized for d > 2. The result is different from the ML lattice point, which minimizes ∥∥G · b− y∥∥2, where both criteria give the same point for weak noise but may give different solutions for strong noise. This may explain the experiments where the iterative decoder is successful in decoding the ML point for the AWGN channel near channel capacity, but fails in quantization or shaping applications where the effective noise is much stronger. These results also show that the iterative decoder converges to impulses at the coordinates of a single lattice point. It was then explained that analyzing the amplitudes of consistent Gaussians is not sufficient, so these results can not be regarded as a complete convergence analysis. However, the analysis gave a set of necessary conditions on H , and also led to useful insights to the convergence process. V. CODE DESIGN A. Choosing the Generating Sequence We shall concentrate on magic square LDLC, since they have inherent diversity of the nonzero elements in each row and column, which was shown above to be beneficial. It still remains to choose the LDLC generating sequence h1, h2, ...hd. Assume that the algorithm converged, and each PDF has a peak at the desired value. When the periodic functions are multiplied at a variable node, the correct peaks will then align. We would like that all the other peaks will be strongly attenuated, i.e. there will be no other point where the peaks align. This resembles the definition of the least common multiple (LCM) of integers: if the periods were integers, we would like to have their LCM as large as possible. This argument suggests the sequence {1/2, 1/3, 1/5, 1/7, 1/11, 1/13, 1/17, ...}, i.e. the reciprocals of the smallest d prime numbers. Since the periods are 1/h1, 1/h2, ...1/hd, we will get the desired property. Simulations have shown that increasing d beyond 7 with this choice gave negligible improvement. Also, performance was improved by adding some “dither” to the sequence, resulting in {1/2.31, 1/3.17, 1/5.11, 1/7.33, 1/11.71, 1/13.11, 1/17.55}. For d < 7, the first d elements are used. An alternative approach is a sequence of the form {1, �, �, ..., �}, where � << 1. For this case, every variable node will receive a single message with period 1 and d − 1 messages with period 1/�. For small �, the period of these d − 1 messages will be large and multiplication by the channel Gaussian will attenuate all the unwanted replicas. The single remaining replica will attenuate all the unwanted replicas of the message with period 1. A convenient choice is � = 1√ , which ensures that α = d−1 < 1, as required by Theorem 1. As an example, for d = 7 the sequence will be {1, 1√ B. Necessary Conditions on H The magic square LDLC definition and convergence analy- sis imply four necessary conditions on H: 1) |det(H)| = 1. This condition is part of the LDLC definition, which ensures proper density of the lattice points in Rm. If |det(H)| 6= 1, it can be easily normalized by dividing H by n |det(H)|. Note that practically we can allow |det(H)| 6= 1 as long as |det(H)| ≈ 1, since n |det(H)| is the gain factor of the transmitted codeword. For example, if n = 1000, having |det(H)| = 0.01 is acceptable, since we have |det(H)| = 0.995, which means that the codeword has to be further amplified by 20 · log10(0.995) = 0.04 dB, which is negligible. Note that normalizing H is applicable only if H is non-singular. If H is singular, a row and a column should be sequentially omitted until H becomes full rank. This process may result in slightly reducing n and a slightly different row and column degrees than originally planned. 2) α < 1, where α i=2 h . This guarantees expo- nential convergence rate for the variances (Theorem 1). Choosing a smaller α results in faster convergence, but we should not take α too small since the steady state variance of the wide variable node messages, as well as the steady state error of the mean values of these messages, increases when α decreases, as discussed in Section IV-C. This may result in deviation of the decoded codeword from the ML codeword, as discussed in Section IV-D. For the first LDLC generating sequence of the previous subsection, we have α = 0.92 and 0.87 for d = 7 and 5, respectively, which is a reasonable trade off. For the second sequence type we have α = d−1 3) All the eigenvalues of H̃ must have magnitude less than 1, where H̃ is defined in Theorem 2. This is a necessary condition for convergence of the mean values of the narrow messages. Note that adding random signs to the nonzero H elements is essential to fulfill this necessary condition, as explained in Section IV-C. 4) All the eigenvalues of F must have magnitude less than 1, where F is defined in Theorem 3. This is a necessary condition for convergence of the mean values of the wide messages. Interestingly, it comes out experimentally that for large code- word length n and relatively small degree d (e.g. n ≥ 1000 and d ≤ 10), a magic square LDLC with generating sequence that satisfies h1 = 1 and α < 1 results in H that satisfies all these four conditions: H is nonsingular without any need to omit rows and columns, n |det(H)| ≈ 1 without any need for normalization, and all eigenvalues of H̃ and F have magnitude less than 1 (typically, the largest eigenvalue of H̃ or F has magnitude of 0.94 − 0.97, almost independently of n and the choice of nonzero H locations). Therefore, by simply dividing the first generating sequence of the previous subsection by its first element, the constructed H meets all the necessary conditions, where the second type of sequence meets the conditions without any need for modifications. C. Construction of H for Magic Square LDLC We shall now present a simple algorithm for constructing a parity check matrix for a magic square LDLC. If we look at the bipartite graph, each variable node and each check node has d edges connected to it, one with every possible weight h1, h2, ...hd. All the edges that have the same weight hj form a permutation from the variable nodes to the check nodes (or vice versa). The proposed algorithm generates d random permutations and then searches sequentially and cyclically for 2-loops (two parallel edges from a variable node to a check node) and 4-loops (two variable nodes that both are connected to a pair of check nodes). When such a loop is found, a pair is swapped in one of the permutations such that the loop is removed. A detailed pseudo-code for this algorithm is given in Appendix VII. VI. DECODER IMPLEMENTATION Each PDF should be approximated with a discrete vector with resolution ∆ and finite range. According to the Gaussian Q-function, choosing a range of, say, 6σ to both sides of the noisy channel observation will ensure that the error probability due to PDF truncation will be ≈ 10−9. Near capacity, σ2 ≈ , so 12σ ≈ 3. Simulation showed that resolution errors became negligible for ∆ = 1/64. Each PDF was then stored in a L = 256 elements vector, corresponding to a range of size 4. At the check node, the PDF fj(x) that arrives from variable node j is first expanded by hj (the appropriate coefficient of H) to get fj(x/hj). In a discrete implementation with resolution ∆ the PDF is a vector of values fj(k∆), k ∈ Z. As described in Section V, we shall usually use hj ≤ 1 so the expanded PDF will be shorter than the original PDF. If the expand factor 1/|hj | was an integer, we could simply decimate fj(k∆) by 1/|hj |. However, in general it is not an integer so we should use some kind of interpolation. The PDF fj(x) is certainly not band limited, and as the iterations go on it approaches an impulse, so simple interpolation methods (e.g. linear) are not suitable. Suppose that we need to calculate fj((k + �)∆), where −0.5 ≤ � ≤ 0.5. A simple interpolation method which showed to be effective is to average fj(x) around the desired point, where the averaging window length lw is chosen to ensure that every sample of fj(x) is used in the interpolation of at least one output point. This ensures that an impulse can not be missed. The interpolation result is then 2lw+1 i=−lw fj((k − i)∆), where lw = d1/|hj |e The most computationally extensive step at the check nodes is the calculation the convolution of d − 1 expanded PDF’s. An efficient method is to calculate the fast Fourier transforms (FFTs) of all the PDF’s, multiply the results and then perform inverse FFT (IFFT). The resolution of the FFT should be larger than the expected convolution length, which is roughly Lout ≈ L · i=1 hi, where L denotes the original PDF length. Appendix VIII shows a way to use FFTs of size 1/∆, where ∆ is the resolution of the PDF. Usually 1/∆ << Lout so FFT complexity is significantly reduced. Practical values are L = 256 and ∆ = 1/64, which give an improvement factor of at least 4 in complexity. Each variable node receives d check node messages. The output variable node message is calculated by generating the product of d−1 input messages and the channel Gaussian. As the iterations go on, the messages get narrow and may become impulses, with only a single nonzero sample. Quantization effects may cause impulses in two messages to be shifted by one sample. This will result in a zero output (instead of an impulse). Therefore, it was found useful to widen each check node message Q(k) prior to multiplication, such that Qw(k) = i=−1Q(k + i), i.e. the message is added to its right shifted and left shifted (by one sample) versions. VII. COMPUTATIONAL COMPLEXITY AND STORAGE REQUIREMENTS Most of the computational effort is invested in the d FFT’s and d IFFT’s (of length 1/∆) that each check node performs each iteration. The total number of multiplications for t iterations is o n · d · t · 1 · log2( . As in binary LDPC codes, the computational complexity has the attractive property of being linear with block length. However, the constant that precedes the linear term is significantly higher, mainly due to the FFT operations. The memory requirements are governed by the storage of the nd check node and variable node messages, with total memory of o(n · d ·L). Compared to binary LDPC, the factor of L significantly increases the required memory. For example, for n = 10, 000, d = 7 and L = 256, the number of storage elements is of the order of 107. VIII. ENCODING AND SHAPING A. Encoding The LDLC encoder has to calculate x = G ·b, where b is an integer message vector. Note that unlike H , G = H−1 is not sparse, in general, so the calculation requires computational complexity and storage of o(n2). This is not a desirable property because the decoder’s computational complexity is only o(n). A possible solution is to use the Jacobi method [22] to solve H · x = b, which is a system of sparse linear equations. Using this method, a magic square LDLC encoder calculates several iterations of the form: x(t) = b̃− H̃ · x(t−1) (21) with initialization x(0) = 0. The matrix H̃ is defined in Lemma 6 of Section IV-C. The vector b̃ is a permuted and scaled version of the integer vector b, such that the i’th element of b̃ equals the element of b for which the appropriate row of H has its largest magnitude value at the i’th location. This element is further divided by this largest magnitude element. A necessary and sufficient condition for convergence to x = G ·b is that all the eigenvalues of H̃ have magnitude less than 1 [22]. However, it was shown that this is also a necessary condition for convergence of the LDLC iterative decoder (see Sections IV-C, V-B), so it is guaranteed to be fulfilled for a properly designed magic square LDLC. Since H̃ is sparse, this is an o(n) algorithm, both in complexity and storage. B. Shaping For practical use with the power constrained AWGN chan- nel, the encoding operation must be accompanied by shaping, in order to prevent the transmitted codeword’s power from being too large. Therefore, instead of mapping the information vector b to the lattice point x = G · b, it should be mapped to some other lattice point x′ = G · b′, such that the lattice points that are used as codewords belong to a shaping region (e.g. an n-dimensional sphere). The shaping operation is the mapping of the integer vector b to the integer vector b′. As explained in Section II-A, this work concentrates on the lattice design and the lattice decoding algorithm, and not on the shaping region or shaping algorithms. Therefore, this section will only highlight some basic shaping principles and ideas. A natural shaping scheme for lattice codes is nested lattice coding [12]. In this scheme, shaping is done by quantizing the lattice point G · b onto a coarse lattice G′, where the transmitted codeword is the quantization error, which is uni- formly distributed along the Voronoi cell of the coarse lattice. If the second moment of this Voronoi cell is close to that of an n-dimensional sphere, the scheme will attain close-to- optimal shaping gain. Specifically, assume that the information vector b assumes integer values in the range 0, 1, ...M − 1 for some constant integer M . Then, we can choose the coarse lattice to be G′ = MG. The volume of the Voronoi cell for this lattice is Mn, since we assume det(G) = 1 (see 0 0.5 1 1.5 2 2.5 3 3.5 4 distance from capacity [dB] Symbol error rate (SER) for various block lengths 1000 10,000 100,000 Fig. 4. Simulation results Section II-A). If the shape of the Voronoi cell resembles an n- dimensional sphere (as expected from a capacity approaching lattice code), it will attain optimal shaping gain (compared to uncoded transmission of the original integer sequence b). The shaping operation will find the coarse lattice point MGk, k ∈ Zn, which is closest to the fine lattice point x = G · b. The transmitted codeword will be: x′ = x−MGk = G(b−Mk) = Gb′ where b′ = b −Mk (note that the “inverse shaping” at the decoder, i.e. transforming from b′ to b, is a simple modulo calculation: b = b′ mod M ). Finding the closest coarse lattice point MGk to x is equivalent to finding the closest fine lattice point G ·k to the vector x/M . This is exactly the operation of the iterative LDLC decoder, so we could expect that is could be used for shaping. However, simulations show that the iterative decoding finds a vector k with poor shaping gain. The reason is that for shaping, the effective noise is much stronger than for decoding, and the iterative decoder fails to find the nearest lattice point if the noise is too large (see Section IV-D). Therefore, an alternative algorithm has to be used for finding the nearest coarse lattice point. The complexity of finding the nearest lattice point grows exponentially with the lattice dimension n and is not feasible for large dimensions [23]. However, unlike decoding, for shaping applications it is not critical to find the exact nearest lattice point, and approximate algorithms may be considered (see [15]). A possible method [24] is to perform QR decomposition on G in order to transform to a lattice with upper triangular generator matrix, and then use sequential decoding algorithms (such as the Fano algorithm) to search the resulting tree. The main disadvantage of this approach is computational complexity and storage of at least o(n2). Finding an efficient shaping scheme for LDLC is certainly a topic for further research. IX. SIMULATION RESULTS Magic square LDLC with the first gen- erating sequence of Section V-A (i.e. {1/2.31, 1/3.17, 1/5.11, 1/7.33, 1/11.71, 1/13.11, 1/17.55}) were simulated for the AWGN channel at various block lengths. The degree was d = 5 for n = 100 and d = 7 for all other n. For n = 100 the matrix H was further normalized to get n det(H) = 1. For all other n, normalizing the generating sequence such that the largest element has magnitude 1 also gave the desired determinant normalization (see Section V-B). The H matrices were generated using the algorithm of Section V-C. PDF resolution was set to ∆ = 1/256 with a total range of 4, i.e. each PDF was represented by a vector of L = 1024 elements. High resolution was used since our main target is to prove the LDLC concept and eliminate degradation due to implementation considerations. For this reason, the decoder was used with 200 iterations (though most of the time, a much smaller number was sufficient). In all simulations the all-zero codeword was used. Ap- proaching channel capacity is equivalent to σ2 → 1 Section II-A), so performance is measured in symbol error rate (SER), vs. the distance of the noise variance σ2 from capacity (in dB). The results are shown in Figure 4. At SER of 10−5, for n = 100000, 10000, 1000, 100 we can work as close as 0.6dB, 0.8dB, 1.5dB and 3.7dB from capacity, respectively. Similar results were obtained for d = 7 with the second type of generating sequence of Section V-A, i.e. {1, 1√ }. Results were slightly worse than for the first generating sequence (by less than 0.1 dB). Increasing d did not give any visible improvement. X. CONCLUSION Low density lattice codes (LDLC) were introduced. LDLC are novel lattice codes that can approach capacity and be decoded efficiently. Good error performance within ∼ 0.5dB from capacity at block length of 100,000 symbols was demon- strated. Convergence analysis was presented for the iterative decoder, which is not complete, but yields necessary condi- tions on H and significant insight to the convergence process. Code parameters were chosen from intuitive arguments, so it is reasonable to assume that when the code structure will be more understood, better parameters could be found, and channel capacity could be approached even closer. Multi-input, multi-output (MIMO) communication systems have become popular in recent years. Lattice codes have been proposed in this context as space-time codes (LAST) [25]. The concatenation of the lattice encoder and the MIMO channel generates a lattice. If LDLC are used as lattice codes and the MIMO configuration is small, the inverse generator matrix of this concatenated lattice can be assumed to be sparse. Therefore, the MIMO channel and the LDLC can be jointly decoded using an LDLC-like decoder. However, even if a magic square LDLC is used as the lattice code, the concatenated lattice is not guaranteed to be equivalent to a magic square LDLC, and the necessary conditions for convergence are not guaranteed to be fulfilled. Therefore, the usage of LDLC for MIMO systems is a topic for further research. APPENDIX I EXACT PDF CALCULATIONS Given the n-dimensional noisy observation y = x + w of the transmitted codeword x = Gb, we would like to calculate the probability density function (PDF) fxk|y(xk|y). We shall start by calculating fx|y(x|y) = fx(x)fy|x(y|x) fy(y) . Denote the shaping region by B (G will be used to denote both the lattice and its generator matrix). fx(x) is a sum of |G ∩ B| n-dimensional Dirac delta functions, since x has nonzero probability only for the lattice points that lie inside the shaping region. Assuming further that all codewords are used with equal probability, all these delta functions have equal weight of 1|G∩B| . The expression for fy|x(y|x) is simply the PDF of the i.i.d Gaussian noise vector. We therefore get: fx|y(x|y) = fx(x)fy|x(y|x) fy(y) = (22) |G∩B| l∈G∩B δ(x− l) · (2πσ 2)−n/2e− i=1(yi−xi) 2/2σ2 fy(y) = C · l∈G∩B δ(x− l) · e−d 2(l,y)/2σ2 Where C is not a function of x and d2(l, y) is the squared Euclidean distance between the vectors l and y in Rn. It can be seen that the conditional PDF of x has a delta function for each lattice point, located at this lattice point with weight that is proportional to the exponent of the negated squared Euclidean distance of this lattice point from the noisy observation. The ML point corresponds to the delta function with largest weight. As the next step, instead of calculating the n-dimensional PDF of the whole vector x, we shall calculate the n one- dimensional PDF’s for each of the components xk of the vector x (conditioned on the whole observation vector y): fxk|y(xk|y) = (23) xi,i6=k · · · fx|y(x|y)dx1dx2 · · · dxk−1dxk+1 · · · dxn = = C · l∈G∩B δ(xk − lk) · e−d 2(l,y)/2σ2 This finishes the proof of (1). It can be seen that the conditional PDF of xk has a delta function for each lattice point, located at the projection of this lattice point on the coordinate xk, with weight that is proportional to the exponent of the negated squared Euclidean distance of this lattice point from the noisy observation. The ML point will therefore correspond to the delta function with largest weight in each coordinate. Note, however, that if several lattice points have the same projection on a specific coordinate, the weights of the corresponding delta functions will add and may exceed the weight of the ML point. APPENDIX II EXTENDING GALLAGER’S TECHNIQUE TO THE CONTINUOUS CASE In [5], the derivation of the LDPC iterative decoder was simplified using the following technique: the codeword ele- ments xk were assumed i.i.d. and a condition was added to all the probability calculations, such that only valid codewords were actually considered. The question is then how to choose the marginal PDF of the codeword elements. In [5], binary codewords were considered, and the i.i.d distribution assumed the values ’0’ and ’1’ with equal probability. Since we extend the technique to the continuous case, we have to set the continuous marginal distribution fxk(xk). It should be set such that fx(x), assuming that x is a lattice point, is the same as f(x|s ∈ Zn), assuming that xk are i.i.d with marginal PDF fxk(xk), where s = H ·x. This fx(x) equals a weighted sum of Dirac delta functions at all lattice points, where the weight at each lattice point equals the probability to use this point as a codeword. Before proceeding, we need the following property of conditional probabilities. For any two continuous valued RV’s u, v we have: f(u|v ∈ {v1, v2, ..., vN}) = k=1 fu,v(u, vk)∑N k=1 fv(vk) (This property can be easily proved by following the lines of [21], pp. 159-160, and can also be extended to the infinite sum case). Using (24), we now have: f(x|s ∈ Zn) = i∈Zn fx,s(x, s = i)∑ i∈Zn fs(i) f(x)f(s = i|x) = C ′ f(x)δ(x−Gi) (25) where C,C ′ are independent of x. The result is a weighted sum of Dirac delta functions at all lattice points, as desired. Now, the weight at each lattice point should equal the probability to use this point as a codeword. Therefore, fxk(xk) should be chosen such that at each lattice point, the resulting vector distribution fx(x) = k=1 fxk(xk) will have a value that is proportional to the probability to use this lattice point. At x which is not a lattice point, the value of fx(x) is not important. APPENDIX III DERIVATION OF THE ITERATIVE DECODER In this appendix we shall derive the LDLC iterative decoder for a code with dimension n, using the tree assumption and Gallager’s trick. Referring to figure 2, assume that there are only 2 tiers. Using Gallager’s trick we assume that the xk’s are i.i.d. We would like to calculate f(x1|(y, s ∈ Zn), where s = H · x. Due to the tree assumption, we can do it in two steps: 1. calculate the conditional PDF of the tier 1 variables of x1, conditioned only on the check equations that relate the tier 1 and tier 2 variables. 2. calculate the conditional PDF of x1 itself, conditioned only on the check equations that relate x1 and its first tier variables, but using the results of step 1 as the PDF’s for the tier 1 variables. Hence, the results will be equivalent to conditioning on all the check equations. There is a basic difference between the calculation in step 1 and step 2: the condition in step 2 involves all the check equations that are related to x1, where in step 1 a single check equation is always omitted (the one that relates the relevant tier 1 element with x1 itself). Assume now that there are many tiers, where each tier contains distinct elements of x (i.e. each element appears only once in the resulting tree). We can then start at the farthest tier and start moving toward x1. We do it by repeatedly calculating step 1. After reaching tier 1, we use step 2 to finally calculate the desired conditional PDF for x1. This approach suggests an iterative algorithm for the cal- culation of f(xk|(y, s ∈ Zn) for k=1, 2..n. In this approach we assume that the resulting tier diagram for each xk contains distinct elements for several tiers (larger or equal to the number of required iterations). We then repeat step 1 several times, where the results of the previous iteration are used as initial PDF’s for the next iteration. Finally, we perform step 2 to calculate the final results. Note that by conditioning only on part of the check equa- tions in each iteration, we can not restrict the result to the shaping region. This is the reason that the decoder performs lattice decoding and not exact ML decoding, as described in Section III. We shall now turn to derive the basic iteration of the algorithm. For simplicity, we shall start with the final step of the algorithm (denoted step 2 above). We would like to perform t iterations, so assume that for each xk there are t tiers with a total of Nc check equations. For every xk we need to calculate f(xk|s ∈ ZNc , y) = f(xk|s(tier1) ∈ Zck , s(tier2:tiert) ∈ ZNc−ck , y), where ck is the number of check equations that involve xk. s(tier1) = H(tier1) ·x denotes the value of the left hand side of these check equations when x is substituted (H(tier1) is a submatrix of H that contains only the rows that relate to these check equations), and s(tier2:tiert) relates in the same manner to all the other check equations. For simplicity of notations, denote the event s(tier2:tiert) ∈ ZNc−ck by A. As explained above, in all the calculations we assume that all the xk’s are independent. Using (24), we get: f(xk|s(tier1) ∈ Zck , A, y) = i∈Zck f(xk, s (tier1) = i|A, y)∑ i∈Zck f(s (tier1) = i|A, y) Evaluating the term inside the sum of the nominator, we get: f(xk, s (tier1) = i|A, y) = = f(xk|A, y) · f(s(tier1) = i|xk, A, y) (27) Evaluating the left term, we get: f(xk|A, y) = f(xk|yk) = f(xk)f(yk|xk) f(yk) f(xk) f(yk) − (yk−xk) 2σ2 (28) where f(xk|y) = f(xk|yk) due to the i.i.d assumption. Evaluating now the right term of (27), we get: f(s(tier1) = i|xk, A, y) = f(s(tier1)m = im|xk, A, y) (29) where s(tier1)m denotes the m’th component of s(tier1) and im denotes the m’th component of i. Note that each element of s(tier1) is a linear combination of several elements of x. Due to the tree assumption, two such linear combinations have no common elements, except for xk itself, which appears in all linear combinations. However, xk is given, so the i.i.d assumption implies that all these linear combinations are independent, so (29) is justified. The condition A (i.e. s(tier2:tiert) ∈ ZNc−ck ) does not impact the independence due to the tree assumption. Substituting (27), (28), (29) back in (26), we get: f(xk|s(tier1) ∈ Zck , A, y) = (30) = C · f(xk) · e− (yk−xk) i∈Zck f(s(tier1)m = im|xk, A, y) = = C · f(xk) · e− (yk−xk) · · · · · · ick∈Z f(s(tier1)m = im|xk, A, y) = = C · f(xk) · e− (yk−xk) f(s(tier1)m = im|xk, A, y) where C is independent of xk. We shall now examine the term inside the sum: f(s(tier1)m = im|xk, A, y). Denote the linear combination that s (tier1) m rep- resents by: s(tier1)m = hm,1xk + hm,lxjl (31) where {hm,l}, l = 1, 2...rm is the set of nonzero coefficients of the appropriate parity check equation, and jl is the set of indices of the appropriate x elements (note that the set jl depends on m but we omit the “m” index for clarity of notations). Without loss of generality, hm,1 is assumed to be the coefficient of xk. Define zm l=2 hm,lxjl , such that (tier1) m = hm,1xk + zm. We then have: f(s(tier1)m = im|xk, A, y) = (32) = fzm|xk,A,y(zm = im − hm,1xk|xk, A, y) Now, since we assume that the elements of x are independent, the PDF of the linear combination zm equals the convolution of the PDF’s of its components: fzm|xk,A,y(zm|xk, A, y) = |hm,2| fxj2 |A,y |A, y |hm,3| fxj3 |A,y |A, y · · ·~ |hm,rm | fxjrm |A,y hm,rm |A, y Note that the functions fxji |y xji |A, y are simply the output PDF’s of the previous iteration. Define now pm(xk) = fzm|xk,A,y(zm = −hm,1xk|xk, A, y) (34) Substituting (32), (34) in (30), we finally get: f(xk|s(tier1) ∈ Zck , A, y) = (35) = C · f(xk) · e− (yk−xk) pm(xk − This result can be summarized as follows. For each of the ck check equations that involve xk, the PDF’s (previous iteration results) of the active equation elements, except for xk itself, are expanded and convolved, according to (33). The convolution result is scaled by (−hm,1), the negated coefficient of xk in this check equation, according to (34), to yield pm(xk). Then, a periodic function with period 1/|hm,1| is generated by adding an infinite number of shifted versions of the scaled convolution result, according to the sum term in (35). After repeating this process for all the ck check equations that involve xk, we get ck periodic functions, with possibly different periods. We then multiply all these functions. The multiplication result is further multiplied by the channel Gaussian PDF term e− (yk−xk) 2σ2 and finally by f(xk), the marginal PDF of xk under the i.i.d assumption. As discussed in Section III, we assume that f(xk) is a uniform distribution with large enough range. This means that f(xk) is constant over the valid range of xk, and can therefore be omitted from (35) and absorbed in the constant C. As noted above, this result is for the final step (equivalent to step 2 above), where we determine the PDF of xk according to the PDF’s of all its tier 1 elements. However, the repeated iteration step is equivalent to step 1 above. In this step ,we assume that xk is a tier 1 element of another element, say xl, and derive the PDF of xk that should be used as input to step 2 of xl (see figure 2). It can be seen that the only difference between step 2 and step 1 is that in step 2 all the check equations that involve xk are used, where in step 1 the check equation that involves both xk and xl is ignored (there must be such an equation since xk is one of the tier 1 elements of xl). Therefore, the step1 iteration is identical to (35), except that the product does not contain the term that corresponds to the check equation that combines both xk and xl. Denote fkl(xk) = f(xk|s(tier1 except l) ∈ Zck−1, A, y) (36) We then get: fkl(xk) = C · e− (yk−xk) m 6=ml pm(xk − ) (37) where ml is the index of the check equation that combines both xk and xl. In principle, a different fkl(xk) should be calculated for each xl for which xk is a tier 1 element. However, the calculation is the same for all xl that share the same check equation. Therefore, we should calculate fkl(xk) once for each check equation that involves xk. l can be regarded as the index of the check equation within the set of check equations that involve xk. We can now formulate the iterative decoder. The decoder state variables are PDF’s of the form f (t)kl (xk), where k = 1, 2, ...n. For each k, l assumes the values 1, 2, ...ck, where ck is the number of check equations that involve xk. t denotes the iteration index. For a regular LDLC with degree d there will be nd PDF’s. The PDF’s are initialized by assuming that xk is a leaf of the tier diagram. Such a leaf has no tier 1 elements, so fkl(xk) = f(xk) · f(yk|xk). As explained above for equation (35), we shall omit the term f(xk), resulting in initialization with the channel noise Gaussian around the noisy observation yk. Then, the PDF’s are updated in each iteration according to (37). The variable node messages should be further normalized in order to get actual PDF’s, such that −∞ fkl(xk)dxk = 1 (this will compensate for the constant C). The final PDF’s for xk, k = 1, 2, ...n are then calculated according to (35). Finally, we have to estimate the integer valued informa- tion vector b. This can be done by first estimating the codeword vector x from the peaks of the PDF’s: x̂k = argmaxxk f(xk|s (tier1) ∈ Zck , A, y). Finally, we can esti- mate b as b̂ = bHx̂e. We have finished developing the iterative algorithm. It can be easily seen that the message passing formulation of Section III-A actually implements this algorithm. APPENDIX IV ASYMPTOTIC BEHAVIOR OF THE VARIANCES RECURSION A. Proof of Lemma 3 and Lemma 4 We shall now derive the basic iterative equations that relate the variances at iteration t + 1 to the variances at iteration t for a magic square LDLC with dimension n, degree d and generating sequence h1 ≥ h2 ≥ ... ≥ hd > 0. Each iteration, every check node generates d output mes- sages, one for each variable node that is connected to it, where the weights of these d connections are ±h1,±h2, ...,±hd. For each such output message, the check node convolves d − 1 expanded variable node PDF messages, and then stretches and periodically extends the result. For a specific check node, denote the variance of the variable node message that arrives along an edge with weight ±hj by V j , j = 1, 2, ...d. Denote the variance of the message that is sent back to a variable node along an edge with weight ±hj by Ṽ j . From (2), (3), we get: i 6=j i (38) Then, each variable node generates d messages, one for each check node that is connected to it, where the weights of these d connections are ±h1,±h2, ...,±hd. For each such output message, the variable node generates the product of d − 1 check node messages and the channel noise PDF. For a specific variable node, denote the variance of the message that is sent back to a check node along an edge with weight ±hj by (t+1) j (this is the final variance of the iteration). From claim 2, we then get: (t+1) i 6=j From symmetry considerations, it can be seen that all mes- sages that are sent along edges with the same absolute value of their weight will have the same variance, since the same variance update occurs for all these messages (both for check node messages and variable node messages). Therefore, the d variance values V (t)1 , V 2 , ..., V d are the same for all variable nodes, where V (t)l is the variance of the message that is sent along an edge with weight ±hl. This completes the proof of Lemma 3. Using this symmetry, we can now derive the recursive update of the variance values V (t)1 , V 2 , ..., V d . Substituting (38) in (39), we get: (t+1) m 6=i h2m∑d j 6=m for i = 1, 2, ...d, which completes the proof of Lemma 4. B. Proof of Theorem 1 We would like to analyze the convergence of the nonlinear recursion (4) for the variances V (t)1 , V 2 , ..., V d . This recur- sion is illustrated in (5) for the case d = 3. It is assumed that α < 1, where α = i=2 h . Define another set of variables 1 , U 2 , ..., U d , which obey the following recursion. The recursion for the first variable is: (t+1) where for i = 2, 3, ...d the recursion is: (t+1) h21∑d j=2 h with initial conditions U (0)1 = U 2 = ... = U d = σ It can be seen that (41) can be regarded as the approximation of (4) under the assumptions that V (t)i << V 1 and V σ2 for i = 2, 3, ...d. For illustration, the new recursion for the case d = 3 is: (t+1) (t+1) 2 + h (t+1) 2 + h It can be seen that in the new recursion, U (t)1 obeys a recursion that is independent of the other variables. From (41), this recursion can be written as 1 (t+1) with initial condition U (0)1 = σ 2. Since α < 1, this is a stable linear recursion for 1 , which can be solved to get 1 = σ 2(1− α) 1 1−αt+1 . For the other variables, it can be seen that all have the same right hand side in the recursion (41). Since all are initialized with the same value, it follows that U (t)2 = U 3 = ... = U for all t ≥ 0. Substituting back in (41), we get the recursion (t+1) 2 = αU 2 , with initial condition U 2 = σ 2. Since α < 1, this is a stable linear recursion for U (t)2 , which can be solved to get U (t)2 = σ We found an analytic solution for the variables U (t)i . How- ever, we are interested in the variances V (t)i . The following claim relates the two sets of variables. Claim 3: For every t ≥ 0, the first variables of the two sets are related by V (t)1 ≥ U 1 , where for i = 2, 3, ...d we have i ≤ U Proof: By induction: the initialization of the two sets of variables obviously satisfies the required relations. Assume now that the relations are satisfied for iteration t, i.e. V (t)1 ≥ 1 and for i = 2, 3, ...d, V i ≤ U i . If we now compare the right hand side of the update recursion for 1 (t+1) to that of (t+1) (i.e. (4) to (41)), then the right hand side for 1 (t+1) is smaller, because it has additional positive terms in the denominators, where the common terms in the denominators are larger according to the induction assumption. Therefore, (t+1) 1 ≥ U (t+1) 1 , as required. In the same manner, if we compare the right hand side of the update recursion for 1 (t+1) to that of 1 (t+1) for i ≥ 2, then the right hand side for 1 (t+1) is larger, because it has additional positive terms, where the common terms are also larger since their denominators are smaller due to the induction assumption. Therefore, V (t+1)i ≤ (t+1) i for i = 2, 3, ...d, as required. Using claim 3 and the analytic results for U (t)i , we now have: 1 ≥ U 1 = σ 2(1− α) 1− αt+1 ≥ σ2(1− α) (43) where for i = 2, 3, ...d we have: i ≤ U i = σ 2αt (44) We have shown that the first variance is lower bounded by a positive nonzero constant where the other variances are upper bounded by a term that decays exponentially to zero. Therefore, for large t we have V (t)i << V 1 and i << σ 2 for i = 2, 3, ...d. It then follows that for large t the variances approximately obey the recursion (41), which was built from the actual variance recursion (4) under these assumptions. Therefore, for i = 2, 3, ...d the variances are not only upper bounded by an exponentially decaying term, but actually approach such a term, where the first variance actually approaches the constant σ2(1−α) in an exponential rate. This completes the proof of Theorem 1. Note that the above analysis only applies if α < 1. To illustrate the behavior for α ≥ 1, consider the simple case of h1 = h2 = ... = hd. From (4), (5) it can be seen that for this case, if V (0)i is independent of i, then V independent of i for every t > 0, since all the elements will follow the same recursive equations. Substituting this result in the first equation, we get the single variable recursion (t+1) with initialization V (0)i = σ 2. This recursion is easily solved to get 1 = t+1 or V (t)i = can be seen that all the variances converge to zero, but with slow convergence rate of o(1/t). APPENDIX V ASYMPTOTIC BEHAVIOR OF THE MEAN VALUES RECURSION A. Proof of Lemma 5 and Lemma 6 (Mean of Narrow Mes- sages) Assume a magic square LDLC with dimension n and degree d. We shall now examine the effect of the calculations in the check nodes and variable nodes on the mean values and derive the resulting recursion. Every iteration, each check node generates d output messages, one for each variable node that connects to it, where the weights of these d connections are ±h1,±h2, ...,±hd. For each such output message, the check node convolves d− 1 expanded variable node PDF messages, and then stretches and periodically extends the result. We shall concentrate on the nd consistent Gaussians that relate to the same integer vector b (one Gaussian in each message), and analyze them jointly. For convenience, we shall refer to the mean value of the relevant consistent Gaussian as the mean of the message. Consider now a specific check node. Denote the mean value of the variable node message that arrives at iteration t along the edge with weight ±hj by m j , j = 1, 2, ...d. Denote the mean value of the message that is sent back to a variable node along an edge with weight ±hj by m̃ j . From (2), (3) and claim 1, we get: bk − d∑  (45) where bk is the appropriate element of b that is related to this specific check equation, which is the only relevant index in the infinite sum of the periodic extension step (3). Note that the check node operation is equivalent to extracting the value of mj from the check equation i=1 himi = bk, assuming all the other mi are known. Note also that the coefficients hj should have a random sign. To keep notations simple, we assume that hj already includes the random sign. Later, when several equations will be combined together, we should take it into account. Then, each variable node generates d messages, one for each check node that is connected to it, where the weights of these d connections are ±h1,±h2, ...,±hd. For each such output message, the variable node generates the product of d− 1 check node messages and the channel noise PDF. For a specific variable node, denote the mean value of the message that arrives from a check node along an edge with weight ±hj by m̃ j , and the appropriate variance by Ṽ j . The mean value of the message that is sent back to a check node along an edge with weight ±hj is m (t+1) j , the final mean value of the iteration. From claim 2, we then get: (t+1) i 6=j i /Ṽ 1/σ2 + 1/Ṽ (t)i where yk is the channel observation for the variable node and σ2 is the noise variance. Note that m̃(t)i , i = 1, 2, ..., d in (46) are the mean values of check node messages that arrive to the same variable node from different check nodes, where in (45) they define the mean values of check node messages that leave the same check node. However, it is beneficial to keep the notations simple, and we shall take special care when (46) and (45) are combined. It can be seen that the convergence of the mean values is coupled to the convergence of the variances (unlike the recursion of the variances which was autonomous). However, as the iterations go on, this coupling disappears. To see that, recall from Theorem 1 that for each check node, the variance of the variable node message that comes along an edge with weight ±h1 approaches a finite value, where the variance of all the other messages approaches zero exponentially. According to (38), the variance of the check node message is a weighted sum of the variances of the incoming variable node messages. Therefore, the variance of the check node message that goes along an edge with weight ±h1 will approach zero, since the weighted sum involves only zero-approaching variances. All the other messages will have finite variance, since the weighted sum involves the non zero-approaching variance. To summa- rize, each variable node sends (and each check node receives) d− 1 “narrow” messages and a single “wide” message. Each check node sends (and each variable node receives) d − 1 “wide” messages and a single “narrow” message, where the narrow message is sent along the edge from which the wide message was received (the edge with weight ±h1). We shall now concentrate on the case where the variable node generates a narrow message. Then, the sum in the nominator of (46) has a single term for which Ṽ (t)i → 0, which corresponds to i = 1. The same is true for the sum in the denominator. Therefore, for large t, all the other terms will become negligible and we get: (t+1) j ≈ m̃ 1 (47) where m̃(t)1 is the mean of the message that comes from the edge with weight h1, i.e. the narrow check node message. As discussed above, d − 1 of the d variable node messages that leave the same variable node are narrow. From (47) it comes out that for large t, all these d− 1 narrow messages will have the same mean value. This completes the proof of Lemma 5. Now, combining (45) and (47) (where the indices are arranged again, as discussed above), we get: (t+1) where li, i = 1, 2..., d are the variable nodes that take place in the check equation for which variable node l1 appears with coefficient ±h1. bk is the element of b that is related to this check equation. m(t+1)l1 denotes the mean value of the d− 1 narrow messages that leave variable node l1 at iteration t + 1. m(t)li is the mean value of the narrow messages that were generated at variable node li at iteration t. Only narrow messages are involved in (48), because the right hand side of (47) is the mean value of the narrow check node message that arrived to variable node l1, which results from the convolution of d−1 narrow variable node messages. Therefore, for large t, the mean values of the narrow messages are decoupled from the mean values of the wide messages (and also from the variances), and they obey an autonomous recursion. The mean values of the narrow messages at iteration t can be arranged in an n-element column vector m(t) (one mean value for each variable node). We would like to show that the mean values converge to the coordinates of the lattice point x = Gb. Therefore, it is useful to define the error vector = m(t)−x. Since Hx = b, we can write (using the same notations as (48)): xl1 = hixli Subtracting (49) from (48), we get: (t+1) Or, in vector notation: e(t+1) ≈ −H̃ · e(t) (51) where H̃ is derived from H by permuting the rows such that the ±h1 elements will be placed on the diagonal, dividing each row by the appropriate diagonal element (h1 or −h1), and then nullifying the diagonal. Note that in order to simplify the notations, we embedded the sign of ±hj in hj and did not write it implicitly. However, the definition of H̃ solves this ambiguity. This completes the proof of Lemma 6. B. Proof of Lemma 7 (Mean of Wide Messages) Recall that each check node receives d−1 narrow messages and a single wide message. The wide message comes along the edge with weight ±h1. Denote the appropriate lattice point by x = Gb, and assume that the Gaussians of the narrow variable node messages have already converged to impulses at the corresponding lattice point coordinates (Theorem 2). We can then substitute in (45) m(t)i = xi for i ≥ 2. The mean value of the (wide) message that is returned along the edge with weight ±hj (j 6= 1) is: bk − d∑ hixi − h1m  = (52) h1x1 + hjxj − h1m = xj + x1 −m As in the previous section, for convenience of notations we embed the sign of ±hj in hj itself. The sign ambiguity will be resolved later. The meaning of (52) is that the returned mean value is the desired lattice coordinate plus an error term that is proportional to the error in the incoming wide message. From (38), assuming that the variance of the incoming wide message has already converged to its steady state value σ2(1−α) and the variance of the incoming narrow messages has already converged to zero, the variance of this check node message will be: σ2(1− α) (53) where α = i=2 h . Now, each variable node receives d − 1 wide messages and a single narrow message. The mean values of the wide messages are according to (52) and the variances are according to (53). The single wide message that this variable node generates results from the d − 1 input wide messages and it is sent along the edge with weight ±h1. From (46), the wide mean value generated at variable node k will then be: (t+1) k = (54) (xp(k,j) −m p(k,j) 2(1−α) 1/σ2 + 2(1−α) Note that the x1 and m1 terms of (52) were replaced by xp(k,j) and mp(k,j), respectively, since for convenience of notations we denoted by m1 the mean of the message that came to a check node along the edge with weight ±h1. For substitution in (46) we need to know the exact variable node index that this edge came from. Therefore, p(k, j) denotes the index of the variable node that takes place with coefficient ±h1 in the check equation where xk takes place with coefficient ±hj . Rearranging terms, we then get: (t+1) k = (55) yk(1− α) + xk · α+ xp(k,j) −m p(k,j) (1− α) + α = yk + α(xk − yk) + hj(xp(k,j) −m p(k,j) Denote now the wide message mean value error by e(t)k k −xk (where x = Gb is the lattice point that corresponds to b). Denote by q the difference vector between x and the noisy observation y, i.e. q = y−x. Note that if b corresponds to the correct lattice point that was transmitted, q equals the channel noise vector w. Subtracting xk from both sides of (55), we finally get: (t+1) k = qk(1− α)− p(k,j) If we now arrange all the errors in a single column vector e, we can write: e(t+1) = −F · e(t) + (1− α)q (57) where F is an n× n matrix defined by: Fk,l = if k 6= l and there exist a row r of H for which |Hr,l| = h1 and Hr,k 6= 0 0 otherwise F is well defined, since for a given l there can be at most a single row of H for which |Hr,l| = h1 (note that α < 1 implies that h1 is different from all the other elements of the generating sequence). As discussed above, we embedded the sign in hi for conve- nience of notations, but when several equations are combined the correct signs should be used. It can be seen that using the notations of (57) resolves the correct signs of the hi elements. This completes the proof of Lemma 7. An alternative way to construct F from H is as follows. To construct the k’th row of F , denote by ri, i = 1, 2, ...d, the index of the element in the k’th column of H with value hi (i.e. |Hri,k| = hi). Denote by li, i = 1, 2, ...d, the index of the element in the ri’th row of H with value h1 (i.e. |Hri,li | = h1). The k’th row of F will be all zeros except for the d− 1 elements li, i = 2, 3...d, where Fk,li = Hri,k Hri,li APPENDIX VI ASYMPTOTIC BEHAVIOR OF THE AMPLITUDES RECURSION A. Proof of Lemma 9 From (10), a(t)i is clearly non-negative. From Sections IV- B, IV-C (and the appropriate appendices) it comes out that for consistent Gaussians, the mean values and variances of the messages have a finite bounded value and converge to a finite steady state value. The excitation term a(t)i depends on these mean values and variances according to (10), so it is also finite and bounded, and it converges to a steady state value, where caution should be taken for the case of a zero approaching variance. Note that at most a single variance in (10) may approach zero (as explained in Section IV-B, a single narrow check node message is used for the generation of narrow variable node messages, and only wide check node messages are used for the generation of wide variable node messages). The zero approaching variance corresponds to the message that arrives along an edge with weight ±h1, so assume that Ṽ approaches zero and all other variances approach a non-zero value. Then, V̂ (t)k,i also approaches zero and we have to show that the term , which is a quotient of zero approaching terms, approaches a finite value. Substituting for V̂ (t)k,i , we get: k,1→0 = lim k,1→0  1σ2 + j 6=i = lim k,1→0  Ṽ + 1 + j 6=i = 1 (59) Therefore, a(t)i converges to a finite steady state value, and has a finite value for every i and t. This completes the first part of the proof. We would now like to show that limt→∞ i=1 a i can be expressed in the form 1 (Gb − y)TW (Gb − y). Every variable node sends d− 1 narrow messages and a single wide message. We shall start by calculating a(t)i that corresponds to a narrow message. For this case, d− 1 check node messages take place in the sums of (10), from which a single message is narrow and d − 2 are wide. The narrow message arrives along the edge with weight ±h1, and has variance Ṽ k,1 → 0. Substituting in (10), and using (59), we get: (k−1)d+i →  d∑ j 6=i k,1 − m̃ k,1 − yk Denote x = Gb. The mean values of the narrow check node messages converge to the appropriate lattice point coordinates, i.e. m̃(t)k,1 → xk. From Theorem 3, the mean value of the wide variable node message that originates from variable node k converges to xk+ek, where e denotes the vector of error terms. The mean value of a wide check node message that arrives to node k along an edge with weight ±hj can be seen to approach k,j = xk − ep(k,j), where p(k, j) denotes the index of the variable node that takes place with coefficient ±h1 in the check equation where xk takes place with coefficient ±hj . For convenience of notations, we shall assume that hj already includes the sign (this sign ambiguity will be resolved later). The variance of the wide variable node messages converges to σ2(1 − α), so the variance of the wide check node message that arrives to node k along an edge with weight ±hj can be seen to approach Ṽ (t)k,j → σ2(1 − α). Substituting in (60), and denoting q = y − x, we get: (k−1)d+i →  d∑ j 6=i ep(k,j) σ2(1− α) (xk − yk)  =  11− α j 6=i e2p(k,j) + q2k  (61) Summing over all the narrow messages that leave variable node k, we get: (k−1)d+i → (62)  d− 2 e2p(k,j) + (d− 1)q2k To complete the calculation of the contribution of node k to the excitation term, we still have to calculate a(t)i that corresponds to a wide message. Substituting m̃(t)k,j → xk − ep(k,j), k,j → σ2(1− α), V̂ (t)k,1 → σ 2(1− α) in (10), we get: (k−1)d+1 → j=l+1 ep(k,l) − h1hj ep(k,j) σ2(1− α) xk − h1hl ep(k,l) − yk Starting with the first term, we have: j=l+1 ep(k,l) − h1hj ep(k,j) = (64) ep(k,l) − ep(k,j) e2p(k,l) + e2p(k,j) − 2 ep(k,l)ep(k,j) e2p(k,j) − ep(k,j) e2p(k,j) − (F · e) where F is defined in Theorem 3 and (F · e)k denotes the k’th element of the vector (F · e). Note that using F solves the sign ambiguity that results from embedding the sign of ±hj in hj for convenience of notations, as discussed above. Turning now to the second term of (63): xk − h1hl ep(k,l) − yk = (65) e2p(k,l) + q2k + 2qkep(k,l) e2p(k,l) + αq2k + 2qk (F · e)k = e2p(k,l) + αq2k + 2qk[(1− α)qk − ek] = e2p(k,l) + (2− α)q2k − 2qkek where we have substituted F e→ (1− α)q − e, as comes out from Lemma 7. Again, using F resolves the sign ambiguity of hj , as discussed above. Substituting (64) and (65) back in (63), summing the result with (62), and rearranging terms, the total contribution of variable node k to the asymptotic excitation sum term is: (k−1)d+i → 2σ2(1− α) e2p(k,j)+ (66) d+ 1− α q2k − 2σ2(1− α) (F e)2k − Summing over all the variable nodes, the total asymptotic excitation sum term is: (k−1)d+i → (d− 1)2 2σ2(1− α) ‖e‖2 + (67) d+ 1− α ∥∥q∥∥2 − 1 2σ2(1− α) ‖F e‖2 − Substituting e = (1 − α)(I + F )−1q (see Theorem 3), we finally get: qTW q (68) where: = (1− α)(I + F )−1 (d− 1)2I − F TF (I + F )−1+ +(d+ 1− α)I − 2(1− α)(I + F )−1 (69) From (10) it can be seen that i=1 a i is positive for every nonzero q. Therefore, W is positive definite. This completes the second part of the proof. Since a(t)i is finite and bounded, there exists ma such that |a(t)i | ≤ ma for all 1 ≤ i ≤ nd and t > 0. We then have: i=1 a (d− 1)2j+2 nd ·ma (d− 1)2j+2 n ·ma (d− 2) Therefore, for d > 2 the infinite sum will have a finite steady state value. This completes the proof of Lemma 9. APPENDIX VII GENERATION OF A PARITY CHECK MATRIX FOR LDLC In the following pseudo-code description, the i, j element of a matrix P is denoted by Pi,j and the k’th column of a matrix P is denoted by P:,k. # Input: block length n, degree d, nonzero elements {h1, h2, ...hd}. # Output: a magic square LDLC parity check matrix H with generating sequence {h1, h2, ...hd}. # Initialization: choose d random permutations on {1, 2, ...n}. Arrange the permutations in an d× n matrix P such that each row holds a permutation. c = 1; # column index loopless columns = 0; # number of consecutive # columns without loops # loop removal: while loopless columns < n changed permutation = 0; if exists i 6= j such that Pi,c = Pj,c # a 2-loop was found at column c changed permutation = i; # if there is no 2-loop, look for a 4-loop if exists c0 6= c such that P:,c and P:,c0 have two or more common elements # a 4-loop was found at column c changed permutation = line of P for which the first common element appears in column c; if changed permutation 6= 0 # a permutation should be modified to # remove loop choose a random integer 1 ≤ i ≤ n; swap locations c and i in permutation changed permutation; loopless columns = 0; # no loop was found at column c loopless columns = loopless columns+ 1; # increase column index c = c+ 1; if c > n c = 1; # Finally, build H from the permutations initialize H as an n× n zero matrix; for i = 1 : n for j = 1 : d HPj,i,i = hj · random sign; APPENDIX VIII REDUCING THE COMPLEXITY OF THE FFT CALCULATIONS FFT calculation can be made simpler by using the fact that the convolution is followed by the following steps: the convolution result p̃j(x) is stretched to pj(x) = p̃j(−hjx) and then periodically extended to Qj(x) = i=−∞ pj (see (3)). It can be seen that the stretching and periodic exten- sion steps can be exchanged, and the convolution result p̃j(x) can be first periodically extended with period 1 to Q̃j(x) =∑∞ i=−∞ p̃j (x+ i) and then stretched to Qj(x) = Q̃j(−hjx). Now, the infinite sum can be written as a convolution with a sequence of Dirac impulses: Q̃j(x) = p̃j (x+ i) = p̃j(x) ~ δ(x+ i) (70) Therefore, the Fourier transform of Q̃j(x) will equal the Fourier transform of p̃j(x) multiplied by the Fourier transform of the impulse sequence, which is itself an impulse sequence. The FFT of Q̃j(x) will therefore have several nonzero values, separated by sequences of zeros. These nonzero values will equal the FFT of p̃j(x) after decimation. To ensure an integer decimation rate, we should choose the PDF resolution ∆ such that an interval with range 1 (the period of Q̃j(x)) will contain an integer number of samples, i.e. 1/∆ should be an integer. Also, we should choose L (the number of samples in Q̃j(x)) to correspond to a range which equals an integer, i.e. D = L ·∆ should be an integer. Then, we can calculate the (size L) FFT of p̃j(x) and then decimate by D. The result will give 1/∆ samples which correspond to a single period (with range 1) of Q̃j(x). However, instead of calculating an FFT of length L and im- mediately decimating, we can directly calculate the decimated FFT. Denote the expanded PDF at the convolution input by f̃k, k = 1, 2, ...L (where the expanded PDF is zero padded to length L). To generate directly the decimated result, we can first calculate the (size D) FFT of each group of D samples which are generated by decimating f̃k by L/D = 1/∆. Then, the desired decimated result is the FFT (of size 1/∆) of the sequence of first samples of each FFT of size D. However, The first sample of an FFT is simply the sum of its inputs. Therefore, we should only calculate the sequence (of length 1/∆) gi = k=0 f̃i+k/∆, i = 1, 2, ...1/∆ and then calculate the FFT (of length 1/∆) of the result. This is done for all the expanded PDF’s. Then, d− 1 such results are multiplied, and an IFFT (of length 1/∆) gives a single period of Q̃j(x). With this method, instead of calculating d FFT’s and d IFFT’s of size larger than L, we calculate d FFT’s and d IFFT’s of size L/D = 1/∆. In order to generate the final check node message, we should stretch Q̃j(x) to Qj(x) = Q̃j(−hjx). This can be done by interpolating a single period of Q̃j(x) using interpolation methods similar to those that were used in Section VI for expanding the variable node PDF’s. ACKNOWLEDGMENT Support and interesting discussions with Ehud Weinstein are gratefully acknowledged. REFERENCES [1] C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, pp. 379-423 and pp. 623-656, July and Oct. 1948. [2] P. Elias, “Coding for noisy channels,” in IRE Conv. Rec., Mar. 1955, vol. 3, pt. 4, pp. 37-46. [3] C. E. Shannon, “Probability of error for optimal codes in a Gaussian channel,” Bell Syst. Tech. J., vol. 38, pp. 611-656, 1959. [4] R. E. Blahut, Theory and Practice of Error Control Codes. Addison- Wesley, 1983. [5] R. G. Gallager, Low-Density Parity-Check Codes. Cambridge, MA: MIT Press, 1963. [6] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error- correcting coding and decoding: Turbo codes,” Proc. IEEE Int. Conf. Communications, pp. 1064–1070, 1993. [7] R. de Buda, “The upper error bound of a new near-optimal code,” IEEE Trans. Inform. Theory, vol. IT-21, pp. 441-445, July 1975. [8] R. de Buda, “Some optimal codes have structure,” IEEE J. Select. Areas Commun., vol. 7, pp. 893-899, Aug. 1989. [9] T. Linder, Ch. Schlegel, and K. Zeger, “Corrected proof of de Buda‘s Theorem,” IEEE Trans. Inform. Theory, pp. 1735-1737, Sept. 1993. [10] H. A. Loeliger, “Averaging bounds for lattices and linear codes,” IEEE Trans. Inform. Theory, vol. 43, pp. 1767-1773, Nov. 1997. [11] R. Urbanke and B. Rimoldi, “Lattice codes can achieve capacity on the AWGN channel,” IEEE Trans. Inform. Theory, pp. 273-278, Jan. 1998. [12] U. Erez and R. Zamir, “Achieving 1/2 log(1 + SNR) on the AWGN channel with lattice encoding and decoding,” IEEE Trans. Inf. Theory, vol. 50, pp. 2293-2314, Oct. 2004. [13] A. R. Calderbank and N. J. A. Sloane, “New trellis codes based on lattices and cosets,” IEEE Trans. Inform. Theory, vol. IT-33, pp. 177- 195, Mar. 1987. [14] G. D. Forney, Jr., “Coset codes-Part I: Introduction and geometrical classification,” IEEE Trans. Inform. Theory, pp. 1123-1151, Sept. 1988. [15] O. Shalvi, N. Sommer and M. Feder, “Signal Codes,” proceedings of the Information theory Workshop, 2003, pp. 332–336. [16] O. Shalvi, N. Sommer and M. Feder, “Signal Codes,” in preparation. [17] A. Bennatan and D. Burshtein, “Design and analysis of nonbinary LDPC codes for arbitrary discrete-memoryless channels,” IEEE Transactions on Information Theory, volume 52, no. 2, pp. 549–583, February 2006. [18] J. Hou, P. H Siegel, L. B Milstein and H. D Pfister, “Capacity approaching bandwidth efficient coded modulation schemes based on low density parity check codes,” IEEE Transactions on Information Theory, volume 49, pp. 2141–2155, Sept. 2003. [19] J. H. Conway and N. J. Sloane, Sphere Packings, Lattices and Groups. New York: Springer, 1988. [20] G. Poltyrev, “On coding without restrictions for the AWGN channel,” IEEE Trans. Inform. Theory, vol. 40, pp. 409-417, Mar. 1994. [21] A. Papoulis, Probability, Random variables and Stochastic Processes. McGraw Hill, second edition, 1984. [22] Y. Saad, Iterative Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematic (SIAM), 2nd edition, 2003. [23] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger, “Closest point search in lattices,” IEEE Trans. Inf. Theory, vol. 48, pp. 2201-2214, Aug. 2002. [24] N. Sommer, M. Feder and O. Shalvi, “Closest point search in lattices using sequential decoding,” proceedings of the International Symposium on Information Theory (ISIT), 2005, pp. 1053–1057 . [25] H. El Gamal, G. Caire and M. Damen, “Lattice coding and decoding achieve the optimal diversity-multiplexing tradeoff of MIMO channels,” IEEE Trans. Inf. Theory, vol. 50, pp. 968–985, Jun. 2004.

0704.1318

The Haunted Halos of Andromeda and Triangulum: A panorama of galaxy formation in action

Draft version October 24, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 THE HAUNTED HALOS OF ANDROMEDA AND TRIANGULUM: A PANORAMA OF GALAXY FORMATION IN ACTION R. Ibata , N. F. Martin , M. Irwin , S. Chapman , A. M. N. Ferguson , G. F. Lewis , A. W. McConnachie Draft version October 24, 2018 ABSTRACT We present a deep photometric survey of the Andromeda galaxy, conducted with the wide-field cameras of the CFHT and INT telescopes. The surveyed area covers the inner 50 kpc of the galaxy and the Southern quadrant out to a projected distance of ∼ 150 kpc. A survey extension to M33 at > 200 kpc probes the interface between the halos of these two galaxies. This survey is the first systematic panoramic study of this very outermost region of galaxies. We detect a multitude of large- scale structures of low surface brightness, including several streams. Significant variations in stellar populations due to intervening stream-like structures are detected in the inner halo along the minor axis. This, together with the fact that the light profile between 0◦.5 < R < 1◦.3 follows the exponential “extended disk”, is particularly important in shedding light on the mixed and sometimes conflicting results reported in previous studies. Two new relatively luminous (MV ∼ −9) dwarf galaxies And XV and XVI are found in the study; And XVI is a particularly interesting specimen being located 270 kpc in front of M31, towards the Milky Way. Underlying the many substructures that we have uncovered lies a faint, smooth and extremely extended halo component, reaching out to 150 kpc, whose stellar populations are predominantly metal-poor. This is consistent with recent claims based on spectroscopy of a small sample of stars. We find that the smooth halo component in M31 has a radially-decreasing profile that can be fit with a Hernquist model of immense scale radius ∼ 55 kpc, almost a factor of 4 larger than theoretical predictions. Alternatively a power-law with ΣV ∝ R −1.91±0.11 can be fit to the projected profile, similar to the density profile in the Milky Way. If it is symmetric, the total luminosity of this structure is ∼ 109 L⊙, again similar to the stellar halo of the Milky Way. This vast, smooth, underlying halo is reminiscent of a classical “monolithic” model and completely unexpected from modern galaxy formation models where stars form in the most massive subhalos and are preferentially delivered into the inner regions of the galaxy. Furthermore, over the region surveyed, the smooth stellar halo follows closely the profile of the dark matter distribution predicted from earlier kinematic analyses. M33 is also found to have an extended metal-poor halo component, which can be fit with a Hernquist model also of scale radius ∼ 55 kpc. These extended slowly-decreasing halos will provide a challenge and strong constraints for further modeling. Subject headings: galaxies: individual (M31) — galaxies: individual (M33) — galaxies: structure — galaxies: evolution — Local Group 1. INTRODUCTION The outskirts of galaxies hold fundamental clues about their formation history. It is into these regions that new material continues to arrive as part of their on-going assembly, and it was also into these regions that ma- terial was deposited during the violent interactions in the galaxy’s distant past. Moreover, the long dynami- cal timescales for structures beyond the disk ensure that 1 Observatoire Astronomique, Universit de Strasbourg, CNRS, 11 rue de l’universit, 67000 Strasbourg, France 2 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany 3 Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA, U.K. 4 Institute for Astronomy, University of Edinburgh, Royal Ob- servatory, Blackford Hill, Edinburgh, UK EH9 3HJ 5 Institute of Astronomy, School of Physics, A29, University of Sydney, NSW 2006, Australia 6 Department of Physics and Astronomy, University of Victoria, Victoria, B.C., V8W 3P6, Canada ∗ Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France- Hawaii Telescope (CFHT) which is operated by the National Re- search Council (NRC) of Canada, the Institute National des Sci- ences de l’Univers of the Centre National de la Recherche Scien- tifique of France, and the University of Hawaii. the debris of accreted material takes a very long time to be erased by the process of phase mixing, which in turn means that we can hope to detect many of these signatures of formation as coherent spatial structures (Johnston, Hernquist & Bolte 1996). Much theoretical effort has been devoted in re- cent years to understanding the fine-scale structure of galaxies (Abadi et al. 2003; Bullock & Johnston 2005; Abadi et al. 2006), as researchers realized that cosmo- logical models could be tested not only with the clas- sical large-scale probes such as galaxy clusters, fila- ments and voids, but also with observations on galac- tic and sub-galactic scales (Freeman & Bland-Hawthorn 2002). Indeed, it is precisely in these latter re- gions that the best constraints on cosmology are expected to be put (Springel, Frenk & White 2006) in the coming decades. Λ-CDM cosmologies, in particular, are now sufficiently well developed the- oretically (e.g., Bullock, Kravtsov & Weinberg 2001; Bullock & Johnston 2005) that the Local Group provides a means of directly testing and constraining these theo- ries, by observing the profiles of density, age, and metal- licity of the structure and substructure predicted to be http://arxiv.org/abs/0704.1318v1 found in the outer parts of galaxy disks and in galaxy halos. 1.1. The Andromeda galaxy Andromeda, like the Milky Way, is a canonical galaxy, and a laboratory for examining in close detail many of the astrophysical processes that are investigated in the more distant field. Studying Andromeda and Triangu- lum in the Local Group has the advantage that it affords us a view free from the problems that plague Galactic studies due to our position within the Milky Way, yet their location within the Local Group allows us to re- solve and study individual stars and deduce population properties in incomparably greater detail than is possible in distant systems. Andromeda is the closest giant spiral galaxy to our own, and the only other giant galaxy in the Local Group. In many ways Andromeda is the “sister” to the Milky Way, having very similar total masses (including the dark matter, Evans et al. 2000; Ibata et al. 2004), having shared a common origin, and probably sharing the same ultimate fate when they finally merge in the distant fu- ture. However, there are significant differences between these “twins”. M31 is slightly more luminous than the Milky Way, it has a higher rotation speed, and a bulge with higher velocity dispersion. M31 possesses a globular cluster system with ∼ 500 members, approximately three times more numerous than that of the Milky Way. The disk of Andromeda is also much more extensive, with a scale-length of 5.9±0.3 kpc (R-band value corrected for a distance of 785 kpc, Walterbos & Kennicutt 1988) com- pared to 2.3±0.1 for the Milky Way (Ruphy et al. 1996); but which is currently forming stars at a lower rate than the Galaxy (Avila-Reese, Firmani & Hernández 2002; Walterbos & Braun 1994). There are indications that the Milky Way has undergone an exceptionally low amount of merging and has unusually low specific an- gular momentum, whereas M31 appears to be a much more normal galaxy in these respects (Hammer et al. 2007). Though possibly the consequence of low-number statistics, it is tempting to attribute significance to the fact that Andromeda has a compact elliptical (M32) and three dwarf elliptical galaxies (NGC 205, NGC 147, NGC 185) among its entourage of satellites, and no star- forming dwarf irregulars (dIrrs) within 200 kpc, whereas the Milky Way has no ellipticals but two dIrrs. How- ever, it is perhaps in their purported halo populations that the differences between the two galaxies are most curious and most interesting. 1.2. Comparing the halos of Andromeda and the Milky A large number of studies of the Milky Way halo (e.g., Ryan & Norris 1991; Chiba & Beers 2000, and references therein), have revealed that this structure is very metal- poor, with a median 〈[Fe/H]〉 = −1.6. It has a high veloc- ity dispersion, with (U, V,W ) values in the solar neigh- borhood of (141 km s−1 : 106 km s−1 : 94 km s−1), and a small prograde rotation of 30 – 50 km s−1. There is broad agreement that the stellar halo is flattened with b/a ∼ 0.6 (e.g., Morrison et al. 2000; Yanny et al. 2000; Chen et al. 2001; Siegel et al. 2002), though there are in- dications that the distribution becomes spherical beyond 15 – 20 kpc (Chiba & Beers 2000). The volume density profile and extent of this struc- ture have been harder to pin down. This is perhaps not surprising given the patchy sky coverage of most stud- ies, since current expectations are that the stellar halo is significantly lumpy (Bullock & Johnston 2005). The stellar volume density is generally modeled as ρ(r) ∝ r−α, and recent studies (Wetterer & McGraw 1996; Morrison et al. 2000; Yanny et al. 2000; Ivezic et al. 2000; Siegel et al. 2002; Vivas & Zinn 2006) have found values of the exponent ranging from α = 3.55 ± 0.13 (Chiba & Beers 2000) to α = 2.5±0.3 (Chen et al. 2001), with a general consensus of ρ(r) ∝ r−3. Note that in ex- ternal systems, where we observe the projected density, ρ(r) ∝ r−3 would correspond to Σ(R) ∝ R−2. Recent wide-field studies have gone a long way in improving our knowledge of the radial extent of the Milky Way halo. Using the SDSS database, Yanny et al. (2000) were able to follow A-colored stars in the halo to ∼ 25 kpc, and blue-straggler candidates out to ∼ 50 kpc. From the same survey, Ivezic et al. (2000) followed the profile of RRLyrae candidates, and found a sharp drop in the star-counts between 50 – 60 kpc, though this discon- tinuity in density has since been found to be due to the intervening stream of the Sgr dwarf galaxy (Ibata et al. 2001c). From VLT spectroscopy of 34 faint A-stars se- lected from the SDSS, Clewey et al. (2005) were able to show that the stellar halo extends out to at least 100 kpc, although again a sub-sample of their stars appears to be associated to the stream of Carbon stars emanating from the Sgr dwarf (Ibata et al. 2001a). Several other stud- ies have found evidence for further lumpy structures in the halo (e.g., Vivas & Zinn 2006; Martin et al. 2007a; Belokurov et al. 2007, and references therein). It has been believed for many years that M31 pos- sesses a stellar halo that is fundamentally different to that deduced from the above and earlier observations in the Milky Way. The first deep CCD studies by Mould & Kristian (1986) in a field in the inner halo of M31 found a surprisingly high mean metallicity of 〈[M/H]〉 = −0.6. While the surface brightness profile measured along the minor axis from integrated light (Pritchet & van den Bergh 1994) is consistent with a de Vaucouleurs R1/4-law out to R = 20 kpc, quite unlike the power-law behavior deduced for the halo of the Milky Way. Both the de Vaucouleurs profile and the high metallicity are suggestive of an active merger history at the time of halo (or bulge) formation. The existence of the metal-rich halo population was confirmed by several subsequent studies; notably among these the wide-field (0.16 deg2) photometric study by Durrell, Harris & Pritchet (2001) in a location 20 kpc out along the minor axis. In addition to the main 〈[M/H]〉 = −0.5 component, Durrell, Harris & Pritchet (2001) also discovered that 30-40% of of the stars at this location belong to a metal-poor population. The surface density of the metal-poor sub-sample falls off rapidly as Σ(R) ∝ R−5.25±0.63, but slower than the Σ(R) ∝ R−6.54±0.59 relation for the metal-rich sub- sample. These results were later complemented by the same authors with a minor axis field at R = 30 kpc (Durrell, Harris & Pritchet 2004) , which showed essen- tially identical abundance properties to their 20 kpc field, leading them to conclude that the outer halo shows little or no radial metallicity gradient. As an alternative to the above “wide-field” ap- proach, Bellazzini et al. (2003) analyzed a set of 16 HST/WFPC2 fields with much deeper photometry, mostly in and around the M31 disk, but with some fields extending out to a distance of 35 kpc. Through- out this area they detect the previously-discussed dom- inant metal-rich component with [Fe/H] ∼ −0.6, but also an additional high metallicity component with [Fe/H] ∼ −0.2. Interestingly, they found that the frac- tion of metal-poor stars is constant from field to field, though metal-rich stars are enhanced in regions contain- ing substructure, especially along the extended path of the Giant Stream (Ibata et al. 2001b). The inclusion of kinematic information has been ex- tremely useful, but has also added another dimension of complexity to the puzzle. Reitzel & Guhathakurta (2002) analyzed a sample of 29 stars in a field at R = 19 kpc on the minor axis, and found the mean metallicity to be in the range 〈[M/H]〉 = −1.9 to −1.1, dependent on calibration and sample selection issues, but significantly lower than the results deduced from the above photomet- ric analyses. A wider-field view was obtained by Chapman et al. (2006), who sampled the halo at 54 locations between 10 – 70 kpc, isolating 827 out of a sample of ∼ 104 stars as having kinematics consistent with being halo mem- bers. The population was found to have 〈[Fe/H]〉 ∼ −1.4 with a dispersion of 0.2 dex, indicating that kinematic se- lection reveals a halo similar to that of the Milky Way un- derneath the “halo” substructures, which in many cases are metal-rich, and in general cannot have halo-like kine- matics. The (central) velocity dispersion of 152 km s−1 deduced from the sample, is also comparable to that of the Milky Way. In an impressive effort of finding needles in a haystack, Kalirai et al. (2006b) and Gilbert et al. (2006) extended the kinematic coverage out to 165 kpc, and claim a de- tection of the halo at R > 100 kpc based on a sample of 3 stars. To minimize contamination they implemented a complex non-linear algorithm to assign likelihoods to the observed stars, and as the algorithm was trained on the inner region of M31, the biasses for the outer halo population are not well known. 1.3. The Triangulum galaxy If Andromeda is the twin of the Milky Way, the Triangulum galaxy (M33) with a mass ∼ 10 times lower than either of these two giants, is their little sister. M33 is the third brightest galaxy in the Lo- cal Group (MV = −18.9), and probably a satellite of M31. The relatively undisturbed optical appearance of M33 places strong constraints on the past interaction of these two galaxies (Loeb et al. 2005), though it should be noted that the gaseous component is extremely warped (Rogstad, Wright & Lockhart 1976). The early CCD study of the halo of M33 by Mould & Kristian (1986) claimed an inner halo compo- nent with a more “normal” metallicity (〈[M/H]〉 = −2.2) than deduced for M31. In reality however, this field lies within the disk of M33 and does not probe the “halo”, as we show below in §9. Further progress in understanding the elusive halo component of this galaxy was only achieved recently. In their kinematic study of star clusters in M33, Chandar et al. (2002) find evidence for two sub-populations, with old clusters showing evi- dence for a large velocity dispersion, which they inter- pret as the sign of a halo population. Further signs of this halo component were detected in the spectroscopic study of McConnachie et al. (2006) with Keck/DEIMOS, who distinguished halo field stars from stars in the disk via their kinematics, and deduce a mean metallicity for the halo component of 〈[Fe/H]〉 = −1.5, with a narrower spread of abundance than the disk stars. 1.4. Halos of more distant disk galaxies Due to their extremely faint nature the halos of spiral galaxies beyond the Local Group have been extremely challenging to observe. A major advance in detecting extra-planar light in distant galaxies was made by stack- ing 1047 edge-on spiral galaxies observed in the SDSS (Zibetti, White & Brinkman 2004). The resulting stack showed a flattened (c/a ∼ 0.6) distribution with a power- law density profile ρ(r) ∝ r−3, similar to the properties of the halo of the Milky Way deduced from the stud- ies reviewed above. This structure could be detected out to approximately 10 exponential scalelengths of the disk (i.e., approximately 25 kpc for the case of the Milky Way). An analogous structure was also detected di- rectly from the surface brightness around a single isolated galaxy in an ultra-deep HST survey (Zibetti & Ferguson 2004). Extra-planar populations have also been detected via star-counts of resolved RGB populations in nearby (< 10Mpc) galaxies from deep HST imaging. Notable among these is the survey of Mouhcine et al. (2005a,b), who employed WFPC2 to survey 8 nearby spirals. Their fields probed the minor axis halo out to R = 13 kpc. Interestingly, they find a correlation between galaxy lu- minosity and the metallicity of the extra-planar popula- tion, with low luminosity galaxies containing metal-poor stars with a narrow abundance spread, while luminous galaxies contain metal-rich stars and a wide abundance spread. Their results for galaxies of similar luminosity to M31 are in good agreement with the metallicity distri- bution of minor axis fields in Andromeda at 10 – 20 kpc. However, as we will show below, the minor axis fields in M31, from which most of the information on the “halo” or “spheroid” is derived, do not directly probe that component. Furthermore, as we have reviewed above, kinematically-selected halo stars in M31 display a simi- lar metallicity to genuine halo stars in the Milky Way (Chapman et al. 2006). These considerations suggest that the Mouhcine relation is caused by small structures accreted into the inner regions of the halo, and which are largely supported by rotation, rather than random motions. The correlation of the metallicity of the extra- planar stars with galaxy luminosity found by Mouhcine et al. may then simply reflect that more massive host galaxies are able to accrete larger dwarf galaxies which themselves have a higher metallicity. Nevertheless, we stress that all of these observations beyond the Local Group are derived from regions close to the centre of the galaxy, and there is concern that con- tamination from other components, such as streams or a warped disk could be affecting the observations. Extend- ing further out in radius, as we will do in this contribu- tion, will allow us to eliminate this uncertainty. But most importantly it will allow us to examine a different region of the halo, one that is less dominated by the remnants of massive accretions. 1.5. Theoretical motivation Several theoretical studies have been undertaken in re- cent years to attempt to understand and reproduce the above observations and to make useful predictions for the next generation of surveys. Bullock & Johnston (2005) implemented a hybrid N- body plus semi-analytic approach. Their simulations provide very high spatial resolution compared to the other studies discussed below, which they achieve by concentrating on each merger event in turn, with the rest of the galaxy modeled with analytic (but time vary- ing) potentials. The drawback of this method is that the dynamical evolution of the system is not fully self- consistent, and star-formation is implemented with em- pirical recipes. They find that the present-day density profile of stars within 10 kpc of a Milky Way or M31- like galaxy should be shallow, ρ(r) ∝ r−1, steepen- ing to ρ(r) ∝ r−4 beyond 50 kpc, resembling a Hern- quist profile with scale radius of ∼ 15 kpc. They also find that the bulk of the stars that constitute a stellar halo were formed more than 8Gyr ago, with most of these stars originating from massive accretions (Mvir > 2×1010M⊙). Beyond 30 kpc, substructure begins to pre- dominate in their simulations, and they find that most of the stars beyond this radius arrived after the last major merger. The problem of stellar halo formation was also tackled by Renda et al. (2005), who used a chemodynamical code to treat self-consistently gravity, gas dynamics, radiative cooling, star formation and chemical enrichment. The drawback of this approach was a very much lower spa- tial resolution compared to Bullock & Johnston (2005). Renda et al. (2005) find a large (∼ 1 dex) spread in the mean metallicity of halos of galaxies of a given (final) luminosity, where the large variations in the metallicity distribution between their galaxy models is related to the diversity in the galactic mass assembly history. This is somewhat at odds with the finding that M31 and the Milky Way have underlying halos of similar metallicity (Chapman et al. 2006). They also find that a more ex- tended assembly history gives more massive stellar halos, and a higher halo surface brightness. Yet another approach was adopted by Abadi et al. (2006), who undertook SPH simulations that follow the gas evolution in a small sample of galaxy models form- ing in a ΛCDM cosmology. Overcooling early on leads to large spheroid component in their simulations, though they claim that the insensitivity of the halo parameters to the final stellar halo mass implies that their simu- lations are also applicable to Milky Way-like systems. In their models stars formed in situ in the galaxy are all confined to the inner luminous region, while accreted stars dominate beyond 20 kpc, and are the main popula- tion contributing to the spheroid. The stellar surface density profile is very similar in all their simulations, and has Σ(R) ∝ R−2.3 at r ∼ 20 kpc, steepening to Σ(R) ∝ R−2.9 at r ∼ 100 kpc, and steepening further to at Σ(R) ∝ R−3.5 near the virial radius. Furthermore, they find that the stellar halo is a mildly triaxial struc- ture (〈c/b〉 = 0.90, 〈c/a〉 = 0.84, with no obvious align- ment of the triaxial halo with the angular momentum vector of the galaxy. Old stars disrupted in the early history of the galaxy are ejected into highly eccentric and energetic orbits during close perigalactic passages, and it is these stars that primarily populate the outer halo. Complementary studies using pure N-body simula- tions were undertaken by Diemand et al. (2005) and Gauthier, Dubinkski & Widrow (2006). Diemand et al. (2005) focus on the evolution of high density peaks in cosmological simulations that formMilky Way-like galax- ies. Their result of relevance to the present study is the asymptotic density profile of these peaks in the galaxy simulation: they find that the outer profile behaves as ρ(r) ∝ r−3.26 for 1σ peaks, steepening to ρ(r) ∝ r−4.13 for 2.5σ and ρ(r) ∝ r−5.39 for 4σ peaks. In contrast, Gauthier, Dubinkski & Widrow (2006) simulate the evo- lution of satellites around a fully-formed M31-like galaxy, with the satellites modeled as a collection of NFW den- sity profiles (Navarro, Frenk & White 1997). They do not consider star-formation in the satellites, but instead identify the 10% most bound particles as tracers of the stars in the satellite. They predict that disrupted satel- lites give rise to a halo luminosity profile that falls as ρ(r) ∝ r−3.5 at large radii. Since massive satellites cor- respond to rare overdense peaks in cosmological simu- lations, the difference in the profile slope compared to Diemand et al. (2005) suggests that taking the full cos- mological evolution of the host galaxy into account is important. 1.6. Purpose of the present study In this contribution we are building upon an ear- lier wide-field survey of Andromeda with the Wide Field Camera camera at the Isaac Newton Telescope (Ibata et al. 2001b; Ferguson et al. 2002; Irwin et al. 2005). This panoramic survey covered the entirety of the disk and inner halo of the galaxy out to ∼ 55 kpc (see Fig. 1), which combined with follow-up kinematics from Keck/DEIMOS (Ibata et al. 2004, 2005; Chapman et al. 2006) and deep HST/ACS photometry in selected fields (Ferguson et al. 2005; Faria et al. 2007) opened up a new violent vision of an apparently normal disk galaxy. We found that M31 possesses of order half a dozen substructures, probably debris fragments from merg- ing galaxies that have not yet lost all spatial coher- ence (Ferguson et al. 2002, 2005); that it is surrounded by a vast rotating disk-like structure, extending out to ∼ 40 kpc (Ibata et al. 2005); that it contains a giant stel- lar stream of width greater than the diameter of the disk of the Milky Way and > 100 kpc long (Ibata et al. 2001b; McConnachie et al. 2003); and that underlying all of this substructure there is a kinematically hot, metal- poor halo (Chapman et al. 2006). Thus the inner halo region covered by the INT survey is completely contaminated by these various structures. Indeed it was a surprising result of that survey that it is necessary to observe at much larger radius to obtain a clear measurement of the accretion rate, the incidence of sub-structures, the stellar mass of the accreted objects, and the global properties of the halo. We therefore em- barked on the deep imaging campaign of the outer halo presented in this contribution, undertaken with Mega- Fig. 1.— The coverage of our large panoramic survey of M31 with the INT camera, in standard coordinates (ξ, η). The inner ellipse represents a disk of inclination 77◦ and radius 2◦ (27 kpc), the approximate end of the regular HI disk. The outer ellipse shows a 55 kpc radius ellipse flattened to c/a = 0.6, and the major and minor axis are indicated with straight lines out to this ellipse. This map is constructed from a total of 164 INT/WFC individual pointings. Cam, a state-of-the-art wide-field camera at the CFHT. One of the main aims of the present survey was to in- vestigate the prediction of CDM cosmology that upward of 500 satellites reside in the halo of a galaxy like M31 (Klypin et al. 1999; Moore et al. 1999). The possibility remains that many dwarf galaxies are being missed in current surveys. However we defer all discussion of this issue to a companion paper (Martin et al. 2007b). The layout of this paper is as follows. In §2 we first present the photometric data and data processing. The color-magnitude distribution of detected sources is dis- cussed in §3, and their spatial distribution in §4. The re- sulting maps of the stellar populations of interest are pre- sented in §5, continuing in §6 with the detected streams and other spatial substructures, and in §7 with the prop- erties of the outer halo. The radial profiles of the stellar populations in M31 are analyzed in §8. A short discus- sion of the properties of the halo of M33 are presented in §9. Finally in §10 we discuss the implications of our findings and compare to previous studies, and draw con- clusions in §11. Throughout this work, we assume a distance of 785 kpc to M31 (McConnachie et al. 2005). We also adopt the convention of using R to denote projected radius, s an elliptical projected radius, and r a three-dimensional dis- tance or radius. 2. OBSERVATIONS 2.1. INT observations The Wide Field Camera (WFC) of the Isaac Newton Telescope (INT) was used in four observing runs between 1998 and 2003 to map the Andromeda galaxy over the area displayed in Fig. 1. The observations were taken with the V and i filters, with exposures of 1200 sec and 900 sec, respectively, in each of these two bandpasses. The data were obtained in dark skies, with typical seeing of 1′′. A total of 164 individual fields were observed, each Fig. 2.— The main area surveyed with the CFHT MegaCam instrument. As we describe below, the image stability over the field of view of the camera varied slightly from one year to another. We therefore show the year that the field was observed in by a color code: red, green and black mark fields obtained in 2003, 2004 and 2005–2006, respectively. The field T6, centered on M33, was observed in primarily in 2004, with some data in 2003. The offset fields colored turquoise mark the positions of the short exposure fields. In the case of field H13, we also display the layout of the 36 CCDs. The meaning of the ellipses centered on M31 is described in Fig. 1. covering an “L”-shaped region of 0.33 deg2. A small ∼ 5% overlap between adjacent fields was adopted to ensure a homogenous photometric survey. The images were processed by the Cambridge Astro- nomical Survey Unit (CASU) pipeline (Irwin & Lewis 2001), in an identical manner to that described in Ségall et al. (2006). This includes corrections for bias, flat-fielding, and for the fringing pattern. The software then proceeds to detect sources, and measures their pho- tometry, the image profile and shape. Based upon the information contained in the curve of growth, the algo- rithm classifies the objects into noise detections, galaxies, and probable stars. (For comparison to previous studies using this classification algorithm, throughout this paper we adopt as stars those objects that have classifications of either -1 or -2 in both colors; this corresponds to stars up to 2σ from the stellar locus). 2.2. CFHT observations The survey of the inner halo of M31 with the INT was complemented with a deeper survey with the CFHT MegaCam wide-field camera to probe the outer reaches of the halo of this galaxy. MegaCam consists of a mosaic of 36 2048 × 4612 pixel CCDs, covering a 0.96◦ × 0.94◦ field, with a pixel scale of 0.187 arcsec/pixel. The greater photometric depth and field-of-view achievable with this instrument makes it particularly powerful in such regions of extremely low surface density of stars. The g and i- band filters were used, totalling 5 × 290 sec of exposure per field in each passband. Figure 2 displays the survey fields, while Fig. 3 shows this area in relation to the en- vironment around M31. The survey comprises 89 deep fields, observed in service mode over the 2003 to 2006 sea- sons. We chose a tiling pattern with no overlap between the deep fields, using instead short (45 sec) exposures in g Fig. 3.— The survey region (irregular blue polygon) is overlaid on a schematic diagram of M31 and surrounding Local Group structure. Note that the survey extension along the M31 minor axis reaches M33 and therefore probes the halos of both these disk galaxies. In addition to the ellipses reproduced from Fig. 1, the two concentric (dashed-line) circles show projected radii of 100 kpc and 150 kpc. A grid in Galactic longitude and latitude has been marked. The extinction over the surveyed region, interpolated from the maps of Schlegel et al. (1998) is also shown. and i to establish a consistent photometric level over the survey. These short exposure images were taken offset by half a field size in the right-ascension and declination directions. The fields were observed in photometric con- ditions in good seeing conditions (typically better than 0′′.8). In addition, the two inner halo fields marked H11 and H13 were retrieved from the CFHT archive. These g and i-band images are somewhat deeper that the main survey fields with exposures of 5 × 289 sec in each pass- band. A further field centered on M33 (marked field T6 in Fig. 2) was obtained from the archive. After elimina- tion of frames with poorer seeing (> 1′′) or CCD con- troller problems, 37 g-band frames and 32 i-band frames were combined, for a total of 18306 sec in the g-band and 19165 sec in the i-band. The solid angle covered by the INT survey corresponds to a projected area of ∼ 9500 kpc2 at the distance of M31 (∼ 7400 kpc2 not overlapping with the MegaCam survey), while the MegaCam survey area subtends 1.6× 104 kpc . This vast area encompasses several previously known structures, as we show in Fig. 3. These are the dwarf galaxies M32, NGC 205, And I, And II (though we miss its center), And III, And IX, as well as the new discoveries from this work: And XI, And XII, And XIII, all discussed in (Martin et al. 2006), and And XV, and And XVI presented below. We also mark the positions of the known globular clusters in the MegaCam region: GC 5, GC 6, EC 4 (Mackey et al. 2006, 2007), and GC- M06 (Martin et al. 2006). In addition to the INT fields and the 92 contiguous MegaCam fields, we consider below two additional fields, which will be used as background references: a compari- son field taken for a study of the Draco dwarf spheroidal (dSph) galaxy (field D7 of Ségall et al. 2006, located at ℓ = 81◦.5, b = 34◦.9), and the field D3 of the Legacy Sur- vey of the CFHT (CFHTLS). The observations on the Draco dSph comparison field had slightly different expo- sure times to those taken for the M31 survey (950 s in g and 1700 s in i), though similar image quality. From the public release data of the CFHTLS field D3 (located at ℓ = 96◦.3, b = 59◦.7), we selected a subset of the best seeing frames, totaling 2702 s in the g-band and 4520 s in the i-band. The MegaCam data were pre-processed by CFHT staff using the “Elixir” pipeline; which accomplishes the usual bias, flat and fringe corrections, and also determines the photometric zero-point of the observations. These im- ages were then processed by the Cambridge Astronomical Survey Unit photometry pipeline in an identical manner to that described above for the INT data. Using the mul- tiple overlaps between deep and shallow fields we correct the photometric solution provided by the “Elixir” algo- rithm (by up to ∼ 0.5 mag), finding a global solution over all 92 deep fields that has an RMS scatter of 0.02 mags. Using observations of the Draco dwarf spheroidal galaxy for which we had both INT-WFC and CFHT- MegaCam data in the (V,i) and (g,r,i) bandpasses, re- spectively, we determined colour transformations to put the INT (Vega-calibrated) photometry onto the Mega- Cam AB photometric system. The advantage of using the Draco field is that the region has also been covered by the Sloan Digital Sky Survey (SDSS), providing an external check to the photometry. Note that the Mega- Cam (g, i) bands are not identical to the SDSS (g′, i′), though the conversions between these two systems have been determined by the CFHT staff. We refer the in- terested reader to Ségall et al. (2006) for further details. The conversion between INT (V,i) and MegaCam (g,i) were found to be: iMC = iINT − 0.105 , gMC = 0.030 + 1.400× (V − i)INT + iMC for (V − i)INT < 1.3 , 0.491 + 1.046× (V − i)INT + iMC for (V − i)INT > 1.3 . In order to enable the construction of maps over the combined area of the INT and CFHT surveys, we con- verted the INT photometry to (g,i) using these relations. The conversion appears to be adequately accurate, judg- ing from the photometry of bright stars (with magnitudes in the range 18 < g < 20 and 18 < i < 20) in the large overlap region between the two surveys: the RMS scatter around zero offset was found to be < 0.02 mags in both bands. Given the huge area of the survey it is necessary to be aware of variations in the interstellar extinction which will affect the depth of the photometry. In Fig. 3 the surveyed area is superimposed on a map of the extinc- tion derived from Schlegel et al. (1998); the maximum i- band extinction over the halo region observed with Mega- Cam is Ai = 0.27 mags, with a mean of Ai = 0.1 mags. Thus the extinction is neither very high nor very vari- able, though we nevertheless correct for it using the Schlegel et al. (1998) maps. In all the discussion below, g0 and i0 will refer to extinction-corrected magnitudes. 3. COLOR-MAGNITUDE DISTRIBUTION OF SOURCES As well as encompassing a large fraction of the halo of M31, the survey also intersects a substantial volume of the foreground Milky Way. This is clearly seen in Fig. 4.— The combined CMD of the MegaCam survey fields of M31, except fields T5 and T6 which are excluded because they are dominated by stars from M33 (including young stars in the disk), and fields 6, H11, and H13 which are close to the M31 disk. The fiducial RGBs correspond to, from left to right, NGC 6397, NGC 1851, 47 Tuc, NGC 6553, which have metallicity of [Fe/H] = −1.91, −1.29, −0.71, and −0.2, respec- tively. The sequences have been shifted to a distance modulus of (m−M)0 = 24.47. The dashed-line rectangles show the regions selected to probe the foreground Galactic halo (blue) and Galactic disk (red). Fig. 4, where we show the combined color-magnitude distribution of all stars in the deep MegaCam fields of the main survey, except for fields T5 and T6, close to M33, and fields 6, H11 and H13 close to M31. Prominent at (g − i)0 > 1.5 and i0 < 23 is the sequence of Galactic disk dwarfs; the vertical sequence is the result of low- mass stars accumulating in a narrow color range, yet be- ing seen over a large range in distance along the line of sight. In addition, on the blue side of this diagram, at (g − i)0 < 0.8 and i0 < 23, resides the Galactic halo se- quence. Usually, this is seen as a smooth vertical struc- ture, due to stars at or close to the main-sequence turnoff at increasing distance through the Galactic halo. Curi- ously, however in these fields towards M31 the sequence bifurcates — indicating that the Galactic “halo” is not spatially smooth along this line of sight. This issue is explored in detail in a companion article (Martin et al. 2007a). The stellar populations of immediate interest to this study are revealed by the red giant branch (RGB) stars that span the globular cluster fiducial sequences that have been overlaid on the CMD. The bluemost and redmost sequence correspond to clusters of metallicity Fig. 5.— The left and right panels show the distributions of photometric uncertainty in g0 and i0, respectively, together with simple exponential fits (red lines). Some fields have slightly better photometry than others, giving rise to the inhomogenous aspect at faint magnitudes. [Fe/H] = −1.91 and [Fe/H] = −0.2, respectively, so the survey is sensitive to stars of a wide range of abun- dance. At the limiting magnitude of i0 ∼ 24.5, the sur- vey can in principle detect horizontal branch stars (see Martin et al. 2006), though of course the contamination at these magnitudes, mostly from unresolved background galaxies and noise artifacts, is very large. Nevertheless down to i0 ∼ 24.0 the photometric quality remains excel- lent, as we show in Fig. 5, with δi < 0.1 mag. There are substantial variations of stellar populations between fields, as we demonstrate in Fig. 6. Here, panel ‘a’ displays the CMD of field 46, which lies in a dense area of the so-called “Giant Stream” (Ibata et al. 2001b), and clearly contains a numerous population of RGB sources with a wide spread of metallicity. Panel ‘b’ shows the photometry of field 106 in the far outer halo; no obvious RGB is discernible visually in this diagram, though as we shall see later in §7, the combination of this with several other outer fields does allow a detection of the stellar halo of M31. For comparison, we also display the CMDs of the reference fields near the Draco dSph (panel ‘c’) and the CFHTLS field D3 (panel ‘d’). The photometric depth of the survey clearly varies slightly from field to field (note that the images from which the CMDs in panels ‘a’ and ‘b’ were constructed had identical exposure times). The data taken in the 2005 and 2006 runs (of which panel ‘b’ is an example) were very homogenous in depth, whereas the earlier 2003 and 2004 runs were more patchy. It is likely that the improvement in the 2005 and 2006 seasons was a result of the correction of the detector plane tilt ††, allowing a uniform focus to be achieved over the 0◦.96× 0◦.94 field of view. (For comparison to Fig. 6, in Fig. 7 we show the color-magitude distribution of sources classified as galaxies). Though the globular cluster RGB ridge-lines shown in Figs. 4 and 6 are useful to show the behavior of known stellar populations, the set of 4 templates is too sparse to allow accurate comparisons to be made with the distant †† See http://www.cfht.hawaii.edu/News/Projects/MPIQ/ Fig. 6.— The upper panels show sample CMDs of point-sources in the MegaCam survey. The panel ‘a’ is for field 46, in a dense region within the giant stream, while panel ‘b’ is for field 106, in the outer halo. The lower panels correspond to the comparison fields: ‘c’ lies near the Draco dSph, while ‘d’ is contructed from the CFHTLS field D3. As in Fig. 4, the lines in panel ‘a’ are the RGB ridge-lines of globular clusters of metallicity (from left to right) [Fe/H] = −1.91, −1.29, −0.71, and −0.2. The dense grouping of objects with −0.5 < (g − i)0 < 1.5 are mostly due to misclassified compact galaxies. M31 population. Instead we chose to adopt the Padova isochrones (Girardi et al. 2004), which conveniently have been calculated in the Sloan passbands. Figure 8 shows the isochrones we used, converted into the MegaCam photometric system, which were chosen for a population age of 10Gyr. For each star in the survey, a photomet- ric metallicity was calculated by interpolating between the RGB curves. The assumption that the stellar pop- ulations have an age of 10Gyr over the entirety of the survey is clearly incorrect (Brown et al. 2006b), but this is probably a reasonable estimate for the majority of the stars at large radius. As we have shown in Fig. 3, the region surveyed with MegaCam includes several known sources. For compari- son to the populations we will encounter below, we dis- play their CMD structure in Fig. 9. 4. SPATIAL DISTRIBUTION OF SOURCES Although the MegaCam camera covers a large area, there are large inter-CCD gaps in the mosaic, that were not filled by our chosen dithering pattern with 5 sub- exposures. These gaps are partially filled by the short exposures, but of course reach to a much shallower lim- iting depth. These inter-CCD gaps are seen in Fig. 10, http://www.cfht.hawaii.edu/News/Projects/MPIQ/ Fig. 7.— As Fig. 6, but for sources classified as galaxies by the image analysis algorithm. Fig. 8.— The Padova isochrones superimposed on the CMD of field 47. The isochrone models are all for 10Gyr, and [Fe/H] metal- licity (from left to right) of −3 (actually Z = 0), −2.3, −1.7, −1.3, −0.7, −0.4, 0.0 and +0.2. The continuous line part of each of these curves corresponds to the RGB, while the horizontal branch and asymptotic giant branch are indicated with dashed lines. Fig. 9.— CMDs of known satellite galaxies in the MegaCam survey region. The Padova isochrones from Fig. 8 are repro- duced here. For M33 we show the sources within an annulus between 1◦ and 2◦ while for And II, And III and the remaining dwarfs, we show the sources within a circular region of 12′, 6′, and 12′, respectively. For the purposes of overlaying the isochrones, we adopt the following distance moduli. M33: 24.54 ± 0.06; And II: 24.07±0.06 (both from McConnachie et al. 2004a; And III: 24.37± 0.07 (McConnachie et al. 2005); while for And XI, XII and XIII (Martin et al. 2006) we assume the distance modulus of M31: 24.47± 0.07 (McConnachie et al. 2005). which shows the stellar density in one of the MegaCam fields. Another problem that is not limited to the Mega- Cam data are the “halos” of bright stars that effectively render useless certain regions of the detector mosaic. The effect of these halos is also illustrated in Fig. 10. Both the gaps and bright star holes could easily be accounted for in the analysis of the surface density, by simply correcting for the missing area. However, we found this approach to be somewhat unsatisfactory when making maps of spatial resolution smaller than the area of the bright star “ha- los”. Instead we chose to replace the affected areas with nearby counts: the inter-CCD gaps were filled with the detections of the CCD immediately to the South, while the bright star halos were filled with detections either to the East or West of the hole (depending on the location of the field edge or other nearby bright stars). Figure 10 shows an example of the procedure adopted. A further problem was that in several fields observed in 2003 the data for CCD 4 of the MegaCam mosaic was absent due to a CCD controller malfunction. For these fields, which comprise fields 48, 63, 77, 92, H11, H13, T2, T3, T4 and T5, we copied over the sources from CCD 3, adjacent on the mosaic. All the sources that were added artificially in these various ways were flagged. The final catalog contains a total of 19 million sources. However, many of these sources are foreground and back- ground contaminants, so we must assess their numbers and distribution before being able to analyze the dis- tribution of genuine M31 stars. In Fig. 11 we show the spatial distribution of Galactic disk dwarf stars with 1.5 < (g−i)0 < 3.0 and 15.0 < i0 < 19.5; from an inspec- tion of Fig. 4 it can be seen that these stars are located at brighter magnitudes than the tip of the M31 red giant branch (RGB) and should therefore be an almost pure Galactic sample. Figure 11 shows that this is not entirely Fig. 10.— As an example of our correction technique for the effect of bright stars, we show in the left-hand panel the distribution of stellar sources in field 70, a field containing several unusually bright stars. The two horizontal gaps are due to a physical gap between the first two and the last two rows of detectors on the mosaic camera. The lower source density at ξ ∼ −1◦.1, η ∼ −5◦.9 is due to a bright star “halo”. In the right-hand panel, we show the corrected counts in this region, where the stars in the affected region have been deleted, and replaced with artificial sources (red points) that were copied from adjacent areas of the sky. Fig. 11.— The distribution of stars within the color-magnitude selection box 1.5 < (g − i)0 < 3.0 and 15.0 < i0 < 19.5, which outside of the inner regions of M31 and M33, which contain blue loop and AGB stars, gives a clean sample of Milky Way disk dwarf stars. The map is a linear representation of the star counts, with pixels of size 0◦.1× 0◦.1. correct, as a strong enhancement of sources is seen in the inner regions of M31 and M33, due to the presence of blue loop stars and asymptotic giant branch (AGB) stars in the disks of those galaxies. Ignoring these disk regions, we detect a smooth gradient towards the Galactic plane in the North, with no obvious structures. In addition to the Galactic disk dwarfs, there is some contamination from distant bright main-squence halo stars, as we showed in Fig. 4. We select a representa- tive sample of this population by choosing stars within Fig. 12.— As Fig. 11, but showing the distribution of Milky Way halo stars over the survey region, selected within the color- magnitude box 0.0 < (g − i)0 < 0.8 and 20.0 < i < 22.5. The concentration of sources inside the disks of M31 and M33 is due to young blue main sequence stars in those galaxies. the box 0.0 < (g − i)0 < 0.8 and 20.0 < i0 < 22.5. The resulting spatial distribution is presented in Fig. 12. The contamination to this sample from the disks of M31 and M33 is not at all surprising, as young blue supergiant stars in these galaxies will fall into this color-magnitude selection box. However, excluding a 2◦ and 1◦ circle around M31 and M33, respectively, shows the remain- ing Galactic population to have a very uniform density over the survey region. A further source of contaminants are background galaxies. Most of these are readily identifiable from their image parameters, though there will be some distant compact galaxies that are unresolved with the typical depth and seeing achieved in this survey. The map of the sources classified as galaxies by the algorithm is displayed in Fig. 13. Apart from the usual filamentary signature of large-scale structure there is no apparent correlation with either the Milky Way, Andromeda or M33, beyond the disks of these latter two galaxies (where some sources are classified as being extended due to image crowding). The colour-magnitude distribution of these contaminants is displayed in Fig. 7 for four selected fields. These re- solved galaxies are approximately as numerous as the point-sources in the dense Giant Stream fields, but be- come up to 6 times more numerous than point-sources in the outer halo fields. Clearly a small error in image classification towards fainter magnitudes could have a significant repercussion in the measured density of point- sources. We return to this issue below. 4.1. Foreground subtraction We had envisaged using the MegaCam comparison fields presented in Fig. 6 to subtract off the background counts, however since the Galactic contamination varies substantially from these fields to our M31 fields of in- terest, and even varies significantly over the main area of this vast survey, we decided to investigate whether Galactic models could be used instead to predict the contamination more reliably. To this end we tessellated the survey area with 0◦.5 × 0◦.5 bins, and generated sim- Fig. 13.— As Fig. 11, but showing the distribution of objects clas- sified as extended sources over the survey region. Due to the high source density in the disks of M31 and M33, some point sources are blended and are classified as galaxies by the photometry software. A pixel size of 0◦.05× 0◦.05 has been used. Fig. 14.— The map of the fractional residuals between the Galac- tic disk selection previously presented in Fig. 11, and the Besançon model predictions (calculated as (Data −Model)/Model for each 0◦.5 × 0◦.5 bin). Ignoring a 2◦ circle around M31 and a 1◦ circle around M33, the average difference is less than 2%. ulated catalogues using the Besançon Galactic popula- tions model. All stellar populations in the model with i-band magnitudes between 15 < i0 < 26 were accepted. To reduce shot noise in the randomly generated catalogs, at each spatial bin we simulated a 10 times larger solid angle, and later corrected the density maps for this fac- tor. Finally, the artificial photometry was convolved with the observed magnitude-dependent uncertainty function (from Fig. 5). We were impressed to discover the accuracy to which the Besançon model predicts the starcounts towards our fields. For the Galactic disk sample selected with 1.5 < (g−i)0 < 3.0 and 15.0 < i0 < 19.5 (red dashed-line box in Fig. 4), whose observed spatial distribution was presented previously in Fig. 11, the Besançon model cor- Fig. 15.— The luminosity function of point sources in the color range 0.8 < (g− i)0 < 1.8 for the sample fields shown previously in Figs. 6 and 7: field 46 (a), field 106 (b), the Draco dSph comparison field (c) and the CFHTLS field D3 (d). The observed luminosity functions are shown in black, while the red lines show the Besançon model predictions. In panel ‘a’ the stellar populations of the Giant Stream cause the large increase in counts beyond i0 = 21. The correspondence between observations and model in panels ‘b’ and ‘c’ is excellent, though there is a significant departure in panel ‘d’. A limiting g-band magnitude of g0 < 25.5 was imposed to data and models alike. rectly predicts the observed counts over the survey area to better than 2%. The fractional residuals between the observations and the model are shown in Fig. 14. Evidently the Besançon model has the correct ingredi- ents to reproduce very accurately the Galactic disk star- counts towards these fields around M31. However, we need to investigate the model further before we can use it with confidence. The color-magnitude region that is of particular interest to us, is the region where the RGB of M31 has its greatest contrast over the contaminants. We will return to this in more quantitative detail later, when we discuss the matched filter method, yet a visual inspec- tion of Fig. 4 shows that the color interval will be approx- imately in the range 0.8 < (g− i)0 < 1.8, where we avoid the bulk of the Galactic disk contamination, and also the faint blue contaminants, which are most likely unresolved background galaxies. In Fig. 15 we display the observed luminosity function in this color interval (in black), as well as the corresponding Besançon model predictions (in red) for the two representative fields and the two ref- erence fields that we presented previously in Figs. 6 and 7. The correspondence is excellent from i0 = 15 down to i0 = 20.0, with Kolmogorov-Smirnof (KS) test proba- Fig. 16.— The color-magnitude distribution of sources from the Besançon model for the MegaCam comparison fields is shown in panel (a), where the model predictions have been smoothed with the observational errors in Fig. 5. The corresponding observed dis- tribution is given in panel (b). Clearly, in reality the stellar popu- lations have a much wider color spread than the model predicts. To alleviate this problem we have introduced an additional smoothing to the model, as detailed in the text. In panel ‘c’ the ratio of the luminosity function in the color range 2.0 < (g − i)0 < 3.0 of the model (red) and the data (black) is used to compute an empiri- cal completeness correction, which applied to the color-magnitude data, gives the distribution shown in panel ‘d’. (A g-band limit of g0 = 25.5 has been imposed throughout). bility that the observations are drawn from the model of greater than 10% for all four fields. In panel ‘a’ the ob- servations depart strongly from the model for i > 21, this is however completely expected, as the field contains the RGB of Andromeda at these magnitudes. Panel ‘b’ is for field 106 in the outer halo, and panel ‘c’ is the Draco dSph comparison field; in both cases the model predictions are extremely close to what is observed: the KS test over the range 15 < i0 < 24 gives 27% and 9% probability, respec- tively, that the observed and modeled distributions are identical, and the total counts agree to within better than 2σ. However, for the CFHTLS field D3, shown in panel ‘d’, the Besançon model predictions over the full range 15 < i0 < 24, do not accurately match the observations (KS-test probability < 0.01%). This failure towards the direction (ℓ = 96◦.3, b = 59◦.7), is likely due to a slightly inaccurate model of the Galactic halo component, or due to local deviations from a globally correct halo model. Despite this shortcoming, we consider these comparisons to have been very encouraging. The Besançon model predicts reasonably well the details of the star counts to- wards our two comparison fields, and it predicts perfectly well the star counts in the outer halo field (panel ‘b’). Very similar results were found upon widening the color range to 0.5 < (g − i)0 < 1.8, to include the bluest RGB stars of interest. Given the variations in the luminosity function that are clearly visible in Fig. 15, it is evidently better to use the model to subtract off the expected con- tamination rather than use a comparison field located at a different Galactic latitude and longitude. This is true even for relatively nearby fields: the difference in the pre- dicted luminosity function of foreground stars in panels ‘a’ and ‘b’ is substantial. The excellent agreement between the observations and the model predictions in panels ‘b’ and ‘c’ of Fig. 15 is somewhat surprising given the fact that we did not apply any incompleteness corrections to the model, and have not corrected for contaminating background unre- solved galaxies. We chose not to perform artificial star completeness tests for this survey as it would have been a prohibitively expensive undertaking, and refer instead to a previously computed comparison between MegaCam and Hubble Space Telescope photometry from the center of the Draco dSph. As we show in Fig. 2 of Ségall et al. (2006), the completeness of MegaCam down to i = 24 from data of similar exposure time is greater than 80%. Note however, that this completeness was calculated in a relatively crowded central field of the Draco dSph (not the Draco comparison field shown in panel ‘c’ of Figs. 6, 7 and 15), and is therefore likely to be substantially worse than what we face in the almost empty fields in the outer halo of M31. Despite these successes of the Besançon model, it un- fortunately fails to predict the correct color-magnitude distribution. The reason for this is apparent from a visual inspection of panel ‘a’ of Fig. 16, in which we present the predicted color-magnitude distribution over the Mega- Cam fields 93, 105, 106, 115, 120 and 121, which are all located at the outer edge of the survey near a projected radius of 150 kpc (In the analysis below we shall refer to these fields as “background” fields). Comparing this distribution to its observed counterpart in panel ‘b’, we see that the model has features that are too sharp, de- spite the convolution with the photometric uncertainties. This is likely due to the model not containing a realis- tic spread of stellar populations types, in particular the color-magnitude sequences are evidently not as varied in the model as in reality. To alleviate this problem we have introduced an ad- ditional smoothing to the model. From a Gaussian fit to the color distribution of Galactic “halo” and Galactic disk populations in the magnitude range 20 < i0 < 21 (where the sequences are almost vertical in the CMD), we measured the intrinsic FWHM of the observed dis- tributions. By introducing a color-dependent additional Gaussian spread to the model of σ = 0.05+0.075(g− i)0, we find a similar color spread in the halo and disk pop- ulations to the observations. In panel (c) we compare the luminosity function in the color range 2.0 < (g−i)0 < 3.0 in the resulting smoothed model (red) with that of the data. We see an excellent match down to g0 = 23.25, after which the model begins to diverge, due to the effects of incompleteness. We use the ratio of these distributions beyond g0 = 23.25 to correct the model for incompletness; the resulting final model for the background region is displayed in panel ‘d’. The excellent agreement of the Besançon model with our observations to g0 = 23.25, indicates that the number of background galaxies masquerading as point-sources cannot be a substantial fraction of the total counts down to these photometric limits. Beyond this limit, some background galaxy contamination may offset the incom- pleteness, in which case it will be hidden in the empiri- cal completeness correction adopted for the background fields. The Besançon model, smoothed and corrected for in- completeness, as discussed previously, can now be used Fig. 17.— The spatial distribution of the Besançon model (cal- culated for each 0◦.5×0◦.5 bin) over the survey region for two differ- ent color-magnitude selections. Panel ‘a’ is for Galactic stars that have color and magnitude in the region occupied by stars in M31 of metallicity in the range −3.0 < [Fe/H] < +0.2 according to the 10Gyr Padova models. Panel ‘b’ is for the more restricted range −3.0 < [Fe/H] < −0.7. to predict the expected foreground contamination, for stars of color and magnitude that will masquerade as M31 halo stars. In Fig. 17 we show two such predic- tions over the area of the study. The top panel shows the equivalent surface brightness of the star-count model for stars with metallicities −3 < [Fe/H] < +0.2 interpo- lated from the Padova models shifted to the distance of M31. The bottom panel shows a similar map for −3 < [Fe/H] < −0.7, which is substantially fainter than that of panel ‘a’ because this metallicity interval excludes most red stars from the Galactic disk sequence (as can be seen in Fig. 16). To construct Fig. 17 we have converted the predicted Galactic star-counts to an “equivalent surface bright- ness” ΣV in the V-band, as if these contaminants were RGB stars in M31. The motivation for converting the measured star-counts into surface brightness is of course to be able to compare our observations to previous stud- ies and also to theoretical predictions. However the pro- cedure requires some further explanation. Both for the model and for the survey data, we convert the MegaCam g and i-band photometry into the V-band using the color equation above. The resulting V-band luminosities are summed for the stars in a spatial and/or color-magnitude bin, but we must still correct for the fact that we are only observing RGB stars which represent only a fraction of the total luminosity. By comparing the RGB star-counts of And III down to a limiting magnitude of i0 = 23.5 with the integrated magnitude of mV = 14.4± 0.3 of this dwarf galaxy (McConnachie & Irwin 2006), we measure an offset of 2.45 mag. This is consistent, and similar, to the value of 2.3 mag estimated in the same manner by Martin et al. (2006) for a limiting magnitude of i0 = 24. Furthermore, as we shall see below in §8, with this off- set we obtain a good correspondence between the pro- file of metal-poor stars and the V-band surface bright- ness profile derived from integrated light (Irwin et al. 2005). Clearly the uncertainties in this simple correc- tion are large: we are implicitly assuming that the un- derlying population has the same luminosity function as And III for all metallicities. The equivalent surface brightness measurements we shall present below must therefore be interpreted with caution, as they are likely to contain substantial systematic errors. However, the interested reader who may wish to convert these sur- face brightness profiles back to the reliable measure of luminosity-weighted star-counts (to a limiting magnitude of i0 = 23.5) can do so by simply subtracting 2.45 mag. The predicted distributions such as those shown in Fig. 17 are the best means we have to subtract fore- ground contamination from the spatial maps. However, we found that we could improve upon the foreground subtraction in color-magnitude (Hess) diagrams by us- ing the observed color-magnitude distribution in the 6 background fields (93, 105, 106, 115, 120 and 121) ap- propriately scaled according to the model to account for the predicted density variations over the survey. A dif- ferent scaling correction is adopted for each metallicity interval; we show in Fig. 18 an example of the scal- ing factor applied to the stars with colors consistent with being M31 stars with metallicity in the interval −0.7 < [Fe/H] < −0.4, according to the Padova models. The density of contaminants subtracted from the higher latitude fields is more than a factor of two larger than from the lower latitude fields. Panel ‘a’ of Fig. 19 shows the color-magnitude dis- tribution of the MegaCam fields shown previously in Fig. 4, with the contamination removed statistically. The subtracted CMD displays a clear RGB-like popu- lation, with a broad range of metallicity, although the detection of the more metal-rich populations is clearly hampered by the observational g-band limit. In or- der to investigate the luminosity function along this RGB, we select stars with interpolated metallicities in the range −2.3 < [Fe/H] < −0.7 (i.e., between the green and pink isochrones). The result is shown on panel ‘b’, together with a simple fit. A linear fit in log(Counts), is precisely what is expected for an RGB population (Bergbusch & Vandenberg 2001). If this statistical fore- ground subtraction is reliable, over 105 halo RGB stars belonging to M31 are detected over these MegaCam fields. 5. STELLAR POPULATION MAPS Fig. 18.— An example of a map of the density scaling factor applied to the background fields (marked in green) to compensate for the expected variations in foreground stellar populations over the survey. In this case, we have chosen stars with colors between the Padova isochrones of metallicity −0.7 < [Fe/H] < −0.4. Fig. 19.— Panel ‘a’ shows the Hess diagram of the MegaCam fields previously shown in Fig. 4, with foreground and background contamination subtracted by comparison to six background fields as detailed in the text. The Padova isochrone models from Fig. 8 are reproduced to help guide the eye. Panel ‘b’: the luminos- ity function of stars with −2.3 > [Fe/H] > −0.7. Panel ‘c’: the matched filter weight map, trimmed to the color-magnitude region encompassing stars of metallicity −3.0 < [Fe/H] < +0.2. (Both gray-scale maps are shown on a linear scale, with the photome- try limited to g0 < 25.5). Having shown that there is a relatively clean signal of the expected RGB of M31 in the combined data, we now proceed to mapping out these stellar populations. A very powerful technique for revealing a signal buried under heavy contamination is the so-called “Matched Filter” method, which is an optimal search strategy (in a least- squares sense) if one has a precise idea of the properties of the signal and the contamination. The properties could be, for instance, the spatial properties of the population of interest (a characteristic size or shape) as well as those of the contamination. Alternatively (or in addition), one Fig. 20.— Matched-filter map to i0 = 24.5 (i0 = 22.8 over the INT survey region). The artifacts of the MegaCam fields observed in the 2003 and 2004 seasons are clearly seen. A logarithmic scale is used for the representation. Fig. 21.— As Fig. 20, but to the limiting depth of the INT survey (i0 = 22.8 for S/N∼ 10). The map is virtually free of obvious artifacts over the entire region observed with MegaCam. may use the color-magnitude distribution, or whatever other physical properties of these populations that have been measured. To apply the matched filter method one simply weights each datum by the ratio of signal to contamination ex- pected for that datum given its parameters. The re- sulting ensemble of weighted data can then be analyzed in the usual way. However, the advantage this effort has afforded us is that the distribution of weighted data will optimally suppress the contamination, revealing best whatever signal is present. In the particular situation confronting us here, we know the color-magnitude distri- bution of the signal of interest, as we have just presented in panel ‘a’ of Fig. 19, and as discussed above the Mega- Cam “background” fields (93, 105, 106, 115, 120 and 121) give us a reasonable model for the color-magnitude behavior of the contamination in the absence (or near absence) of that signal. The ratio of these two CMD distributions gives the weight matrix, which we show in panel ‘c’ of Fig. 19. Here we have trimmed the weight matrix down to the maximum possible physical inter- val (−3.0 < [Fe/H] < +0.2). Note that, as expected, the greatest weight arises at faint magnitudes in the color range 0.75 < (g − i)0 < 1.5, so of course stars with this photometric property will contribute most strongly in the following matched filter maps. Figure 20 displays a matched filter map over the entire survey region, where we have chosen a limiting magni- tude (i0 = 24.5), a metallicity range (−3.0 < [Fe/H] < 0) and a gray-scale representation to highlight the survey defects. The sky region surveyed by the INT is clearly not as deep as the outer MegaCam region, causing the sharp edge along the MegaCam survey boundary. How- ever, the most important defect visible here are the long horizontal stripes, which are present on the top and bot- tom row of CCDs in the 2003 and 2004 data, but not after the camera refurbishment in 2005. The effect is due to a deterioration of the point spread function (PSF) in those areas, causing stars to appear elliptical and simi- lar to barely-resolved galaxies. We spent a considerable amount of effort adapting our processing software to cor- rect for this effect, but though substantial improvement was obtained compared to the starcounts derived assum- ing a constant PSF, the problem could not be removed entirely, since some galaxies intrinsically have ellipticity and major axis position angle similar to the deformed PSFs. We also attempted to correct the maps by calcu- lating the equivalent of a flat-field for star-counts from the median of many fields. However this was not im- plemented for the maps presented here, as the defects were found to be insufficiently stable, so that the com- puted corrections introduced other artifacts of almost the same amplitude as those they corrected for. Instead, the problem is largely removed by choosing a brighter lim- iting magnitude, and virtually disappears if we adopt i0 = 22.8 as in Fig. 21, the limit of the INT photome- try (Ibata et al. 2001b). Of the remaining artifacts, the most obvious remaining are the handful of shallow INT fields mainly clustered around (ξ = 0◦,η = −3◦) which were observed in conditions of poorer seeing than av- erage, and of course the hole in the star-counts at the center of M31, where the photometry of individual stars broke down due to very high crowding. In Fig. 22 we present the matched filter maps for six different ranges in metallicity. The limiting magnitude over the MegaCam region was chosen to be i0 = 23.5, and we kept a limit of i0 = 22.8 (S/N ∼ 10) for the INT survey, which gives rise to the obvious discontinu- ity around η ∼ −3◦. These maps possess a bewilder- ing amount of information on a large range of spatial scales and surface densities, so it is impossible to dis- play all the information at a given pixel scale or with a given color representation. The diagrams in Fig. 22 have been constructed to show the large-scale distribu- tion of stellar populations in the MegaCam region of the survey, while retaining some sensitivity to small struc- tures such as dwarf galaxies which have scales of a few arcmin; in each row the right-hand panel shows a higher resolution version of the selection in the left-hand panel; the lower-resolution maps are useful for appreciating the large-scale behavior of the diffuse components. We start our discussion with panel ‘b’, which displays the metal- rich selection (−0.7 < [Fe/H] < 0.0). Though noisy, we can discern many features: • The elliptical but irregular distribution of stars with axis ratio ∼ 0.5 and major axis diameter ∼ 5◦ (∼ 70 kpc), containing several previously reported substructures (Ferguson et al. 2002). As we have argued elsewhere (Ibata et al. 2005), this is a giant rotating component which is dominant beyond the end of the classical disk, and possibly the residue of a significant merger that occurred many Gyr ago (Peñarrubia, McConnachie & Babul 2006). • The large (∼ 1◦ diameter) overdensity to the north- east (ξ ∼ 1◦.5, η ∼ 3◦), almost certainly unbound debris (Zucker et al. 2004; Ibata et al. 2005). • The “G1” clump at (ξ ∼ −1◦, η ∼ −1◦.5), a structure surrounding but unrelated to the lumi- nous globular cluster “G1” (Ferguson et al. 2002; Rich et al. 2004; Reitzel, Guhathakurta & Rich 2004; Faria et al. 2007). • The stream-like “Eastern shelf” (Ferguson et al. 2002), at (ξ ∼ 2◦, η ∼ 0◦.5). • A fainter stream on the western side of the galaxy, the “Western shelf” at (ξ ∼ −1◦, η ∼ 0◦.5), and seen in the map of Irwin et al. (2005). • The “Giant Stream” (Ibata et al. 2001b, 2004), which in the INT data appears to be a linear struc- ture stretching from very close to the center of M31 to (ξ ∼ 1◦.5, η ∼ −3◦), but which shows up as a substantially wider structure in the MegaCam sur- vey extending to (ξ ∼ 3◦, η ∼ −6◦). • A previously unknown stream is seen extending be- tween (ξ ∼ 4◦, η ∼ −1◦.5) and (ξ ∼ 3◦, η ∼ −4◦); we will refer to this as “Stream C” in the discussion below. • Vast expanses apparently devoid of stars over most of the Southern half of the survey MegaCam. • A faint diffuse component is detected approxi- mately 4◦ from M33. In panel ‘c’ we show an intermediate metallicity selec- tion (−1.7 < [Fe/H] < −0.70), somewhat “overexposed” to bring out better the fainter structures. In addition to the previously-discussed features, we now notice: • The inner ellipse, attributed to the giant rotating component, has become larger and even more irreg- ular. The more irregular aspect is of course con- sistent with the expected longer mixing times of debris at larger radius. An interesting point is that the distribution appears now to be less flattened, suggesting that this extreme color stretch may be revealing another rounder structure previously hid- den beneath the flattened rotating component. • The dwarf galaxies And II and And III (cf. Fig. 3) become apparent. Fig. 22.— Logarithmic scale matched-filter maps to a limiting magnitude of i0 = 23.5, g0 = 25.5. Low resolution images (0 ◦.2 × 0◦.2 pixels) are shown on the left, high resolution versions (0◦.05× 0◦.05 pixels) on the right-hand column. Fig. 22 — continued.— Logarithmic scale matched-filter maps to a limiting magnitude of i0 = 23.5, g0 = 25.5. Low resolution images (0◦.2× 0◦.2 pixels) are shown on the left, high resolution versions (0◦.05 × 0◦.05 pixels) on the right-hand column. Fig. 23.— Star-count map of the MegaCam region, with the foreground contamination subtracted using the Besançon model. A limiting magnitude of i0 = 23.5 has been adopted. The red, green, blue and pink polygons delineate the regions chosen to sample, respectively, the Giant Stream, the major axis structure, the minor axis stream and the empty outer halo region. • Two strong localized structures, at (ξ ∼ 6◦.23, η ∼ −2◦.89) and (ξ ∼ 6◦.23, η ∼ −8◦.89), which as we will discuss below, are two new dwarf satellite galaxies. • A faint low surface brightness fuzz is detected on the extension of the major axis of M31, out to (ξ ∼ −5◦, η ∼ −7◦), we will refer to this as the “Major axis diffuse structure”. • A strong stream-like structure is detected between (ξ ∼ 3◦, η ∼ −1◦.5) and (ξ ∼ 2◦, η ∼ −2◦.5), which we call “Stream D” below. • A further faint low surface stream-like structure is detected towards (ξ ∼ 6◦, η ∼ −6◦), which we will refer to as “Stream A”. • The extended structure near M33 is stronger. • The region (ξ < 4◦, η < −9◦) remains devoid of stars. The more metal-poor selection in panel ‘d’ (−2.3 < [Fe/H] < −1.1) displays essentially the same properties as in panel ‘c’, except that a considerable amount of localized density spikes are detected, covering one to a few contiguous pixels. Among these are the newly-discovered dwarf galaxies And XI, XII, and XIII (Martin et al. 2006). Panel ‘e’ shows the most metal-poor sample (−3 < [Fe/H] < −1.70). Now the Giant Stream has almost disappeared, and only And II and III are still clearly visible as substructures, yet one also discerns a radial gradient from M31 over the MegaCam survey region. For completeness, in panel ‘a’ we show the most metal-rich selection considered here (0.0 < [Fe/H] < +0.2) in which only the inner disk of M33 and a small portion of the Giant Stream are discernible, while panel ‘f’ shows the map over the full metallicity range. The increased sensitivity with the full metallicity range reveals a further feature on the Fig. 24.— Cartoon of the main structures presented in §5. The circled dots and ‘star’ markers are reproduced from Fig. 3, and show the positions of dwarf galaxies and selected globular clusters, respectively. minor axis with a stream-like structure between (ξ ∼ 5◦, η ∼ −2◦.5) and (ξ ∼ 3◦, η ∼ −5◦), which we will refer to as “Stream B”. The maps displayed in Fig. 22 show the distribution of the matched filter statistic, so the resulting counts are therefore somewhat difficult to interpret directly. The reason for this is primarily that the matched fil- ter method relies on a model of the stellar population that one desires to detect, and the statistic we mea- sure will depend on the assumed luminosity function and how we choose to weight populations of different metal- licity. A secondary reason is that, as discussed above, the foreground Galaxy counts do vary over this vast sur- vey, so the contamination model also varies. For these reasons we also present in Fig. 23 a straightforward sur- face density map, where we have counted up stars in the color-magnitude interval 0.8 < (g − i)0 < 1.8 and 20.5 < i0 < 23.5, and have subtracted off the correspond- ing Besançon model counts over the same area of sky. The main structures previously seen in Fig 21 are nicely confirmed, and which we highlight in Fig. 23, namely the very extended Giant Stream (red polygon), the dif- fuse major axis structure (green polygon), the minor axis stream-like structure (blue polygon), the extended out- skirts of M33, and the voids elsewhere (pink polygon). The advantage of this map is that we can now interpret the physical meaning of the color scale, which is shown with the wedge at right-hand edge of the diagram. Black corresponds to 10−4 RGB stars per square arcsecond down to i0 = 23.5. Using the conversion of star-counts to surface brightness discussed above, the saturated black level translates to ΣV = 30.3mag arcsec In the next section we discuss in more detail the popu- lations that are highlighted in Fig. 23. To ease interpre- tation, in Fig. 24 we show a cartoon of the positions of these populations with respect to the various structures discussed above. Fig. 25.— The spatial distribution of point-sources in a 9′ × 9′ area in the vicinity of And XV (panel ‘a’). The parallel red lines mark the CCD boundaries, though there is no gap at this location due to the adopted dithering pattern. The CMD of the stars within the 2′ circular region is shown in panel ‘b’. Selecting those stars with color and magnitude within the red dashed polygon, yields the spatial distribution shown in panel ‘c’ whose radial profile is given in panel ‘d’. The continuous, dashed and dot-dashed lines in panel ‘d’ are, respectively, a Plummer model, an exponential model, and a King model fit to the profile inside of 5′. 6. SPATIAL SUBTRUCTURES 6.1. Discovery of 2 bright satellites A thorough analysis of these data regarding the inci- dence of low mass satellites around M31 and its impli- cations for galaxy formation theory and cosmology will be presented in a later publication in this series (Martin et al. 2007b). However, we discuss briefly here two new dwarf galaxies which were discovered immediately from simple visual inspection of the starcounts maps. Since the analysis is identical for both objects we include the results for And XVI in brackets. And XV (XVI), located at α0 = 1 h14m18.7s, δ0 = 38◦7′3′′ (α0 = 0 h59m29.8s, δ0 = 32 ◦22′36′′) can be noticed as an obvious enhancement in the matched- filter maps presented previously. In panel ‘a’ of Fig. 25 (Fig. 26), we show the distribution of all detected point sources in a 9′ × 9′ region around the dwarf galaxy. The color-magnitude distribution of the sources within the 2′ (1′.5) radius circle centered at the point of max- imum density is shown in panel ‘b’. A very clear and strong RGB is present. Assuming that the stars out- side of the irregular polygon are contaminants, we pro- ceed to estimate the distance of the structure using the tip of the RGB. We adopt MTRGB = −4.04± 0.12 from Bellazzini, Ferraro & Pancino (2001) for the ab- solute I-band magnitude of the RGB tip, and con- vert into the Landolt system using the color equations above and those given by McConnachie et al. (2004a); this yields a distance modulus of m−M = 24.0± 0.2 (m−M = 23.6± 0.2) or alternatively a distance of 630± 60 kpc (525 ± 50 kpc). With this distance modulus we find a reasonable visual fit to the RGB with a Padova isochrone of metallicity [Fe/H] = −1.1 ([Fe/H] = −1.7). Given the angular distance of 6◦.8 (9◦.5) fromM31, the ob- ject lies at an M31-centric distance of 170 kpc (270 kpc). Fig. 26.— As Fig. 25, but for And XVI. The presence of several bright stars causes the irregular spatial distribution in the left hand panels. With the CMD selection polygon from panel ‘b’, we filter out foreground contamination, which gives the dis- tribution shown in panel ‘c’. The corresponding density profile is given in panel ‘d’, where we have subtracted off a background count determined from an annulus between 10′ and 15′. Fitting the distribution with an exponential profile (dashed line), yields a scale length of 0′.72 ± 0′.03 (0′.53 ± 0′.03), though a Plummer model (solid line) of scale size 1′.2 (0′.9) also fits acceptably well, as does a King (1962) model (dot-dashed line) with core radius of 0′.91 (0′.64) and tidal radius of 5′.7 (4′.3). By integrating the star-counts up to the half-light radius, and correct- ing by 2.45 mag (as above) to account for stars below i0 = 23.5, we estimate a total absolute magnitude of MV = −9.4 (MV = −9.2). And XVI will be a particularly interesting object for further study given its extreme distance from M31, and its location between M31 and the Milky Way, where it presumably has felt a non-negligible perturbation from the potential of our Galaxy. It is also curious that And XV appears to be structurally disturbed and elon- gated, which is suggestive of the action of galactic tides. Yet how this very distant galaxy might have been affected by tides is hard to imagine. (The irregular morphology seen in the distribution of And XVI stars in Fig. 26 is an artifact of nearby bright star “holes”). 6.2. Giant Stream The Giant Stream around M31 has been the subject of numerous studies, due to the fact that it is a nearby in- termediate mass merging event, and that it can be used to measure the potential of M31. The initial discovery in the INT survey (Ibata et al. 2001b) showed the structure to be (in projection) an approximately linear and radial stream, with a metallicity slightly higher than that of 47 Tuc ([Fe/H]− 0.71), and a total absolute magnitude of MV ≈ −14. We probed more fully its extent and the line of sight depth with the CFHT12K (McConnachie et al. 2003), a precursor wide-field camera to MegaCam at the CFHT. These photometric and positional data were then complemented by radial velocities obtained at 4 loca- Fig. 27.— Panel ‘b’ displays the stellar populations in the core of the Giant Stream (sampled in the spatial region shown with a red polygon in panel ‘a’); while panel ‘c’ displays those on the periphery of this structure (dark blue polygon in panel ‘a’). The foreground contamination has been removed from the two Hess diagrams. Fig. 28.— The metallicity distribution functions (with error bars denoting 1σ uncertainties) for the Giant Stream core sample (red) and the envelope sample (blue), as interpolated from the chosen Padova isochrones. Photometric limits of i0 = 23.5 and g0 = 25.5 have been imposed. The background fields, normalized with the Besançon model, have been used to subtract off the expected fore- ground counts in each of the metallicity bins. The two distributions are completely inconsistent with each other to high confidence. tions along the stream with the DEIMOS multi-object spectrograph at the Keck Observatory, which allowed a measurement of the mass of the halo of Andromeda out to 125 kpc (Ibata et al. 2004), and enabled us to develop a model of the orbital path of the stream progenitor. We found the orbit to be highly radial, and predicted that the stream fans out towards the East after passing very close to the nucleus of M31, losing its stream-like spatial coherence. This analysis also posed an interesting puz- zle, which is still unsolved: since the stream is on such a highly destructive radial orbit, how did the progenitor survive until so recently? Subsequently, Guhathakurta et al. (2006) also used Keck/DEIMOS to obtain spectra in one stream field, where they measured a mean metallicity of 〈[Fe/H]〉 = −0.51. The kinematic data sets were reana- lyzed by Font et al. (2006), who undertook N-body simu- lations to attempt to reproduce the stream morphology. They found that the progenitor must have been more massive than 108M⊙, and that the time since its disso- lution is a mere 0.25Gyr. Recently, Fardal et al. (2006) have shown how the fanning-out of the stream into shells to the East and West can be used to place constraints on the galaxy potential. We defer a full re-analysis of the Giant Stream to a subsequent contribution, focussing here on the salient new features that are revealed in the MegaCam survey. An inspection of Fig 22, shows that the Giant Stream extends out to a projected radius of ∼ 100 kpc (the in- ner dashed circle). With the maximum line of sight distance to the stream of 886 ± 20 kpc estimated by McConnachie et al. (2003) (at ξ = 2◦, η = −4◦), this cor- responds to an apocenter distance of ∼ 140 kpc. Though this is further than it had been mapped out before, the possibility that the stream reaches this projected dis- tance was anticipated by one of the orbit models pre- sented in Ibata et al. (2004) (cf. Fig. 4 of that paper). Fig. 29.— Counts in a 1◦-wide East-West band between −4◦.5 < η < −3◦.5 for different metallicity intervals. That particular orbit model, however, does not agree well with the measured line of sight distance gradient, though we note that debris does not exactly follow the orbit of the progenitor. Further detailed modeling is clearly re- quired to understand the dynamics of this stream. The MegaCam data also shows that there are stel- lar populations variations in the stream. We illustrate the evidence for this in Fig. 27, where the colour magni- tude distribution in the core of the Giant Stream (panel ‘b’) is compared to a region on the western periphery of the structure. Both of these spatial selections con- tain stars over a wide range of metallicities, and peak at high mean metallicity, consistent with the mean photo- metric metallicity of 〈[Fe/H] = −0.51〉 measured from a kinematically-selected sample of stars on the periphery of the Giant Stream (Guhathakurta et al. 2006). It is clear from an inspection of this diagram, however, that relative to the outer field the core is lacking the blue stellar popu- lations (around the isochrone with [Fe/H] = −1.3). The concentration of very “metal-rich” stars to the core of the stream can also be seen in Fig. 22 (compare panel ‘a’ to panel ‘c’). We stress here that these red stars need not be as metal-rich as they appear from comparison to these ancient isochrones, due to the well-known age-metallicity degeneracy. While the majority of other “halo” popula- tions studied in this contribution are very likely old, this is not the case for the Giant Stream. In the spectral sample of bright stream stars obtained by Ibata et al. (2004), many targets could be identified as Asymptotic Giant Branch (AGB) stars from their spectral features, which indicates that a fraction of these stars are of in- termediate age. This is consistent also with the deep photometric survey in a Giant Stream field undertaken by Brown et al. (2006b) with the Advanced Camera for Surveys (ACS) on board the HST. They detected a dom- inant population of age ∼ 8Gyr, as well as a younger ∼ 5Gyr component. We will continue to label these red stars as “metal-rich” for the sake of brevity, though the above caveat should be kept in mind. The stellar populations differences can be put on a more quantitative basis by constructing the metallicity distribution functions for the “stream core” and “outer stream” selections; this is displayed in Fig. 28, which shows the striking difference very clearly. The core of the stream clearly has a very large fraction of red stars. Fig. 29 shows the star-counts in different metallicity in- tervals as a function of ξ in a 1◦-wide band between −4◦.5 < η < −3◦.5. The distribution, which peaks near ξ ∼ 1◦.5 for [Fe/H] > −0.4, becomes broader for the metallicity intervals −1.3 < [Fe/H] < −0.4. 6.3. Major axis structure The faint diffuse population detected on the major axis between a projected distance of 50 kpc and 100 kpc (de- lineated with the green polygon in Fig. 23) is a con- spicuous feature of the MegaCam survey. The average surface brightness in this region is ≈ 31 mag arcsec2. The dwarf galaxy And III lies on the edge of this re- gion, so to avoid contamination we remove the data from a 0◦.5 radius circle around And III for the subsequent analysis. The color-magnitude distribution of the area is displayed in panel ‘a’ of Fig. 30, which clearly pos- sesses a well-populated RGB with a dominant population of color similar to the Padova isochrones of metallicity [Fe/H] ∼ −1.3. The corresponding MDF in Fig. 31 (red line) confirms this visual impression. Thus despite the visual impression that the “overex- posed” density map of Fig. 23 gives that the major axis population merges with the Giant Stream, we find that these two stellar populations are very different and likely unrelated. This diffuse low-constrast feature has no clear spatial structure as one would expect of a stream. In- deed, it could be the inner regions of the halo, though it appears not to be a smooth roughly spherical structure since there is an obvious deficit of stars at (ξ ∼ −0◦.6, η ∼ −6◦) compared to (ξ ∼ −3◦, η ∼ −5◦). We re- frain from estimating the total luminosity of the struc- ture, since we have clearly only detected a fraction of the entire object. Additional photometry to the North and West and possibly even kinematics will be needed to understand this structure further. 6.4. Distant minor axis stream ‘A’ In contrast, the structure on the minor axis (delineated with the blue polygon in Fig. 23, which covers 1.7 deg2) at R ∼ 120 kpc is much more confined spatially as can be perceived from an inspection of the matched-filter maps in Fig. 22. Curiously, this population (which we refer to as stream ‘A’ in the discussion below) has a very similar color-magnitude distribution to that of the major axis structure, with a dominant population again just slightly redward of the [Fe/H] = −1.3 Padova isochrone, as can be seen in panel ‘b’ of Fig. 30. The corresponding MDF Fig. 30.— Panel ‘a’ shows a foreground-subtracted Hess diagram of the major axis diffuse population over the region marked out with the green polygon in Fig. 23, while panel ‘b’ presents the foreground-subtracted Hess diagram of the minor axis stream pop- ulation over the region marked out with the blue polygon in Fig. 23. The gray scale wedge on the right shows the count level per CMD bin of size 0.05mag × 0.05mag. Fig. 31.— The metallicity distribution function of the major axis diffuse structure and the minor axis stream ‘A’ population, as derived from the data in Fig. 30. is compared to that of the diffuse major axis feature in Fig. 31. The structure is very faint, with an average surface brightness of ΣV ∼ 31.7± 0.2mag arcsec −2 . Integrating over the blue polygon in Fig. 23, and subtracting the average counts at this radius calculated from the “outer halo” region (contained in the pink polygon), gives a to- tal luminosity of LV sin 2.3 × 10 6 L⊙ (MV ∼ −11.1). If we are detecting the entirety of the stars in the original structure, the progenitor must have been a galaxy simi- lar to the Sculptor dwarf spheroidal (MV = −10.7± 0.5, Irwin & Hatzidimitriou 1995). 6.5. Minor axis streams at R < 100 kpc Figure 32 shows a close-up map of the minor axis region in the proximity of M31 and the Giant Stream. Here we have used the matched-filter technique to detect struc- tures of metallicity in the range−3.0 < [Fe/H] < 0.0, and have chosen a grayscale representation that highlights the three linear structures that appear almost perpendic- ular to the minor axis and merge into the Giant stream. Three arrows have been added to the diagram to indicate the approximate location of these stream-like features, which we denote ‘B’, ‘C’ and ‘D’ in order of increasing declination. The nature of these streams becomes more appar- ent if we investigate the color profile along the minor axis region. We choose to remove the Giant Stream stars by selecting only those point-sources within the yellow-line polygon in Fig. 32, and sum stars perpen- dicular to the minor axis (rather than taking radial bins) so as to enhance the density peaks. The correspond- ing foreground-subtracted surface brightness profiles are shown in Fig. 35, where the blue line shows the metal- poor populations with −3.0 < [Fe/H] < −0.7 and the red line those with −0.7 < [Fe/H] < +0.2. The foreground, as before, is estimated using the Besançon model. As ex- pected, several strong peaks are detected, however, the locations of the peaks in the metal-poor subsample do not coincide with those of the metal-rich subsample, suggest- ing very strong stellar populations differences between Fig. 32.— Matched-filter map of the minor axis populations with metallicity in the range −3.0 < [Fe/H] < 0.0. The map is, as be- fore, a superposition of MegaCam and INT photometry, the differ- ences in depth of which account for the discontinuous density distri- bution. The region surrounded by the yellow polygon encloses the MegaCam area used to investigate the minor axis density profile in Fig. 33. The red, green and blue polygons enclose the stream-like structures labeled, respectively, ‘B’, ‘C’ and ‘D’. (These structures can be appreciated better in panel ‘a’ of Fig. 27). Fig. 33.— The surface density profile along the minor axis, se- lected from the region within the yellow polygon in Fig. 32. The arrows point out significant peaks in the profile. The positions of the labeled peaks correspond to the streams seen in Fig. 32. these stream-like features. This deduction is borne out by the variations in the color-magnitude distributions in adjacent spatial loca- tions. In Fig. 34 we display the Hess diagrams of the stream-like structures enclosed within the green, red and blue polygons of Fig. 32, and also show the stellar pop- ulation between streams ‘B’ and ‘C’. The corresponding MDFs are given in Fig. 35. These data show that stream ‘D’ is a relatively metal-poor structure, while stream ‘C’ is predominantly metal-rich. Curiously, the population contained within gap between streams ‘B’ and ‘C’ has a narrow range of metallicity and is metal-rich. These stream-like structures overlap along the line of Fig. 34.— Background-subtracted Hess diagrams for four adja- cent MegaCam fields near the minor axis. In panels ‘a’, ‘b’, ‘c’ and ‘d’ we display the data for fields 13, 14, 23 and 24, respectively. Fig. 35.— The MDF determined from the four fields of Fig. 36: 13 (black), 14 (red), 23 (green) and 24 (blue). sight (which is why we chose not to extend the stream ‘D’ spatial selection polygon in Fig. 32 up to the north- eastern end of the survey region). A spectacular example of this can be seen in Fig. 36, which shows the CMD of MegaCam field 14, where streams ‘C’ and ‘D’ cohabit over essentially the entirety of the field. Thus, although these stream-like structures appear to merge spatially with the Giant Stream, such that it is tempting at first sight to associate them to that huge structure, their stellar populations properties are so dif- ferent both from each other and from the Giant Stream, that this proposition is untenable. In the present survey these streams or stream-like structures are clearly truncated at the Eastern edge of the dataset, so it is impossible to determine their full ex- tent or nature. Instead, we obtain a first and very rough estimate of their luminosities by integrating within the 3 stream polygons in Fig. 32. In this way we estimate that stream ‘B’, which lies at R ∼ 80 kpc, has a luminosity Fig. 36.— The color-magnitude diagram for field 14, showing the presence of two co-spatial populations with very different RGB tracks. within the red polygon of ∼ 1.0 × 107 L⊙; stream C at R ∼ 60 kpc has ∼ 1.4 × 107 L⊙ in the green polygon; while stream ‘D’ at R ∼ 40 kpc has ∼ 9.5×106 L⊙ in the blue polygon. In estimating these luminosities we have ignored the complex background in this region. Never- theless, these estimates indicate that the progenitors of the streams were sizable dwarf galaxies, likely more lumi- nous than the Fornax dSph. We note that the extended globular cluster (Huxor et al. 2005) EC4 (Mackey et al. 2007) lies within or superimposed on stream ‘C’. 7. THE OUTER HALO The primary reason for undertaking this survey was initially to investigate the large-scale structure of the ha- los of M31 and M33, and to some extent the substruc- tures discussed above are a hindrance for this purpose. In particular, we had not expected the Giant Stream to be as extended and polluting of the inner halo as it turned out to be, and the various “contaminating” streams along the minor axis were a surprise, as we had chosen those fields from the shallower INT survey to probe the surface density profile of the “clean” inner halo. However, there is a relatively empty region of the sur- vey free from obvious substructures towards the South- west. This ∼ 30 deg2 region previously surrounded with a pink polygon in Fig. 23, is reproduced in Fig 37, where we have converted the counts of stars in the var- ious metallicity ranges shown into an equivalent surface brightness. The four white pixels within the polygon in the diagram are pixels discarded from the analysis as they contain the dwarf galaxies And XI, XII, XIII and The equivalent mean surface brightness of the outer halo stars for the full [Fe/H] range given in panel ‘a’ of Fig. 37 is ΣV = 33.0± 0.05mag arcsec −2, where the un- certainty is calculated using Poisson statistics, assuming no uncertainty in the background subtraction. Note that a 2% error in the subtraction (the average difference of the residuals between the Galactic model and observed Galactic disk found in Fig. 14), will incur a 0.25 mag systematic error. However, the rms scatter in the pixel values in Fig. 37 (calculated in counts and then converted Fig. 37.— Background-subtracted maps of the equivalent sur- face brightness in the outer halo region. Panel ‘a’ shows stars in the metallicity range −3.0 < [Fe/H] < +0.2, while panel ‘b’ is re- stricted to −3.0 < [Fe/H] < −0.7, a range which suffers much less from uncertainties in the background correction. into magnitudes) is 1.1 mag; for this calculation, we only took into account those (128) pixels in Fig. 37 for which the surface area correction was less than 10%. The fact that this rms scatter is larger than the 0.2 mag random uncertainty expected from Poisson uncertainties in the total measured star counts, could be due to an intrinsic lumpiness in the star distribution on the 0◦.5× 0◦.5 scale of the pixels in Fig. 37, but we consider it likely that it is largely due to slight variations in observing conditions between fields, and slight variations of image quality over the camera. Panel ‘b’ gives the map for the metal-poor range −3.0 < [Fe/H] < −0.7, which has the advantage of reducing the amount of residual Galactic contamination. The equivalent mean surface brightness for this selection is ΣV = 33.7± 0.08mag arcsec −2. It is pertinent to point out here that the six fields chosen to probe the back- ground all lie within this outer halo region, indeed they are the fields closest to the outer dashed circle segment marking a projected radius of 150 kpc (cf. Fig. 18). The Hess diagram for the outer halo region is shown in Fig. 38, with the foreground subtracted as before. Though noisy, a low contrast RGB population can iden- Fig. 38.— Foreground-subtracted Hess diagram of the outer halo region shown in Fig. 37. tified that is strongest between the [Fe/H] = −1.7 and [Fe/H] = −0.7 isochrones. Clearly for [Fe/H] > −0.7, residual foreground subtraction errors dominate the data. 8. HALO PROFILES Before presenting the radial profiles of the stellar popu- lation present in the survey, we first investigate whether our conversion from star-counts to “equivalent surface brightness” (first presented in §4.1) yields consistent re- sults with previous studies. To this end we compare our measurements to those of Irwin et al. (2005) who ana- lyzed the profile along the minor axis of M31. We at- tempted to reproduce as closely as possible the spatial selection chosen by Irwin et al. (2005), a band between ±0◦.5 of the minor axis (see their Fig. 1). The MegaCam survey covers most, but not all, of this area (there is a small gap near ξ ∼ 1◦.5, η ∼ −1◦.5, as can be seen in Fig. 32, for example). In panel ‘a’ of Fig 39 the black dots mark the surface brightness measurements from in- tegrated light by Irwin et al. (2005), while the blue his- togram shows the MegaCam profile. Though the mea- surements from integrated light end at R = 0◦.5, just before the beginning of the MegaCam survey, there is good consistency between these two profiles. In panel ‘b’ the black dots now show the star-counts profile of the blue RGB selection of Irwin et al. (2005) converted into an equivalent surface brightness. This V-band profile was determined from a color cut in the INT (V,i) system, designed to select metal poor stars. Fig. 39.— Panel ‘a’ compares the surface brightness profile from integrated light (black points) deduced from the INT survey by Irwin et al. (2005), with the converted star-counts derived from the present survey in a ±0◦.5 band around the minor axis of M31. The color variations as a function of radius are attributable to substruc- tures with different stellar populations intersecting this area. The blue RGB star-counts profile of (Irwin et al. 2005) is compared in panel ‘b’ to the metal-poor MegaCam selection in the same spatial region. The differences between these curves at r < 2◦.5 are likely due to the fact that the two stellar selections, though similar, are not identical. For r > 2◦.5 the Irwin et al. (2005) profile decreases sharply due to over-subtraction of foreground contaminants in that analysis. Panel ‘c’ is similar to panel ‘a’ but the profile is derived over a wider minor axis area (contained within the yellow polygon of Fig. 32). Here we have chosen not to adopt that approach, relying on interpolation between Padova isochrones. This dif- ference in stellar populations must account for some of the differences between the two profiles. However, the shape of the Irwin et al. (2005) profile at large radius drops rapidly unlike the MegaCam profile derived from the same region. This effect is due to the foreground subtraction method chosen by Irwin et al. (2005), who selected fields within 4◦ of M31 to probe and remove the contaminating foreground populations. With hind- sight this is clearly not appropriate given that the present MegaCam data shows that the halo is very extended, and has a rather flat profile. However, out to R ∼ 2◦.5 the INT and MegaCam profiles agree very well. To complement the profiles derived from the narrow 1◦ band shown in panels ‘a’ and ‘b’, we present in panel ‘c’ the surface brightness profile derived from data over the wider minor axis area enclosed within the yellow polygon in Fig. 32. This is of course less noisy at large radius. The various peaks in the profile correspond to the locations Fig. 40.— Panel ‘a’ shows the radial profile of all the stellar structures present in the MegaCam survey to a limiting magnitude of i0 = 23.5. The points show the V-band surface brightness profile from integrated light, as derived by Irwin et al. (2005). The profiles colored in blue and red show the MegaCam data for the metal-poor and metal-rich selections, respectively. Panel ‘b’ gives the “metal- licity” distribution of this entire region (and down to i0 = 23.5), derived from the stellar color by comparison to 10Gyr old Padova isochrone models. The corresponding background-subtracted Hess diagram is shown in panel ‘c’. of the stream-like structures discussed above. Having shown that the minor axis profile is consis- tent with previous measurements in the inner regions (for R < 2◦.5), we now proceed to determine the radial trend of the halo populations over the full survey area. The large amount of substructure detected in the maps above means that the result we find will depend sensitively on what populations we decide to include or reject in the analysis. We therefore adopt a pragmatic approach, tak- ing in turn various population selections, which may be helpful when comparing these data to cosmological sim- ulations. We begin by showing the profile of all stellar popu- lations present in the survey down to a limiting mag- nitude of i0 = 23.5 (Fig. 40, panel ‘a’). The counts in each radial bin are derived from averaging over the entire azimuthal coverage of the MegaCam sur- vey, with the foreground subtracted using the Besançon model. The V-band surface brightness profile measured from the integrated light in the INT data (Irwin et al. 2005) is reproduced here with black dots. The pro- files measured from the present MegaCam data are dis- played in blue for −3.0 < [Fe/H] < −0.7, and in red for −0.7 < [Fe/H] < 0.0. As discussed above, we expect the metal-rich selection to be compromised by foreground correction uncertainties, though this is likely only to be an issue at low surface brightness levels where the signal is small. This sample contains all populations – including satel- lites and streams, so the profile is not obvious to inter- pret. However, it transpires that the peak near 4◦ is due to the presence of the Giant Stream at that location. The metal-rich nature of that structure enhances the metal- rich (red) profile in the region between 3◦.5 < R < 6◦, giving the impression that the halo becomes more metal poor at large radius. This is, however, purely an artifact Fig. 41.— As Fig. 40, but removing the inner halo of M31 out to r = 2◦ and that of M33 out to r = 5◦, as well as all known satellites. For the satellites we excised data within 0◦.5 of And II and III, and within 0◦.2 for the remaining satellites in the MegaCam region. of the presence of that one stream. A clear radial decrease is detected in the surface bright- ness of this combined population up to a distance of about R ∼ 10◦, where it begins to rise again towards M33. Given that M31 and M33 lie at approximately the same Heliocentric distance (McConnachie et al. 2005), this is a spectacular demonstration that the stellar halos of the two galaxies actually pass through each other like ghostly bodies. Panels ‘b’ and ‘c’ show the MDF and background sub- tracted Hess diagram of these stellar populations. In this situation the distributions are overwhelmingly dom- inated by stars close to M31 and in the disk of M33. In Fig. 41 we repeat this analysis, after removing large areas around the inner halos of the two main galaxies and their known satellites. Clearly the tiny bound satel- lites found within the MegaCam survey do not have a significant effect on the global surface brightness profile. However the Giant Stream does have a large effect, and the MDF and Hess diagram in panels ‘b’ and ‘c’ are dom- inated by that population (compare to panels ‘b’ and ‘c’ of Fig. 27). Removing the Giant Stream in addition to the inner halo and bound satellites, reveals a fascinating profile (Fig. 42). We find a very flat decrease as a function of radius, visually resembling an exponential profile in the log-linear diagram of panel ‘a’. Moving outwards from 2◦ to 5◦.5 the offset between the metal-poor and the metal- rich profile remains approximately constant. This data comes primarily from the minor axis area previously pre- sented in Fig 34 (the region within the yellow polygon). At a radius of R ∼ 5◦.5 the metal-rich population drops significantly, and again appears to mimic the metal-poor profile out to R ∼ 7◦. The fact that the metal-poor and metal-rich profiles track each other fairly well in each of these two radial ranges, suggests that the mix of stellar populations present does not change considerably over each range. Whether the drop at R ∼ 5◦.5 reflects a real change in stellar populations at this radius (75 kpc) remains to be confirmed. Fig. 42.— As Fig 41, but with the additional removal of the Giant Stream, as contained within the red polygon of Fig. 23. Fig. 43.— As Fig. 42, but containing only diffuse stellar popula- tions not identified as streams. Finally, we show in Fig. 43 the result of removing all of the identified structures from the survey, leaving only the widely-distributed diffuse population behind. For this we have excised the inner halos of M31 and M33, as well as the satellites as detailed previously. We have also re- moved the areas within the red, green and blue polygons in Fig. 23, and the region contained within the yellow polygon in Fig. 32. As can be appreciated from panel ‘c’ of Fig. 43, we cannot have much confidence in the metal-rich selection, and correspondingly the red profile of panel ‘a’ is very uncertain. However, the metal-poor profile appears fairly smooth. Indeed, the outer halo profile appears remarkably flat in log-linear representation, essentially an exponential function. The blue dashed line in Fig. 44 shows an exponential model fit to the outer halo data (blue his- togram); we find an extremely long exponential scale length of hR = 46.8 ± 5.6 kpc. We also show a pro- jected Hernquist model fit (blue dot-dashed line) to these data, a model choice motivated by the simulations of Fig. 44.— The radial profile from the metal-poor selection of the diffuse outer halo (previously shown in Fig. 43) is displayed with the blue histogram. The blue dashed line is an exponential fit to these data with hR = 46.8 ± 5.6 kpc, while the blue dot-dashed line is a Hernquist model with scale length of 53.5± 0.2 kpc. The black histogram reproduces the metal-poor minor axis profile of panel ‘c’ of Fig. 39. We reject the data below R = 30 kpc, as the profile is dominated by the inner R1/4 de Vaucouleurs profile in this region (Pritchet & van den Bergh 1994; Irwin et al. 2005). As we have shown, between 35 < R < 90 kpc there are copious stream- like substructures on the minor axis, so we reject these regions as well. The best exponential model fit to the remaining data (marked with red points) is shown with a black dashed line, and has hR = 31.4± 1.0 kpc. The black dot-dashed line shows the best fit Hernquist model, which has a scale length of 53.7±0.1 kpc. The red line shows a power-law model fit to these data, which has an exponent of 1.91± 0.11. In addition, with the green line, we show the NFW model halo mass profile fit by Ibata et al. (2004) to the kinematics of the Giant Stream, with an offset (arbitrarily) chosen to fit the outer halo data. The virial radius of this model is 191 kpc. (Bullock & Johnston 2005); the best model has a scale radius of 53.5±0.2 kpc, more that a factor of 3 larger than predicted by Bullock & Johnston (2005). The black his- togram in Fig. 44 reproduces the metal-poor minor axis profile from panel ‘c’ of Fig. 39. Recall that this minor axis selection contains the stream-like structures ‘B’, ‘C’ and ‘D’, so it does not represent the underlying halo. Nevertheless, beyond R = 6◦.5 there was no obvious sub- structure in that region of the halo, and we see that the profile from the minor axis agrees reasonably well with that deduced from the “outer halo”. For R > 6◦.5 the minor axis profile appears slightly higher than the “outer halo” profile. It is possible that this may reflect the real geometry of the halo, the differ- ence would be consistent with the halo being a slightly prolate structure. We do not favor this interpretation, however. The copious substructures seen at R < 80 kpc testify to the dominance of stochastic accretion events in the halo. Given this, its seems more natural to postu- late that the variation in the profile that we see here is another consequence of this messy merging process. If such a thing as a smooth dynamically relaxed halo exists underneath all of the substructure, it cannot have a hole, so the interval 30 < R < 35 kpc is a good place to probe the upper limit to the radial profile in the in- ner region. We therefore fit models to the data in that region and also at R > 90 kpc (the data points used are marked red in Fig. 44). The best-fit exponential Fig. 45.— The radial surface brightness profile for stars with −3.0 < [Fe/H] < −0.7 is shown for the minor axis data (black), and for three sub-samples of the outer halo region: −7◦ < ξ < −1◦ (in red), −1◦ < ξ < 2◦ (in green), and 2◦ < ξ < 7◦ (in blue). The similar radial decease indicates that an underlying halo population is present in all these samples, which are separated by up to 150 kpc. model to these minor axis data (black dashed line) has hR = 31.4 ± 1.0 kpc; while the best fit projected Hern- quist model (black dot-dashed line) has a scale radius of 35.7 ± 0.1 kpc. We also fit a power-law model, and find that an exponent of 1.91 ± 0.11 is preferred. Thus we find again a similar slow decline and a long scale length. This is a very important, and rather unexpected re- sult, and therefore deserves to be checked carefully. In Fig. 45 we have split the “outer halo” sample into three sub-samples (contained within the regions red: −7◦ < ξ < −1◦, green: −1◦ < ξ < 2◦ and blue: 2◦ < ξ < 7◦); the same slow decline with radius is seen in each sub- sample, and in the minor axis sample shown in black, indicating that we are not simply detecting the effects of some localized substructure: approximately 150 kpc sep- arate the red and black profiles! It is possible that the signal arises from an incorrect subtraction of the Galactic contamination. Since the density of stars decreases away from the Galactic plane, which also happens to be the direction away from the centre of M31, an insufficient subtraction of the contaminants could leave a residual that decreases with R as observed. Furthermore, the Galactic disk has an exponential profile, which would naturally explain the observed decline. To examine this possibility we recalculate the surface brightness profiles as before, selecting on metallicity, but this time in ad- dition using a draconian color-magnitude selection. We limit the data to i0 < 22.5 and retain stars only in the color interval 0.8 < (g−i)0 < 1.8. An inspection of panel ‘b’ of Fig. 16 reveals that this selection avoids the bulk of the Galactic disk and halo. The results are shown in Fig. 46, and reassuringly they are qualitatively and quan- titatively identical to the previous selection with deeper data and the full color interval. The predicted behav- ior of the Galactic foreground contamination (with this same color-magnitude selection) is also shown in Fig. 46 (turquoise line). The profile of the contamination is nearly flat in this log-linear representation, so contam- Fig. 46.— As Fig. 44, but for stars restricted to the small color- magnitude region 0.8 < (g − i)0 < 1.8 and i0 < 22.5, to ensure a minimal contamination from the Galactic halo and disk. Since this selection is for the purpose of verification only, we make no attempt to calibrate the absolute surface brightness values; hence the ordinate includes an unknown constant. The exponential fit to the outer halo (blue dashed line) has hR = 48.8 ± 8.8 kpc, while the Hernquist fit (blue dot-dashed line) has a scale radius of 53.6 ± 0.3 kpc. The black histogram is the metal-poor minor axis selection, also constrained to the narrow color-magnitude re- gion. The exponential fit to these data (black dashed line) has hR = 32.5 ± 1.5 kpc, while the Hernquist model has a scale ra- dius of 53.9 ± 0.1 kpc. The power-law fit to these same data (red line) has an exponent of 1.85±0.16. For comparison, we also show the profile of the Galactic foreground as predicted by the Besançon model (turquoise line). The same color-magnitude selection is used as for the observed profiles, although we show here the model pre- diction over the entire MegaCam survey area (not just the “outer halo” or minor axis regions). The model predicts a decrease in the foreground contamination with radial distance over the survey region, but it is essentially flat compared to the observed decrease in the M31 populations. Fig. 47.— The expected spread in distance modulus as a function of projected radius if the underlying halo component falls off as ρ(r) ∝ r−2.91. The dashed line shows the distance modulus to M31, while the full lines shows the limit of r = 191 kpc (the virial radius estimated by Ibata et al. 2004). The dashed and dot-dashed lines mark the region enclosing 50% and 90% of the stars. Fig. 48.— Matched filter map (logarithmic representation) to search for structures around M33 constituted of stars with metal- licity in the range −3.0 < [Fe/H] < 0.0. A limiting magnitude of i0 = 23.5 was used. The two red ellipses mark elliptical radii of s = 0◦.5 and s = 0◦.75 around M33. The pink square at ξ = 0◦.30, η = −0◦.24 marks the location of the “halo” field of Mould & Kristian (1986), which is in fact clearly probing the disk of the galaxy. ination cannot account for the observed profile. Thus a slow decline with an exceeding long scale length for the outer halo population is a robust result of this survey. This slow decline has important consequences on the detectability of halo populations. In particular one may worry about the distance spread in the halo, whether we are able to detect stars on the far side of M31, and the corresponding spread in the CMD. Assuming an ρ(r) ∝ r−2.91 profile, we display in Fig. 47 the ex- pected spread as a function of projected radius. We see that even with this extended profile, the distance spread should be relatively modest, ∼ 0.5 mag. 9. M33 The South-eastern corner of the survey extends out to the Triangulum galaxy, M33. The motivation for this part of the study was to attempt to investigate the in- terface region between the halos of M33 and M31. Four fields were positioned along the extension of the minor axis of M31, as shown in Fig. 48, connecting to the archival data centered on the disk of M33. The map reveals clearly the very regular outer disk of M33, as well as the presence of an extended component out to ∼ 3◦, possibly the stellar “halo” of this galaxy. A more detailed discussion of the structural and stellar popula- tions properties of M33 based upon a much wider survey conducted with the INT will be presented in a compan- ion paper (Ferguson et al. 2007, in prep.). We note here that a previous claimed detection of the stellar halo component of this galaxy (Mould & Kristian 1986), was in reality studying the outer disk (their field is marked with a pink square in Fig. 48). We adopted the geometry of the model of McConnachie et al. (2006) for the disk of M33, namely a position angle of 23◦ and an inclination of 53◦.8. The outer red dashed ellipse in Fig. 49 shows the corresponding elliptical radius s = 0◦.75, approximately Fig. 49.— The radial profile as a function of elliptical coordinate distance from M33, in 3 color-magnitude selection regions corre- sponding to locations between Padova isochrones. We truncate the “metal-rich” profile (which is more heavily affected by Galac- tic foreground contamination), where the noise begins to dominate. where the disk appears to truncate in this diagram. As we have mentioned before, the applicability of the isochrones to estimate metallicity is only justified in re- gions composed of old stars, so the “metallicity” profiles displayed in Fig. 49 must be interpreted with extreme caution. Here we show the trends as a function of ellipti- cal coordinate s for three different CMD bins, as shown. The data interior to s = 0◦.5 is severely affected by crowd- ing, and we therefore neglect that region. In the region to 0◦.75 < s < 1◦, the blue selection becomes more pro- nounced with increasing radius relative to the other two selections, indicating strong radial variations in the stel- lar populations. The exponential profile of the inner disk ends changes abruptly at s ∼ 0◦.9 into an apparently flat distribution for 1◦ < s < 2◦.5. Fitting the profiles in the interval 1◦ < s < 2◦.5 with an exponential function gives exceedingly long scale-lengths, or even rising profiles. The spatial extent of the MegaCam survey around M33 is very limited, so it is impossible to construct a global model for the extended outer component. Thus it is not clear whether the appropriate geometry for calculating the profiles is spherical or ellipsoidal. If we adopt a spher- ical coordinate as in Fig. 50, the profile of the extended component for the selection −3.0 < [Fe/H] < −0.7 seems more reasonable, as it descends monotonically apart from a bump at 1◦.6. Fitting the data between 1◦ < R < 4◦ (but reject- ing the bin at 1◦.6) yields a scale length of 18 ± 1 kpc for an exponential model, or alternatively a scale radius of 55 ± 2 kpc for a projected Hernquist model. These scale lengths are surprisingly large, reminiscent of the large values measured above for the outer halo of M31. Curiously, the central surface brightness of the extrapo- lated exponential models are rather similar too. In M33 the model has ΣV (0) = 29.7 ± 0.1, while in M31 the two exponentials fit in Fig 44 bracket this value with ΣV (0) = 30.6 ± 0.3 and ΣV (0) = 29.0 ± 0.06 (taking the metallicity selection −3.0 < [Fe/H] < −0.7 for both objects). We stress here that the detection of a halo component around M33 gives further confirmation that Fig. 50.— The radial light profile in M33 as a function of the radial coordinate r. We display a fitted exponential model with scale length 18±1 kpc and a projected Hernquist model with scale radius 55± 2 kpc. Fig. 51.— Background-subtracted Hess diagrams in four selected regions near M33. In panel ‘a’, we show the region 0◦.5 < s < 0◦.75, panel ‘b’ is for 0◦.75 < s < 1◦, panel ‘c’ is for 1◦ < r < 2◦ and panel ‘d’ is for 2◦ < r < 3◦. the M31 detection is not due to errors in the foreground subtraction, since the foreground contamination profile has the opposite slope as a function of galactic radial distance in the M33 survey fields compared to the M31 fields. The bump in the surface brightness profile at 1◦.6 is (just) visible as a faint arc on the map in Fig. 48, but we are unsure of the reality of the structure, since it is a very faint feature and only extends over one field. Further imaging is required to determine whether this is a substructure in the halo of M33 or not. Finally, we show in Fig. 51 the progression in the stel- lar populations as we move from the outer disk into the halo component. Of particular interest is the difference between panels ‘b’ at the disk edge (0◦.75 < s < 1◦) with panel ‘c’ (1◦ < R < 2◦) in the halo. The “halo” compo- nent contains a higher proportion of blue stars, compared to the broader distribution in panel ‘b’. 10. DISCUSSION 10.1. The underlying halo The analysis presented above in §8 indicates that un- derneath the many substructures that we have uncovered in M31 lurks an apparently smooth and extremely ex- tended halo. A similar structure is also detected in M33. By “smooth” what we mean here is not necessarily that the component is perfectly spatially smooth, but instead that any substructures that may be present are below detectability with the current survey. The detectability threshold is a function of radius, but it corresponds to approximately 1 mag arcsec−2 brighter than the smooth background over spatial scales >∼ 1 deg The existence of a stellar halo component which ap- pears smooth at these surface brightness levels is com- pletely unexpected given recent numerical models that implement recipes for star-formation in merging CDM subhalos (Bullock & Johnston 2005; Abadi et al. 2006). Those models predict that the light at large radius is confined to arcs, shells and streams, with essentially no smoothly-distributed stars beyond ∼ 50 kpc in a Milky- Way (or M31) analogue. The reason for this is that dynamical times at large distances from the galaxy are extremely long, so material has not had anywhere near enough time to mix. The more recent the accretion, in general the more spatially confined the stars should be. Given these considerations, one would expect a smooth component to be made in the early violent phases of galaxy formation, and since the disk is a fragile structure (Toth & Ostriker 1992), the formation of the structure would have had to have occurred before the formation of the thin disk. This scenario still poses problems how- ever, since the proto-Andromeda at z ∼ 2 would have been much less massive than it is today, so the extreme distances of these halo stars — most likely beyond the virial radius of the galaxy at that redshift — are hard to explain. Interestingly, the radial profile of this smooth halo component in M31 is similar to what is deduced for the Milky Way. As we have reviewed in §1.2, in the case of the Milky Way, current data probe the halo well up to r ∼ 20 kpc, we have reasonable constraints up to r ∼ 50 kpc, but beyond that distance the information is very scanty indeed. However, at least up to r = 50 kpc, and given variations from study to study (which are prob- ably due to halo substructures) the density can be ap- proximated by ρ(r) ∝ r−3. For instance, the study of Siegel et al. (2002), which made use of good distance estimates to halo stars found ρ(r) ∝ r−2.75±0.3. Simi- larly, analysis of the RRLyrae sample of Vivas & Zinn (2006) yielded ρ(r) ∝ r−2.7±0.1 or ρ(r) ∝ r−3.1±0.1, depending on model assumptions of the shape of the halo. This is completely consistent with the present Σ(R) ∝ R−1.91±0.11 fit to the minor axis selection in In modern galaxy formation simulations stars are formed only within the most massive sub-haloes that merge to form a galaxy. This is because star-formation recipes used in the simulations impose a threshold in gas density below which stars cannot form, bas- ing this condition on observed correlations between Hα emission and gas surface density in galaxy disks (Kennicutt 1989). Furthermore, those satellites that were not massive enough to accrete sufficient gas be- fore the epoch of reionization are expected not to have been able to form stars subsequent to that epoch (Bullock, Kravtsov & Weinberg 2000). Dynamical fric- tion acts more strongly upon the most massive subhalos, making them fall rapidly into the potential well, where they become disrupted and their contents mixed into the evolving galaxy. Because of this, stars accreted from sub- halos are expected to have a more rapidly falling pro- file than the dark matter, as we have reviewed in §1.5, with the light profile falling as r−4 or steeper. Neverthe- less, this prediction does not appear to hold out. If dark matter is distributed according to the “Universal” NFW profile (Navarro, Frenk & White 1997), the density pro- file in the outer regions of the halo will be ρ(r) ∝ r−3, consistent with what we have measured from the stars. This suggests that stars in these tenuous outer reaches of giant galaxies trace the dark matter. We stress here that the present analysis of M31 is based on a dataset that is much more spatially extensive than has been possible for the Milky Way. We have covered substantially more than a quarter of the halo of M31. In comparison, even the SDSS studies of Yanny et al. (2000); Ivezic et al. (2000) or Chen et al. (2001) covered only 1% of the sky. Another measure of the halos of these two galaxies that we may now compare is their total luminosity. Integrat- ing the lower of the two exponential profiles shown in Fig. 44 out to 140 kpc, gives a conservative lower limit to the smooth halo of LV ∼ 2.2 × 10 8L⊙. We estimate an upper limit by integrating the power-law up to the virial radius (which we take to be 191 kpc), assuming that the halo density inside 0.5 kpc is constant; this yields a value of LV ∼ 1.3× 10 9L⊙. For the Milky Way, we esti- mate the total luminosity by assuming a Solar Neighbor- hood V-band luminosity of halo stars of 22300L⊙/ kpc (Morrison 1993); for a density law ρ(r) ∝ r−3, inte- gration out to 50 kpc gives LV ∼ 7 × 10 8L⊙ or alter- natively LV ∼ 1.2 × 10 9L⊙ for ρ(r) ∝ r −3.5 (follow- ing Robin et al. 2003 we also assume that the density of the halo is constant in the inner 0.5 kpc). These esti- mates both for M31 and the Milky Way are very crude, but taken at face value they indicate that the stellar halo of M31 is very similar in total luminosity to that of the Milky Way. Thus it appears that previous esti- mates (e.g. Reitzel, Guhathakurta & Gould 1998) who reported that the halo in M31 is ∼ 10 times denser than that of the Milky Way apply only to the inner regions of the galaxy, where contamination from the large bulge, extended disk and intervening substructures are clearly a concern. As reviewed above, Chapman et al. (2006) were able to detect the true inner halo of M31 by observing mostly major axis fields where halo stars have a very different kinematic signature to other components. At radii be- tween 10 and 70 kpc, the halo component was found to have a mean metallicity of [Fe/H ∼ −1.4. This is con- sistent with the photometric estimate derived for the outer halo component in Fig. 43 over the radial range 75 < R < 140 kpc, and suggests that the halo has a small or negligible metallicity gradient. This result pro- Fig. 52.— Spectroscopically-observed fields. Kalirai et al. (2006b) fields are shown in red, Chapman et al. (2006) fields in green. Many of these pointing were chosen without knowledge of the underlying populations, so only now is it possible to properly interpret the spectroscopic results. vides further support for the case of a smooth monolithic halo formed in a single merging event. 10.2. Comparison to Kalirai et al. (2006b) Our discovery of a smooth very extended halo component covering the entire southern quadrant of Andromeda was anticipated by the kinematic study of Kalirai et al. (2006b). These authors used the Keck/DEIMOS spectrograph to survey a number of fields in this region of the sky, targeting known dwarf galax- ies as well as “empty” halo fields. The position of the fields presented in Kalirai et al. (2006b) are shown with red dots in Fig. 52, green dots mark the positions of fields observed with this instrument by our own group (Ibata et al. 2004, 2005; Chapman et al. 2006). The Kalirai et al. (2006b) fields marked ‘d2’ and ‘d3’ being located on the satellites And II and III, are not of relevance to the current discussion. But for many of the remaining of their fields our present panoramic survey is invaluable, as it allows one to identify the stellar popu- lations that study actually targeted. In particular, their fields “m6” were placed on the edge of stream ‘B’, while their fields ‘a13’ and ‘b15’ lie on the extended cocoon of the Giant Stream. Likewise, in Chapman et al. (2006) we serendipitously targeted streams ‘C’ (fields F25 and F26) and ‘D’ (field F7). Thus we see that only fields ‘m8’ and ‘a19’ were tar- geted in regions where we can be sure that no substruc- ture was present, while field ‘m11’ lies outside of the cur- rent survey region. In these fields, Kalirai et al. (2006b) report 1 probable M31 halo star in ‘m8’, 4 stars in ‘a19’, and 3 stars in ‘m11’. Are these counts consistent with our results? We nor- malize with respect to the Kalirai et al. (2006b) field ‘a0’ at 30 kpc, where we deduce ΣV ∼ 30mag arcsec −2. In that field 67 halo stars were detected in observations over 3 spectroscopic masks (i.e. 3 subfields were observed). Whereas in their field ‘m11’ at 165 kpc, where a mild extrapolation from our survey region gives ΣV ∼ 34 – 35mag arcsec−2, 3 halo stars were detected using 4 spec- troscopic masks. We therefore expect 40 to 100 times lower stellar density in ‘m11’ compared to ‘a0’, that is, we expect 0-2 stars to be detected in the 4 masks ob- served in field ‘m11’ (taking the best-case scenario that all available halo stars were observed and correctly clas- sified). This is then consistent with the sample of 3 halo stars that were reported by Kalirai et al. (2006b) in field ‘m11’. We note however, that their field lies ∼ 4◦ from M33, where we have found that the halos of M31 andM33 overlap, and are approximately of equal surface bright- ness. Though it is dangerous to draw conclusions from such a minuscule sample, one out of the 3 halo stars in m11 has a velocity of −150 km s−1, and is highly unlikely to belong to M31, but could be perfectly consistent with being a member of the halo of M33. Likewise, in field ‘m8’ we expect 2.5 stars, while in field ‘a19’ we expect 2.2 stars, consistent with the number of stars detected spectroscopically. In summary, despite the very small number of stars in their sample, and despite the probable contamination from M33 in their most distant (and interesting) field, we take the results of Kalirai et al. (2006b) as confirma- tion that a smooth extended stellar halo is present in M31 out to at least 150 kpc. We note in passing that Kalirai et al. (2006b) estimate the photometric metal- licity of their outer halo sample (R > 60 kpc) to be 〈[Fe/H]〉 = −1.26± 0.1. Although this is apparently con- sistent with the MDF shown in Fig. 43, their sample is almost entirely dominated by “contamination” from substructure, which as we have shown above, in predom- inantly metal-poor. 10.3. Shape of the smooth stellar halo As reviewed in §1.2, most studies of the halo of the Milky Way find that this component is oblate in- terior to r ∼ 20 kpc, with flattening b/a ∼ 0.6. Studies of the halo component in external galax- ies, be it from a medianed stack of edge-on spirals (Zibetti, White & Brinkman 2004), or from an individ- ual edge-on galaxy (Zibetti & Ferguson 2004) find an identical measurement of b/a ∼ 0.6, within roughly the same radius. The data we have presented on M31 do not allow us to make any statement about the halo flatten- ing in the same volume, and it is very hard to imagine that such a measurement will be possible in the fore- seeable future given the difficulty of disentangling bulge, disk and halo in the inner regions of M31. Previous mea- surements of the flattening of M31 in this region (e.g. Pritchet & van den Bergh 1994: a/b = 0.55 ± 0.05 at 10 kpc), give an indication of the shape of the total light distribution, but do not constrain the shape of the halo. However, we believe we have been able to identify the main substructures beyond a distance of R = 6◦.5, giv- ing a relatively uncontaminated measurement of the den- sity profile beyond that radius. We find, however, that the minor axis profile is higher than the profile from the broad region we have termed “outer halo” and which lies closer to the major axis. This allows us to firmly reject an oblate halo with b/a ∼ 0.6 at these distances, and suggests instead the possibility that the halo is prolate, with c/a >∼ 1.3. Further data in other quadrants is re- quired to assess the reliability of this estimate. However, in any case, the shape of the outer halo of M31 is mani- festly different to that of the inner halos of other galaxies observed to date. 10.4. Substructures Every step we have taken in obtaining a wider view of Andromeda has awarded us with new discoveries in the form of previously unknown substructure. The large area surveyed with MegaCam in the present contribution has continued this trend showing new dwarf galaxies, and several diffuse stellar populations in the form of arcs, streams or shell segments. These structures testify that accretion and therefore galaxy buildup is still continuing to the present time. Of the substructures that are present in the survey region the Giant Stream is by far the most significant. The data presented in §6.2 shows that the Giant Stream is a long cigar-shaped structure made up of metal rich, or young, stars with a metal-poor envelope or cocoon, possibly ∼ 3◦ wide. This lack of homogeneity of the stellar populations in the Giant Stream indicates that so far the system has not been fully mixed during the course of the tidal disruption process, so it is likely a dynamically very young stream. The requirement that the center and the cocoon remain spatially distinct will likely provide very useful additional constraints for the modeling of the system. We count up the Giant Stream stars to i0 = 23.5, and as before use And III to normalize the total luminosity. (We caution the reader again that using And III as a ref- erence introduces a large uncertainty into the luminosity estimate). Integrating within the red polygon shown in Fig 23 (and removing a 0◦.5 circle around both And I and And III), and subtracting off the expected foreground from the Besançon model, we find LV ∼ 1.5 × 10 (MV ∼ −15.6) over this region. This corresponds to ap- proximately a tenth of the luminosity of M33, and given that the MegaCam region only probes a fraction of the total stream, it is plausible that the progenitor of the Giant Stream was initially a galaxy of similar luminos- ity to M33. The width of the stream appears consistent with this possibility, though of course it must have been broadened in the merging process. The core and cocoon dichotomy support further the analogy with a dwarf disk galaxy like M33. Indeed, the metal-poor cocoon may be the remnant of a vestigial halo. It will be interesting to conduct new simulations in which a small disk galaxy is accreted by M31. This luminosity of the Giant Stream, measured from the southern quadrant, is between a factor of 1 and a factor of 10 less luminous than that of the total smooth halo component estimated above. This indicates that the Giant Stream is a very significant, probably the largest, merging event into the halo that has ever taken place in Andromeda. If merging dwarf galaxies are responsible for contributing globular clusters into halos, one should therefore expect to find a commensurate number of halo globular clusters with kinematics compatible the Giant Stream and its extension. In Fig. 53 we present an RGB image of the survey region, in which the red, green and blue channels contain, respectively, the matched filter maps for metal-rich (−0.7 < [Fe/H] < 0.0), inter- mediate (−1.7 < [Fe/H] < −0.7) and metal-poor (−3.0 < [Fe/H] < −1.7) stars. This image shows the striking differences in stellar populations of the halo Fig. 53.— RGB color composite map, in which red shows stars with −0.7 < [Fe/H] < 0.0, green shows −1.7 < [Fe/H] < −0.7 and blue shows −3.0 < [Fe/H] < −1.7. To render the inner region within the 4◦ ellipse easier to interpret, we have removed the MegaCam data from that region. Dwarf satellite galaxies, being essentially the only structures with a strong metal-poor population appear blue on this map. The differences in stellar populations between the Giant Stream and the several minor axis streams can be seen as striking differences in color. At the center of the galaxy we have added to scale an image of the central regions of M31 constructed from Palomar sky survey plates. substructures we have identified in this survey. Even though the Giant Stream remains the most significant accretion, many more smaller systems are being ac- creted. M31 is evidently still leading a colorful life assimilating its small neighbors. We see also that halo formation is evidently a stochas- tic process. The halo profile and detailed properties of the halo can therefore be expected to differ from galaxy to galaxy depending on the amount of substructure and merging debris that is present. This makes it all the more surprising that the profile of the smooth halo discussed above resembles well that of the Milky Way, suggesting that the reason for this is an underlying similarity in the mass distributions, which is independent of the detailed assembly history. 10.5. The inner minor axis The several streams detected on the minor axis from ∼ 6◦.5 all the way into the edge of the disk are partic- ularly important in that they shed light on the numer- ous previous studies (reviewed in §1) made in this region because it has been considered “clean halo” for many years. Indeed, it is not obvious that there exists a region of “clean halo” in the inner galaxy. This is demonstrated in Fig. 54, which shows a RGB color composite similar to Fig. 53, but using only INT data and with a smaller pixel scale. The variations in stellar populations are ap- parent as color differences, and one can readily see that the G1 clump and NE structure have a different distribu- tion of stellar populations to the Giant Stream and the two “shelves” to the East and West (the figure caption Fig. 54.— RGB color composite map, as Fig. 53, but for the INT data within the 4◦ ellipse (the blue area around the center of the image is an artifact of crowding in certain central fields). We again see the presence of many streams and structures that have been discussed in earlier articles by our group. This RGB image, however, shows vividly the differences and similarities in the stellar populations of these structures. In particular, one notices that the color of the Giant Stream is similar to that of the two “shelves” (at ξ ∼ 2◦, η ∼ 0◦.5 and ξ ∼ −1◦.5, η ∼ 0◦.5) that appear on this map (see Ferguson et al. 2002). Other structures, such as the diffuse NE structure (ξ ∼ 1◦.5, η ∼ 2◦.5) and the G1 clump (ξ ∼ −1◦.5, η ∼ −1◦.7) possess a different distribution of stellar populations. This diagram also allows one to understand the nature of popula- tions seen at various distances along the minor axis. It is clear that at R ∼ 10 kpc on the minor axis the dominant stellar population is that of the extended messy ellipsoidal structure that we have shown previous is a giant rotating component (Ibata et al. 2005). Beyond that radius out to R ∼ 20 kpc we discern a stellar popula- tion with the same color as the Giant Stream. The contours show the approximate location of ΣV = 27, 28 and 29 mag/arcsec 2. The locations of the ACS fields of Brown et al. (2003, 2006a,b, 2007) are indicated with purple squares (the ACS field sizes have been exaggerated for display purposes). states their location). The contours in Fig. 54 show the iso-luminosity surfaces derived from star-counts for stars with −3.0 < [Fe/H] < 0.0, with contour separation of 1 mag/arcsec2 (the levels correspond approximately to ΣV = 27, 28 and 29 mag/arcsec 2). It is immediately apparent from this diagram that at a projected radius from R ∼ 10 kpc to R ∼ 20 kpc on the minor axis, the dominant component is a large irregular ellip- soidal structure whose major axis size extends out to R ∼ 40 kpc. We have shown previously from kinematics in many fields around the galaxy that this is an extended rotating disk-like component (Ibata et al. 2005). Thus, although the surface brightness profile on the minor axis follows approximately an R1/4 law out to 1◦.4, or 19 kpc (Pritchet & van den Bergh 1994; Irwin et al. 2005), it is unlikely that the bulge itself extends out to those radii. Indeed the bulge in near infrared wavelengths is a relatively compact structure that dominates out to Fig. 55.— The black points in both panels reproduce the V-band minor axis surface brightness profile from Irwin et al. (2005). The radial interval 8 < R < 18 kpc (marked with red points in panel ‘a’) is clearly almost straight in this log-linear representation. Fitting the data in this region with an exponential function (dashed line), yields a scale length of 3.22 ± 0.02 kpc (where the uncertainty is the formal error on the fit). We also indicate the regions where the various components are dominant. Recent analysis of the 2MASS 6X imaging data of M31 shows a high-contrast bulge that domi- nates the near infrared light out to ∼ 2.6 kpc on the major axis (Beaton et al. 2007). The bumps in the disk-dominated region (at projected radii between 2 kpc <∼ R ∼ 6 kpc on the minor axis) are due to spiral arms and the star-forming ring. The dashed line in panel ‘b’ shows a de Vaucouleurs model fit using an effective ra- dius of Re = 0 ◦.1 as found by Pritchet & van den Bergh (1994), equivalent to 1.4 kpc (Irwin et al. 2005), which overestimates the starcounts between 1◦ < R < 1◦.5. ∼ 2.6 kpc on the major axis (Beaton et al. 2007). It is therefore pertinent in the current context to review the evidence for the R1/4 law profile. In Fig. 55 we repro- duce the V-band minor axis profile from Irwin et al. (2005); in the interval 8 < R < 18 kpc the light profile is actually remarkably similar to an exponential function with a scale length of 3.22 kpc. We stress that this exponential behavior is not confined to the minor axis data alone: it is present with the same density profile (and normalization) at all azimuth angles (see Fig. 3 of Ibata et al. 2005). In contrast, the de Vaucouleurs profile of Pritchet & van den Bergh (1994), shown in panel ‘b’ of Fig. 55, over-predicts the counts in the radial range 1◦ < R < 1◦.5. The “extended disk” component was found to have an intrinsic scale length of 6.6± 0.4 kpc (Ibata et al. 2005), and to follow an exponential profile out to ∼ 40 kpc (after which the profile flattens out). For the minor axis scale length of 3.22 kpc to be consistent with that intrinsic scale length, the inclination of the outer disk would have to be 60◦.8, very close to the value of 64◦.7 estimated by Ibata et al. (2005). Furthermore, the intrinsic break at 40 kpc (deprojected) would correspond to 1◦.4 on the minor axis, exactly where it is seen. If one wishes to adhere to the previously-held assump- tion that the minor axis is dominated out to R ∼ 20 kpc by an immense R1/4-law “bulge” or “spheroid”, it re- quires a considerable stretch of credibility. It means that this “spheroid” has to be substantially flattened to be consistent with the contours of Fig. 54; the “spheroid” must have an exponential-like profile between (depro- jected) radii of 15 <∼ R ∼ 40 kpc at all azimuth angles; and it must be rotationally-supported, but with a ro- tation rate almost as fast as that of the H I disk. We therefore judge that the “extended disk” picture is a far more likely and less contrived model. This confirms the visual impression of Fig 54: in the distance range 10 <∼ R ∼ 15 kpc the minor axis profile is dominated by a disk-like population, with only minor contribution from the bulge or spheroid. Since we now understand the kinematic and chemical behavior of the “extended disk” from observations close to the major axis (where stars of different components may be more easily distinguished by their differences in kinematics), we can use these insights to interpret the radial variation in the properties of the stellar popula- tions on the minor axis. Interior to ∼ 0◦.2 on the minor axis the dominant population will clearly be the bulge; further out between 0◦.2 < R < 0◦.4, the normal disk contributes in a non-negligible fashion to the profile, as noted by Irwin et al. (2005); then from 0◦.5 < R < 1◦.3 the extended disk component becomes dominant; finally beyond 1◦.5 the underlying smooth halo becomes impor- tant, though spatially confined streams dominate at var- ious locations. Consequently, one should also expect strong radial variations in metallicity and kinematics. The kinematics on the minor axis in particular will be complex, and dif- ficult to disentangle, since all populations have the same mean velocity and their velocity distributions overlap. Going out from the center one should therefore expect to find the bulge, with high metallicity and high velocity dispersion; then in the bulge plus disk region, a wide metallicity range, but a narrower velocity dispersion; then with the addition of the extended disk, the mean metallicity should decrease towards [Fe/H] ∼ −0.9± 0.2, and the velocity distribution should contain a signifi- cant fraction of stars in a peak with dispersion in the range 20 km s−1 to 50 km s−1 (Ibata et al. 2005); then the halo component should appear with [Fe/H] ∼ −1.4 and with a large velocity dispersion of σv ∼ 140 km s −1 at R = 20 kpc, decreasing outwards (Chapman et al. 2006). In addition to these smooth structures one will find the multiple streams detected (and not yet detected!) in this area, which as we have shown can have quite different stellar populations, but which are likely to be dominated by the metal-rich Giant Stream. The velocity distribu- tion of these streams in a small field will in general be a narrow velocity spike of dispersion ∼ 10 km s−1. How- ever, we stress that the minor axis is a very complex region interior to ∼ 30 kpc, with a complex mix of many stellar populations, each component overlapping consid- erably with the others in terms of radial velocity, metal- licity, spatial location, color-magnitude structure, etc. This finding that the minor axis region between 8 <∼ R ∼ 20 kpc is dominated by the extended disk, and not bulge, halo or spheroid as has been assumed in nu- merous earlier articles, goes a long way towards clarifying the diverse and confusing results that have been deduced from observations in this region. In particular, it helps interpret the findings of Brown et al. (2003, 2006a,b, 2007). These authors obtained ultra-deep HST/ACS photometry in two minor axis fields, a Giant stream field, and a field at the edge of the NE disk, in order to determine ages of the underlying populations via main- sequence turnoff fitting (field locations are shown with purple squares in Fig. 54). Their two minor axis fields lie at projected radii of R = 11 and 21 kpc. Due to the rea- sons detailed above, their “spheroid” field at R = 11 kpc probes a location which is dominated by the extended disk population. From their photometry in this region they deduce a best fitting stellar populations model that has 〈[Fe/H]〉 = −0.6 and 〈age〉 = 9.7Gyr. Brown et al. (2006b) dismiss the possibility that the field is related to the extended disk partly on the grounds that the field lies at a de-projected distance of 51 kpc, yet any small warping of the plane of the galaxy, such as we de- duced in (Ibata et al. 2004), invalidates this argument. The remaining argument is the velocity dispersion mea- surement of ∼ 80 km s−1, which appears high for the ex- tended disk (σv ∼ 50 km s −1), until one considers the mix of components that must be present at this location. Further out on the minor axis at R ∼ 20 kpc one can discern a diffuse component that appears of the same red hue as the Giant Stream with this color representa- tion. This is clearly a metal-rich region, and possibly re- lated to the extension of the “NE shelf” of Ferguson et al. (2002), itself the likely continuation of the orbit of the Giant Stream (Ibata et al. 2004). Indeed, Ferguson et al. (2005) showed that the Giant Stream and NE shelf are connected on the basis of near identical stellar popu- lations to 3 magnitudes below the horizontal branch. With hindsight it is therefore not surprising that the R = 21 kpc field of Brown et al. (2006b) contains in- termediate age stars that have a distribution of stellar populations essentially identical to that of their Giant Stream field (which is itself on the outskirts of the “ex- tended disk” region). Fig. 53 also suggests that their NE disk field is also a complex mixture of disk, extended disk, and possibly metal-rich debris from the Giant Stream. We note also in passing that the geometry of the minor axis populations has important consequences for microlensing studies in M31 (e.g., Calchi Novati et al. 2005). With most of the stellar populations previously assumed to lie in the spheroid, being confined primarily in a disk, we predict a much lower self-lensing rate. 10.6. Kinematics of substructures The above discussion also clarifies some previous claims for the existence of kinematic substructure around M31. In a field at R = 19 kpc, Reitzel & Guhathakurta (2002) find four metal-rich stars in their sample with similar radial velocity of ≈ −340 km s−1, which they in- terpreted as evidence for accretion debris. This position lies within the diffuse region that has stellar populations similar to Giant Stream (Fig. 54), so the kinematic sub- structure in the Reitzel & Guhathakurta (2002) sample is likely related to that structure. Further kinematic substructure in this region was found by Kalirai et al. (2006a) who in studying the kinematics of the Giant Stream find a secondary kine- matic peak R = 20 kpc with v = −417 km s−1 and σv ≈ 16 km s −1. The location of this field (H13s) lies at ξ = 0◦.29,η = −1◦.53, clearly within the ellipsoidal con- tours in Fig. 54, and furthermore the expected mean ve- locity of the “extended disk” model of Ibata et al. (2005) predicts v = −381 ± 22 km s−1 in this field. The veloc- ity dispersion of the cold component is also similar to what has been found in certain regions of the extended disk (e.g., 17 km s−1 in field F3 of Ibata et al. 2005). We speculate therefore that the cold kinematic structure in field H13s is clumpy structure of the edge of the “ex- tended disk”. Most recently Gilbert et al. (2007) have presented a kinematic survey of several fields along the minor axis of M31. They detect kinematic substructure in three fields, with dispersions of 55.5+15.6−12.7 km s −1 (R = 12 kpc) 51.2+24.4−15.0 km s −1 (R = 13 kpc) and 10.6+6.9−5.0 km s −1 (R = 18 kpc). It is probable that the two structures of velocity dispersion ∼ 50 km s−1 are also related to the “extended disk” component. The large de-projected distances they deduce along the minor axis (51 – 83 kpc) are acutely dependent on the assumption of constant inclination of the disk, which as we have shown is not supported by the data (Ibata et al. 2005). In particular, the R = 12 and 13 kpc fields of Gilbert et al. (2007) lie in the distance regime where the extended disk is dominant in Fig. 55. The cold kinematic component observed in their R = 18 kpc field is likely related to the Giant Stream for the same reason as is the cold kinematic structure in the Reitzel & Guhathakurta (2002) sample. 11. CONCLUSIONS This article has presented a deep panoramic view of the Andromeda galaxy and part of the Triangulum galaxy. Though it is not the deepest external galaxy survey ever undertaken, nor the most extended, we have for the first time covered a substantial fraction of a galaxy out to a substantial fraction of the virial radius to sufficient depth to detect several magnitudes of the red giant branch and with sufficient photometric accuracy to estimate stellar metallicity. To our knowledge this is the first deep wide- field view of the outermost regions of galaxies. The new CFHT data presented here are combined with an earlier survey of the inner regions of M31 (s <∼ 55 kpc) taken with the INT (Ibata et al. 2001b; Ferguson et al. 2002; Irwin et al. 2005). We summarize below the main findings from these surveys. • A huge amount of confusion in the literature has arisen from assuming that the minor axis region between projected radii of 0◦.5 < R < 1◦.3 (7 kpc < R < 18 kpc) is representative of the spheroid. We have shown here that it is not. Instead it is likely to be a complex mix of stellar populations, dom- inated over much of this radial range by the “ex- tended disk”. Many of the previous claims that the spheroid or stellar halo of M31 is very differ- ent to that of the Milky Way were based upon a comparison of the properties of genuine Milky Way halo stars to those of stars in M31 in quite different components. • Beyond the inner (∼ 20 kpc) disk, Andromeda con- tains a multitude of streams, arcs, shells and other irregular structures. Some of these structures ap- pear to be related (they have a similar mix of stellar populations) others are manifestly due to separate accretion events. • The largest of these structures, the Giant Stream, is very luminous, possessing LV ∼ 1.5×10 8L⊙ over the region surveyed with MegaCam. This body dominates the luminosity budget of the inner halo, and once it becomes fully mixed, may double the luminosity of the smooth underlying halo. This ongoing accretion event must be among the most significant the halo has suffered since its initial for- mation. • Ignoring regions with obvious substructure, we find that the remaining area of the survey exhibits a smooth metal-poor stellar halo component. This structure need not be perfectly spatially smooth, but the intrinsic inhomogeneities are below the sen- sitivity of this study. The smooth halo is vast, ex- tending out to the radial limit of the survey, at 150 kpc. The profile of this component can be mod- eled with a Hernquist profile as suggested by sim- ulations, but the resulting scale radius of ∼ 55 kpc is almost a factor of 4 larger than modern halo for- mation simulations predict. A power-law profile with Σ(R) ∝ R−1.91±0.11 (i.e. ρ(r) ∝ r−2.91±0.11) can also be fit to the data. Simulations predicted a sharp decline in the power law exponent beyond the central regions of the galaxy to ρ(r) ∝ r−4 or ρ(r) ∝ r−5. This is not observed. Instead, and unexpectedly, the stellar profile mirrors closely the expected profile of the dark matter. • Since dynamically young accretion events give rise to arcs and streams, and because dynamical times are very long in the outer reaches of the halo, the smoothness of the component over huge areas of the outskirts of the galaxy suggests that the com- ponent is very old. It therefore seems plausible that the structure was formed in a cataclysmic merging event early in the history of the galaxy, probably before the formation of the fragile disk. • The outer halo of M31 (R >∼ 80 kpc) is not oblate. On the contrary, the stellar distribution appears to be slightly prolate with c/a >∼ 1.3, though we judge that a reliable measurement of this parameter will require further data in other quadrants. • Both the density profile of the smooth halo in M31 and its total luminosity (∼ 109 L⊙) are very simi- lar to the Milky Way. Their metallicity and kine- matic properties also resemble each other closely (Chapman et al. 2006; Kalirai et al. 2006a). This is somewht surprising if halo formation is a stochas- tic process as suggested by simulations (see, e.g. the discussion in Renda et al. 2005). • Lumping all stellar populations together, we de- tect a stellar population gradient in the survey such that the more metal-rich populations are more cen- trally concentrated, consistent with the predictions of Bullock & Johnston (2005). However, this is al- most entirely due to the presence of the metal-rich Giant Stream “contaminating” the inner halo. • An extended slowly-decreasing halo is also detected around M33. Fitting this distribution with a Hern- quist model gives a scale radius of ∼ 55 kpc, es- sentially identical to that of M31, though we cau- tion that the poor azimuthal coverage of the survey around M33 makes this result sensitive to uniden- tified substructures and to assumptions about the geometry of the halo. • The stellar halos of M31 and M33 touch in projec- tion, and are probably passing through each other. The kinematics of stars in this overlap region will be fascinating to analyze, though large samples will probably be needed to disentangle the structures. • Two new dwarf satellite galaxies of M31, And XV and And XVI, are presented, which together with those reported in a previous contribution (Martin et al. 2006), brings the number of new satellites detected in the MegaCam survey region up to five. Follow-up studies are currently under- way to understand the nature of these objects and those of lower S/N satellite candidates found in the survey. Many questions remain open. What is the radial de- pendence of the metallicity and stellar populations in the smooth component? Is there a discontinuity in proper- ties between the inner halo and the outer halo similar to the simulations of Abadi et al. (2003, 2006), reflecting native and immigrant stars? It will be very interesting to extend the survey out to the virial radius of the Galaxy and verify whether the correlation between the observed stellar profile and the expected dark matter surface density continues to that radius. Further photometric coverage to the East of the minor axis will also be helpful to study fully the mor- phology and extent of the stream-like structures detected from R = 30 to ∼ 120 kpc and to determine whether these objects are indeed streams, and so make plausible judgements about their origin and evolution and com- pare them to theoretical predictions of the formation of the outer halo. This panorama of the outer fringes of Andromeda and Triangulum has shown that halos are truly misnamed: they are in reality dark galactic graveyards, full of the ghosts of galaxies dismembered in violent clashes long ago. Other, even more ancient remnants, have lost all memory of their original form, and in filling these haunted halos with the faintest shadow of their former brilliance, they follow faithfully the dark forces to which they first succumbed. The true nature of this most som- bre of galactic recesses is finally beginning to be revealed. ACKNOWLEDGMENTS This study would not have been possible without the excellent support of staff at the CFHT telescope, and the careful and meticulous observations performed in queue mode. RI wishes to thank Annie Robin for allowing us privileged access to the Besançon model via UNIX scripts which greatly facilitated the construction of the foreground model, and also many thanks to Michele Bel- lazzini for helpful comments on this work. 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0704.1319

Using conceptual metaphor and functional grammar to explore how language used in physics affects student learning

Using conceptual metaphor and functional grammar to explore how language used in physics affects student learning David T. Brookes Department of Physics; Loomis Laboratory of Physics; 1110 West Green St.; Urbana, IL 61801-3080 Eugenia Etkina The Graduate School of Education; 10 Seminary Place; New Brunswick, NJ 08901 This paper introduces a theory about the role of language in learning physics. The theory is developed in the context of physics students’ and physicists’ talking and writing about the subject of quantum mechanics. We found that physicists’ language encodes different varieties of analogi- cal models through the use of grammar and conceptual metaphor. We hypothesize that students categorize concepts into ontological categories based on the grammatical structure of physicists’ language. We also hypothesize that students over-extend and misapply conceptual metaphors in physicists’ speech and writing. Using our theory, we will show how, in some cases, we can explain student difficulties in quantum mechanics as difficulties with language. PACS numbers: 01.40.Fk;01.40.Ha;03.65.-w I. INTRODUCTION A. Our Starting Point The goal of this paper is to present a theoretical frame- work explaining the role of spoken and written language in physics. This framework can be used to probe how physicists represent their ideas in language and more im- portantly, to understand how physics students interpret language they read and hear. We will use the frame- work to understand the types of meaning students may construct from language and the sorts of difficulties they may encounter in trying to construct meaning from the language that they read and hear in physics. We are going to suggest that there are some student difficulties that may be recognized primarily as difficulties with lan- guage. Below we present two initial theoretical points that will help the reader understand the role of language in learning and communicating physics. 1. Language as a Representation We will adopt Jay Lemke’s view that the primary ac- tivity that students encounter and participate in, in a physics course, is representing [1]. They encounter many different representations of physics ideas: graphs, equa- tions, tables, pictures, diagrams, and words. These repre- sentations of physics ideas are each by themselves incom- plete. It takes an act of assimilating, coordinating, and moving between many different representations in order to create understanding. Therefore one of the first abil- ities students have to develop is the ability to represent ideas and physical processes in different ways and move between representations. Physicists are conscious of the role of equations and graphs in their reasoning. Less attention, however, has been paid to language as a repre- sentation of knowledge and ideas in physics. Our starting point will be to treat language as a legitimate representa- tion of physical ideas and processes. Physicists are aware that some student difficulties may be caused by confusing language (see for example, [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]), but only a relatively small amount of research has been done in this area [12, 13, 14]. 2. Information and Communication Reddy [15] has suggested that people construct mean- ing from the words that they hear, based on their prior knowledge and experience. For example, If you ask some- one: “Are you sad?” And they respond with a “1”: What would this response mean? By itself it means nothing, it is simply a signal. Imagine that you and your partner have a list of possible responses: either 1, 2 or 3. This is called a “repertoire” of responses. After the two of you have established a repertoire of responses you need to assign a meaning to them. Say you two agree before hand that 1 = “yes,” 2 = “no,” and 3 = “unable to give a definite answer.” Now you both have established a shared repertoire of acceptable signals, plus a shared code. You and your partner have the means to com- municate. This example shows that by itself the signal is meaningless. A recipient has to construct the mean- ing using a commonly understood repertoire and a pre- viously shared code (shared a priori between sender and receiver). From the above discussion, it follows that meaning can- not be directly passed, conveyed or in any way trans- ported from the instructor to the student. The teacher has to help the student construct meaning by elaborat- ing the code. Students can then use this code to decode the words that the instructor uses. For example, when a physicist says “the electron is in the ground state,” she means that the electron has a particular energy. How- ever, if the students do not share the code for the word http://arXiv.org/abs/0704.1319v1 “state” as the energy state, they may construct a spatial interpretation from the same statement. B. Overview of the paper In Section II we will explain our theoretical frame- work in the context of the data we gathered. First, we will describe our language data sources (QM text- books, interviews with physics professors, and videos of QM students working on QM problems.). To be able to explain how language works in physics, we found it necessary to introduce the formal categories of analogy, metaphor, grammar and ontology. We will elaborate how analogy, metaphor, grammar and ontology fit together to describe physicists’ language when they “speak and write physics”. Finally, we will present two hypotheses about how one can use our theoretical framework to understand how students are interpreting the language that they read and hear in a physics class. In Section III we will return to the interviews with physics professors and the QM textbooks and code the language used. By looking for patterns of usage that can be described and explained by the theoretical framework we have developed, we will show how this framework is applicable for understanding how physicists use language in their reasoning process. In Section IV we will describe how we tested the ap- plicability of our theoretical framework for understand- ing students’ reasoning and learning in physics. We will present two case studies from our video data of physics students working on QM problems. In Section VI we will explore some future directions that this research on language in physics could proceed. II. THEORETICAL FRAMEWORK A. Introduction Our theory was developed from a number of sources of data: (1) Interviews with 5 physics professors. (2) Orig- inal QM papers from Born [16] and Schrödinger [17], as well as an analysis from Goldstein [18] of how Schrödinger developed the wave equation. (3) A selection of older and more modern, popular introductory quantum mechanics textbooks [19, 20, 21, 22, 23, 24]. (4) Two physics stu- dent homework study groups. We began by comparing the way Schödinger and Born wrote about their ideas with the way modern textbooks and physics professors wrote and spoke about the same ideas. This led us to define two separate patterns of language used to express ideas in QM: 1. The first pattern was language used by the in- ventors of QM. They tended to use cautious and figurative language. Ideas were often expressed as comparisons of the form “X is like Y in cer- tain respects.” They made analogies explicit and cautioned against overextending or misinterpreting these analogies. 2. The second pattern we observed was language used to communicate already established knowledge of QM. (Language used by physics professors and modern QM textbooks.) This language was char- acterized by statements of fact with little if any ref- erence to the original analogies on which the ideas were based. These two patterns of language lead us to investigate the role of analogy and metaphor in describing physicists’ language. In our data of students working on their QM home- work problems, we focussed our attention on episodes when students stopped calculating, and engaged in an activity that could loosely be described as sense-making. In these episodes it appeared to us as if students under- stood the physical ideas, but they were confused about the language used to express the physical ideas. We hy- pothesized that students were confused by the figurative language that physicists used to describe their ideas. We will justify this claim in Section IV. B. Analogical Models Encoded as Metaphors in Physics 1. Metaphors in Physics Language Lakoff and Johnson [25] have hypothesized that hu- man language and the human conceptual system are largely made up of unconscious conceptual metaphors. We have extended this idea to physics by suggesting that physicists speak and write using conceptual metaphors. For example, physicists talk about “diffraction of elec- trons” and a “wave equation for the electron.” Both phrases suggest the conceptual metaphor the electron is a wave. Conceptual metaphors are often unconscious metaphors and seldom made explicit. They have become quite literal, losing their figurative origin through their unconscious and frequent use. For a more complete dis- cussion of what a metaphor is and how it differs from analogy and simile, we refer the interested reader to [26]. Other excellent discussions of the theoretical status of, and issues surrounding, metaphor and analogy may be found in [27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. 2. Types of Analogies Encoded as Metaphors Researchers have shown that physicists use analogical models to construct new ideas [37, 38]. These analogical models become, in time, encoded linguistically as concep- tual metaphors [39, 40]. The way physicists talk about already established knowledge is different than the way they talk about new ideas they are trying to comprehend themselves. We will take this idea further. From the primary data (textbooks, original papers, and interviews with physics professors), we have identified three types of analogical model that metaphors encode. These can be classified by their origin and function: 1. Current analogical models: For example, Schrödinger based his wave equation on an anal- ogy to wave optics [17, 18]. The corresponding metaphorical system is the electron is a wave and is spoken about by modern physicists in terms such as “electron interference,” “electron diffrac- tion,” “wave equation,” and so on. 2. Defunct analogical models: It is often the case in physics that older models, whose limitations have been experimentally exposed and supplanted by better models, live on in the language of physics. The caloric theory of heat lives on in phrases which reflect the heat is a fluid metaphor. For exam- ple, “heat flows from object A to object B.” Physi- cists use these metaphorical pictures when they rea- son. We will elaborate this point further below. 3. Descriptive analogies: For example an anal- ogy between a physical valley and a potential en- ergy graph. The metaphor is potential energy graphs are water wells. Examples of how the metaphor is used in language are: “potential well,” “potential step,” “energy level,” “ground state,” and so on. We will identify metaphors that encode analogies 1 to 3 by identifying the base of the analogy [35]. We will use the idea that conceptual metaphors borrow terms from the base of the analogy and apply these words directly to the target concept. For example, if we look at the matter- wave analogy in QM, we can consider that a water wave or an electro-magnetic wave is the prototypical example which will serve as the “base” of the analogy. Thus words such as “interfere,” “polarize,” “diffract,” and “wave” are used in the context of “an electron.” Such examples will be identified as instances of the electron is a wave metaphor. 3. Features and Functions of Metaphors in Physics We hypothesize that physicists unconsciously prefer to speak and write in metaphors because metaphors have certain features and functions that are advantageous to them. The features and functions of these metaphorical systems are listed below with examples from interview data with physics professors. Feature 1: Conceptual metaphors encode analogies. They encode a more deep and complex piece of knowledge which is the completely elaborated analogy. That elabo- ration as an analogical model is, however, tacit amongst the community who use the metaphor and associated model regularly. Function: Physicists are able to use these metaphor- ical systems to reason productively about a particular situation or problem. For example, the electron is a wave metaphor can be used productively to explain the Heisenberg uncertainty principle: “I often think of it. . . in terms of Fourier transforms and the reciprocity between the bandwidth of the channel and the length of the signal pulse that can be detected.” (Prof Note the use of words from the base domain of electro- magnetic waves: “Fourier transforms,” “bandwith,” and “signal pulse” in particular. Even defunct analogies (type 2) represent productive modes of thought for physicists. There is a class of prob- lems for which it is quite adequate to talk about heat as a fluid. For example, when there is no work being done on or by the thermodynamic system, it is satisfac- tory to think of heat flowing into or out of the system and that the change in temperature of the system is directly proportional to the amount of heat gained or lost. Feature 2: Metaphorical systems are partial in na- ture. This means that more than one metaphorical sys- tem is needed to fully understand a physical concept. Function: We observed that physicists switch easily and unconsciously between one system and another de- pending on the type of question that is asked. For exam- ple, in the following extract, Prof. D switches back and forth between particle and wave metaphor to describe the process of electrons passing through a Young’s double slit apparatus. “Of course in any one experiment,. . . you will not observe. . . an interference pattern on the screen [wave metaphor] — if all you do is to scatter one electron [particle metaphor]. The intensities are just too low [wave metaphor]. . . . you have to have a large number of elec- trons [particle metaphor], you have to have a beam of electrons [wave metaphor]. And each electron will contribute a little piece of the in- tensity that you see on that screen [particle metaphor]. What I envisage is. . . a beam of electrons which can be represented by a plane wave [wave metaphor]. . . ” Feature 3: Metaphors involve the use of the verb “is” rather than “is like.” Metaphors are grammatically iden- tifying relational processes, i.e., they are grammatically equivalent to statements of category membership. Function: We hypothesize that metaphor reflects a par- ticular aspect of an expert physicist’s thought process. The use of metaphor itself rather than simile is signifi- cant. Irrespective of deep philosophical discussions about what is “real,” it seems apparent that physicists them- selves need to assert something stronger than “like” — they need to assert “is” in their own reasoning process. We suggest that this is a fundamental trait of how knowl- edge is generated and represented in physics. It is signifi- cant because such assertions may often conceal the vague or partial nature of metaphor itself. For example, Prof. D provided the following response to the question: What happens to a single electron when it passes through a Young’s double slit apparatus? “. . . [to] understand that experiment, you’ve got to forget about the idea that an electron is a particle. It is not a particle in that context, it behaves like a wave. So you just think of it as a plane wave [our emphasis] advancing on the two slits, and the interference between the two. . . outgoing beams, just using Huy- gen’s principle, leads to the. . . interference pattern. . . ” Note that comparison, “it behaves like a wave,” is fol- lowed directly by, “just think of it as a plane wave.” Feature 4: The apparatus of language constrains the ways physicists can talk about physical phenomena and therefore constrains the types of models that can be rep- resented in language. Function: Descriptive analogies (type 3) encoded as metaphors also represent ways of speaking about/describing physical systems. This is very important because there is a limit on what can be represented with language. Such metaphors also give abstract concepts and quantities a grounding in physical reality and physical experience. Consider for example, the modern physicist’s view of energy. Physicists can define energy as a state function yet can physicists speak literally about energy as a state function? Our hypothesis is that it is simply impossible to come up with grammatical constructions that convey the meaning of energy as a state function. The very best locutions are “energy flows into the system,” or “process X caused the kinetic energy of the system to increase.” In both these cases, metaphorically, energy is being spo- ken of as matter and the system as a container of energy. (This is suggested particularly by the use of the preposi- tions “into” and “of.”) It is no coincidence that these two locutions are identical to examples given by Lakoff and Johnson [25]. The authors describe similar metaphori- cal patterns in how humans (in English at least) encode physical processes and events as movement of substances into and out of containers. Physicists are aware of the limitations of their lan- guage. When asked about what is oscillating in a quan- tum mechanical wave, one professor responded: Prof B: “The problem is you’re trying to shoe- horn a phenomenon into ordinary everyday English language, and I think the problem is with the language, not with the phenomenon. So, if you ask me to explain it in English, I think English has limitations which make it impossible to give a satisfactory explanation in English. But, I don’t have to understand it in English. I mean, I think I sort of know what’s going on. At least I have realized the limitations in English and, it doesn’t bother C. Ontological Underpinnings 1. Introduction It has been suggested that humans divide the world into ontological categories of matter, processes and men- tal states [41]. In this section we will show that this idea can be applied to models in physics. The ele- ments of a physical model: the objects or systems of objects, interaction laws, force laws, state laws etc., may be mapped to the ontological categories of matter, processes, and physical states. In cognitive linguistics, Lakoff and Johnson [25] have shown that systems of con- ceptual metaphors are based on ontological metaphors. These ontological metaphors often give abstract concepts an existence as concrete objects or things. To unite these two views and systematize our linguistic analysis, we hy- pothesize that ontological metaphors in physics language are realized as grammatical metaphors. Functional gram- marians have suggested [42] that the elements of a sen- tence can be divided into participants (nouns or noun groups), processes (verbs or verb groups), and circum- stances (generally adverbial or prepositional phrases). In order to unify the metaphorical and grammatical views, we have suggested [26, 43] that grammatical partici- pants should be mapped to the ontological category of matter, and grammatical processes represent ontological processes. Ontological physical states also have unique grammatical representations, through the use of gram- matical location. 2. A Lexical Ontology We hypothesize that the concepts in a physical or ana- logical model can be arranged into an ontological tree similar to the one proposed by Chi et al. [41]. It is nec- essary to modify Chi et al.’s ontology tree to accommo- date one missing category: namely physical states. (See Fig. 1.) Etkina et al. have suggested that physical models can be broken up into a taxonomy of (1) models of objects, (2) models of interactions between objects, (3) models of systems of objects, and (4) models of processes that the objects/system undergoes [44]. In addition to their taxonomy, we are going to suggest that there are two classes of physical variables that describe a system or the objects in it. These are (5) physical properties of FIG. 1: A revised ontology tree based on [41] objects (such as mass and charge), and (6) state vari- ables that describe a configuration of the system (e.g., position, momentum) or state functions defined over a system configuration (e.g., energy, entropy). We will now show how Etkina et al.’s model taxonomy can be mapped into the ontology tree shown in Fig. 1. This mapping is shown in Table I. Physical properties such as mass and charge should be considered properties of objects classified in the matter category. The categorization of concepts in physics into an on- tology tree (as shown in Table I), will be termed a lexical ontology. For example, physicists generally agree that energy is a state function, while heat and work are pro- cesses by which energy is transfered into or out of a sys- tem. Thus a lexical ontology refers definitions of physics concepts into matter, processes, and states that physicists would agree with as a community. 3. Grammar and Ontology Although physicists can agree on the meaning of terms, how do they represent the ontology of physics concepts with language? We suggest that every physical model described in lan- guage has an ontology and that this ontology is encoded in the grammar of the sentence. This grammatical on- tology can be either literal or figurative (metaphorical). If the lexical ontology matches the grammatical ontology then the sentence is literal. If the lexical ontology does not match the grammatical ontology of the same term in a given sentence, then a grammatical metaphor is present. We suggest that these metaphors may be consistently identified by using the grammatical/ontological analysis elaborated below. For an introduction to the methods of functional grammar, we refer the reader to [42]. Consider for example, “John [agent] kicked [process] the ball [medium].” Here “John” and “the ball” are gram- matical participants, functioning grammatically as ob- jects or matter. We also recognize that “John” and “the ball” are naturally defined as matter in some sense. Thus the grammatical ontology and lexical ontology match. There is nothing metaphorical in this sentence. Consider now for example, “heat [medium] flows [process] from the environment to the gas.” In this sentence a physicist would recognize heat to define a process of movement of energy into the system (lexical ontology). But grammat- ically “heat” is functioning as a participant, namely heat is the matter that is flowing. In this case the grammati- cal function of the term “heat” and the lexical ontology of “heat” contradict each other. The sentence is therefore metaphorical. We are going to propose the following mapping from grammar to the ontology tree shown in Fig. 1: Gram- matical participants should be mapped into the ontolog- ical category of matter. Participants can immediately be separated into living and non-living ontological subcate- gories: Beneficiary, agent, and medium (as it participates in an action process, such as “a force [medium] acts [pro- cess]”), can all be thought of as living entities. Range and medium (as it participates passively in an event process such as “heat [medium] flows [process]”) can be thought of as non-living entities. Certain parts of circumstantial elements can also be mapped to the matter category. In the example, “. . . the incident particles [medium] will be. . . partially transmit- ted [process] through the potential-well region [location].” “the potential-well region” could be classified as non- living matter. However location also functions grammati- cally to make ontological physical states as in “A particle [medium] is [relational process] at coordinates (1,1,1) [lo- cation].” This will be discussed further below. An important type of grammatical process in the dis- course of physics is the relational process. Relational processes are processes of being in that they almost al- ways include some form of the verb “to be.” Relational processes have two modes: identifying and attributional. The identifying mode is a reflexive relationship. For ex- ample, “the neutrino is the lightest known particle.” It makes sense to say “the lightest known particle is the neu- trino.” The attributional mode denotes category mem- bership and is not reflexive. For example: “An electron is a lepton.” We hypothesize that physical states (as expressed in physicists’ language) are commonly comprised of identi- fying relational processes where the second identifier is missing and replaced by a grammatical circumstance of location. Typical examples are: “The electron is in the ground state,” “the particle is at such and such coordi- nates.” Such sentences very often involve a grammatical TABLE I: Table illustrating how Etkina’s model taxonomy successfully maps into Chi’s (modified) ontology Ontological category Matter Process State Ontological sub- category Non-living Event Procedure and Constraint- based Interaction Physical State Taxonomy element objects system interaction causal laws, state laws state variables, state functions metaphor. Ontologically location is mapped to some sort of physical object or matter, this often conflicts with the lexical ontology. These grammatical metaphors corre- spond directly to the ontological metaphors of Lakoff and Johnson in choice of preposition: “in” implies container, “at” implies point location in either time or space, “on” implies surface. We believe that it is also no coincidence that these statements have a grammatical structure iden- tical to those of mental states. For example, in English we say, “I am in love,” or “I am in trouble,” or “I am in a state of confusion” etc. It seems to us that physicists have borrowed this metaphor wholesale and blended it with the notion of a physical state, to create a way of speaking about physical states. Ontological processes that describe the behavior of a physical system, are realized in speech by grammatical material processes. Relational processes realize either physical states as shown above, or denote some compo- nent of the model in the sense of category membership. In grammar there are two types of material process: ac- tion and event. These two types of process can be used to distinguish between living and non-living matter. The entire mapping from grammar to ontological cat- egory is summarized in Table II below. TABLE II: Summary of the mapping between grammar and ontological category Grammatical function Ontological category (If X functions grammat- ically as. . . ) −→ (. . . classify X ontologi- cally as. . . ) Agent −→ Matter:living Beneficiary −→ Matter:living Medium (action process) −→ Matter:living Medium (event process) −→ Matter:non-living Role −→ Matter:non-living Objects in Location −→ Matter:non-living Process −→ Process Manner −→ Process D. Summary The theoretical framework is summarized in Fig. 2 be- Consider, for example, the caloric theory of heat. This theory of thermodynamics began as an analogy to a FIG. 2: Summary of the role of analogy, metaphor, ontology and grammar weightless fluid in the late eighteenth century. Over time the elements of this theory became encoded in the lan- guage of physics as a conceptual metaphor. For example, physicists today still say “heat flows from object A to ob- ject B,” and talk about the “heat capacity” of an object. Phrases and sentences such as these are evidence of the conceptual metaphor heat is a fluid in physicists’ lan- guage. For physicists, speaking about heat is a fluid is a productive mode of reasoning as long as there is no work being done on or by the thermodynamic system. The ap- plicability and limitations of talking about heat as a fluid are communally well understood. The analogy between heat and a fluid has an underlying ontology of matter (the heat fluid), processes (the movement of heat from one object to another), and states (the amount of heat in an object — indicated by the object’s temperature). This ontology is encoded in the grammar of each sen- tence used to speak or write about the thermodynamic system. In the example, “heat flows from object A to object B,” heat is a grammatical participant, while the grammatical process is “flows”. Object A and B are parts of grammatical location. Implicitly, the amount of heat in object A or object B indicates the current state of the system. In the modern thermodynamic model, the onto- logical matter is the atoms or molecules in the system, the processes that the system undergoes are heating and work (processes of energy transfer), and the state of the system is represented by the energy or entropy of the sys- tem for a given configuration of the molecules. Note how the modern ontology is in direct conflict with the caloric ontology of thermodynamics. Speaking about heat as matter is therefore a grammatical metaphor. It is gram- matical metaphors like this that underpin the conceptual metaphor heat is a fluid. E. Student Difficulties, Student Learning The central question of our paper is: What is the in- terplay between the linguistic representations that physi- cists use and students’ learning and students’ difficulties? We will narrow this down to two hypotheses regarding the role of language and learning in physics. These are elaborated in Sections II E 1 and II E 2 below. 1. Student Difficulties Interpreting Metaphors Students struggle to see the applicability and limita- tions of analogies that they encounter. We suggest the same applies to metaphorical language that they hear and read. To comprehend a metaphor people construct an ad hoc category [32, 33]. This means that a state- ment of the form “X is Y” has to be interpreted through the formation of a new shared category (an ad hoc cate- gory) of which Y is a prototypical member. For example, to comprehend a metaphor such as the electron is a smeared paste, the reader has to come up with an ad hoc category shared by both entities. A physicist who understands the quantum mechanical behavior of an elec- tron, might suggest an ad hoc category of “things that don’t have a well-defined location.” There is no guaran- tee that a student will come up with the same classifica- tion. We hypothesize that students often come up with an ad hoc category that is inappropriate to a given situa- tion. This inappropriate categorization is at the heart of their difficulties. These difficulties may manifest them- selves as “misconceptions” or student difficulties. We predict that students will overextend and misapply key aspects of metaphorical systems in physics. Instances where metaphors are overextended or taken too literally will be connected with their faulty reasoning. In order to test these ideas it is first necessary to iden- tify if there are really coherent systems of conceptual metaphors in the way physicists speak and write. In Sec- tion III we will show some of the interview data with physics professors that lead us to see that this view of lan- guage was really applicable to the discourse of physics. In Section IV we will consider examples of student difficul- ties in QM that we can explain as examples of metaphor- ical overextension. 2. Students’ Ontological Confusion Previously, Chi et al. have shown that many student “misconceptions” are based on students’ incorrect onto- logical classification of physics concepts [41, 46, 47]. For example, physicists classify heat is a process, but students reason with it as if it were matter. We want to propose an extension of this idea. From our data it appears that physicists reason about physics by co-ordinating multiple descriptions of a particular phe- nomenon. These descriptions may possess different or conflicting ontological properties. For example, there are times when physicists talk about QM phenomena in terms of waves (a process description) whereas there are other times when physicists prefer to talk about a QM phenomenon in terms of particles (a matter descrip- tion). Physicists are good at co-ordinating these different and sometimes conflicting descriptions. Physicists under- stand when a wave or a particle description work and how use them appropriately in their reasoning. Students learn these descriptions by listening to and reading what their teachers say and write. Our sec- ond hypothesis is that students are failing to co-ordinate appropriately the many different descriptions that they learn from physicists’ language. For example, physicists often describe the “potential energy graph” in QM in terms of physical objects (well, barrier, etc.), endowing the graph with the properties of a physical object. Stu- dents, hearing this language, also learn to think of the graph as a physical object. However, students are often unaware of when this picture is appropriate or inappro- priate. Thus students may attach inappropriate ontolog- ical properties to the idea of the graph as a physical ob- ject. We hypothesize that this process leads to patterns of student reasoning that researchers sometimes interpret as “misconceptions”. In Section IV we will test this hypothesis by analyz- ing an example of students solving a QM problem and consider several studies from the PER literature. III. METAPHORS IN QUANTUM MECHANICS We will trace two metaphorical systems in QM from their origins as analogies through to modern language that physicists use to speak and write about their ideas. These two systems are (1) the potential well metaphor, and (2) the Bohmian metaphor. Both grammatical and metaphorical analyses will serve together to illustrate a number of claims made in Sec- tion II. The claims are: (1) Coherent systems of metaphors exist in physicists’ language. (2) Systems of metaphors encode historical analogies. (3) The language encodes a representation or representations of a phys- ical model that has an underlying ontology of matter, processes and states. (4) Physicists use these linguistic representations to reason productively about certain phe- nomena. The data for the linguistic analysis will come from two sources. The first is the interview study with physics professors referred to in Section II. These professors were all native English speakers We asked them to describe and explain various ideas in QM such as the Heisenberg uncertainty principle, or how they would respond to a student who asked, “what is oscillating in a QM wave?” The interview study consisted of five subjects. The full set of interview questions may be obtained by request from the authors. The second source of data is a selection of QM textbooks [19, 20, 21, 22, 23, 24]. A. The Potential Well Metaphor 1. Original Descriptive Analogy “Because of the Pauli exclusion principle, the electrons must be spread over the available states; but they settle down to the states of lowest energy, so that as more electrons are added, the energy levels in the band fill up like a bucket fills with water.” [48] In this example Peierls makes the analogy explicit. The way he uses it shows that this analogy has a descriptive 2. Analysis of Modern Language Grammatical and ontological analysis: When physicists speak of “potential well” and “energy level,” they give energy an existence as water. When physicists speak about quantum particles “leaking through a barrier,” they give the quantum particles an existence as water. When physicists speak of a “potential well,” “potential step,” “potential barrier,” “confinement,” “trap” etc. . . they give the potential energy graph an existence as a physical object. This ontology is encoded in the grammar. Samples of textbook writing and physicists’ talk from interview data and accompanying grammatical analysis are provided in Table III. From the data we have studied, this selection of talk and writing of physicists (Table III) is representative of the type of language associated with the potential well metaphor. One can see a clear pattern of gram- mar that can be mapped to the ontological categories of matter and processes. This is shown in Table IV below. The grammatical analysis can tell us more than what the objects and processes are in the metaphorical model. It shows us that the potential well metaphor consists of two objects, the particle/wave function, and the poten- tial energy graph, that function as separate grammatical participants. They interact with each other via a number of possible processes such as “tunnel through,” “reflects,” and so on. Thus the common grammatical structure of the potential well metaphor contradicts the conven- tional view of the potential energy as a property of the particle or the system. Metaphorical analysis The ontology encoded in the grammar describes the basic objects and processes of the physical model. To understand more subtle properties of these objects, and their interactions, we need to apply a metaphorical analysis. For this, we need to identify the base of domains of various analogs that go into making up the potential well metaphor. In this section we will analyze an additional sample of clauses and sentences from a selection of popular intro- ductory quantum mechanics textbooks [19, 20, 21, 22, 23]. We will identify each metaphor that makes up the metaphorical system, and present sample examples of its use by physicists. Additional examples may be found in [26].) Where necessary, we will identify the analogi- cal base from where the words have been “borrowed” to create the metaphor. • Metaphor: The potential energy graph is a physical object or physical/geographical feature. Examples:“The perfectly rigid box, represented by a rectangular potential well with infinitely high walls, is an ideally simple vehicle for introducing the mathematics of quantum systems.” [21] “Scattering from a ‘cliff’.” [23] “. . . even for a total energy of the particle less than the maximum height of the potential hill. . . ” [19] “What are the classical wave analogs for particle reflection at a potential down-step and a potential up-step?” [21] Base: The words “box,” “well,” “hole,” “cliff,” and “hill” are borrowed from the category of physical objects or physical/geographical features. • Metaphor: The previous metaphor the poten- tial energy graph is a physical object or physical/geographical feature entails an- other metaphor: The “walls” of the well or bar- rier correspond to a physical height above the ground. In other words, energy is a vertical spatial dimension in the Earth’s gravitational field. The potential energy graph is a phys- ical object or physical/geographical fea- ture metaphor builds on this spatial metaphor. Examples:“It is instructive to consider the effect on the eigenfunctions of letting the walls of the square well become very high. . . ” [20]. Prof A: “. . . your zero point energy is going to go up and up.” “. . .ψ1, which carries the lowest energy, is called the ground state. . . ” [23] Base: The words “high,” “up,” “lowest,” and “ground” all suggest an analogy between the verti- cal axis of the potential energy graph and a vertical spatial dimension on the Earth’s surface. • Metaphor: The potential energy graph is a container. The potential energy graph “con- tains” or “traps” either the wave function, the par- ticle or the energy of the particle. TABLE III: Samples of physicists’ speech and writing for grammatical analysis. Sample of physicist’s speech or writing Simplified exerpt with analysis “In both cases, a classical particle of total energy E. . .moves back and forth between the boundaries.” [21] a classical particle [medium] moves back and forth [pro- cess:event] between the boundaries [circumstance:location]. “. . . when you have a confined system, . . . [the width of the box is] going to set the scale for what . . . the magnitude of the energy is, so as you confine it [the particle] more and more, your zero point energy is going to go up and up.” - Prof. A, interview study . . . your zero point energy [medium] is going to go up and up [process:event]. “. . . it has been seen that potential barriers can reflect parti- cles that have sufficient energy to ensure transmission clas- sically.” [19] . . . potential barriers [agent] can reflect [process:event] parti- cles [medium]. . . “This [wave] packet would move classically, being reflected at the wall. . . ” [22] This wave packet [medium] is reflected [process:event] at the wall [circumstance:location]. “The α-particle then ‘tunnels through’ the barrier. . . ” [19] The α-particle [medium] then ‘tunnels through’ [process] the barrier. . . [range] “. . . they [α particles] start out with the energy E inside the nucleus and ‘leak’ through the potential barrier.”[24] α particles [medium] leak through [process:event] the poten- tial barrier [range]. “. . . the phenomenon of tunneling. . . allows the particle to ‘leak’ through any finite potential barrier. . . ” [23] The particle [medium] leaks through [process] any finite po- tential barrier [range]. TABLE IV: Ontology of the potential well metaphor Matter Process QM/classical particles, wave packet, energy, energy walls, energy barrier, potential barrier, barrier moves, reflect(s), tunnels through, leaks through Examples: “The exponential decrease of the wave function outside the square well for the second energy state is less rapid than is the correspond- ing decrease for the lowest energy state as indi- cated. . . ” [21] “Inside the well where V (x) = −V0. . . ” [23] “. . .bound [energy] states in the. . . well” [21] Base: Words such as “well,” “confined,” “bottle,” and “bound” all suggest an analogy to some sort of container. Physicists also make a distinction between “in/inside” and “outside” the well: Such adverbial phrases also indicate the presence of the container metaphor. Various other elaborated metaphors are built on this ba- sic set. Examples of their usage may be found in [26]. • The potential energy graph is a barrier. • The potential energy graph is a hard container/barrier or the potential energy graph is a semi-hard container/barrier • The particle, the wave packet, and the energy are all given an ontological status of matter. More specifically: – QM particles are hard objects. – The wave packet is a soft or breakable object – QM particles are a fluid – The energy is a fluid See Table V for a summary of the metaphorical map- ping from the domain of physical/geographical features to the domain of quantum systems that involve an inter- action between two or more objects. 3. Productive Modes How do physicists piece together the gram- mar/ontology of the potential well metaphor? How do they use the associated imagery to reason productively about quantum systems? From the dis- course of professors and textbooks we have identified the presence of productive modes for the potential well metaphor. We present five examples below: Squeezing: Squeezing the walls of the well forces the water upwards, thereby raising and spacing out the “en- ergy levels.” Example: Prof. A: “. . . when you have a confined sys- tem, . . . [the width of the well is] going to set the scale for TABLE V: Summary of the metaphorical mapping between the base domain of physical/geographical features and the tar- get domain of interacting QM systems Base domain Target domain Physical/geographical features Interacting QM systems Physical or geographical features → Potential energy graph Vertical height of physi- cal/geographical feature → Magnitude of energy at a point/region on the poten- tial energy graph. Hardness or softness of a → “Height” of the potential en- ergy graph Container with top face → “Trapping” of QM particles, “bound” states Billiard ball → QM particle in some circumstances Soft or breakable objects → QM wave function or wave packet Fluid → QM particle in some circum- stances, or the energy of the particle/system Ball bounding off a wall → Reflection of QM particle Tunneling/penetration → Process by which a QM particle “passes through” a seemingly solid “barrier” Leaking → Process by which a QM par- ticle “escapes” from a QM “container” what . . . the magnitude of the energy is, so as you confine it more and more, your zero point energy is going to go up and up.” Stacking: Matter takes up space. Filling up the well/bucket can be used to understand the behavior of fermions. Example: We already observed Peierls make this anal- ogy explicit[48]. The following is an example of Prof. A. using it: “if you have fermions then. . . you have to keep stacking the fermions into levels which get more and more elevated in energy. . . ” Tunneling/leaking: The potential energy graph be- haves as a physical container or barrier, preventing the escape of the particle. This leads to the ideas of “tunnel- ing” or “leaking.” When reasoning productively, physi- cists recognize that a higher or wider barrier means less probability of tunneling. Examples: Feynman et al. write: “. . . they [α-particles] start out with the energy E inside the nucleus and ‘leak’ through the potential barrier.”[24] Griffiths writes: “If the barrier is very high and/or very wide (which is to say, if the probability of tunneling is very small), then the coefficient of the exponentially increasing term (C) must be small. . . ” [23]. Reflecting/scattering: The wave function or particle is reflected by or scatters off a hard barrier. Example: “For this reason, the rectangular potential barrier simulates, albeit schematically, the scattering of a free particle from any potential.” [22] A way of speaking: It is difficult to come up with realistic physical systems of quantum mechanical parti- cles without resorting to lengthly descriptions. (See [21] for examples.) By separating the QM particle from its potential energy graph, physicists are able to talk easily about the particle interacting with an external object (its own potential energy graph). Example: During one of the interviews we asked a pro- fessor to describe the process of trapping and cooling atoms to absolute zero. DTB: “Are the atoms going to jump out, are you not going to be able to trap them?” Prof. E: “No, of course not, you’d just go down to the lowest eigenstate. I mean, I don’t know how they were trapped in the first place, but suppose you had them in a square well for example.” 4. Summary We have tried to illustrate how grammar and metaphor work together to encode the features of a particular de- scriptive model. Each aspect is necessary and the gram- matical and metaphorical analysis together serve to illu- minate features that each individual analysis cannot do on its own. Fig. 3 presents a visual summary of how the po- tential well metaphor is structured. The poten- tial well metaphor is an example of a metaphor- ical system made up of three ontological metaphors, two of which are encoded in the grammar (the po- tential energy graph is a physical object or physical/geographical feature, and the parti- cle/wave function/energy is a physical object or matter), and one which can only be identified by looking at the imagery (energy is a vertical spatial dimension). Other metaphors such as the potential energy graph is a container or the potential en- ergy graph is a hard barrier build on and elab- orate this ontology. At the sentence level, we can see how productive modes of reasoning are formed by intro- ducing grammatical processes through which the particle or wave function interacts with its own potential energy graph. These productive modes are squeezing, stacking tunneling/leaking, and reflecting/scattering. Physicists also use the potential energy graph is a physical object metaphor as a substitute term (or metonym) for the actual physical QM system. FIG. 3: Summary of the metaphorical system and its usage by physicists. B. The Bohmian Metaphor 1. Introduction The potential well metaphor, as a linguistic repre- sentation, has many of the characteristics of a physical model as described by Etkina et al. [44]. The language describes objects with properties, and processes by which those objects interact with each other. In contrast, the Bohmian metaphor has almost none of those character- istics. It seems to exist in the language of physics solely as a way of speaking. It is an interesting case because it is easy to identify the metaphor, but not the original analogy. Therefore, in this section, we will present the linguistic analysis first and the study of the analogy on which it is based, second. Although we have called it the “Bohmian” metaphor in honor of David Bohm, who advocated the Bohmian interpretation of QM, the entry into the language of physics can be traced back much earlier than this. 2. Modern Language The Bohmian metaphor is identified in language by words and phrases that suggest that the wave function or quantum state is a container that contains the quantum mechanical particle. There are only two metaphors that make up the Bohmian metaphor: • Metaphor: The wave function/quantum state is a container. Examples: Noun groups such as “wave packet” or “envelope function” indicate an analogy to a container. • Metaphor: The QM particle is a physi- cal object contained inside the wave func- tion/quantum state. Examples: This is suggested by prepositional phrases such as “in the ground state” in sentences such as “The electron is in the ground state.” Connection to grammar: In the Bohmian metaphor the wave function or quantum state is conceived of as a container that has a particle as a separate entity inside it. The language is based on two sources. The first source is an analogy to Einstein’s ghost field idea (see Section III B 3 below) but the second source is language itself. Cognitive linguists hypothe- size that mental states are spoken about in language metaphorically as containers [25]. For example, if one is depressed one can say, “I am [relational process] in a state of depression [location].” Such statements seem to all have the same grammatical structure, namely a relational process followed by circumstance of location. It seems as if ontological physical states are expressed by an identical grammatical structure: such as, “the electron is [relational process] in the ground state [location].” It seems as if physicists have unconsciously borrowed this grammar that expresses mental states in every day experience and used it to express physical states in physics. As mentioned in Section II, the metalingual apparatus that we have for realizing physical states in language appears to be extremely limited. The states are locations metaphor, supported by this unique grammatical structure, is one of these limited means of expression. A statement about the physical location of an object within another object would be classified in the ontological category of matter if taken literally. Metaphorically a statement such as “the electron is in the ground state,” is a statement about the energy of a quantum system, and physicists recognize energy as a state function. There is a clear ontological conflict between the literal interpretation of the statement and the meaning that is intended. This leads us to hypothesize that such statements will cause students confusion and may lead to difficulties. 3. The Original Analogy In the case of the potential well metaphor, the analogy on which it is based, is relatively well under- stood. The Bohmian metaphor is easy to identify, but the original analogy is not well known. If our frame- work is correct and language is built on analogy then an original analogy should exist in the mainstream QM lit- erature. We started searching the original QM papers in the hope that we would find some explicit reference to the idea that the wave function could contain the parti- cle inside it. Remarkably, we found such a reference in a paper by Max Born, published in 1926 [16]. “Neither of these two views seem satisfactory to me. [Heisenberg’s interpretation of the wave function and the Schrödinger/deBroglie interpretation of the wave function] I would like to attempt here a third interpretation and test its applicability to collision processes. I thereby pin my hopes on a comment of Ein- stein’s regarding the relationship between the wave field and light quanta. He says roughly that the waves may only be seen as guid- ing [showing] the way for corpuscular light quanta, and he spoke in the same sense of a “ghost field.” This determines the probability that one light quantum, which is the carrier of energy and momentum, chooses a particular [definite] path. The field itself, however, does not have energy or momentum.” [16] [Trans- lation by D.T.B.] There are several remarkable features about this pas- sage from Born: • Firstly, it lays out the Bohmian interpretation of quantum mechanics twenty-five years or more be- fore Bohm proposed the same idea, and one year before deBroglie’s attempt at a “pilot wave” the- • Secondly, when Born says “Neither of these two views seem satisfactory to me,” he is referring to (1) the Heisenberg interpretation of QM which Born describes as “an exact description of the pro- cesses in space and time are principally impossi- ble,” and (2) the Schrödinger/deBroglie interpreta- tion which Born summarizes: “He tries to construct wave groups which have relatively small dimensions in all directions and should, as it seems, directly represent moving corpuscles.” Born is cautioning against overly literal interpretations of (1) an anal- ogy to a classical particle (Heisenberg’s approach), or (2) an analogy to a physical wave (Schrödinger’s approach). Born suggests that both views lead to untenable positions in the physical interpretation of QM and introduces a third model which is essen- tially a hybrid of the wave and particle analogies. Born makes, an analogy to Einstein’s interpreta- tion of light waves and light quanta and applies it to particles with non-zero mass. Born’s mode of reasoning appears to be metaphor- ical as well as analogical. He makes an analogy to Einstein’s view of the electromagnetic field as a ghost field, but he does not suggest that the wave function is “like a guiding field.” Rather, he ex- presses Einstein’s idea directly as “. . . the waves may only be seen as guiding the way for corpus- cular light quanta. . . ” [our emphasis]. For Born to interpret the wave function as a probability distri- bution, he felt it necessary to blend together a wave picture and a particle picture with real particles who have definite trajectories determined proba- bilistically by the wave function. Lakoff and Núñez refer to such a mental construct as a metaphorical blend [28] after the conceptual blend of Fauconnier and Turner [31]. • Thirdly, Born is aware of the limitations of the metaphorical picture he has introduced. In blend- ing a wave and particle picture into a model that looks and feels like a statistical ensemble, Born cau- tions about taking this “Bohmian” picture too lit- erally when he writes: “However, the proposed the- ory is not in accordance with the consequences of the causal determinism of single events.” [16] 4. Productive Modes One of the difficulties with QM is the question of how to speak about quantum processes meaningfully. We sug- gest that the Bohmian metaphor permits a partial so- lution to this problem. Although Born’s suggestion (in- tepreting the wave function as a pilot wave) never made it to the mainstream of physics, the associated language is now ubiquitous and used productively by physicists as we will show in the following example: D.T.B.: “. . . if you wanted to think about how an electron propagates. . . It wouldn’t be sen- sible to talk about it as a wave, you would think more as a particle?” Prof C: “. . . you can think of it as a plane wave. Yeah, . . . in an envelope function which makes it into a wave packet.” More examples may be found in [26]. IV. STUDENT DIFFICULTIES A. The Potential Well Metaphor A group of four junior students in their first QM course, were video taped while working on their QM homework problems. All students were native English speakers. The discussion we present is centered around a problem from French and Taylor[21]. The question was: “What are the classical wave analogs for particle reflection at a potential down-step and a potential up-step?” Notice here the potential well metaphorical system serving a specific function: namely, it describes the shape of the potential energy graph (“potential down-step”). S1: Well, there wouldn’t be reflection in par- ticle physics on a down-step right? Or even, I don’t think even on an up-step. . . S3: No, there’s reflection on an up-step, total reflection. S1: Not classical though, right? S2: Not if its less than the energy though. S1: It just slows it down. In this opening exchange we can observe S1 talking at cross purposes with S2 and S3. S2 and S3 seem to be imagining a classical particle approaching the step and bouncing back (later dialogue show that they do not re- ally shift from this literal view of the situation), while S1 seems to be thinking of a wave approaching with energy greater than the energy of the step. As we see later, S1 is reasoning from picture of a surface water wave passing over a step in a river or sea bed. S1: Not quite sure what the wave analogs would be. If I had to guess I’d say it would be like sound, like those things that male cheerleaders have, like big cones. S4: Megaphones? S1: Yeah. ’Cause I think, you know,. . . basically a step up or step down in resistance. But I am not quite sure what we are supposed to say about that. This is the first example of an analog from S1. It is interesting that S1 sees the key as a change in resistance (at the end of the first exchange S1 says “It just slows it down”), yet he still is the one who proposes a phys- ical form (consistent with the ontology of the graph as a physical object) surrounding the medium rather than a change in the medium itself (which would represent a more obvious change in resistance for the wave). S2: So they’re saying that there would be re- flection on a potential up-step like a. . . S1: Yeah, just like a sound, or a water wave or something. S1: Um, well ’cause I know on a potential up- step,. . . like if you just had. . . water and you had, you know, deeper part and a shallower part, and you had a wave, some of it would reflect back. Here S1 applied the metaphor of a physical object again, and proposes a second analog based on the physi- cal form of the graph rather than a change in “density” or “tension” of the medium. Actually, a physical step on a river bed could be a valid example if S1 connected it to a model of how the resistance experienced by a surface wave attenuates with the depth of the water. He does not, and this explains his uncertainty below. S1: So that’s not too hard to see. But like, I would guess that the same thing would hap- pen if you had a down-step, but that’s not something like I really, I could vouch for. Like I think they’re looking for stuff that like most people know. S2: Is that what its saying? Its coming at it with every energy, like continuous energies, like around the step? S2’s statement is interesting. The use of “at” and “around” are examples of grammatical location and sug- gest the metaphor: the step is a physical object. S1 shows he is still on the right track when he says: S1: I think they’re just asking for like, examples from. . . in real life from when a wave. . . goes into a space of less resistance and has reflection back. S4: So in classical what would happen at a potential down-step? S1: A potential down-step? S2: It would just keep going. . . S1: . . . It would just speed up. At a potential up-step it would just slow down. 1. Discussion One alternative hypothesis to explain the difficulties presented above could be that the students are unable to interpret the physical meaning of the potential en- ergy graph or are simply not understanding the situation. However, S1’s ability to interpret potential energy graphs correctly and articulate the key to the analogy discounts this hypothesis. The data show that his inability to come up with a productive analog must be based on other fac- tors. Our framework explains how S1 is distracted by ap- plying an overly literal interpretation of the potential well metaphor in an inappropriate situation. Possibly, a way of talking (i.e., describing the potential graph as a “step”) is affecting students’ reasoning. Our analysis (Section IV above) shows that the students in this group are searching in the category of “physical objects” for an analogy, in accordance with the underlying ontological metaphor the potential energy graph is a phys- ical object rather than searching in a more produc- tive category. Other researchers have also noticed that QM students tend to pick 2-d gravitational analogs when asked to come up with physical examples of 1-d potential energy graphs [49, 50]. As a control we posed the same problem to the profes- sors in the interview study. They all responded that an analogy of an electron beam scattering off of a potential down step is light traveling from a medium with greater index of refraction to a medium with a lesser index of refraction. When asked why changing optical media was a good analog, most were unable to explain, but contin- ued to elaborate their answer. Only one professor was able to explain why this was a good analogy. Prof. E: “I know because we’ve thought about these things before and its just been classified in that category.” This state- ment suggests that physicists are able to automatically search for an analog in a category of analogous processes rather than analogous objects. It may also suggest that physicists’ ideas have become so tightly bound into larger conceptual units that professors are unable to break down their reasoning into smaller parts again. We have shown how physics professors can use metaphorical systems to reason productively in certain situations while students take the same representation and apply it too literally and inappropriately in other situations. Strange ideas like the megaphone make sense if we understand the underlying ontology of the graph, spoken of as a physical object. We think that the ex- ample of student discourse presented above is a typical example of students’ difficulties arising from linguistic representations. 2. “Robust Misconceptions” Related to the Potential Well Metaphor Are there “robust misconceptions” in QM? The char- acteristics of a robust misconception are that it must be (a) present before instruction, (b) common to a signifi- cant percentage of students in a particular class, and re- producible in form and structure across different classes at different institutions in different contexts, and (c) re- sistant to instruction. (See [51] for example.) Although research on students’ understanding of QM is in its infancy, it appears that students do have spe- cific difficulties that have the characteristics of a robust misconception. One emerging example is presented in Table VI. It has been observed that students think that a QM particle loses energy when it tunnels through a bar- rier. McKagan et al., who studied this example, freely use the word “misconception” in their paper [52]. A consistent pattern of reasoning is presented in Ta- ble VI. This pattern contains the following two elements: (1) It takes energy for a particle to tunnel through a barrier. (2) Making the barrier wider or higher means that the particle loses more energy/expends more effort when tunneling through it. Morgan et al. speculate that the difficulty may come from either (a) intuitive classical ideas about a particle passing though a barrier, or (b) physicists tend to draw the potential energy graph and the wave function superimposed. Thus a decaying wave- function amplitude may be confused with a decrease in energy. McKagan et al., however, noticed something interest- ing in their study. In interviews, they discovered that students do not see the potential energy graph as repre- senting the potential energy of the particle in question. They see it rather as some external object with which the particle interacts. The authors describe an example from their interviews: “When pressed, he said that the ‘bump’ was ‘the external energy that the electron inter- acts with’ and insisted that it was not the po- tential energy of the electron itself, in spite of the fact that it was explicitly labeled as such in the previous question.” The authors speculate that statements like “a particle in a potential” may be the cause of this problem. Our analysis supports this idea and provides an expla- nation for the underlying causes of this student difficulty. The problem is much more widespread than just phrases like “a particle in a potential.” As we pointed out in Section III A 2, many statements that fall under the cate- gory of the potential well metaphor, tend to separate the particle or wave function from its potential energy graph in the grammar of the sentence. Most often the particle/wave function functions grammatically as the medium while the potential energy “barrier” functions as either the range, or circumstance of location. The two grammatical participants then interact with each other by a grammatical process such as “tunnels through” or “is reflected.” We hypothesize that the language is the primary source of the students’ model. Graphical repre- sentations (such as the superposition of the energy graph and the wave-function) and classical intuitions build on and extend this basic model, leading to the idea that energy is lost in the tunneling process. TABLE VI: Selected examples of the “exhaustion” misconception: Summary from three studies. Authors’ summary and explanation Sample student responses used to justify this explanation. Lei Bao [49] interviewed ten students over two semesters. Three responded with the incorrect idea that a quantum mechanical particle loses energy when it tunnels through a potential barrier. Bao observed that all three students gave similar re- sponses. Mike: “. . . less energy so the amplitude will be reduced,. . . Amplitude is reduced because energy is lost in the passage [our emphasis]. . . ” Jeffrey Morgan et al. [53] found that all six students that they interviewed thought that the particle lost energy when it went through a potential barrier. Two of the students had completed a senior level QM course and four had completed a sophomore level introductory QM course Selena: “Uh, because it requires energy to go through this barrier.” Jack: “. . . when the particle of some . . . energy, encounters a potential barrier, there is a possibility. . . that a particle will actually just go straight on through, losing energy as it does so, and come out on the other side. . . at a lower energy. . . ” McKagan et al. [52] gave a conceptual test to a group of engineering majors (N = 68) and physics majors (N = 64) after they had completed a modern physics course. One of the questions probed students’ understanding of tunneling processes. On this question 24% of the engineering majors were able to answer correctly and 38% of the physics majors were able to answer correctly. No interview samples were provided, but the authors sum- marize the student responses as follows: “all students who selected answers A, B, or E [more than 50% for both en- gineers and physicists] argued that since energy was lost in tunneling, making the barrier wider and/or higher would lead to greater energy loss.” 3. Summary The example of the potential well metaphor illus- trates how the language used to describe certain QM sys- tems may pose extraordinary difficulties, especially if stu- dents are not aware of how and why metaphorical terms are being used. The metaphorical language, grounded in the classical world, may encourage students to associate extra (classical) properties to the QM system as they try to coordinate these new representations with their prior understanding of the world. These over-extensions of the representation seem to be the source of their difficulties. B. The Bohmian Metaphor As part of our study, two senior undergraduate physics majors (in their second QM course) agreed to be video- taped while working on their QM homework together. Both were native English speakers. In this particular ses- sion S1 and S2 were working on a problem worked out in class by the lecturer that they did not understand. The question may be expressed as follows: “Given an electron in the ground state of an infinite square well of width L. The walls are suddenly moved apart so that the width of the well becomes 2L. What is the probability that the electron is in the ground state of the new system?” The two students working on the problem understood the sudden approximation, they calculated the overlap integral and got a numerical answer which was reason- able. Then S1 stopped and pondered that his answer made no sense. He argued that his answer should be zero. A discussion with the observer (D.T.B.) followed. S1: But I am still confused about what I was. . . saying about if there is a probability FIG. 4: Wave function of the electron in the sudden approx- imation that it is in the [sic] first ground state — it seems to say that the particle can be where it is not. D.T.B.: Why do you say that? S1: Because we know that the wave function looks like this [points to a sketch similar to Fig. 4] — Oh, so its not the probability of it being in the ground state really. . . I think the probability is really. . . I mean, we know that its in this state [points to sketch similar to Fig. 4] so it can’t be in the ground state. So it’s zero [the probability]. The discussion circled around this theme for some time. S1 was concerned that if the particle was “in the ground state” of the new well, it would permit the parti- cle to exist outside of the [-L/2,L/2] region of its initial wave function. The wave function limits where the par- ticle can be, but to say the electron is “in the ground state of the new well” does not suddenly permit it to ex- ist outside of the [-L/2,L/2] region; it is simply a state- ment about measuring the energy of the electron. We believe that the linguistic framework we have developed provides both a reasonable and parsimonious explana- tion for S1’s difficulties. The prepositional phrase,“in the ground state,” is functioning grammatically as a location. S1’s argument, that the probability should be zero, draws specifically on the location metaphor. He says, “it [the original question] seems to say that the particle can be where it is not.” This statement suggests that he is view- ing the question as a question about the location of the particle. In other words, he is interpreting the phrase “in the ground state” literally rather than figuratively. This difficulty with the Bohmian metaphor remains undocumented in the physics education research liter- ature. However, a physics professor who teaches un- dergraduate quantum mechanics, reported in a private conversation that he observed the identical difficulty amongst his students with the same sudden approxima- tion problem. V. CONCLUSION We have shown that coherent systems of metaphors exist in physicists’ language. We have shown that physi- cists use these metaphorical systems in their language to speak and reason productively about QM systems. They are able to invoke many different metaphorical sys- tems, sometimes with apparently conflicting ontologies, depending on the situation they are trying to describe. At the same time, physicists appear to understand the applicability and limitations of their metaphorical lan- guage in each situation. We have also shown how these metaphorical systems can be identified with systematic use of both grammatical and metaphorical analysis. And we have shown how the elaborated metaphors build on the underlying ontology encoded in the grammar. In some cases, it seems that physicists have appro- priated conceptual metaphors from language to express their ideas. The example with Born and the Bohmian metaphor shows how a new idea in physics comes out of a blending of older ideas into a metaphorical blend. Like- wise the final product of the language is (in this case) a blend between an analogy to Einstein’s ghost field and also already existing structures in language that are nor- mally used to describe ontological mental states. We have presented two case studies of groups of stu- dents struggling with and being confused by overly lit- eral interpretations of the metaphorical language they encounter in QM. The context of QM is particularly con- vincing because it is difficult to argue that students enter their QM course with preconceptions or misconceptions about QM based on personal experience. Many of the difficulties observed, appear after instruction. It seems more plausible to hypothesize that these difficulties are related to the way in which physical ideas are presented during instruction itself. We have presented one example (the exhaustion mis- conception) of a documented common conceptual diffi- culty that students have with QM and how we can ac- count for their näıve model with the linguistic framework we have developed. Physicists understand that higher barrier means a slower rate of tunneling or leaking. In contrast, students think that the particles get tired. The underlying issue is use and misuse of the metaphorical picture. VI. FUTURE DIRECTIONS We feel that further research on the role of language in learning physics needs to examine more carefully the in- structional implications of language as a legitimate repre- sentation of knowledge and ideas in physics. For example, how can we make students more aware of the presence of physical models encoded in the metaphorical language that we use? Can students be encouraged to think about the applicability and limitations of different metaphor- ical pictures [40]? Instead of allowing students to say, “the electron is trapped in a square well,” unchallenged, maybe the most important question to ask students is, “what do you mean, what is this ‘square well’ you are talking about?” As a corollary, maybe we should en- courage students to ask us, “what do you mean?” when we use a metaphor such as the electron is a wave without justifying why it is applicable and when it is not. Does it matter how we ask questions of our students? If we phrase a question with different grammar or differ- ent metaphors, do students respond differently? There maybe occasions when the way in which the question is asked is obscuring the real physical understanding that students have. There is some preliminary evidence that this may indeed be the case [54]. It seems to us that if we think of language as a repre- sentation and recognize its unique difficulties, we should put more effort into helping students become comfort- able with this representation. Future research could fo- cus on student difficulties in other areas of physics that may be related to the language that students hear in the physics classroom. (See [13] for example.) In some cases difficulties may be related to linguistic models that stu- dents have developed prior to instruction. We suggest that some student difficulties may be attempts to nego- tiate the meaning and applicability of different linguistic models. Awareness of such linguistic difficulties would help teachers to facilitate their students’ learning. More research on this idea is needed. There is one major aspect of cognitive linguistics that we have not attempted to apply to the field physics edu- cation research in this paper. This is the idea of concep- tual blending [31]. Conceptual blending may provide a complementary account of many of the ideas in this pa- per. Conceptual blending also has an added advantage in that it could account for online meaning construction in terms of the blending of metaphors. This may bet- ter account for “local” or “personal” ways of expression observed among individual professors and students. The dynamics of blending may also be useful for answering questions about how we can make students more aware of the myriad of models encoded by the metaphors in physicists’ language. We think that this may be a fruit- ful line of inquiry in future work. Acknowledgments We would like to thank the following people for their help with this paper: L. Atkins, H. Brookes, G. Horton, Y. Lin, J. Mestre, E. Redish, M. Sindel, A. Van Heuvelen, A. Zech. [1] J. L. Lemke (2004), presentation given at a conference in Barcelona, URL http://www-personal.umich.edu/∼jaylemke/papers/barcelon.htm. 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Shen, Poetics Today 13, 771 (1992). [34] A. Tversky, Psychological Review 84, 327 (1977). [35] D. Gentner, Cognitive Science 7, 155 (1983). [36] R. Boyd, in Metaphor and Thought, edited by A. Ortony (Cambridge University Press, Cambridge, England, 1993), pp. 481 – 532, 2nd ed. [37] M. B. Hesse, Models and analogies in science (University of Notre Dame Press, Notre Dame, Indiana, 1966). [38] N. J. Nersessian, in Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathemat- ics, edited by D. B. Malament (Open Court, Chicago and La Salle, Illinois, 2002), pp. 129 – 166. [39] C. R. Sutton, Leicester University School of Education Occasional Papers (1978). [40] C. R. Sutton, Journal of Research in Science Teaching 30, 1215 (1993). [41] M. T. H. Chi, J. D. Slotta, and N. de Leeuw, Learning and Instruction 4, 27 (1994). [42] M. A. K. Halliday, An Introduction to Functional Gram- mar (Edward Arnold, London, 1985). [43] D. T. Brookes, G. K. Horton, A. 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0704.1320

Supersymmetry versus Gauge Symmetry on the Heterotic Landscape

April 2007 Supersymmetry versus Gauge Symmetry on the Heterotic Landscape Keith R. Dienes1∗, Michael Lennek1†, David Sénéchal2‡, Vaibhav Wasnik1§ 1Department of Physics, University of Arizona, Tucson, AZ 85721 USA 2Département de Physique, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1 Canada Abstract One of the goals of the landscape program in string theory is to extract in- formation about the space of string vacua in the form of statistical correlations between phenomenological features that are otherwise uncorrelated in field the- ory. Such correlations would thus represent predictions of string theory that hold independently of a vacuum-selection principle. In this paper, we study statistical correlations between two features which are likely to be central to any potential description of nature at high energy scales: gauge symmetries and spacetime supersymmetry. We analyze correlations between these two kinds of symmetry within the context of perturbative heterotic string vacua, and find a number of striking features. We find, for example, that the degree of spacetime supersymmetry is strongly correlated with the probabilities of realizing certain gauge groups, with unbroken supersymmetry at the string scale tending to fa- vor gauge-group factors with larger rank. We also find that nearly half of the heterotic landscape is non-supersymmetric and yet tachyon-free at tree level; indeed, less than a quarter of the tree-level heterotic landscape exhibits any supersymmetry at all at the string scale. ∗ E-mail address: [email protected] †E-mail address: [email protected] ‡E-mail address: [email protected] §E-mail address: [email protected] http://arxiv.org/abs/0704.1320v1 1 Introduction Recent developments in string theory suggest that there exists a huge “landscape” of self-consistent string vacua [1]. The existence of this landscape is of critical im- portance for string phenomenology since the specific low-energy phenomenology that can be expected to emerge from string theory depends critically on the particular choice of vacuum state. Detailed quantities such as particle masses and mixings, and even more general quantities and structures such as the choice of gauge group, num- ber of chiral particle generations, and the magnitude of the supersymmetry-breaking scale, can be expected to vary significantly from one vacuum solution to the next. Thus, in the absence of some sort of vacuum-selection principle, it is natural to deter- mine whether there might exist generic string-derived statistical correlations between different phenomenological features that would otherwise be uncorrelated in field the- ory [2]. In this way, one can still hope to extract phenomenological predictions from string theory. To date, there has been considerable work in this direction [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]; for recent reviews, see Ref. [13]. Collectively, this work addresses questions ranging from the formal (such as the finiteness of the number of string vacua and the methods by which they may be efficiently scanned and classified) to the phenomenological (such as the value of the cosmological constant, the scale of supersymmetry breaking, and the statistical prevalence of the Standard Model gauge group and three chiral generations). In this paper, we shall undertake a statistical study of the correlations between two phenomenological features which are likely to be central to any description of na- ture at high energy scales: spacetime supersymmetry and gauge symmetry. Indeed, over the past twenty years, a large amount of theoretical effort has been devoted to studying string models with N=1 spacetime supersymmetry. However, it is im- portant to understand the implications of choosing N=1 supersymmetry over other classes of string models (such as models with N=2 or N=4 supersymmetry, or even non-supersymmetric string models) within the context of the landscape. Moreover, since N=1 supersymmetry plays a huge role in current theoretical efforts to extend the Standard Model, we shall also be interested in understanding the statistical preva- lence of spacetime supersymmetry across the landscape and the degree to which the presence or absence of supersymmetry affects other phenomenological features such as the choice of gauge group and the resulting particle spectrum. In this paper, we shall investigate such questions within the context of the het- erotic string landscape. There are several reasons why we shall focus on the heterotic landscape. First, heterotic strings are of tremendous phenomenological interest in their own right; indeed, these strings the framework in which most of the original work in string phenomenology was performed in the late 1980’s and early 1990’s. Second, heterotic strings have internal constructions and self-consistency constraints which are, in many ways, more constrained than those of their Type I (open) coun- terparts. Thus, they are likely to exhibit phenomenological correlations which differ from those that might be observed on the landscape of, say, intersecting D-brane models or Type I flux vacua. Finally, in many cases these perturbative supersym- metric heterotic strings are dual to other strings (e.g., Type I orientifold models) whose statistical properties are also being analyzed in the literature. Thus, analysis of the perturbative heterotic landscape, both supersymmetric and non-supersymmetric, might eventually enable statistical tests of duality symmetries across the entire string landscape. The first statistical study of the heterotic landscape appeared in Ref. [8]. This study, which focused exclusively on the statistical properties of non-supersymmetric (N=0) tachyon-free heterotic string vacua, was based on a relatively small data set of four-dimensional heterotic string models [14] which were randomly generated using software originally developed in Ref. [15]. Since then, there have been several ad- ditional statistical examinations of certain classes of N=1 supersymmetric heterotic strings [10, 11]. Together, such studies can therefore be viewed as providing heterotic analogues of the Type I statistical studies reported in Refs. [5, 6, 7]. Although the study we shall undertake here is similar in spirit to that of Ref. [8], there are several important differences which must be highlighted. First, as discussed above, we shall be focusing here on the effects of spacetime supersymmetry. Thus, we shall be examining models with all levels of spacetime supersymmetry (N=0, 1, 2, 4), not just non-supersymmetric models, and examining how the level of spacetime su- persymmetry correlates with gauge symmetry. Second, the current study is based on a much larger data set consisting of approximately 107 heterotic string models which was newly generated for this purpose using an update of the software originally devel- oped in Ref. [15]. This data set is thus approximately two orders of magnitude larger than that used for Ref. [8], and represents literally the largest set of distinct heterotic string models ever constructed. Indeed, for reasons we shall discuss in Sect. 3, we believe that data sets of this approximate size are probably among the largest that can be generated using current computer technology. But perhaps most importantly, because our heterotic-string data set was newly generated for the purpose of this study, we are able to quote results that take into account certain subtleties concerning so-called “floating correlations”. As discussed in Ref. [9], the problem of floating correlations is endemic to investigations of this type, and reflects the fact that not all physically distinct string models are equally likely to be sampled in any random search through the landscape. This thereby causes statistical correlations to “float” as a function of sample size. In Ref. [9], several methods were developed that can be used to overcome this problem, and it was shown through explicit examples that these methods allow one to extract correlations and statistical distributions which are not only stable as a function of sample size, but which also differ significantly from those which would have been näıvely apparent from a direct counting of generated models. We shall therefore employ these techniques in the current paper, extracting each of our statistical results in such a way that they represent stable correlations across the entire heterotic landscape we are examining. As with most large-scale statistical studies of this type, there are several lim- itations which must be borne in mind. First, our sample size is relatively small, consisting of only ∼ 107 distinct models. However, although this number is miniscule compared with the numbers of string models that are currently quoted in most land- scape discussions, we believe that the statistical results we shall obtain are stable as a function of sample size and would not change significantly as more models are added to the data sample. We shall discuss this feature in more detail in Sect. 3. Indeed, as mentioned above, data samples of the current size are likely to be the largest possible given current computer technology. Second, the analysis in this paper shall be limited to correlations between only two phenomenological properties of these models: their low-energy gauge groups, and their levels of supersymmetry. More detailed examinations of the particle spectra of these models will be presented in Ref. [16]. Finally, the models we shall be discussing are stable only at tree level. For exam- ple, the models with spacetime supersymmetry continue to have flat directions which have not been lifted. Even worse, the non-supersymmetric models (even though tachyon-free) will generally have non-zero dilaton tadpoles and thus are not stable beyond tree level. Despite these facts, each of the string models we shall be studying represents a valid string solution at tree level, satisfying all of the necessary string self-consistency constraints. These include the requirements of worldsheet confor- mal/superconformal invariance, modular-invariant one-loop and multi-loop ampli- tudes, proper spacetime spin-statistics relations, and physically self-consistent layers of sequential GSO projections and orbifold twists. Thus, although such models may not represent the sorts of truly stable vacua that we would ideally like to be study- ing, it is reasonable to hope that any statistical correlations we uncover are likely to hold even after vacuum stabilization. Indeed, since no stable perturbative non- supersymmetric heterotic strings have yet been constructed, this sort of analysis is currently the state of the art for large-scale statistical studies of this type, and mirrors the situation on the Type I side, where state-of-the-art statistical analyses [5, 6, 7] have also focused on models which are only stable at tree level. Eventually, once the heterotic model-building technology develops further and truly stable vacua can be analyzed, it will be interesting to compare those results with these in order to ascertain the degree to which vacuum stabilization might affect these other phe- nomenological properties. This paper is organized as follows. In Sect. 2, we describe the class of models that we shall be examining in this paper. In Sect. 3, we summarize our method of analysis which enables us to overcome the problem of floating correlations in order to extract statistically meaningful correlations. In Sect. 4, we present our results concerning the prevalence of spacetime supersymmetry across the heterotic landscape, and in Sect. 5 we present our results concerning correlations between spacetime supersymmetry and gauge groups. Finally, our conclusions are presented in Sect. 6. 2 The models The models we shall be examining in this paper are similar to those studied in Ref. [8]. Specifically, each of the vacua we shall be examining in this paper represents a weakly coupled critical heterotic string compactified to four large (flat) spacetime dimensions. In general, such a string may be described in terms of its left- and right- moving worldsheet conformal field theories (CFT’s). For a string in four dimensions, these must have central charges (cR, cL) = (9, 22) in order to enforce worldsheet conformal anomaly cancellation, and must exhibit conformal invariance for the left- movers and superconformal invariance for the right-movers. While any such CFT’s may be considered, in this paper we shall focus on those string models for which these internal worldsheet CFT’s may be taken to consist of tensor products of free, non-interacting, complex (chiral) bosonic or fermionic fields. As discussed in Ref. [8], this is a huge class of models which has been discussed and analyzed in many different ways in the string literature. On the one hand, taking these worldsheet fields as fermionic leads to the so-called “free-fermionic” construc- tion [17] which will be our primary tool throughout this paper. In the language of this construction, different models are achieved by varying (or “twisting”) the boundary conditions of these fermions around the two non-contractible loops of the worldsheet torus while simultaneously varying the phases according to which the con- tributions of each such spin-structure sector are summed in producing the one-loop partition function. However, alternative but equivalent languages for constructing such models exist. For example, we may bosonize these worldsheet fermions and construct “Narain” models [18, 19] in which the resulting complex worldsheet bosons are compactified on internal lattices of appropriate dimensionality with appropriate self-duality properties. Furthermore, many of these models have additional geomet- ric realizations as orbifold compactifications with appropriately chosen Wilson lines; in general, the process of orbifolding is quite complicated in these models, involving many sequential layers of projections and twists. All of these constructions generally overlap to a large degree, and all are capable of producing models in which the cor- responding gauge groups and particle contents are quite intricate. Nevertheless, in all cases, we must ensure that all required self-consistency constraints are satisfied. These include modular invariance, physically sensible GSO projections, proper spin- statistics identifications, and so forth. Thus, each of these vacua represents a fully self-consistent string solution at tree level. In order to efficiently survey the space of such four-dimensional string-theoretic vacua, we implemented a computer search based on the free-fermionic spin-structure construction [17]. Details of this study are similar to those of the earlier study de- scribed in Ref. [8], and utilize an updated version of the model-generating software that was originally written for Ref. [15]. In our analysis, we restricted our attention to those models for which our real worldsheet fermions can always be uniformly paired to form complex fermions, and therefore it was possible to specify the boundary con- ditions (or spin-structures) of these real fermions in terms of the complex fermions directly. We also restricted our attention to cases in which the worldsheet fermions exhibited either antiperiodic (Neveu-Schwarz) or periodic (Ramond) boundary con- ditions around the non-contractible loops of the torus. Of course, in order to build a self-consistent string model in this framework, these boundary conditions must sat- isfy tight constraints. These constraints are necessary in order to ensure that the one-loop partition function is modular invariant and that the resulting Fock space of states can be interpreted as arising through a physically sensible projection from the space of all worldsheet states onto the subspace of physical states with proper space- time spin-statistics. Thus, within a given string model, it is necessary to sum over appropriate sets of untwisted and twisted sectors with different boundary conditions and projection phases. Our statistical analysis consisted of an examination of over 107 distinct vacua in this class. Essentially, each set of fermion boundary conditions and GSO projec- tion phases was chosen randomly in each sector, subject only to the required self- consistency constraints. However, in our statistical sampling, we placed essentially no limits on the complexity of the orbifold twisting (i.e., in the free-fermionic lan- guage, we allowed as many as sixteen linearly independent basis vectors). Thus, our statistical analysis included models of arbitrary intricacy and sophistication. We also made use of techniques developed specifically for analyzing string models generated in random searches, allowing for the mitigation of many of the effects of bias which are endemic to studies of this sort. As part of our study, we generated string models with all degrees of spacetime supersymmetry (N=0, 1, 2, 4) that can arise in four dimensions. For N=0 models, we further demanded that supersymmetry be broken without introducing tachyons. Thus, the N=0 vacua are all non-supersymmetric but tachyon-free, and can be con- sidered as four-dimensional analogues of the ten-dimensional SO(16)× SO(16) het- erotic string [20] which is also non-supersymmetric but tachyon-free. However, other than this, we placed no requirements on other possible phenomenological properties of these vacua such as their possible gauge groups, numbers of chiral generations, or other aspects of the particle content. We did, however, require that our string construction begin with a supersymmetric theory in which the supersymmetry may or may not be broken by subsequent orbifold twists. (In the language of the free- fermionic construction, this is tantamount to demanding that our fermionic boundary conditions include a superpartner sector, typically denoted W1 or V1.) This is to be distinguished from a potentially more general class of models in which supersymme- try does not appear at any stage of the construction. This is merely a technical detail in our construction, and we do not believe that this ultimately affects our results. As with any string-construction method, the free-fermionic formalism contains numerous redundancies in which different choices of worldsheet fermion boundary conditions and/or GSO phases lead to identical string models in spacetime. Indeed, a given unique string model can have many different representations in terms of worldsheet constructions. For this reason, we judged string vacua to be distinct based on their spacetime characteristics — i.e., their low-energy gauge groups and massless particle content. SUSY class # distinct models N=0 (tachyon-free) 4 946 388 N=1 3 772 679 N=2 492 790 N=4 1106 Total: 9 212 963 Table 1: The data set of perturbative heterotic strings analyzed in this paper. For each level of supersymmetry allowed in four dimensions, we list the number of corresponding distinct models generated. As discussed in the text, models are judged to be distinct based on their spacetime properties (e.g., gauge groups and particle content). All non- supersymmetric models listed here are tachyon-free and thus are four-dimensional analogs of the SO(16) × SO(16) string model in ten dimensions. Given this, our ultimate data set of heterotic strings is as described in Table 1. Note that all non-supersymmetric models listed in Table 1 are tachyon-free, and thus are stable at tree level. We should mention that while generating these models, we also generated over a million distinct non-supersymmetric tachyonic vacua which are not even stable at tree level. We therefore did not include their properties in our analysis, and recorded their existence only as a way of gauging the overall degree to which the tree-level heterotic string landscape is tachyon-free. Also note that as the level of supersymmetry increases, the number of distinct models in our sample set decreases. This reflects the fact that relatively fewer of these models exist, so they become more and more difficult to generate. This will be discussed further in Sects. 3 and 4. Of course, the free-fermionic construction realizes only certain points in the full model space of self-consistent heterotic string models. For example, since each world- sheet fermion is nothing but a worldsheet boson compactified at a specific radius, a larger (infinite) class of models can immediately be realized through a bosonic for- mulation by varying these radii away from their free-fermionic values. However, this larger class of models has predominantly only abelian gauge groups and rather lim- ited particle representations. Indeed, the free-fermionic points typically represent precisely those points at which additional (non-Cartan) gauge-boson states become massless, thereby enhancing the gauge symmetries to become non-abelian. Thus, the free-fermionic construction naturally leads to precisely the set of models which are likely to be of direct phenomenological relevance. We should note that it is also possible to go beyond the class of free-field string models altogether, and consider models built from more complicated worldsheet CFT’s (e.g., Gepner models). One could even go beyond the model space of crit- ical string theories, and consider non-critical strings and/or strings with non-trivial background fields. Likewise, we may consider heterotic strings beyond the usual per- turbative limit. However, although such models may well give rise to phenomenolo- gies very different from those that emerge in free-field constructions, their spectra are typically very difficult to analyze and are thus not amenable to an automated statistical investigation. 3 Method of analysis Each string model-construction technique provides a mapping between a space of internal parameters and a corresponding physical string model in spacetime. In the case of closed strings, for example, such internal parameters might include compact- ification moduli, boundary-condition phases, Wilson-line coefficients, or topological quantities specifying Calabi-Yau manifolds; in the case of open strings, by contrast, they might include D-brane dimensionalities and charges, wrapping numbers or in- tersection angles, fluxes, and the vevs of moduli fields. Regardless of the construction technique at hand, however, there is a well-defined procedure through which one can derive the spectrum and couplings of the corresponding model in spacetime. Given this, one generally conducts a random search through the space of models by randomly choosing self-consistent values of these internal parameters, and then deriving the physical properties of the corresponding string models. Questions about statistical correlations are then addressed in terms of the relative abundances of models that emerge with different spacetime characteristics. Indeed, if {α, β, γ, ...} denote these different spacetime characteristics (or different combinations of these characteristics), then we are generally interested in extracting ratios of population abundances of the form Nα/Nβ, where Nα and Nβ are the numbers of models which exhibit physical characteristics α and β across the landscape as a whole. Clearly, we cannot survey the entire landscape, and thus we are forced to at- tempt to extract such ratios with relatively limited information. In particular, let us assume that our search has consisted of analyzing D different randomly generated sets of internal parameters, ultimately yielding a set of different models in spacetime exhibiting varying physical characteristics. Let Mα(D) denote the number of distinct models which are found which exhibit characteristic α. Our natural tendency is then to attempt to associate Mα(D) Mβ(D) (3.1) for some sufficiently large value of D. While this relation might not hold exactly for relatively small values of D, the expectation is that we might be able to reach sufficiently large values of D for which we might hope to extract reasonably accurate predictions for Nα/Nβ. Unfortunately, as has recently been discussed in Ref. [9], Eq. (3.1) does not gen- erally hold for any reasonable value of D (short of exploring the full landscape). Indeed, the violations of this relation are striking, even in situations in which sizable fractions of the landscape are explored, and will ultimately doom any attempt at extracting population fractions in this manner. In the remainder of this section, we shall first explain why Eq. (3.1) fails. We shall then summarize the methods which were developed in Ref. [9] for circumventing these difficulties, and which we will be employing in the remainder of this paper. As stated above, each string model-construction technique provides a mapping between a space of internal parameters and a physical string model in spacetime. However, this mapping is not one-to-one, and there generally exists a huge redun- dancy wherein a single physical string model in spacetime can have multiple realiza- tions or representations in terms of internal parameters. For this reason, the space of internal parameters is usually significantly larger than the space of obtainable distinct models. The failure of this mapping to be one-to-one is critical because any random statis- tical study of the string landscape must ultimately take the form of a random explo- ration of the space of internal parameters that lead to these models. First, one must randomly choose a self-consistent configuration of internal parameters; only then can one derive and tabulate the spacetime properties of the corresponding model. But then we are faced with the question of determining whether spacetime models with multiple internal realizations should be weighted more strongly in our statistical analysis than models with relatively few realizations. In other words, we must decide whether our landscape measure should be based on internal parameters (wherein each model is weighted according to its number of internal realizations) or based on spacetime properties (wherein each physically distinct model is weighted equally regardless of the number of its internal realizations). If we were to base our landscape measure on internal parameters, then these redundancies would not represent problems; they would instead become vital ingre- dients in our numerical analysis. However, if we are to perform statistics in the space of models in a physically significant way, it is easy to see that we are forced to count distinct models rather than distinct combinations of internal parameters. The reason for this is as follows. In many cases, these redundancies arise as the result of world- sheet symmetries (e.g., mirror symmetries), and even though such symmetries may be difficult to analyze and eliminate analytically for reasonably complicated models, their associated redundancies are similar to the redundancies of gauge transforma- tions and do not represent new physics. In other cases, such redundancies are simply reflections of the failures or limitations of a particular model-construction technique; once again, however, they do not represent new physics, but rather reflect a poor choice of degrees of freedom for our internal parameters, or a mathematical difficulty or inability to properly define their independent domains. Finally, such redundancies can also emerge because entirely different model-construction techniques can often lead to identical models in spacetime. Thus, two landscape researchers using dif- ferent construction formalisms might independently generate random sets of models which partially overlap, but once again this does not mean that the models which are common to both sets should be double-counted when their statistical results are merged. Indeed, in all of these cases, redundancies in the mapping between inter- nal parameters and spacetime properties do not represent differences of physics, but rather differences in the description of that physics. We thus must use spacetime characteristics (rather than the parameters internal to a given string construction) as our means of counting and distinguishing string models. Many of these ideas can be illustrated by considering the E8×E8 heterotic string in ten dimensions. As is well known, this string model can be represented in many ways: as a ZZ2 orbifold of the SO(32) supersymmetric string, as a ZZ2 × ZZ2 orbifold of the non-supersymmetric SO(32) heterotic string, and so forth. Likewise, this model can be realized through an orbifold construction, through a free-fermionic construc- tion, through a bosonic lattice construction, and through other constructions as well. Yet, there is only a single E8 × E8 string model in ten dimensions. It is therefore necessary to tally distinct string models, and not distinct internal formulations, when performing landscape calculations and interpreting their results. Unfortunately, this redundancy inherent in the mapping between internal param- eters and their corresponding string models implies that in any random exploration of the space of models, certain string models are likely to be sampled much more frequently than other models. Thus, one must filter out this effect by keeping a record of each distinct model that has already been sampled so that each time an additional model is generated (i.e., each time there is a new “attempt”), it can be compared against all previous models and discarded if it is not new. Although this is a memory-intensive and time-consuming process which ultimately limits the sizes of the resulting data sets that can be generated using current automated technology, this filtering can successfully be employed to eliminate model redundancies. However, there remains the converse problem: because some models strongly dominate the random search, others effectively recede and are therefore extremely difficult to reach. They therefore do not tend to show up during the early stages of a random search, and tend to emerge only later in the search process after the dominant models have been more fully tallied. Indeed, as the search proceeds into its later stages, it is only the models with “rare” characteristics which increasingly tend to be generated, precisely because those models with “common” characteristics will have already been generated and tabulated. Thus, the proportion of models with “rare” characteristics tends to evolve rather dramatically as a function of time through the model-generation process. This type of bias is essentially unavoidable, and has the potential to seriously distort the values of any numerical correlations that might be extracted from a ran- dom search through the landscape. In particular, as discussed in Ref. [9], this type of bias generally causes statistical correlations to “float” or evolve as a function of the sample size of models examined. Moreover, since one can ultimately explore only a limited portion of the landscape, there is no opportunity to gather statistics at the endpoint of the search process at which these correlations would have floated to their true values. This, then, is the problem of floating correlations. Fortunately, as discussed in Ref. [9], there are several statistical methods which can be used in order to overcome this difficulty. These methods enable one to extract statistical correlations and distributions which are stable as a function of sample size and which, with some reasonable assumptions, represent the statistical results that would be obtained if the full space of models could be explored. We shall now describe the most important of these methods, since we shall be using this technique throughout the rest of this paper. In general, a model search proceeds as follows. One randomly generates a self- consistent set of internal parameters, and calculates the properties of the correspond- ing string model. One then compares this model against all models which have pre- viously been generated: if the model is distinct, it is recorded and saved; if it is redundant, it is discarded. One then repeats this process. Early in the process, most attempts result in new distinct models because very few models have already been found. However, as the search proceeds, an increasing fraction of attempts fail to produce new models. This rise in the ratio of attempts per new model indicates that the space of models is becoming more and more explored. Thus, attempts per model can be used as a measure of how far into the full space of corresponding models our search has penetrated. Therefore, if we are interested in extracting the ratio Nα/Nβ for two physical characteristics α and β, as discussed above Eq. (3.1), the solution is not to extract this ratio through Eq. (3.1) because such a relation assumes that the spaces of α- models and β-models are being penetrated at exactly the same rates during the random search process. Rather, the solution [9] is to keep a record not only of the models generated as the search proceeds, but also of the cumulative average attempts per model that are needed in order to generate these models. We then extract the desired ratio Nα/Nβ through a relation of the form Mα(dα) Mβ(dβ) Mα(dα) Mβ (dβ) (3.2) where dα and dβ respectively represent the numbers of attempts that resulted in α- models and β-models, regardless of whether the models in each class were distinct. Thus, we must essentially perform two independent search processes, one for α- models and one for β-models, and we terminate these searches only when they have each reached the same degree of penetration as measured through their respective numbers of attempts per model dα/Mα. The value of Nα/Nβ obtained in this way should then be independent of the chosen reference value of dα/Mα for sufficiently large dα/Mα. This method of extracting Nα/Nβ is discussed more fully in Ref. [9], where the derivation and limitations of this method are outlined in detail. Of course, in the process of randomly generating string models, we cannot nor- mally control whether a random new model is of the α- or β-type. Both will tend to be generated together, as part of the same random search. Thus, our procedure requires that we completely disregard the additional models of one type that might be generated in the process of continuing to generate the required, additional models of the other type. This is the critical implication of Eq. (3.2). Rather than let our model-generating procedure continue for a certain duration, with statistics gathered at the finish line as in Eq. (3.1), we must instead establish two separate finish lines for our search process, one for α-models and one for β-models. Of course, these finish lines are not completely arbitrary, and must be chosen such they correspond to the same relative degree of penetration of the α- and β-model spaces. Indeed, these finish lines must be balanced so that they correspond to points at which the same ratio of attempts per model has been reached. However, these finish lines will not generally coincide with each other, which requires that some data actually be disregarded in order to extract meaningful statistical correlations. As discussed in Ref. [9], Eq. (3.2) will enable us to extract a value for the ratio Nα/Nβ which is stable as a function of sample size only when the biases within the α-model space are the same as those within the β-model space. In such cases, we can refer to the physical characteristics α and β as being in the same universality class. However, for a given model-generation method (such as the free-fermionic construction which we shall be employing in this paper), it turns out that many physical characteristics of interest {α, β, ...} have the property that they are in the same universality class. In the rest of this paper, correlations for physical quantities will be quoted only when the physical characteristics being compared are in the same universality class. The above method is then used in order to extract these correlations. 4 Supersymmetry on the heterotic landscape In this section, we begin our analysis of the structure of the heterotic string landscape. In so doing, we shall also provide an explicit example of the method described in Sect. 3. Our focus in this section is to determine the extent to which string models with different levels of unbroken supersymmetry (N=0, 1, 2, 4) populate the tree-level four-dimensional heterotic landscape. ForN=0 models, we shall further distinguish between models which are tachyon-free at tree level, and those which are tachyonic. Note that these characteristics are all mutually exclusive and together span the entire landscape of heterotic string models in four dimensions. Thus, our goal is to achieve nothing less than a partitioning of the full set of tree-level heterotic string models according to their degrees of supersymmetry. (We stress that this analysis will be the only case in which unstable tachyonic N=0 string models will be considered in this paper.) We will then proceed in Sect. 5 to examine questions related to correlations between the numbers of unbroken supersymmetry generators and the corresponding gauge groups. The landscape of four-dimensional heterotic strings is a relatively large and com- plex structure. It may therefore be useful, as an initial step, to quickly recall the much smaller “landscape” of ten-dimensional heterotic strings. In ten dimensions, the maximal allowed supersymmetry is N=1, and thus our tree-level ten-dimensional landscape may be partitioned into only three categories: N=1 models, N=0 tachyon- free models, and N=0 tachyonic models. Note that since the N=0 tachyonic models are not even stable at tree level, the tree-level “landscape” actually consists only of models in the first two categories. However, for convenience, in this section we shall use the word “landscape” to describe the full set of heterotic vacuum solutions regardless of stability. SUSY class % of 10D landscape % of reduced 10D landscape N=0 (tachyonic) 66.7 62.5 N=0 (tachyon-free) 11.1 12.5 N=1 22.2 25.0 Table 2: Classification of the ten-dimensional tree-level heterotic “landscape” as a function of the number of spacetime supersymmetries and the presence/absence of tachyons at tree level. As always, models are judged to be distinct based on their gauge groups and parti- cle contents. The full ten-dimensional heterotic landscape consists of nine distinct string models, while the landscape of models accessible through our random search methods is reduced by one model. In either case, we see that two thirds of the tachyon-free portion of the ten-dimensional landscape is supersymmetric. Thus unbroken supersymmetry tends to dominate the “landscape” consisting of ten-dimensional models which are stable at tree level. As is well known [21], the full set of D = 10 heterotic strings consists of nine distinct string models: two are supersymmetric [these are the SO(32) and E8 × E8 models], one is non-supersymmetric but tachyon-free [this is the SO(16) × SO(16) string model [20]], and six additional models are non-supersymmetric and tachyonic. Expressed as proportions of a full ten-dimensional heterotic landscape, we therefore find the results shown in the middle column of Table 2. It is important to note, however, that not all of these models would be realizable through the methods we shall be employing in this paper (involving a construction in which all degrees of freedom are represented in terms of complex worldsheet fermions). Indeed, one of the tachyonic non-supersymmetric models exhibits rank-reduction and thus would not be realizable in a random search of the sort we shall be conducting. Statistics for the corresponding “reduced” landscape of accessible models are therefore listed along the third column of Table 2; these are the statistics which will form the basis for future comparisons. Note that in either case, the tachyon-free portion of the ten-dimensional landscape is dominated by supersymmetric models. This suggests that breaking supersymmetry without introducing tachyons is relatively difficult in ten dimensions. Our goal is to understand how this picture changes after compactification to four dimensions. Towards this end, one procedure might be to randomly generate a large set of string models, and see how many models one obtains of each type after a certain fixed time as elapsed. However, as discussed in Sect. 3, these percentages will generally float or evolve as a function of the total number of models examined. This behavior is shown in Fig. 1, and we see that while the non-supersymmetric per- centages seem to be floating towards greater values, the supersymmetric percentages seem to be floating towards lesser values. Figure 1: The numbers of distinct string models exhibiting different amounts of spacetime supersymmetry, plotted as functions of the total number of distinct string models examined. Models exhibiting N=4 supersymmetry are too few to appear on this figure. As discussed in Sect. 3, it is easy to understand the reason for this phenomenon. Clearly, as we continue to generate models randomly, an ever-increasing fraction of these models consists of models without supersymmetry. This in turn suggests that at any given time, we have already discovered a greater fraction of the space of su- persymmetric models than non-supersymmetric models. This would explain why it becomes increasingly more difficult to randomly generate new, distinct supersym- metric models as compared with non-supersymmetric models, and why their relative percentages show the floating behavior illustrated in Fig. 1. How then can we extract meaningful information? As discussed in Sect. 3, the remedy involves keeping track of not only the total numbers of distinct models found in each supersymmetric class, but also the total number of attempts which yielded a model in each class, even though such models were not necessarily new. This information is shown in Table 3 for our total sample of >∼ 10 7 models. SUSY class # distinct models # attempts avg. attempts/model N=0 (tachyonic) 1 279 484 3 810 838 2.98 N=0 (tachyon-free) 4 946 388 18 000 000 3.64 N=1 3 772 679 24 200 097 6.41 N=2 492 790 13 998 843 28.41 N=4 1106 6 523 277 5 898.08 Total: 10 492 447 66 533 055 6.34 Table 3: This table expands on Table 1 by including the numbers of attempts to gener- ate models in each class as well as the corresponding average numbers of attempts per distinct model. We also include information about the attempts which resulted in non- supersymmetric models whose spectra are tachyonic at tree level. It is apparent that the number of attempts per model increases rather dramatically as the level of supersymmetry increases, indicating that our heterotic string sample has penetrated further into the spaces of models with greater numbers of supersymmetries than into those with fewer. As we see from Table 3, the number of required attempts per model increases dra- matically with the level of supersymmetry. This in turn implies, for example, that although we may have generated many fewer distinct N=4 models thanN=1 models, the full space of N=4 models has already been penetrated much more fully than the space of N=1 models. Thus, as we continue to generate more models, it should be- come relatively easier to generate non-supersymmetric models than supersymmetric models. If true, this would imply that the relative proportion of non-supersymmetric models should increase as we continue to generate more models, while the relative proportion of supersymmetric models should decrease. This is, of course, exactly what we have already seen in Fig. 1. In order to extract final information concerning the relative sizes of these spaces, the procedure outlined in Sect. 3 instead requires that we do something different, and compare the numbers of distinct models generated in each class at those points in our model-generating process when their corresponding numbers of attempts per model are equal . It is only in this way that we can overcome the effects of floating correlations and extract stable relative percentages which do not continue to evolve as functions of the total sample size. For example, let us consider the relative numbers of N=1 and N=2 models. Although we see from Table 3 that our full sample of >∼ 10 7 models contains approx- imately 7.66 times as many N=1 models as N=2 models, this is not the relative size of their corresponding model spaces because the N=2 space of models has already been explored more fully than the N=1 model space, with 6.41 attempts per N=1 model compared with 28.41 attempts per N=2 model. However, at an earlier point in our search, we found that it took an average of approximately 6.41 attempts to generate a new, distinct N=2 model: this occurred when we had generated only ap- proximately 90 255 models with N=2 supersymmetry. This suggests that the space of N=1 models is actually 3772679/90255 ≈ 41.8 times as large as the space of N=2 models. Moreover, we can verify that this ratio is actually stable as a function of sample size. For example, at an even earlier point in our search when we had generated only ≈ 2.22×106 N=1 models, we found that an average of 3.64 attempts were required to generate a new, distinct N=1 model. However, this same average number of attempts per model occurred in our N=2 sample when we had generated only ≈ 53 000 N=2 models. Thus, once again, the N=1 and N=2 model spaces appear to have a size ratio of ≈ 41.8 : 1. In this way, by comparing total numbers of models examined at equal values of attempts per model, we can extract the relative sizes of the spaces of models with differing degrees of supersymmetry and verify that these results are stable as functions of sample size (i.e., stable as functions of the chosen value of attempts per model). Our results are shown in Table 4. As far as we can determine, the percentages quoted in Table 4 represent the values to which the percentages in Fig. 1 would float if we could analyze what is essentially the full landscape. However, short of examining the full landscape, we see that there is no single point at which these percentages would simultaneously appear in any finite extrapolation of Fig 1. Instead, it is only by comparing the numbers of models obtained at different points in our analysis that the true ratios quoted in Table 4 can be extracted. SUSY class % of heterotic landscape N=0 (tachyonic) 32.1 N=0 (tachyon-free) 46.5 N=1 20.9 N=2 0.5 N=4 0.003 Table 4: Classification of the four-dimensional tree-level heterotic landscape as a function of the number of unbroken spacetime supersymmetries and the presence/absence of tachyons at tree level. This table is thus the four-dimensional counterpart of Table 2, which quoted analogous results for ten dimensions. Relative to the situation in ten dimensions, we see that compactification to four dimensions tends to favor breaking all spacetime supersymmetries without introducing tachyons at tree level. Table 4 thus represents our final partitioning of the tree-level four-dimensional landscape according to the amount of supersymmetry exhibited. There are several rather striking facts which are evident from these results: • First, we see that nearly half of the heterotic landscape is non-supersymmetric and yet tachyon-free. • Second, we see that the supersymmetric portion of the heterotic landscape ap- pears to account for less than one-quarter of the full four-dimensional heterotic landscape. • Finally, models exhibiting extended (N ≥ 2) supersymmetries are exceedingly rare, representing less than one percent of the full landscape. Of course, we stress once again that these results hold only for the tree-level landscape, i.e., models which are stable at tree level only. It is not clear whether these results would persist after full moduli stabilization. However, assuming that they do, these results lead to a number of interesting conclusions. The first conclusion is that the properties of the tachyon-free heterotic landscape as a whole are statistically dominated by the properties of string models which do not have spacetime supersymmetry. Indeed, the N=0 string models account for over three-quarters of this portion of the heterotic string landscape. The fact that the N=0 string models dominate the tachyon-free portion of the landscape suggests that breaking supersymmetry without introducing tachyons is actually favored over preserving supersymmetry for this portion of the landscape. Indeed, we expect this result to hold even after full moduli stabilization, unless an unbroken supersymmetry is somehow restored by stabilization. The second conclusion which can be drawn from these results is that the super- symmetric portion of the landscape is almost completely comprised of N=1 string models. Indeed, only 2% of the supersymmetric portion of the heterotic landscape has more than N=1 supersymmetry. This suggests that the correlations present for the supersymmetric portion of the landscape can be interpreted as the statistical correlations within the N=1 string models, with the N=2 correlations represent- ing a correction at the level of 2% and the N=4 correlations representing a nearly negligible correction. It is natural to ask what effects are responsible for this hierarchy. As was dis- cussed in Sect. 3, two string models are considered distinct if any of their spacetime properties are found to be different. Two models which have the same number of unbroken spacetime supersymmetries must therefore differ in other features, such as their gauge groups and particle representations. Thus, if there exist more models with one level of supersymmetry than another, this must mean that there are more string-allowed configurations of gauge groups and particle representations with one level of supersymmetry than the other. Indeed, given the results of Table 4, our expectation is that increasing the level of supersymmetry will have the effect of de- creasing the number of distinct models with a given gauge group, and possibly even the range of allowed gauge groups. We shall test both of these expectations explicitly in Sect. 5. 5 Supersymmetry versus gauge groups Within the heterotic string, worldsheet self-consistency conditions arising from the requirements of conformal anomaly cancellation, one-loop and multi-loop modular in- variance, physically sensible GSO projections, etc., impose many tight constraints on the allowed particle spectrum. These constraints simultaneously affect not only the spacetime Lorentz structure of the theory (such as is involved in spacetime super- symmetry), but also the internal gauge structure of the theory. Thus, it is precisely within the context of string theory that we expect to find correlations between super- symmetries and gauge symmetries — features which would otherwise be uncorrelated in theories based on point particles. In general, these correlations can lead to certain tensions in a given string con- struction. Models exhibiting large numbers of unbroken supersymmetries may be expected to have relatively rigid gauge structures, and vice versa. There are two spe- cific types of correlations which we shall study. First, we shall analyze how the degree of supersymmetry affects the range of possible allowed gauge groups. For example, in extreme cases it may occur that certain gauge symmetries may not even be allowed for certain levels of spacetime supersymmetry. Second, even within the context of a fixed gauge group, we can expect the degree of spacetime supersymmetry to affect the range of allowed particle representations which can appear at the massless level. In other words, the number of distinct string models with a given fixed gauge group may be highly sensitive to the degree of spacetime supersymmetry. Some of these features are already on display in the ten-dimensional heterotic “landscape”. For example, no gauge group is shared between those ten-dimensional models with supersymmetry and those without. Moreover, in each case, there is only a single model with each allowed gauge group. Thus, in ten dimensions, the specification of the level of supersymmetry (and/or the gauge group) is sufficient to completely fix the corresponding particle spectrum. Clearly, in four dimensions, things will be far more complex. In particular, we shall study three correlations in this section: • First, we shall focus on the number of allowed gauge groups as a function of the degree of supersymmetry. We shall also study gauge-group multiplicities — i.e., the probabilities that there exist distinct string models with the same gauge group but different particle spectra. This will be the focus of Sect. 5.1. • Second, as a function of the degree of supersymmetry, we shall investigate “shatter” — i.e., the degree to which our total (rank-22) gauge group is “shat- tered” into distinct irreducible factors, or equivalently the average rank of each irreducible gauge-group factor. This will be the focus of Sect. 5.2. • Finally, as a function of the degree of supersymmetry, we shall study the prob- abilities of realizing specific (combinations of) gauge-group factors in a given string model. This will be the focus of Sect. 5.3. As we shall see, these studies will find deep correlations which ultimately reflect the string-theoretic tension between supersymmetry and the string consistency condi- tions. 5.1 Numbers and multiplicities of unique gauge groups We begin by studying the total numbers of distinct gauge groups which can be realized as a function of the number of unbroken supersymmetries in a given string model. To do this, one direct approach can might be to classify models according to their numbers of unbroken spacetime supersymmetries, and tabulate the numbers of distinct gauge groups which appear as functions of the total number of models in each class. As we continue to generate more and more models, we then obtain the results shown in Fig. 2. It is evident from Fig. 2 that for a fixed sample size, models with more unbroken supersymmetries tend to exhibit larger numbers of distinct gauge groups, or equiv- alently smaller numbers of model multiplicities per gauge group. For example, we see from Fig. 2 that when each class of models has reached a sample size of 500 000 models, the tachyon-free N=0 models have a greater multiplicity per gauge group than N=1 models by an approximate factor ≈ 1.4, while the N=2 models have a smaller multiplicity per gauge group than the N=1 models by an approximate factor ≈ 0.8. However, it is easy to understand this behavior. As the level of supersym- metry increases, there are more constraints on the possible particle spectra that can emerge for a given gauge group. This in turn implies that there are likely to be fewer ways for two models with the same gauge group to be distinct, which in turn implies that there is a greater chance that distinct models will be forced to exhibit distinct gauge groups. Thus, models exhibiting greater amounts of supersymmetry are likely, on average, to exhibit greater numbers of gauge groups amongst a fixed number of models. Of course, as also evident from Fig. 2, the multiplicity of distinct models per gauge group exhibits a strong, floating dependence on the sample size. Therefore, in order to extract a stable ratio of multiplicity ratios — one which presumably represents the values of these ratios when extrapolated to the full landscape — we must employ the methods described in Sect. 3. We then obtain the results shown in the middle column of Table 5. Using these results in conjunction with the corresponding ratios of landscape magnitudes in Table 4, we can also calculate the relative numbers of Figure 2: Numbers of distinct gauge groups obtained as functions of the number of dis- tinct string models generated. Each curve corresponds to models with a different num- ber of unbroken spacetime supersymmetries, with N=0 signifying models which are non- supersymmetric but tachyon-free. We see that for a fixed sample size, models with more unbroken supersymmetries tend to exhibit a larger number of distinct gauge groups. (Note that models with N=4 supersymmetry are too few to be shown in this plot.) distinct gauge groups realizable within each SUSY class of models. These results are shown in the final column of Table 5. Note that in each case, these quantities are quoted as ratios relative to their N=1 values; this represents the most detailed information that can be extracted using the methods of Sect. 3. We see from Table 5 that both the average multiplicities per gauge group and the total numbers of realizable gauge groups are monotonically decreasing functions of the number of unbroken supersymmetries. While this is to be expected on the basis of the arguments described above, we must realize that our class of N=0 models does not consist of all non-supersymmetric models, but merely those which are tachyon- free. Thus, the requirement of avoiding tachyons could have turned out to be more stringent than the requirement of maintaining an unbroken supersymmetry, at least as far as generating a variety of gauge groups is concerned. This is indeed what happens in the ten-dimensional landscape, where there are fewer realizable gauge groups for non-supersymmetric tachyon-free models than for models with N=1 supersymmetry. avg. multiplicity # of realizable SUSY class per gauge group gauge groups N=0 (tachyon-free) 1.65 1.35 N=1 1.00 1.00 N=2 0.89 0.03 Table 5: The average relative multiplicities (distinct models per gauge group) and total numbers of realizable gauge groups, evaluated for heterotic string models with N = 0, 1, 2 unbroken spacetime supersymmetries. In each case, these quantities are normalized to their N=1 values. However, the results in Table 5 indicate that the opposite is true in D = 4. Note that in Table 5, we do not quote results for the N=4 portion of the heterotic landscape because the absolute numbers of models in this class are so small that no stable numerical results can be extracted relative to the other levels of supersymmetry. However, it is worth noting that literally each N=4 model in our sample has a unique gauge group, so the absolute (rather than relative) gauge-group multiplicity in the N=4 case is exactly 1.000. This only reinforces our general observation that increased levels of supersymmetry reduce the gauge-group multiplicity; indeed, we now see that the case of maximal supersymmetry appears to result in the minimal allowed gauge- group multiplicity. It is likely that this result can be proven analytically for the N=4 landscape as a whole. 5.2 Shatter/average rank Having studied the numbers of different possible gauge groups, we now turn our attention to the gauge groups themselves. Once again, our goal is to study how these gauge groups depend on the presence or absence of spacetime supersymmetry. To begin the discussion, our focus in this section will be on what we call “shat- ter” [8]. Recall that the heterotic string models we are considering all have gauge groups with total rank 22. This stretches from models with gauge group SO(44) all the way down to models with gauge groups of the form U(1)n × SU(2)22−n with potentially all values of n in the range 0 ≤ n ≤ 22. Following Ref. [8], we shall define the “shatter” for a given string model as the number of distinct irreducible gauge- group factors into which its total rank-22 gauge group has been shattered. Note that for this purpose, factors of SO(4) ∼ SU(2)× SU(2) contribute two units to shatter. Since the total rank of the gauge group is fixed at 22 for such models, this means that shatter is also a measure of the average rank of the individual group group factors, with 〈rank〉 = 22/shatter. Roughly speaking, shatter can also be taken as a measure of the degree of complexity needed for the construction of a given string model, with increasingly smaller individual gauge-group factors tending to require increasingly many non-overlapping sequences of orbifold twists and Wilson lines. Given this definition of shatter, we may then calculate the distribution of shatter across the landscape of heterotic strings. We may calculate, for example, the relative probabilities that models with certain levels of shatter emerge across the landscape, and ask how these probability distributions vary with the amount of spacetime su- persymmetry present in the model. Our results are shown in Fig. 3. Once again, we stress that our raw data tends to evolve significantly as a function of the sample size of models considered. It is therefore necessary to employ the techniques described in Sect. 3 in order to extract stable results which should apply across the landscape as a whole. In practice, this requires a difficult and time-consuming process in which each of the data points shown in Figs. 3 for N=0, 1, 2 has individually been extracted through the limiting procedure described in Sect. 3. Only then is an entire “curve” constructed for each level of supersymmetry, as shown. For theN=4 case, by contrast, our sample size is too small to permit stable results to be extracted. However, the fact that the attempts per model count in Table 3 is so large for the N=4 models suggests that our N=4 sample has already explored a significant fraction (and perhaps even most) of the corresponding landscape. The N=4 curve in Fig. 3 thus represents a direct tally of our N=4 sample set. As evident from Fig. 3, certain features of these plots are independent of the level of spacetime supersymmetry. These therefore represent general trends which hold across the entire tachyon-free heterotic string landscape. For example, one general trend is a strong preference for models with relatively high degrees of shatter and correspondingly small average ranks for individual gauge-group factors — models exhibiting shatters near or in the teens clearly dominate. On the other hand, this preference for highly shattered gauge groups does not appear to extend to the limit of completely shattered models with shatter=22; indeed, the set of models with only rank-one gauge-group factors seems to represent a fairly negligible portion of the landscape regardless of the degree of supersymmetry. This indicates that most models in this class have gauge groups which contain at least one factor of rank greater than one.∗ Another universal trend implied by (though not explicitly shown in) Fig. 3 is that string models with shatters of less than four accrue relatively little measurable amount of probability. Even in the N=4 case, these models are thus actually quite rare across the landscape as a whole. In some sense, this too is to be expected, since ∗ Of course, we stress that this conclusion applies only for models in the free-fermionic class. In general, it is always possible to deform away from the free-fermionic limit by adjusting the internal radii of the worldsheet fields away from their free-fermionic values; in such cases, we expect all gauge symmetries to be broken down to U(1)22. However, as noted earlier, the free-fermionic points typi- cally represent precisely those points at which additional (non-Cartan) gauge-boson states become massless, thereby enhancing the gauge symmetries to become non-abelian. Thus, as discussed more fully in Sect. 2, the free-fermionic construction naturally leads to precisely the set of models which are likely to be of direct phenomenological relevance. Figure 3: The absolute probabilities of obtaining distinct four-dimensional heterotic string models with different numbers of unbroken supersymmetries, plotted as functions of the degree to which their gauge groups are “shattered” into separate irreducible factors. The total value of the points (the “area under the curve”) in each case is 1. Here N=0 refers to models which are non-supersymmetric but tachyon-free. there are many more ways of breaking a large gauge symmetry through orbifolds and non-trivial Wilson lines than of preserving it. Despite these universal features, we see that spacetime supersymmetry neverthe- less does have a significant effect on the shapes of these curves. In this regard, there are two features to note. First, we observe that as the degree of unbroken supersymmetry increases, the range of probable shatter values also tends to increase, with probability shifting from models with high shatters to models with lower shatters. This is especially noticeable when comparing the distribution of theN=2 andN=4 models with those of theN=0 and N=1 models. These results indicate that models exhibiting smaller amounts of shatter (i.e., models whose gauge-group factors have larger individual average ranks) become somewhat more probable as the level of supersymmetry increases. Ultimately, this correlation between unbroken supersymmetry and unbroken gauge symmetry emerges since both have their underlying origins in how our orbifold twists and Wilson lines are chosen. Second, and perhaps more unexpectedly, we see that the degree of supersymmetry also affects the overall profiles of these curves. While the N=0 curve is relatively smooth, exhibiting a single peak at shatter=20, these curves begin to experience even/odd oscillations as the degree of supersymmetry increases, with odd values of shatter significantly favored over even values when supersymmetry is present. The origins of this phenomenon are less apparent, and perhaps lie in the modular invari- ance and anomaly cancellation constraints which correlate the orders of the allowed twists leading to self-consistent string models. Interestingly, this even/odd behav- ior continues into the N=4 case, although these oscillations are significantly less pronounced and flip sign, with evens now dominating over odds. One notable feature of the N=4 curve is its approximate reflection symmetry around shatter=10. It is unclear whether this is an exact symmetry which holds in situations with maximal supersymmetry, or whether this is merely an accident. 5.3 Specific gauge-group factors Finally, we turn to an analysis of the probabilities of realizing individual gauge- group factors. Just how likely is it, say, that a randomly chosen heterotic string model will exhibit an SU(3) factor in its gauge group, and how does this probability correlate with the spacetime supersymmetry of the model? Just as with previous questions, addressing this issue requires a detailed analysis along the lines discussed in Sect. 3. This is because the probabilities of realizing different gauge-group factors also float quite strongly as a function of sample size. As dramatic illustration of this fact, let us restrict our attention to models with N=1 spacetime supersymmetry and calculate the probability that a given model will exhibit an SU(3) gauge-group factor as a function of the number of models we have examined. We then obtain the result shown in Fig. 4, and it is clear that the Figure 4: The percentage of distinct four-dimensionalN=1 supersymmetric heterotic string models exhibiting at least one SU(3) gauge-group factor, plotted as a function of the number of models examined for the first 1.25 million models. We see that as we generate further models, SU(3) gauge-group factors become somewhat more ubiquitous — i.e., the fraction of models with this property floats. One must therefore account for this floating behavior using the methods described in Sect. 3 in order to extract meaningful information concerning the relative probabilities of specific gauge-group factors. percentage of models with SU(3) gauge-group factors floats rather significantly as a function of the sample size. Indeed, on the basis of this information alone, it would be quite impossible to determine the final value to which this curve might float. Just as with previous examples, this floating behavior ultimately occurs because models with SU(3) gauge-group factors are relatively difficult to generate using the construction methods we are employing; thus, they tend to emerge in increasing numbers only after other models are exhausted. As discussed more fully in Ref. [9], this does not imply that there are fewer of these models or that our construction method cannot ultimately reach them— all we can conclude is that they are less likely to be generated in a random search than other models, and thus they tend to emerge only later in the search process. Indeed, as we shall shortly see, models with SU(3) gauge-group factors actually tend to dominate the landscape. Therefore, in order to extract meaningful results, we again employ the methods discussed in Sect. 3. We then obtain the final percentages quoted in Table 6. We observe, in particular, that the probability of models with N=1 supersymmetry ex- hibiting at least one SU(3) gauge-group factor has actually risen all the way to 98%. The fact that this probability has floated from nearly 55% to 98% only reinforces the importance of the analysis method presented in Sect. 3, and illustrates the need to properly account for floating correlations when quoting statistical results for such studies. gauge group N=0 N=1 N=2 N=4 U1 99.9 94.5 68.4 89.6 SU2 62.46 97.4 64.3 60.9 SU3 99.3 98.0 93.0 45.1 SU4 14.46 30.0 39.0 53.5 SU5 16.78 43.5 66.3 33.8 SU>5 0.185 1.7 10.6 73.0 SO8 0.482 1.6 6.2 21.1 SO10 0.084 0.2 1.6 18.7 SO>10 0.005 0.038 0.77 7.5 E6,7,8 0.0003 0.03 0.16 11.5 Table 6: Percentages of heterotic string models exhibiting specific gauge-group factors as functions of their spacetime supersymmetry. Here SU>5 and SO>10 collectively indicate gauge groups SU(n) and SO(2n) for any n > 5, while N refers to the number of unbroken supersymmetries at the string scale. Note that the N=0 models are all tachyon-free. As we see from Table 6, supersymmetry can have quite sizable effects upon the probability of realizing specific groups. However, there are some general trends that hold for the full heterotic landscape. These trends include: • A preference for SU(n + 1) over SO(2n) groups for each rank n. Even though these two groups have the same rank, it seems that SU groups are more common than the SO groups for all levels of supersymmetry. • Groups with smaller rank are much more common than groups with larger rank. Once again, this also appears to hold for all levels of supersymmetry. • Finally, the gauge-group factors comprising Standard-Model gauge group GSM ≡ SU3 × SU2 × U1 are particularly common, much more so than those of any of its grand-unified extensions. As we found in Sect. 4, the N=0 string models dominate the tachyon-free portion of the heterotic landscape. Similarly, the N=1 string models are the dominant part of the supersymmetric portion of the landscape. Nevertheless, it is interesting to gauge entire SUSY group landscape subset U1 98.00 93.89 SU2 73.22 96.62 SU3 98.85 97.88 SU4 19.42 30.21 SU5 25.37 44.03 SU>5 0.73 1.92 SO8 0.87 1.71 SO10 0.13 0.23 SO>10 0.02 0.06 E6,7,8 0.01 0.03 Table 7: Percentage of heterotic string models exhibiting specific gauge-group factors, quoted across the entire landscape of tachyon-free models (both supersymmetric and non- supersymmetric) as well as across only that subset of models with at least N≥1 spacetime supersymmetry. These results are derived from those of Table 6 using the landscape weight- ings in Table 4. examine the gauge-group probabilities across both of these portions of the landscape. These probabilities are easy to calculate by combining the results in Tables 4 and 6, leading to the results shown in Table 7. Several features are immediately apparent from Table 7. First, gauge groups with larger ranks appear to be favored more strongly across the supersymmetric subset of the landscape than across the tachyon-free landscape as a whole. Since each of our heterotic string models in this class has a gauge group of fixed total rank, this preference for higher-rank gauge groups necessarily comes at the price of sacrificing smaller-rank gauge groups. Indeed, it often happens that this preference for larger-rank gauge groups actually precludes the appearance of any small-rank gauge groups whatsoever. Interestingly, the supersymmetric portion of the landscape seems to sacrifice U(1) primarily and SU(3) to a lesser extent. This is in contrast to SU(2), which is actually more strongly favored in the supersymmetric portion of the landscape than in the general tachyon-free landscape as a whole. Second, the level of supersymmetry also appears to affect the probability dis- tributions across the different possible gauge-group factors. The supersymmetric portion of the landscape has a much greater representation of the large rank groups. This suggests that the constraints placed on the string spectrum in order to preserve spacetime supersymmetry also have the effect of favoring larger gauge symmetries, a fact already noted in Sect. 5.2. In other words, there tends to be a decrease in the gauge-group multiplicity for highly shattered gauge groups which consist of only very small gauge-group factors, and thus the larger-rank gauge groups make up a larger proportion of the whole landscape. Indeed, this effect is particularly acute for that subset of the landscape exhibiting maximal N=4 supersymmetry, where the larger-rank SU gauge groups are particularly well represented. 6 Discussion In this paper, we have examined both the prevalence of spacetime supersymmetry across the heterotic string landscape and the statistical correlations between the appearance of spacetime supersymmetry and the gauge structure of the corresponding string models. Somewhat surprisingly, we found that nearly half of the heterotic landscape is non-supersymmetric and yet tachyon-free at tree level; indeed, less than a quarter of the tree-level heterotic landscape exhibits any supersymmetry at all at the string scale. Moreover, we found that the degree of spacetime supersymmetry is strongly correlated with the probabilities of realizing certain gauge groups, with unbroken supersymmetry at the string scale tending to favor gauge-group factors with larger rank. There are several extensions to these results which are currently under investiga- tion. For example, we would like to understand how the presence of supersymmetry affects the statistical appearance of the entire composite Standard-Model gauge group GSM ≡ SU3 ×SU2 ×U1, and not merely the appearance of its individual factors. We would also like to understand how the presence or absence of supersymmetry affects other features which are equally important for the overall architecture of the Stan- dard Model: these include the appearance of three chiral generations of quarks and leptons, along with a potentially correct set of gauge couplings and Yukawa couplings. This work has already been completed, and will be reported shortly [16]. Despite this progress, such studies have a number of intrinsic limitations which must continually be borne in mind. A number of these have been emphasized by us in recent articles (see, e.g., the concluding sections of Refs. [8, 9]) and will not be repeated here. However, other limitations are particularly relevant for the results we have quoted here and thus deserve emphasis. First, we must continually bear in mind that our study has been limited to models in which rank-cutting is absent. Thus, all of the four-dimensional heterotic string models we have examined exhibit a fixed maximal rank=22. This has the potential to skew the statistics of the different gauge-group factors. For example, it is possible that gauge-group factors with smaller ranks might be over-represented in this sample simply because the appearance of such groups is often the only way in which a given model can precisely saturate the total rank bound. By contrast, for models which can exhibit rank-cutting, this saturation would not be needed and it is therefore possible that lower-rank groups are consequently less abundant. A second limitation of this study stems from the nature of performing random search studies in general. In Sect. 3, we summarized several methods by which the problematic issue of floating correlations can be transcended, and this paper has provided several examples of not only the need for such methods but also of the means by which they are implemented. As more fully discussed in Ref. [9], such problems are going to arise — and such methods are going to be necessary — whenever we attempt to extract statistical correlations from a large data set to which our computational access is necessarily limited. However, despite the apparent success of such methods, it is always a logical possibility that there exists a huge pool of string models with non-standard physical characteristics remaining just beyond our computational power to observe. The existence of such a pool of models would completely change the nature of our statistical results, to an extent which is essentially unbounded, yet we may miss this completely because of limited computational power. Although this becomes increasingly unlikely as our search through the landscape becomes larger and increasingly sophisticated, this nevertheless always remains a logical possibility which cannot be discounted. But finally, perhaps the most serious limitation of our study is the fact that we are analyzing the statistical properties of string models which are not necessarily stable beyond tree level. Indeed, since none of our non-supersymmetric string models has a vanishing one-loop cosmological constant, these models in particular necessarily have non-zero dilaton tadpoles at one-loop order and thus become unstable. Even our su- persymmetric models have flat directions which have not been lifted. Thus, as we have stressed throughout this paper, the “landscape” we have examined in this paper is at best a tree-level one. Despite this fact, however, it is important to realize that these models do represent self-consistent string solutions at tree level. Specifically, these models satisfy all of the constraints needed for worldsheet conformal/superconformal invariance, modular-invariant one-loop and multi-loop amplitudes, proper spacetime spin-statistics relations, and physically self-consistent layers of sequential GSO pro- jections and orbifold twists. Indeed, since no completely stable perturbative heterotic strings have yet been constructed, this sort of analysis is currently the state of the art for large-scale statistical studies of this type. This mirrors the situation on the Type I side, where state-of-the-art statistical analyses have also focused on models which are only stable at tree level. Nevertheless, we are then left with the single over-arching question: to what extent can we believe that the results we have found for this “tree-level” landscape actually apply to the true landscape that would emerge after all moduli are stabilized? The answer to this question clearly depends on the extent to which the statistical correlations we have uncovered here are likely to hold even after vacuum stabilization. A priori , this is completely unknown. However, one surprising result of this paper is the observation that the string self-consistency requirements themselves — even merely at tree-level — do not preferentially give rise to supersymmetric solutions at the string scale. Indeed, as we discussed in Sect. 4, less than a quarter of the tree- level heterotic landscape appears to exhibit any supersymmetry at all at the string scale. Thus, breaking supersymmetry without introducing tachyons is actually sta- tistically favored over preserving supersymmetry, even at the string scale and even when the requirements of avoiding tachyons are implemented. Observations such as these tend to shift the burden of proof onto the SUSY enthusiasts, and perhaps reframe the question to one in which we might ask whether an unbroken supersym- metry is somehow restored by modulus stabilization. This seems unlikely, especially since most modern methods of modulus stabilization rely on breaking rather than introducing supersymmetry. In either case, however, this shows how the results of such studies — even though limited to only the tree-level landscape — can have the power to dramatically reframe the relevant questions. Indeed, once the technology for building heterotic string models develops further and truly stable vacua can be statistically analyzed in large quantities, it will be interesting to compare the statis- tical properties of those vacua with these in order to ascertain the degree to which vacuum stabilization might affect these other phenomenological properties. 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D 34, 3794 (1986). Introduction The models Method of analysis Supersymmetry on the heterotic landscape Supersymmetry versus gauge groups Numbers and multiplicities of unique gauge groups Shatter/average rank Specific gauge-group factors Discussion